Subgroups — C₃₁×#62003 #21518762

Order 7936.

2126 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3517	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3392	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3359	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3332	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3330	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3274	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3272	C₂	1	3	yes	maximal, complemented
11	C₃₁×#3251	C₂	1	3	yes	maximal, complemented
12	C₃₁×#3207	C₂	1	3	yes	maximal, complemented
13	C₃₁×#3105	C₂	1	3	yes	maximal, complemented
14	C₃₁×#3036	C₂	1	3	yes	maximal, complemented
15	C₃₁×#2991	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₃₁	—	2	1	no	—
17	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	6	yes	—
18	C₃₁×#557	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#553	—	2	3	no	—
20	C₃₁×#550	—	2	1	no	—
21	C₃₁×#550	C₂²	1	1	yes	characteristic
22	C₃₁×#550	C₂²	1	3	yes	complemented
23	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
24	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#548	—	2	1	no	—
26	C₃₁×#547	C₂²	1	3	yes	—
27	C₃₁×#547	C₂²	1	3	yes	complemented
28	C₃₁×#546	—	2	3	no	—
29	D₄²×C₃₁	C₂²	1	3	yes	—
30	C₃₁×#540	—	2	3	no	—
31	C₃₁×#532	C₂²	1	3	yes	—
32	C₃₁×#532	C₂²	1	3	yes	complemented
33	C₃₁×#529	C₂²	1	3	yes	complemented
34	C₃₁×#525	—	2	3	no	—
35	C₃₁×#520	—	2	1	no	—
36	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
38	C₃₁×#493	C₂²	1	3	yes	complemented
39	C₃₁×#492	C₂²	1	3	yes	complemented
40	C₃₁×#487	C₂²	1	3	yes	complemented
41	C₃₁×#486	C₂²	1	3	yes	complemented
42	C₃₁×#484	C₂²	1	3	yes	complemented
43	C₃₁×#481	C₂²	1	3	yes	complemented
44	C₃₁×#478	C₂²	1	1	yes	characteristic
45	C₃₁×#478	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×#468	C₂²	1	6	yes	complemented
50	C₃₁×#463	C₂²	1	3	yes	—
51	C₃₁×#463	C₂²	1	3	yes	complemented
52	C₃₁×#463	C₂²	1	6	yes	complemented
53	C₃₁×#461	C₂²	1	3	yes	—
54	C₃₁×#461	C₂²	1	3	yes	complemented
55	C₃₁×#460	C₂²	1	3	yes	complemented
56	C₃₁×#459	C₂²	1	3	yes	—
57	C₃₁×#459	C₂²	1	3	yes	complemented
58	C₃₁×#459	C₂²	1	3	yes	complemented
59	C₃₁×#450	C₂²	1	3	yes	—
60	C₃₁×#449	C₂²	1	6	yes	complemented
61	C₃₁×#449	C₂²	1	6	yes	complemented
62	C₃₁×#447	C₂²	1	3	yes	—
63	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
64	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
65	C₄×SD₁₆×C₃₁	C₂²	1	3	yes	—
66	C₄×SD₁₆×C₃₁	C₂²	1	3	yes	complemented
67	C₄×D₈×C₃₁	C₂²	1	3	yes	—
68	C₄×D₈×C₃₁	C₂²	1	3	yes	complemented
69	C₃₁×#436	C₂²	1	3	yes	complemented
70	C₃₁×#436	C₂²	1	3	yes	complemented
71	C₃₁×#431	C₂²	1	3	yes	complemented
72	C₃₁×#423	C₂²	1	3	yes	complemented
73	C₃₁×#417	C₂²	1	6	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	—	4	8	no	—
79	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
80	C₂×#41×C₃₁	C₂³	1	6	yes	—
