Subgroups — C₃₁×#61465 #21518224

Order 7936.

2156 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3483	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3362	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3354	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3332	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3325	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3274	C₂	1	3	yes	maximal, complemented
11	C₃₁×#3184	C₂	1	3	yes	maximal, complemented
12	C₃₁×#3115	C₂	1	3	yes	maximal, complemented
13	C₃₁×#3109	C₂	1	3	yes	maximal, complemented
14	C₃₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
16	C₂²×#41×C₃₁	—	2	1	no	—
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	3	yes	—
18	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#555	—	2	3	no	—
20	C₃₁×#555	C₂²	1	3	yes	complemented
21	C₃₁×#550	—	2	1	no	—
22	C₃₁×#550	C₂²	1	1	yes	characteristic
23	C₃₁×#550	C₂²	1	3	yes	complemented
24	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
25	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#548	—	2	1	no	—
27	C₃₁×#547	C₂²	1	3	yes	—
28	C₃₁×#546	—	2	3	no	—
29	D₄²×C₃₁	C₂²	1	3	yes	—
30	C₃₁×#532	—	2	3	no	—
31	C₃₁×#532	C₂²	1	3	yes	—
32	C₃₁×#532	C₂²	1	3	yes	complemented
33	C₃₁×#525	—	2	3	no	—
34	C₄×#41×C₃₁	—	2	1	no	—
35	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×#41×C₃₁	C₂²	1	3	yes	complemented
37	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
38	C₃₁×#498	C₂²	1	3	yes	complemented
39	C₃₁×#493	C₂²	1	3	yes	complemented
40	C₃₁×#486	C₂²	1	3	yes	complemented
41	C₃₁×#486	C₂²	1	3	yes	complemented
42	C₃₁×#482	C₂²	1	6	yes	complemented
43	C₃₁×#481	C₂²	1	6	yes	complemented
44	C₃₁×#476	C₂²	1	3	yes	—
45	C₃₁×#476	C₂²	1	3	yes	complemented
46	C₃₁×#475	C₂²	1	3	yes	complemented
47	C₃₁×#474	C₂²	1	3	yes	complemented
48	C₃₁×#469	C₂²	1	6	yes	complemented
49	C₃₁×#469	C₂²	1	6	yes	complemented
50	C₃₁×#460	C₂²	1	3	yes	complemented
51	C₃₁×#459	C₂²	1	3	yes	—
52	C₃₁×#459	C₂²	1	3	yes	complemented
53	C₃₁×#459	C₂²	1	3	yes	complemented
54	C₃₁×#449	C₂²	1	6	yes	complemented
55	C₃₁×#447	C₂²	1	6	yes	—
56	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
57	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
58	C₃₁×#442	C₂²	1	3	yes	—
59	C₃₁×#442	C₂²	1	6	yes	complemented
60	C₃₁×#442	C₂²	1	3	yes	complemented
61	C₃₁×#440	C₂²	1	1	yes	characteristic
62	C₃₁×#440	C₂²	1	3	yes	complemented
63	C₄×SD₁₆×C₃₁	C₂²	1	3	yes	—
64	C₄×SD₁₆×C₃₁	C₂²	1	3	yes	complemented
65	C₄×D₈×C₃₁	C₂²	1	3	yes	—
66	C₄×D₈×C₃₁	C₂²	1	3	yes	complemented
67	C₃₁×#436	C₂²	1	3	yes	complemented
68	C₃₁×#436	C₂²	1	3	yes	complemented
69	C₃₁×#432	C₂²	1	3	yes	complemented
70	C₃₁×#428	C₂²	1	3	yes	complemented
71	C₃₁×#418	C₂²	1	3	yes	—
72	C₃₁×#417	C₂²	1	3	yes	complemented
73	C₄×M₁₆×C₃₁	C₂²	1	3	yes	complemented
74	C₂×#41×C₃₁	—	2	1	no	—
75	C₂×#41×C₃₁	—	2	1	no	—
76	C₂×#41×C₃₁	—	2	3	no	—
77	C₂×#41×C₃₁	—	2	3	no	—
78	C₂×#41×C₃₁	—	2	3	no	—
79	C₂×#41×C₃₁	—	4	8	no	—
80	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
81	C₂×#41×C₃₁	C₂³	1	3	yes	—
82	C₂²×Q₈×C₃₁	—	2	1	no	—
83	C₂²×D₄×C₃₁	—	2	3	no	—
84	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
85	C₂³×C₄×C₃₁	—	2	3	no	—