81	C₂²×Q₈×C₃₁	—	2	1	no	—
82	C₂²×D₄×C₃₁	—	2	3	no	—
83	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
84	C₂³×C₄×C₃₁	—	2	3	no	—
85	C₃₁×AGL(1,ℤ/8)	—	4	12	no	—
86	C₂×SD₁₆×C₃₁	C₂³	1	3	yes	—
87	C₂×SD₁₆×C₃₁	C₂³	1	3	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
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	6	yes	—
93	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
94	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
95	C₃₁×#127	—	2	3	no	—
96	C₃₁×#127	C₂³	1	3	yes	—
97	C₃₁×#127	C₂³	1	3	yes	—
98	C₃₁×#126	—	2	3	no	—
99	C₃₁×#125	—	2	3	no	—
100	C₃₁×#125	C₂³	1	1	yes	characteristic
101	C₃₁×#124	—	2	3	no	—
102	C₃₁×#124	C₂³	1	3	yes	—
103	C₃₁×#123	—	2	3	no	—
104	C₃₁×#123	—	2	6	no	—
105	C₃₁×#123	—	2	6	no	—
106	C₃₁×C₂²⋊Q₈	—	2	3	no	—
107	C₃₁×C₂²⋊Q₈	—	2	3	no	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
109	C₃₁×#121	—	2	3	no	—
110	C₃₁×#121	—	2	3	no	—
111	C₃₁×#121	—	2	3	no	—
112	C₃₁×#121	—	2	3	no	—
113	C₃₁×#121	—	2	3	no	—
114	C₃₁×#121	—	2	3	no	—
115	C₃₁×#121	—	2	3	no	—
116	C₃₁×#121	—	2	6	no	—
117	C₃₁×#121	C₂³	1	3	yes	—
118	C₃₁×#121	C₂³	1	6	yes	—
119	C₃₁×#120	—	2	6	no	—
120	C₄×Q₈×C₃₁	—	2	1	no	—
121	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
122	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
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	—	2	3	no	—
129	C₄×D₄×C₃₁	—	2	3	no	—
130	C₄×D₄×C₃₁	—	2	3	no	—
131	C₄×D₄×C₃₁	D₄	1	3	yes	—
132	C₄×D₄×C₃₁	C₂³	1	3	yes	—
133	C₄×D₄×C₃₁	C₂³	1	3	yes	—
134	C₄×D₄×C₃₁	C₂³	1	3	yes	—
135	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
136	C₄×D₄×C₃₁	C₂³	1	3	yes	—
137	C₄×D₄×C₃₁	C₂³	1	3	yes	—
138	C₃₁×#117	—	2	3	no	—
139	C₃₁×#117	D₄	1	3	yes	—
140	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
141	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
143	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
144	C₃₁×#107	C₂³	1	3	yes	—
145	C₃₁×#107	C₂³	1	1	yes	characteristic
146	C₃₁×#106	C₂³	1	3	yes	—
147	C₃₁×#106	C₂³	1	1	yes	characteristic
148	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
152	C₃₁×#103	C₂³	1	3	yes	—
153	C₃₁×#103	C₂³	1	6	yes	—
154	C₃₁×#103	C₂³	1	3	yes	—
155	C₃₁×#102	C₂³	1	3	yes	—
156	C₃₁×#102	C₂³	1	6	yes	—
157	C₃₁×#102	C₂³	1	1	yes	characteristic
158	C₃₁×#102	C₂³	1	3	yes	—
159	C₃₁×#102	C₂³	1	3	yes	—
160	C₃₁×#102	C₂³	1	1	yes	characteristic
161	C₃₁×#102	C₂³	1	3	yes	—
162	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
163	C₃₁×C₂²⋊C₈	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	—	4	12	no	—
169	#41×C₃₁	—	4	6	no	—
170	#41×C₃₁	—	4	6	no	—
171	#41×C₃₁	—	4	12	no	—
172	#41×C₃₁	—	4	4	no	—
173	#41×C₃₁	—	4	4	no	—
174	C₂×Q₈×C₃₁	—	2	3	no	—
175	C₂×Q₈×C₃₁	—	4	4	no	—
176	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
178	C₂×D₄×C₃₁	—	2	6	no	—
179	C₂×D₄×C₃₁	—	2	6	no	—
180	C₂×D₄×C₃₁	—	2	3	no	—
181	C₂×D₄×C₃₁	—	2	6	no	—
182	C₂×D₄×C₃₁	—	2	3	no	—
183	C₂×D₄×C₃₁	—	2	3	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	—	4	3	no	—
188	C₂×D₄×C₃₁	—	4	12	no	—
189	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
190	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
191	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
192	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₃₁	—	2	3	no	—
194	C₂²×C₄×C₃₁	—	2	3	no	—
195	C₂²×C₄×C₃₁	—	2	1	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	6	no	—
200	C₂²×C₄×C₃₁	—	4	12	no	—
201	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
202	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
203	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
204	SD₁₆×C₃₁	—	2	6	no	—
205	SD₁₆×C₃₁	—	4	3	no	—
206	SD₁₆×C₃₁	—	4	6	no	—
207	D₈×C₃₁	—	2	6	no	—
208	D₈×C₃₁	—	4	3	no	—
209	D₈×C₃₁	—	4	6	no	—
210	M₁₆×C₃₁	—	4	6	no	—
211	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
213	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
214	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
215	C₄⋊C₄×C₃₁	—	2	3	no	—
216	C₄⋊C₄×C₃₁	—	2	3	no	—
217	C₄⋊C₄×C₃₁	—	2	3	no	—
218	C₄⋊C₄×C₃₁	—	2	1	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	1	yes	characteristic
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	3	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	—	2	3	no	—
239	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
240	C₄²×C₃₁	—	2	3	no	—
241	C₄²×C₃₁	C₂×D₄	1	3	yes	—
242	C₄²×C₃₁	C₂×D₄	1	3	yes	—
243	C₄²×C₃₁	C₂⁴	1	3	yes	—
244	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
245	#62003	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂³×C₃₁	—	2	1	no	—
247	C₂³×C₃₁	—	2	3	no	—
248	C₂³×C₃₁	—	2	3	no	—
249	C₂³×C₃₁	—	4	6	no	—
250	C₂³×C₃₁	—	4	6	no	—
251	C₂³×C₃₁	—	8	3	no	—
252	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
253	Q₈×C₃₁	—	2	3	no	—
254	Q₈×C₃₁	—	2	3	no	—
255	Q₈×C₃₁	—	4	6	no	—
256	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
257	D₄×C₃₁	—	2	6	no	—
258	D₄×C₃₁	—	2	3	no	—
259	D₄×C₃₁	—	2	3	no	—
260	D₄×C₃₁	—	2	1	no	—
261	D₄×C₃₁	—	2	3	no	—
262	D₄×C₃₁	—	2	3	no	—
263	D₄×C₃₁	—	4	6	no	—
264	D₄×C₃₁	—	4	12	no	—
265	D₄×C₃₁	—	4	6	no	—
266	D₄×C₃₁	—	4	6	no	—
267	D₄×C₃₁	—	8	3	no	—
268	D₄×C₃₁	—	8	3	no	—
269	D₄×C₃₁	—	8	3	no	—
270	D₄×C₃₁	—	8	6	no	—
271	D₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
272	C₂×C₄×C₃₁	—	2	6	no	—
273	C₂×C₄×C₃₁	—	2	3	no	—
274	C₂×C₄×C₃₁	—	2	3	no	—
275	C₂×C₄×C₃₁	—	2	1	no	—
276	C₂×C₄×C₃₁	—	2	3	no	—
277	C₂×C₄×C₃₁	—	2	1	no	—
278	C₂×C₄×C₃₁	—	2	1	no	—
279	C₂×C₄×C₃₁	—	2	3	no	—
280	C₂×C₄×C₃₁	—	4	3	no	—
281	C₂×C₄×C₃₁	—	4	2	no	—
282	C₂×C₄×C₃₁	—	4	6	no	—
283	C₂×C₄×C₃₁	—	4	6	no	—
284	C₂×C₄×C₃₁	—	4	6	no	—
285	C₂×C₄×C₃₁	—	4	6	no	—
286	C₂×C₄×C₃₁	—	4	6	no	—
287	C₂×C₄×C₃₁	—	8	6	no	—
288	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
291	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
292	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