86	C₃₁×AGL(1,ℤ/8)	—	2	12	no	—
87	C₂×SD₁₆×C₃₁	C₂³	1	3	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×D₈×C₃₁	C₂³	1	3	yes	—
90	C₂×D₈×C₃₁	C₂³	1	3	yes	—
91	C₂×D₈×C₃₁	C₂³	1	6	yes	—
92	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
94	C₃₁×C₄⋊Q₈	—	2	3	no	—
95	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
96	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
97	C₃₁×#127	—	2	3	no	—
98	C₃₁×#127	C₂³	1	3	yes	—
99	C₃₁×#127	C₂³	1	3	yes	—
100	C₃₁×#124	—	2	6	no	—
101	C₃₁×#124	—	2	3	no	—
102	C₃₁×#124	—	2	3	no	—
103	C₃₁×#123	—	2	6	no	—
104	C₃₁×C₂²⋊Q₈	—	2	3	no	—
105	C₃₁×C₂²⋊Q₈	—	2	3	no	—
106	C₃₁×#121	—	2	3	no	—
107	C₃₁×#121	—	2	3	no	—
108	C₃₁×#121	—	2	3	no	—
109	C₃₁×#121	—	2	3	no	—
110	C₃₁×#121	—	2	3	no	—
111	C₃₁×#121	—	2	3	no	—
112	C₃₁×#121	—	2	6	no	—
113	C₃₁×#121	C₂³	1	6	yes	—
114	C₃₁×#121	C₂³	1	6	yes	—
115	C₃₁×#120	—	2	6	no	—
116	C₄×Q₈×C₃₁	—	2	1	no	—
117	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
118	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
119	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
120	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
121	C₄×D₄×C₃₁	—	2	3	no	—
122	C₄×D₄×C₃₁	—	2	3	no	—
123	C₄×D₄×C₃₁	—	2	3	no	—
124	C₄×D₄×C₃₁	—	2	3	no	—
125	C₄×D₄×C₃₁	—	2	3	no	—
126	C₄×D₄×C₃₁	—	2	3	no	—
127	C₄×D₄×C₃₁	—	2	3	no	—
128	C₄×D₄×C₃₁	D₄	1	3	yes	—
129	C₄×D₄×C₃₁	C₂³	1	3	yes	—
130	C₄×D₄×C₃₁	C₂³	1	3	yes	—
131	C₄×D₄×C₃₁	C₂³	1	3	yes	—
132	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
133	C₄×D₄×C₃₁	C₂³	1	3	yes	—
134	C₄×D₄×C₃₁	C₂³	1	3	yes	—
135	C₄×D₄×C₃₁	C₂³	1	3	yes	—
136	C₃₁×#117	—	2	3	no	—
137	C₃₁×#117	—	2	3	no	—
138	C₃₁×#117	D₄	1	3	yes	—
139	C₃₁×#117	C₂³	1	3	yes	—
140	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
141	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
142	C₂×C₄²×C₃₁	—	2	3	no	—
143	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
144	C₃₁×#107	C₂³	1	1	yes	characteristic
145	C₃₁×#107	C₂³	1	3	yes	—
146	C₃₁×#106	C₂³	1	3	yes	—
147	C₃₁×#106	C₂³	1	3	yes	—
148	C₃₁×#106	C₂³	1	6	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
151	C₃₁×#103	C₂³	1	1	yes	characteristic
152	C₃₁×#103	C₂³	1	3	yes	—
153	C₃₁×#102	C₂³	1	3	yes	—
154	C₃₁×#102	C₂³	1	3	yes	—
155	C₃₁×#102	C₂³	1	1	yes	characteristic
156	C₃₁×#102	C₂³	1	3	yes	—
157	C₃₁×#102	C₂³	1	3	yes	—
158	C₃₁×#102	C₂³	1	3	yes	—
159	C₃₁×#102	C₂³	1	6	yes	—
160	C₃₁×#102	C₂³	1	6	yes	—
161	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
162	C₃₁×#97	C₂³	1	1	yes	characteristic
163	C₃₁×#97	C₂³	1	3	yes	—
164	C₄×C₈×C₃₁	C₂³	1	3	yes	—
165	C₄×C₈×C₃₁	C₂³	1	3	yes	—
166	C₂⁴×C₃₁	—	2	3	no	—
167	#41×C₃₁	—	2	4	no	—
168	#41×C₃₁	—	2	6	no	—
169	#41×C₃₁	—	4	3	no	—
170	#41×C₃₁	—	4	12	no	—
171	#41×C₃₁	—	4	12	no	—
172	#41×C₃₁	—	4	4	no	—
173	#41×C₃₁	—	4	4	no	—
174	#41×C₃₁	—	8	6	no	—
175	C₂×Q₈×C₃₁	—	2	3	no	—
176	C₂×Q₈×C₃₁	—	2	3	no	—
177	C₂×Q₈×C₃₁	—	4	4	no	—
178	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂×D₄×C₃₁	—	2	6	no	—
181	C₂×D₄×C₃₁	—	2	3	no	—
182	C₂×D₄×C₃₁	—	2	6	no	—
183	C₂×D₄×C₃₁	—	2	6	no	—
184	C₂×D₄×C₃₁	—	2	3	no	—
185	C₂×D₄×C₃₁	—	2	3	no	—
186	C₂×D₄×C₃₁	—	2	3	no	—
187	C₂×D₄×C₃₁	—	2	3	no	—
188	C₂×D₄×C₃₁	—	2	3	no	—
189	C₂×D₄×C₃₁	—	4	3	no	—