293	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
294	C₈×C₃₁	—	2	3	no	—
295	C₈×C₃₁	—	2	3	no	—
296	C₈×C₃₁	—	2	1	no	—
297	C₈×C₃₁	—	2	1	no	—
298	#3517	C₂×C₃₁	1	1	yes	characteristic, complemented
299	D₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#3392	C₂×C₃₁	1	1	yes	characteristic, complemented
301	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#3359	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#3358	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#3332	C₂×C₃₁	1	3	yes	complemented
305	#3330	C₂×C₃₁	1	3	yes	complemented
306	#3274	C₂×C₃₁	1	3	yes	complemented
307	#3272	C₂×C₃₁	1	3	yes	complemented
308	#3251	C₂×C₃₁	1	3	yes	complemented
309	#3207	C₂×C₃₁	1	3	yes	complemented
310	#3105	C₂×C₃₁	1	3	yes	complemented
311	#3036	C₂×C₃₁	1	3	yes	complemented
312	#2991	C₂×C₃₁	1	1	yes	characteristic, complemented
313	C₂²×C₃₁	—	2	3	no	—
314	C₂²×C₃₁	—	2	3	no	—
315	C₂²×C₃₁	—	2	2	no	—
316	C₂²×C₃₁	—	2	3	no	—
317	C₂²×C₃₁	—	4	1	no	—
318	C₂²×C₃₁	—	4	1	no	—
319	C₂²×C₃₁	—	4	3	no	—
320	C₂²×C₃₁	—	4	3	no	—
321	C₂²×C₃₁	—	4	3	no	—
322	C₂²×C₃₁	—	4	3	no	—
323	C₂²×C₃₁	—	4	3	no	—
324	C₂²×C₃₁	—	4	3	no	—
325	C₂²×C₃₁	—	8	12	no	—
326	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
327	C₄×C₃₁	—	2	1	no	—
328	C₄×C₃₁	—	2	3	no	—
329	C₄×C₃₁	—	2	3	no	—
330	C₄×C₃₁	—	2	1	no	—
331	C₄×C₃₁	—	4	1	no	—
332	C₄×C₃₁	—	4	3	no	—
333	C₄×C₃₁	—	4	1	no	—
334	C₄×C₃₁	—	4	1	no	—
335	C₄×C₃₁	—	4	3	no	—
336	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
337	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
338	C₄×C₃₁	#585	1	1	yes	characteristic
339	C₂²×#41	—	2	1	no	—
340	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	6	yes	—
341	#557	C₂²×C₃₁	1	1	yes	characteristic, complemented
342	#553	—	2	3	no	—
343	#550	—	2	1	no	—
344	#550	C₂²×C₃₁	1	1	yes	characteristic
345	#550	C₂²×C₃₁	1	3	yes	complemented
346	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
347	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
348	#548	—	2	1	no	—
349	#547	C₂²×C₃₁	1	3	yes	—
350	#547	C₂²×C₃₁	1	3	yes	complemented
351	#546	—	2	3	no	—
352	D₄²	C₂²×C₃₁	1	3	yes	—
353	#540	—	2	3	no	—
354	#532	C₂²×C₃₁	1	3	yes	—
355	#532	C₂²×C₃₁	1	3	yes	complemented
356	#529	C₂²×C₃₁	1	3	yes	complemented
357	#525	—	2	3	no	—
358	#520	—	2	1	no	—
359	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
361	#493	C₂²×C₃₁	1	3	yes	complemented
362	#492	C₂²×C₃₁	1	3	yes	complemented
363	#487	C₂²×C₃₁	1	3	yes	complemented
364	#486	C₂²×C₃₁	1	3	yes	complemented
365	#484	C₂²×C₃₁	1	3	yes	complemented
366	#481	C₂²×C₃₁	1	3	yes	complemented
367	#478	C₂²×C₃₁	1	1	yes	characteristic
368	#478	C₂²×C₃₁	1	3	yes	complemented
369	#475	C₂²×C₃₁	1	3	yes	complemented
370	#474	C₂²×C₃₁	1	3	yes	complemented
371	#469	C₂²×C₃₁	1	6	yes	complemented
372	#468	C₂²×C₃₁	1	6	yes	complemented
373	#463	C₂²×C₃₁	1	3	yes	—
374	#463	C₂²×C₃₁	1	3	yes	complemented
375	#463	C₂²×C₃₁	1	6	yes	complemented
376	#461	C₂²×C₃₁	1	3	yes	—
377	#461	C₂²×C₃₁	1	3	yes	complemented
378	#460	C₂²×C₃₁	1	3	yes	complemented
379	#459	C₂²×C₃₁	1	3	yes	—
380	#459	C₂²×C₃₁	1	3	yes	complemented
381	#459	C₂²×C₃₁	1	3	yes	complemented
382	#450	C₂²×C₃₁	1	3	yes	—
383	#449	C₂²×C₃₁	1	6	yes	complemented
384	#449	C₂²×C₃₁	1	6	yes	complemented
385	#447	C₂²×C₃₁	1	3	yes	—
386	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
387	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
388	C₄×SD₁₆	C₂²×C₃₁	1	3	yes	—
389	C₄×SD₁₆	C₂²×C₃₁	1	3	yes	complemented
390	C₄×D₈	C₂²×C₃₁	1	3	yes	—
391	C₄×D₈	C₂²×C₃₁	1	3	yes	complemented
392	#436	C₂²×C₃₁	1	3	yes	complemented
393	#436	C₂²×C₃₁	1	3	yes	complemented
394	#431	C₂²×C₃₁	1	3	yes	complemented
395	#423	C₂²×C₃₁	1	3	yes	complemented
396	#417	C₂²×C₃₁	1	6	yes	complemented
397	C₂×C₃₁	—	4	3	no	—
398	C₂×C₃₁	—	4	2	no	—
399	C₂×C₃₁	—	8	3	no	—
400	C₂×C₃₁	—	8	3	no	—
401	C₂×C₃₁	#3488	1	1	yes	characteristic
402	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
403	C₂×C₃₁	#3584	1	1	yes	characteristic
404	C₂×#41	—	2	1	no	—
405	C₂×#41	—	2	1	no	—
406	C₂×#41	—	2	3	no	—
407	C₂×#41	—	2	3	no	—
408	C₂×#41	—	4	8	no	—
409	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
410	C₂×#41	C₂³×C₃₁	1	6	yes	—
411	C₂²×Q₈	—	2	1	no	—
412	C₂²×D₄	—	2	3	no	—
413	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
414	C₂³×C₄	—	2	3	no	—
415	AGL(1,ℤ/8)	—	4	12	no	—
416	C₂×SD₁₆	C₂³×C₃₁	1	3	yes	—
417	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
418	C₂×SD₁₆	C₂³×C₃₁	1	3	yes	—
419	C₂×D₈	C₂³×C₃₁	1	3	yes	—
420	C₂×D₈	C₂³×C₃₁	1	6	yes	—
421	C₂×D₈	C₂³×C₃₁	1	3	yes	—
422	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
423	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
424	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
425	#127	—	2	3	no	—
426	#127	C₂³×C₃₁	1	3	yes	—
427	#127	C₂³×C₃₁	1	3	yes	—
428	#126	—	2	3	no	—
429	#125	—	2	3	no	—
430	#125	C₂³×C₃₁	1	1	yes	characteristic
431	#124	—	2	3	no	—
432	#124	C₂³×C₃₁	1	3	yes	—
433	#123	—	2	3	no	—
434	#123	—	2	6	no	—
435	#123	—	2	6	no	—
436	C₂²⋊Q₈	—	2	3	no	—
437	C₂²⋊Q₈	—	2	3	no	—
438	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
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	3	no	—
446	#121	—	2	6	no	—
447	#121	C₂³×C₃₁	1	3	yes	—
448	#121	C₂³×C₃₁	1	6	yes	—
449	#120	—	2	6	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	D₄×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	#117	—	2	3	no	—
469	#117	D₄×C₃₁	1	3	yes	—
470	C₂×C₄⋊C₄	—	2	3	no	—
471	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
472	C₂×C₂²⋊C₄	—	2	3	no	—
473	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
474	#107	C₂³×C₃₁	1	3	yes	—
475	#107	C₂³×C₃₁	1	1	yes	characteristic
476	#106	C₂³×C₃₁	1	3	yes	—
477	#106	C₂³×C₃₁	1	1	yes	characteristic
478	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
479	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