190	C₂×D₄×C₃₁	—	4	12	no	—
191	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
192	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
193	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
194	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	—	2	3	no	—
196	C₂²×C₄×C₃₁	—	2	3	no	—
197	C₂²×C₄×C₃₁	—	2	3	no	—
198	C₂²×C₄×C₃₁	—	2	3	no	—
199	C₂²×C₄×C₃₁	—	2	1	no	—
200	C₂²×C₄×C₃₁	—	2	3	no	—
201	C₂²×C₄×C₃₁	—	2	3	no	—
202	C₂²×C₄×C₃₁	—	2	3	no	—
203	C₂²×C₄×C₃₁	—	4	12	no	—
204	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
205	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
206	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
207	SD₁₆×C₃₁	—	2	6	no	—
208	SD₁₆×C₃₁	#41	1	4	yes	complemented
209	D₈×C₃₁	—	2	6	no	—
210	D₈×C₃₁	—	2	6	no	—
211	D₈×C₃₁	—	4	6	no	—
212	M₁₆×C₃₁	—	2	6	no	—
213	M₁₆×C₃₁	—	4	3	no	—
214	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
215	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
216	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
217	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₄⋊C₄×C₃₁	—	2	3	no	—
219	C₄⋊C₄×C₃₁	—	2	1	no	—
220	C₄⋊C₄×C₃₁	—	2	3	no	—
221	C₄⋊C₄×C₃₁	—	2	3	no	—
222	C₄⋊C₄×C₃₁	—	2	3	no	—
223	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
224	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
225	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
226	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
227	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
228	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
229	C₂²⋊C₄×C₃₁	—	2	3	no	—
230	C₂²⋊C₄×C₃₁	—	2	1	no	—
231	C₂²⋊C₄×C₃₁	—	2	3	no	—
232	C₂²⋊C₄×C₃₁	—	2	3	no	—
233	C₂²⋊C₄×C₃₁	—	2	3	no	—
234	C₂²⋊C₄×C₃₁	—	2	6	no	—
235	C₂²⋊C₄×C₃₁	—	2	3	no	—
236	C₂²⋊C₄×C₃₁	—	2	3	no	—
237	C₂²⋊C₄×C₃₁	—	2	6	no	—
238	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
239	C₄²×C₃₁	—	2	1	no	—
240	C₄²×C₃₁	—	2	3	no	—
241	C₄²×C₃₁	—	2	3	no	—
242	C₄²×C₃₁	C₂×D₄	1	3	yes	—
243	C₄²×C₃₁	C₂×D₄	1	3	yes	—
244	C₄²×C₃₁	C₂⁴	1	3	yes	—
245	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
246	#61465	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂³×C₃₁	—	2	1	no	—
248	C₂³×C₃₁	—	2	3	no	—
249	C₂³×C₃₁	—	2	3	no	—
250	C₂³×C₃₁	—	4	6	no	—
251	C₂³×C₃₁	—	4	6	no	—
252	C₂³×C₃₁	—	8	3	no	—
253	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
254	Q₈×C₃₁	—	2	1	no	—
255	Q₈×C₃₁	—	4	6	no	—
256	Q₈×C₃₁	—	8	3	no	—
257	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
258	Q₈×C₃₁	C₂×#41	1	2	yes	complemented
259	D₄×C₃₁	—	2	6	no	—
260	D₄×C₃₁	—	2	3	no	—
261	D₄×C₃₁	—	2	3	no	—
262	D₄×C₃₁	—	2	1	no	—
263	D₄×C₃₁	—	2	3	no	—
264	D₄×C₃₁	—	2	3	no	—
265	D₄×C₃₁	—	4	6	no	—
266	D₄×C₃₁	—	4	6	no	—
267	D₄×C₃₁	—	4	12	no	—
268	D₄×C₃₁	—	4	6	no	—
269	D₄×C₃₁	—	4	6	no	—
270	D₄×C₃₁	—	8	3	no	—
271	D₄×C₃₁	—	8	3	no	—
272	D₄×C₃₁	—	8	6	no	—
273	D₄×C₃₁	C₂×#41	1	2	yes	—
274	C₂×C₄×C₃₁	—	2	1	no	—
275	C₂×C₄×C₃₁	—	2	6	no	—
276	C₂×C₄×C₃₁	—	2	3	no	—
277	C₂×C₄×C₃₁	—	2	3	no	—
278	C₂×C₄×C₃₁	—	2	1	no	—
279	C₂×C₄×C₃₁	—	2	3	no	—
280	C₂×C₄×C₃₁	—	2	1	no	—
281	C₂×C₄×C₃₁	—	2	3	no	—
282	C₂×C₄×C₃₁	—	4	3	no	—
283	C₂×C₄×C₃₁	—	4	6	no	—
284	C₂×C₄×C₃₁	—	4	2	no	—
285	C₂×C₄×C₃₁	—	4	6	no	—