480	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
481	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
482	#103	C₂³×C₃₁	1	3	yes	—
483	#103	C₂³×C₃₁	1	6	yes	—
484	#103	C₂³×C₃₁	1	3	yes	—
485	#102	C₂³×C₃₁	1	3	yes	—
486	#102	C₂³×C₃₁	1	6	yes	—
487	#102	C₂³×C₃₁	1	1	yes	characteristic
488	#102	C₂³×C₃₁	1	3	yes	—
489	#102	C₂³×C₃₁	1	3	yes	—
490	#102	C₂³×C₃₁	1	1	yes	characteristic
491	#102	C₂³×C₃₁	1	3	yes	—
492	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
493	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
494	C₄×C₈	C₂³×C₃₁	1	3	yes	—
495	C₄×C₈	C₂³×C₃₁	1	3	yes	—
496	C₃₁	#62003	1	1	yes	characteristic, direct product factor, Sylow
497	C₂⁴	—	2	3	no	—
498	#41	—	2	4	no	—
499	#41	—	4	12	no	—
500	#41	—	4	6	no	—
501	#41	—	4	6	no	—
502	#41	—	4	12	no	—
503	#41	—	4	4	no	—
504	#41	—	4	4	no	—
505	C₂×Q₈	—	2	3	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
508	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
509	C₂×D₄	—	2	6	no	—
510	C₂×D₄	—	2	6	no	—
511	C₂×D₄	—	2	3	no	—
512	C₂×D₄	—	2	6	no	—
513	C₂×D₄	—	2	3	no	—
514	C₂×D₄	—	2	3	no	—
515	C₂×D₄	—	2	3	no	—
516	C₂×D₄	—	2	3	no	—
517	C₂×D₄	—	2	3	no	—
518	C₂×D₄	—	4	3	no	—
519	C₂×D₄	—	4	12	no	—
520	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
521	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
522	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
523	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
524	C₂²×C₄	—	2	3	no	—
525	C₂²×C₄	—	2	3	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	3	no	—
528	C₂²×C₄	—	2	3	no	—
529	C₂²×C₄	—	2	3	no	—
530	C₂²×C₄	—	2	6	no	—
531	C₂²×C₄	—	4	12	no	—
532	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
533	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
534	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
535	SD₁₆	—	2	6	no	—
536	SD₁₆	—	4	3	no	—
537	SD₁₆	—	4	6	no	—
538	D₈	—	2	6	no	—
539	D₈	—	4	3	no	—
540	D₈	—	4	6	no	—
541	M₁₆	—	4	6	no	—
542	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
543	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
544	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
545	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
546	C₄⋊C₄	—	2	3	no	—
547	C₄⋊C₄	—	2	3	no	—
548	C₄⋊C₄	—	2	3	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	3	no	—
552	C₄⋊C₄	—	2	3	no	—
553	C₄⋊C₄	—	2	3	no	—
554	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
555	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
556	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
557	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
558	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
559	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
560	C₂²⋊C₄	—	2	3	no	—
561	C₂²⋊C₄	—	2	1	no	—
562	C₂²⋊C₄	—	2	3	no	—
563	C₂²⋊C₄	—	2	3	no	—
564	C₂²⋊C₄	—	2	3	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	C₂×D₄×C₃₁	1	6	yes	—
571	C₄²	—	2	3	no	—
572	C₄²	C₂×D₄×C₃₁	1	3	yes	—
573	C₄²	C₂×D₄×C₃₁	1	3	yes	—
574	C₄²	C₂⁴×C₃₁	1	3	yes	—
575	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
576	C₂³	—	2	1	no	—
577	C₂³	—	2	3	no	—
578	C₂³	—	2	3	no	—
579	C₂³	—	4	6	no	—
580	C₂³	—	4	6	no	—
581	C₂³	—	8	3	no	—
582	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
583	Q₈	—	2	3	no	—
584	Q₈	—	2	3	no	—
585	Q₈	—	4	6	no	—
586	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
587	D₄	—	2	6	no	—
588	D₄	—	2	3	no	—
589	D₄	—	2	3	no	—
590	D₄	—	2	1	no	—
591	D₄	—	2	3	no	—
592	D₄	—	2	3	no	—
593	D₄	—	4	6	no	—
594	D₄	—	4	12	no	—
595	D₄	—	4	6	no	—
596	D₄	—	4	6	no	—
597	D₄	—	8	3	no	—
598	D₄	—	8	3	no	—
599	D₄	—	8	3	no	—
600	D₄	—	8	6	no	—
601	D₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
602	C₂×C₄	—	2	6	no	—
603	C₂×C₄	—	2	3	no	—
604	C₂×C₄	—	2	3	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	3	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	3	no	—
610	C₂×C₄	—	4	3	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	6	no	—
613	C₂×C₄	—	4	6	no	—
614	C₂×C₄	—	4	6	no	—
615	C₂×C₄	—	4	6	no	—
616	C₂×C₄	—	4	6	no	—
617	C₂×C₄	—	8	6	no	—
618	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
620	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
621	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
622	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
623	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
624	C₈	—	2	3	no	—
625	C₈	—	2	3	no	—
626	C₈	—	2	1	no	—
627	C₈	—	2	1	no	—
628	C₂²	—	2	3	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	3	no	—
631	C₂²	—	2	3	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	3	no	—
635	C₂²	—	4	3	no	—
636	C₂²	—	4	3	no	—
637	C₂²	—	4	3	no	—
638	C₂²	—	4	3	no	—
639	C₂²	—	4	3	no	—
640	C₂²	—	8	12	no	—
641	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
642	C₄	—	2	3	no	—
643	C₄	—	2	1	no	—
644	C₄	—	2	1	no	—
645	C₄	—	2	3	no	—
646	C₄	—	4	3	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	3	no	—
649	C₄	—	4	1	no	—
650	C₄	—	4	1	no	—
651	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
652	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
653	C₄	C₃₁×#585	1	1	yes	characteristic
654	C₂	—	4	3	no	—
655	C₂	—	4	2	no	—
656	C₂	—	8	3	no	—
657	C₂	—	8	3	no	—
658	C₂	C₃₁×#3488	1	1	yes	characteristic
659	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
660	C₂	C₃₁×#3584	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×#3517	C₂	semidirect
2. D₄×#41×C₃₁	C₂	semidirect
3. C₃₁×#3392	C₂	semidirect
4. C₃₁×#3391	C₂	semidirect
5. C₃₁×#3359	C₂	semidirect
6. C₃₁×#3358	C₂	semidirect
7. C₃₁×#3332	C₂	semidirect
8. C₃₁×#3330	C₂	semidirect
9. C₃₁×#3274	C₂	semidirect
10. C₃₁×#3272	C₂	semidirect
11. C₃₁×#3251	C₂	semidirect
12. C₃₁×#3207	C₂	semidirect
13. C₃₁×#3105	C₂	semidirect
14. C₃₁×#3036	C₂	semidirect
15. C₃₁×#2991	C₂	semidirect
18. C₃₁×#557	C₂²	semidirect