286	C₂×C₄×C₃₁	—	4	6	no	—
287	C₂×C₄×C₃₁	—	4	6	no	—
288	C₂×C₄×C₃₁	—	4	6	no	—
289	C₂×C₄×C₃₁	—	4	6	no	—
290	C₂×C₄×C₃₁	—	4	6	no	—
291	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
293	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
295	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
296	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
297	C₈×C₃₁	—	2	3	no	—
298	C₈×C₃₁	—	2	1	no	—
299	C₈×C₃₁	—	2	3	no	—
300	C₈×C₃₁	C₂×#41	1	2	yes	complemented
301	#3483	C₂×C₃₁	1	1	yes	characteristic, complemented
302	D₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
304	Q₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#3362	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#3358	C₂×C₃₁	1	1	yes	characteristic, complemented
307	#3354	C₂×C₃₁	1	3	yes	complemented
308	#3332	C₂×C₃₁	1	3	yes	complemented
309	#3325	C₂×C₃₁	1	3	yes	complemented
310	#3274	C₂×C₃₁	1	3	yes	complemented
311	#3184	C₂×C₃₁	1	3	yes	complemented
312	#3115	C₂×C₃₁	1	3	yes	complemented
313	#3109	C₂×C₃₁	1	3	yes	complemented
314	#2971	C₂×C₃₁	1	1	yes	characteristic, complemented
315	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	3	yes	complemented
316	C₂²×C₃₁	—	2	3	no	—
317	C₂²×C₃₁	—	2	3	no	—
318	C₂²×C₃₁	—	2	2	no	—
319	C₂²×C₃₁	—	2	3	no	—
320	C₂²×C₃₁	—	4	1	no	—
321	C₂²×C₃₁	—	4	1	no	—
322	C₂²×C₃₁	—	4	3	no	—
323	C₂²×C₃₁	—	4	3	no	—
324	C₂²×C₃₁	—	4	3	no	—
325	C₂²×C₃₁	—	4	3	no	—
326	C₂²×C₃₁	—	4	3	no	—
327	C₂²×C₃₁	—	4	3	no	—
328	C₂²×C₃₁	—	8	12	no	—
329	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
330	C₄×C₃₁	—	2	1	no	—
331	C₄×C₃₁	—	2	3	no	—
332	C₄×C₃₁	—	2	3	no	—
333	C₄×C₃₁	—	2	2	no	—
334	C₄×C₃₁	—	2	1	no	—
335	C₄×C₃₁	—	4	3	no	—
336	C₄×C₃₁	—	4	1	no	—
337	C₄×C₃₁	—	4	1	no	—
338	C₄×C₃₁	—	4	3	no	—
339	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
340	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
341	C₄×C₃₁	#585	1	1	yes	characteristic
342	C₂²×#41	—	2	1	no	—
343	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	3	yes	—
344	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
345	#555	—	2	3	no	—
346	#555	C₂²×C₃₁	1	3	yes	complemented
347	#550	—	2	1	no	—
348	#550	C₂²×C₃₁	1	1	yes	characteristic
349	#550	C₂²×C₃₁	1	3	yes	complemented
350	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
351	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
352	#548	—	2	1	no	—
353	#547	C₂²×C₃₁	1	3	yes	—
354	#546	—	2	3	no	—
355	D₄²	C₂²×C₃₁	1	3	yes	—
356	#532	—	2	3	no	—
357	#532	C₂²×C₃₁	1	3	yes	—
358	#532	C₂²×C₃₁	1	3	yes	complemented
359	#525	—	2	3	no	—
360	C₄×#41	—	2	1	no	—
361	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
362	C₄×#41	C₂²×C₃₁	1	3	yes	complemented
363	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
364	#498	C₂²×C₃₁	1	3	yes	complemented
365	#493	C₂²×C₃₁	1	3	yes	complemented
366	#486	C₂²×C₃₁	1	3	yes	complemented
367	#486	C₂²×C₃₁	1	3	yes	complemented
368	#482	C₂²×C₃₁	1	6	yes	complemented
369	#481	C₂²×C₃₁	1	6	yes	complemented
370	#476	C₂²×C₃₁	1	3	yes	—
371	#476	C₂²×C₃₁	1	3	yes	complemented
372	#475	C₂²×C₃₁	1	3	yes	complemented
373	#474	C₂²×C₃₁	1	3	yes	complemented
374	#469	C₂²×C₃₁	1	6	yes	complemented
375	#469	C₂²×C₃₁	1	6	yes	complemented
376	#460	C₂²×C₃₁	1	3	yes	complemented
377	#459	C₂²×C₃₁	1	3	yes	—
378	#459	C₂²×C₃₁	1	3	yes	complemented
379	#459	C₂²×C₃₁	1	3	yes	complemented
380	#449	C₂²×C₃₁	1	6	yes	complemented
381	#447	C₂²×C₃₁	1	6	yes	—
382	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
383	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
384	#442	C₂²×C₃₁	1	3	yes	—
385	#442	C₂²×C₃₁	1	6	yes	complemented
386	#442	C₂²×C₃₁	1	3	yes	complemented
387	#440	C₂²×C₃₁	1	1	yes	characteristic
388	#440	C₂²×C₃₁	1	3	yes	complemented
389	C₄×SD₁₆	C₂²×C₃₁	1	3	yes	—
390	C₄×SD₁₆	C₂²×C₃₁	1	3	yes	complemented
391	C₄×D₈	C₂²×C₃₁	1	3	yes	—
392	C₄×D₈	C₂²×C₃₁	1	3	yes	complemented
393	#436	C₂²×C₃₁	1	3	yes	complemented
394	#436	C₂²×C₃₁	1	3	yes	complemented
395	#432	C₂²×C₃₁	1	3	yes	complemented
396	#428	C₂²×C₃₁	1	3	yes	complemented
397	#418	C₂²×C₃₁	1	3	yes	—
398	#417	C₂²×C₃₁	1	3	yes	complemented
399	C₄×M₁₆	C₂²×C₃₁	1	3	yes	complemented
400	C₂×C₃₁	—	4	3	no	—
401	C₂×C₃₁	—	4	2	no	—
402	C₂×C₃₁	—	8	3	no	—
403	C₂×C₃₁	—	8	3	no	—
404	C₂×C₃₁	D₄×#41	1	1	yes	characteristic
405	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
406	C₂×C₃₁	#3585	1	1	yes	characteristic
407	C₂×#41	—	2	1	no	—
408	C₂×#41	—	2	1	no	—
409	C₂×#41	—	2	3	no	—
410	C₂×#41	—	2	3	no	—
411	C₂×#41	—	2	3	no	—
412	C₂×#41	—	4	8	no	—
413	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
414	C₂×#41	C₂³×C₃₁	1	3	yes	—
415	C₂²×Q₈	—	2	1	no	—
416	C₂²×D₄	—	2	3	no	—
417	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
418	C₂³×C₄	—	2	3	no	—
419	AGL(1,ℤ/8)	—	2	12	no	—
420	C₂×SD₁₆	C₂³×C₃₁	1	3	yes	—
421	C₂×SD₁₆	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×D₈	C₂³×C₃₁	1	3	yes	—
423	C₂×D₈	C₂³×C₃₁	1	3	yes	—
424	C₂×D₈	C₂³×C₃₁	1	6	yes	—
425	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
426	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
427	C₄⋊Q₈	—	2	3	no	—
428	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
429	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
430	#127	—	2	3	no	—
431	#127	C₂³×C₃₁	1	3	yes	—
432	#127	C₂³×C₃₁	1	3	yes	—
433	#124	—	2	6	no	—
434	#124	—	2	3	no	—
435	#124	—	2	3	no	—
436	#123	—	2	6	no	—
437	C₂²⋊Q₈	—	2	3	no	—
438	C₂²⋊Q₈	—	2	3	no	—
439	#121	—	2	3	no	—
440	#121	—	2	3	no	—
441	#121	—	2	3	no	—
442	#121	—	2	3	no	—
443	#121	—	2	3	no	—
444	#121	—	2	3	no	—
445	#121	—	2	6	no	—
446	#121	C₂³×C₃₁	1	6	yes	—
447	#121	C₂³×C₃₁	1	6	yes	—
448	#120	—	2	6	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
451	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
452	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
453	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
454	C₄×D₄	—	2	3	no	—
455	C₄×D₄	—	2	3	no	—
456	C₄×D₄	—	2	3	no	—
457	C₄×D₄	—	2	3	no	—
458	C₄×D₄	—	2	3	no	—
459	C₄×D₄	—	2	3	no	—
460	C₄×D₄	—	2	3	no	—
461	C₄×D₄	D₄×C₃₁	1	3	yes	—
462	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
463	C₄×D₄	C₂³×C₃₁	1	3	yes	—
464	C₄×D₄	C₂³×C₃₁	1	3	yes	—
465	C₄×D₄	C₂³×C₃₁	1	3	yes	—
466	C₄×D₄	C₂³×C₃₁	1	3	yes	—
467	C₄×D₄	C₂³×C₃₁	1	3	yes	—
468	C₄×D₄	C₂³×C₃₁	1	3	yes	—
469	#117	—	2	3	no	—
470	#117	—	2	3	no	—
471	#117	D₄×C₃₁	1	3	yes	—
472	#117	C₂³×C₃₁	1	3	yes	—
473	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