22. C₃₁×#550	C₂²	semidirect
23. C₃₁×#550	C₂²	semidirect
27. C₃₁×#547	C₂²	semidirect
32. C₃₁×#532	C₂²	semidirect
33. C₃₁×#529	C₂²	semidirect
38. C₃₁×#493	C₂²	semidirect
39. C₃₁×#492	C₂²	semidirect
40. C₃₁×#487	C₂²	semidirect
41. C₃₁×#486	C₂²	semidirect
42. C₃₁×#484	C₂²	semidirect
43. C₃₁×#481	C₂²	semidirect
45. C₃₁×#478	C₂²	semidirect
46. C₃₁×#475	C₂²	semidirect
47. C₃₁×#474	C₂²	semidirect
48. C₃₁×#469	C₂²	semidirect
49. C₃₁×#468	C₂²	semidirect
51. C₃₁×#463	C₂²	semidirect
52. C₃₁×#463	C₂²	semidirect
54. C₃₁×#461	C₂²	semidirect
55. C₃₁×#460	C₂²	semidirect
57. C₃₁×#459	C₂²	semidirect
58. C₃₁×#459	C₂²	semidirect
60. C₃₁×#449	C₂²	semidirect
61. C₃₁×#449	C₂²	semidirect
63. C₈×Q₈×C₃₁	C₂²	semidirect
64. C₈×Q₈×C₃₁	C₂²	semidirect
66. C₄×SD₁₆×C₃₁	C₂²	semidirect
68. C₄×D₈×C₃₁	C₂²	semidirect
69. C₃₁×#436	C₂²	semidirect
70. C₃₁×#436	C₂²	semidirect
71. C₃₁×#431	C₂²	semidirect
72. C₃₁×#423	C₂²	semidirect
73. C₃₁×#417	C₂²	semidirect
245. #62003	C₃₁	direct (N × H)
256. Q₈×C₃₁	AGL(1,ℤ/8)	semidirect
298. #3517	C₂×C₃₁	semidirect
299. D₄×#41	C₂×C₃₁	semidirect
300. #3392	C₂×C₃₁	semidirect
301. #3391	C₂×C₃₁	semidirect
302. #3359	C₂×C₃₁	semidirect
303. #3358	C₂×C₃₁	semidirect
304. #3332	C₂×C₃₁	semidirect
305. #3330	C₂×C₃₁	semidirect
306. #3274	C₂×C₃₁	semidirect
307. #3272	C₂×C₃₁	semidirect
308. #3251	C₂×C₃₁	semidirect
309. #3207	C₂×C₃₁	semidirect
310. #3105	C₂×C₃₁	semidirect
311. #3036	C₂×C₃₁	semidirect
312. #2991	C₂×C₃₁	semidirect
341. #557	C₂²×C₃₁	semidirect
345. #550	C₂²×C₃₁	semidirect
346. #550	C₂²×C₃₁	semidirect
350. #547	C₂²×C₃₁	semidirect
355. #532	C₂²×C₃₁	semidirect
356. #529	C₂²×C₃₁	semidirect
361. #493	C₂²×C₃₁	semidirect
362. #492	C₂²×C₃₁	semidirect
363. #487	C₂²×C₃₁	semidirect
364. #486	C₂²×C₃₁	semidirect
365. #484	C₂²×C₃₁	semidirect
366. #481	C₂²×C₃₁	semidirect
368. #478	C₂²×C₃₁	semidirect
369. #475	C₂²×C₃₁	semidirect
370. #474	C₂²×C₃₁	semidirect
371. #469	C₂²×C₃₁	semidirect
372. #468	C₂²×C₃₁	semidirect
374. #463	C₂²×C₃₁	semidirect
375. #463	C₂²×C₃₁	semidirect
377. #461	C₂²×C₃₁	semidirect
378. #460	C₂²×C₃₁	semidirect
380. #459	C₂²×C₃₁	semidirect
381. #459	C₂²×C₃₁	semidirect
383. #449	C₂²×C₃₁	semidirect
384. #449	C₂²×C₃₁	semidirect
386. C₈×Q₈	C₂²×C₃₁	semidirect
387. C₈×Q₈	C₂²×C₃₁	semidirect
389. C₄×SD₁₆	C₂²×C₃₁	semidirect
391. C₄×D₈	C₂²×C₃₁	semidirect
392. #436	C₂²×C₃₁	semidirect
393. #436	C₂²×C₃₁	semidirect
394. #431	C₂²×C₃₁	semidirect
395. #423	C₂²×C₃₁	semidirect
396. #417	C₂²×C₃₁	semidirect
496. C₃₁	#62003	direct (N × H)
586. Q₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3517	2
D₄×#41×C₃₁	2
C₃₁×#3392	2
C₃₁×#3391	2
C₃₁×#3359	2
C₃₁×#3358	2
C₃₁×#3332	2
C₃₁×#3330	2
C₃₁×#3274	2
C₃₁×#3272	2
C₃₁×#3251	2
C₃₁×#3207	2
C₃₁×#3105	2
C₃₁×#3036	2
C₃₁×#2991	2
#62003	31

Subgroups — C31×#62003 #21518762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62003 #21518762