474	C₂×C₂²⋊C₄	—	2	3	no	—
475	C₂×C₄²	—	2	3	no	—
476	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
477	#107	C₂³×C₃₁	1	1	yes	characteristic
478	#107	C₂³×C₃₁	1	3	yes	—
479	#106	C₂³×C₃₁	1	3	yes	—
480	#106	C₂³×C₃₁	1	3	yes	—
481	#106	C₂³×C₃₁	1	6	yes	—
482	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
483	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
484	#103	C₂³×C₃₁	1	1	yes	characteristic
485	#103	C₂³×C₃₁	1	3	yes	—
486	#102	C₂³×C₃₁	1	3	yes	—
487	#102	C₂³×C₃₁	1	3	yes	—
488	#102	C₂³×C₃₁	1	1	yes	characteristic
489	#102	C₂³×C₃₁	1	3	yes	—
490	#102	C₂³×C₃₁	1	3	yes	—
491	#102	C₂³×C₃₁	1	3	yes	—
492	#102	C₂³×C₃₁	1	6	yes	—
493	#102	C₂³×C₃₁	1	6	yes	—
494	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
495	#97	C₂³×C₃₁	1	1	yes	characteristic
496	#97	C₂³×C₃₁	1	3	yes	—
497	C₄×C₈	C₂³×C₃₁	1	3	yes	—
498	C₄×C₈	C₂³×C₃₁	1	3	yes	—
499	C₃₁	#61465	1	1	yes	characteristic, direct product factor, Sylow
500	C₂⁴	—	2	3	no	—
501	#41	—	2	4	no	—
502	#41	—	2	6	no	—
503	#41	—	4	3	no	—
504	#41	—	4	12	no	—
505	#41	—	4	12	no	—
506	#41	—	4	4	no	—
507	#41	—	4	4	no	—
508	#41	—	8	6	no	—
509	C₂×Q₈	—	2	3	no	—
510	C₂×Q₈	—	2	3	no	—
511	C₂×Q₈	—	4	4	no	—
512	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
513	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
514	C₂×D₄	—	2	6	no	—
515	C₂×D₄	—	2	3	no	—
516	C₂×D₄	—	2	6	no	—
517	C₂×D₄	—	2	6	no	—
518	C₂×D₄	—	2	3	no	—
519	C₂×D₄	—	2	3	no	—
520	C₂×D₄	—	2	3	no	—
521	C₂×D₄	—	2	3	no	—
522	C₂×D₄	—	2	3	no	—
523	C₂×D₄	—	4	3	no	—
524	C₂×D₄	—	4	12	no	—
525	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
526	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
527	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
528	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₂²×C₄	—	2	3	no	—
530	C₂²×C₄	—	2	3	no	—
531	C₂²×C₄	—	2	3	no	—
532	C₂²×C₄	—	2	3	no	—
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	3	no	—
535	C₂²×C₄	—	2	3	no	—
536	C₂²×C₄	—	2	3	no	—
537	C₂²×C₄	—	4	12	no	—
538	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
539	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
540	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
541	SD₁₆	—	2	6	no	—
542	SD₁₆	#41×C₃₁	1	4	yes	complemented
543	D₈	—	2	6	no	—
544	D₈	—	2	6	no	—
545	D₈	—	4	6	no	—
546	M₁₆	—	2	6	no	—
547	M₁₆	—	4	3	no	—
548	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
549	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
550	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
551	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
552	C₄⋊C₄	—	2	3	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	3	no	—
555	C₄⋊C₄	—	2	3	no	—
556	C₄⋊C₄	—	2	3	no	—
557	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
558	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
559	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
560	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
561	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
562	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
563	C₂²⋊C₄	—	2	3	no	—
564	C₂²⋊C₄	—	2	1	no	—
565	C₂²⋊C₄	—	2	3	no	—
566	C₂²⋊C₄	—	2	3	no	—
567	C₂²⋊C₄	—	2	3	no	—
568	C₂²⋊C₄	—	2	6	no	—
569	C₂²⋊C₄	—	2	3	no	—
570	C₂²⋊C₄	—	2	3	no	—
571	C₂²⋊C₄	—	2	6	no	—
572	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
573	C₄²	—	2	1	no	—
574	C₄²	—	2	3	no	—
575	C₄²	—	2	3	no	—
576	C₄²	C₂×D₄×C₃₁	1	3	yes	—
577	C₄²	C₂×D₄×C₃₁	1	3	yes	—
578	C₄²	C₂⁴×C₃₁	1	3	yes	—
579	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
580	C₂³	—	2	1	no	—
581	C₂³	—	2	3	no	—
582	C₂³	—	2	3	no	—
583	C₂³	—	4	6	no	—
584	C₂³	—	4	6	no	—
585	C₂³	—	8	3	no	—
586	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
587	Q₈	—	2	1	no	—
588	Q₈	—	4	6	no	—
589	Q₈	—	8	3	no	—
590	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
591	Q₈	C₂×#41×C₃₁	1	2	yes	complemented
592	D₄	—	2	6	no	—
593	D₄	—	2	3	no	—
594	D₄	—	2	3	no	—
595	D₄	—	2	1	no	—
596	D₄	—	2	3	no	—
597	D₄	—	2	3	no	—
598	D₄	—	4	6	no	—
599	D₄	—	4	6	no	—
600	D₄	—	4	12	no	—
601	D₄	—	4	6	no	—
602	D₄	—	4	6	no	—
603	D₄	—	8	3	no	—
604	D₄	—	8	3	no	—
605	D₄	—	8	6	no	—
606	D₄	C₂×#41×C₃₁	1	2	yes	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	6	no	—
609	C₂×C₄	—	2	3	no	—
610	C₂×C₄	—	2	3	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	3	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	3	no	—
615	C₂×C₄	—	4	3	no	—
616	C₂×C₄	—	4	6	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	6	no	—
619	C₂×C₄	—	4	6	no	—
620	C₂×C₄	—	4	6	no	—
621	C₂×C₄	—	4	6	no	—
622	C₂×C₄	—	4	6	no	—
623	C₂×C₄	—	4	6	no	—
624	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
625	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
626	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
628	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
629	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
630	C₈	—	2	3	no	—
631	C₈	—	2	1	no	—
632	C₈	—	2	3	no	—
633	C₈	C₂×#41×C₃₁	1	2	yes	complemented
634	C₂²	—	2	3	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	3	no	—
637	C₂²	—	2	3	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	3	no	—
641	C₂²	—	4	3	no	—
642	C₂²	—	4	3	no	—
643	C₂²	—	4	3	no	—
644	C₂²	—	4	3	no	—
645	C₂²	—	4	3	no	—
646	C₂²	—	8	12	no	—
647	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
648	C₄	—	2	3	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	1	no	—
651	C₄	—	2	1	no	—
652	C₄	—	2	3	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	3	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	3	no	—
657	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
658	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
659	C₄	C₃₁×#585	1	1	yes	characteristic
660	C₂	—	4	3	no	—
661	C₂	—	4	2	no	—
662	C₂	—	8	3	no	—
663	C₂	—	8	3	no	—
664	C₂	D₄×#41×C₃₁	1	1	yes	characteristic
665	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
666	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3483	C₂	semidirect
2. D₄×#41×C₃₁	C₂	semidirect
3. C₃₁×#3391	C₂	semidirect
4. Q₈×SD₁₆×C₃₁	C₂	semidirect
5. C₃₁×#3362	C₂	semidirect
6. C₃₁×#3358	C₂	semidirect
7. C₃₁×#3354	C₂	semidirect
8. C₃₁×#3332	C₂	semidirect
9. C₃₁×#3325	C₂	semidirect
10. C₃₁×#3274	C₂	semidirect
11. C₃₁×#3184	C₂	semidirect
12. C₃₁×#3115	C₂	semidirect
13. C₃₁×#3109	C₂	semidirect
14. C₃₁×#2971	C₂	semidirect
15. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
18. Q₈²×C₃₁	C₂²	semidirect
20. C₃₁×#555	C₂²	semidirect
23. C₃₁×#550	C₂²	semidirect
24. C₃₁×#550	C₂²	semidirect
32. C₃₁×#532	C₂²	semidirect
36. C₄×#41×C₃₁	C₂²	semidirect
38. C₃₁×#498	C₂²	semidirect
39. C₃₁×#493	C₂²	semidirect
40. C₃₁×#486	C₂²	semidirect
41. C₃₁×#486	C₂²	semidirect
42. C₃₁×#482	C₂²	semidirect
43. C₃₁×#481	C₂²	semidirect
45. C₃₁×#476	C₂²	semidirect
46. C₃₁×#475	C₂²	semidirect
47. C₃₁×#474	C₂²	semidirect
48. C₃₁×#469	C₂²	semidirect
49. C₃₁×#469	C₂²	semidirect
50. C₃₁×#460	C₂²	semidirect
52. C₃₁×#459	C₂²	semidirect
53. C₃₁×#459	C₂²	semidirect
54. C₃₁×#449	C₂²	semidirect
56. C₈×Q₈×C₃₁	C₂²	semidirect
57. C₈×Q₈×C₃₁	C₂²	semidirect
59. C₃₁×#442	C₂²	semidirect
60. C₃₁×#442	C₂²	semidirect
62. C₃₁×#440	C₂²	semidirect
64. C₄×SD₁₆×C₃₁	C₂²	semidirect
66. C₄×D₈×C₃₁	C₂²	semidirect
67. C₃₁×#436	C₂²	semidirect
68. C₃₁×#436	C₂²	semidirect
69. C₃₁×#432	C₂²	semidirect
70. C₃₁×#428	C₂²	semidirect
72. C₃₁×#417	C₂²	semidirect
73. C₄×M₁₆×C₃₁	C₂²	semidirect
208. SD₁₆×C₃₁	#41	semidirect
246. #61465	C₃₁	direct (N × H)
257. Q₈×C₃₁	AGL(1,ℤ/8)	semidirect
258. Q₈×C₃₁	C₂×#41	semidirect
300. C₈×C₃₁	C₂×#41	semidirect
301. #3483	C₂×C₃₁	semidirect
302. D₄×#41	C₂×C₃₁	semidirect
303. #3391	C₂×C₃₁	semidirect
304. Q₈×SD₁₆	C₂×C₃₁	semidirect
305. #3362	C₂×C₃₁	semidirect
306. #3358	C₂×C₃₁	semidirect
307. #3354	C₂×C₃₁	semidirect
308. #3332	C₂×C₃₁	semidirect
309. #3325	C₂×C₃₁	semidirect
310. #3274	C₂×C₃₁	semidirect
311. #3184	C₂×C₃₁	semidirect
312. #3115	C₂×C₃₁	semidirect
313. #3109	C₂×C₃₁	semidirect
314. #2971	C₂×C₃₁	semidirect
315. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
344. Q₈²	C₂²×C₃₁	semidirect
346. #555	C₂²×C₃₁	semidirect
349. #550	C₂²×C₃₁	semidirect
350. #550	C₂²×C₃₁	semidirect
358. #532	C₂²×C₃₁	semidirect
362. C₄×#41	C₂²×C₃₁	semidirect
364. #498	C₂²×C₃₁	semidirect
365. #493	C₂²×C₃₁	semidirect
366. #486	C₂²×C₃₁	semidirect
367. #486	C₂²×C₃₁	semidirect
368. #482	C₂²×C₃₁	semidirect
369. #481	C₂²×C₃₁	semidirect
371. #476	C₂²×C₃₁	semidirect
372. #475	C₂²×C₃₁	semidirect
373. #474	C₂²×C₃₁	semidirect
374. #469	C₂²×C₃₁	semidirect
375. #469	C₂²×C₃₁	semidirect
376. #460	C₂²×C₃₁	semidirect
378. #459	C₂²×C₃₁	semidirect
379. #459	C₂²×C₃₁	semidirect
380. #449	C₂²×C₃₁	semidirect
382. C₈×Q₈	C₂²×C₃₁	semidirect
383. C₈×Q₈	C₂²×C₃₁	semidirect
385. #442	C₂²×C₃₁	semidirect
386. #442	C₂²×C₃₁	semidirect
388. #440	C₂²×C₃₁	semidirect
390. C₄×SD₁₆	C₂²×C₃₁	semidirect
392. C₄×D₈	C₂²×C₃₁	semidirect
393. #436	C₂²×C₃₁	semidirect
394. #436	C₂²×C₃₁	semidirect
395. #432	C₂²×C₃₁	semidirect
396. #428	C₂²×C₃₁	semidirect
398. #417	C₂²×C₃₁	semidirect
399. C₄×M₁₆	C₂²×C₃₁	semidirect
499. C₃₁	#61465	direct (N × H)
542. SD₁₆	#41×C₃₁	semidirect
590. Q₈	C₃₁×AGL(1,ℤ/8)	semidirect
591. Q₈	C₂×#41×C₃₁	semidirect
633. C₈	C₂×#41×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3483	2
D₄×#41×C₃₁	2
C₃₁×#3391	2
Q₈×SD₁₆×C₃₁	2
C₃₁×#3362	2
C₃₁×#3358	2
C₃₁×#3354	2
C₃₁×#3332	2
C₃₁×#3325	2
C₃₁×#3274	2
C₃₁×#3184	2
C₃₁×#3115	2
C₃₁×#3109	2
C₃₁×#2971	2
C₄×C₃₁×AGL(1,ℤ/8)	2
#61465	31

Subgroups — C31×#61465 #21518224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61465 #21518224