Subgroups — C₃₁×#61907 #21518666

Order 7936.

2092 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3522	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3399	C₂	1	6	yes	maximal, complemented
4	C₃₁×#3391	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3366	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3362	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3217	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3205	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3109	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3105	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2984	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	—	2	1	no	—
13	C₃₁×#559	C₂²	1	3	yes	complemented
14	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#554	C₂²	1	3	yes	complemented
16	C₃₁×#550	—	2	1	no	—
17	C₃₁×#550	—	2	3	no	—
18	C₃₁×#550	C₂²	1	2	yes	—
19	C₃₁×#550	C₂²	1	6	yes	—
20	C₃₁×#550	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#548	—	2	1	no	—
22	C₃₁×#547	—	2	3	no	—
23	C₃₁×#542	—	2	3	no	—
24	C₃₁×#532	C₂²	1	3	yes	—
25	C₂×C₃₁×#127	C₂²	1	3	yes	—
26	C₃₁×#529	C₂²	1	3	yes	—
27	C₂×C₃₁×#121	—	2	3	no	—
28	C₃₁×#520	—	2	1	no	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₃₁	C₂²	1	3	yes	—
31	C₃₁×#496	C₂²	1	6	yes	complemented
32	C₃₁×#493	C₂²	1	6	yes	complemented
33	C₃₁×#490	C₂²	1	6	yes	complemented
34	C₃₁×#486	C₂²	1	6	yes	complemented
35	C₃₁×#484	C₂²	1	1	yes	characteristic
36	C₃₁×#481	C₂²	1	3	yes	—
37	C₃₁×#478	C₂²	1	6	yes	complemented
38	C₃₁×#476	C₂²	1	6	yes	complemented
39	C₃₁×#463	C₂²	1	6	yes	complemented
40	C₃₁×#460	C₂²	1	6	yes	complemented
41	C₃₁×#459	C₂²	1	6	yes	complemented
42	C₃₁×#459	C₂²	1	6	yes	complemented
43	C₃₁×#459	C₂²	1	12	yes	complemented
44	C₃₁×#450	C₂²	1	3	yes	—
45	C₃₁×#447	C₂²	1	3	yes	—
46	C₃₁×#447	C₂²	1	6	yes	—
47	C₃₁×#446	C₂²	1	6	yes	complemented
48	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
49	C₃₁×#442	C₂²	1	12	yes	complemented
50	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
51	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
52	C₃₁×#432	C₂²	1	3	yes	—
53	C₃₁×#431	C₂²	1	3	yes	—
54	C₂×#41×C₃₁	—	2	1	no	—
55	C₂×#41×C₃₁	—	2	6	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	—	2	3	no	—
58	C₂×#41×C₃₁	—	4	8	no	—
59	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₃₁	—	2	3	no	—
61	C₂²×D₄×C₃₁	—	2	3	no	—
62	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
63	C₂³×C₄×C₃₁	—	2	3	no	—
64	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
65	C₂×D₈×C₃₁	C₂³	1	6	yes	—
66	C₂×D₈×C₃₁	C₂³	1	12	yes	—
67	C₃₁×C₄⋊Q₈	—	2	3	no	—
68	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
69	C₃₁×#127	—	2	3	no	—
70	C₃₁×#127	—	2	3	no	—
71	C₃₁×#127	C₂³	1	6	yes	—
72	C₃₁×#127	C₂³	1	3	yes	—
73	C₃₁×#127	C₂³	1	6	yes	—
74	C₃₁×#126	—	2	3	no	—
75	C₃₁×#125	C₂³	1	3	yes	—
76	C₃₁×#124	—	2	3	no	—
77	C₃₁×#124	—	2	3	no	—
78	C₃₁×#124	C₂³	1	3	yes	—
79	C₃₁×#123	—	2	3	no	—
80	C₃₁×#123	—	2	6	no	—
81	C₃₁×C₂²⋊Q₈	—	2	3	no	—
82	C₃₁×C₂²⋊Q₈	—	2	6	no	—
83	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×#121	—	2	3	no	—
85	C₃₁×#121	—	2	6	no	—
86	C₃₁×#121	—	2	6	no	—
87	C₃₁×#121	—	2	12	no	—
88	C₃₁×#121	—	2	3	no	—
89	C₃₁×#121	C₂³	1	3	yes	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₃₁	D₄	1	3	yes	—
93	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
94	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
96	C₄×D₄×C₃₁	—	2	3	no	—
97	C₄×D₄×C₃₁	—	2	3	no	—
98	C₄×D₄×C₃₁	—	2	3	no	—
99	C₄×D₄×C₃₁	—	2	3	no	—
100	C₄×D₄×C₃₁	—	2	3	no	—
101	C₄×D₄×C₃₁	C₂³	1	3	yes	—
102	C₄×D₄×C₃₁	C₂³	1	6	yes	—
103	C₄×D₄×C₃₁	C₂³	1	6	yes	—
104	C₄×D₄×C₃₁	C₂³	1	6	yes	—
105	C₃₁×#117	—	2	3	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
107	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
110	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
111	C₃₁×#107	C₂³	1	2	yes	—
112	C₃₁×#106	C₂³	1	6	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
115	C₃₁×#103	C₂³	1	2	yes	—
116	C₃₁×#102	C₂³	1	6	yes	—
117	C₃₁×#102	C₂³	1	6	yes	—
118	C₃₁×#102	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	12	yes	—
120	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
121	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
122	C₃₁×#97	C₂³	1	6	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	C₂⁴×C₃₁	—	2	3	no	—
125	#41×C₃₁	—	4	24	no	—
126	#41×C₃₁	—	4	4	no	—
127	#41×C₃₁	—	4	8	no	—
128	#41×C₃₁	—	8	6	no	—
129	C₂×Q₈×C₃₁	—	2	4	no	—
130	C₂×Q₈×C₃₁	—	2	3	no	—
131	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
132	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂×D₄×C₃₁	—	2	3	no	—
135	C₂×D₄×C₃₁	—	2	3	no	—
136	C₂×D₄×C₃₁	—	2	3	no	—
137	C₂×D₄×C₃₁	—	2	6	no	—
138	C₂×D₄×C₃₁	—	2	6	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	6	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	—	4	6	no	—
143	C₂×D₄×C₃₁	—	4	12	no	—
144	C₂×D₄×C₃₁	—	8	6	no	—
145	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	3	no	—
149	C₂²×C₄×C₃₁	—	2	6	no	—
150	C₂²×C₄×C₃₁	—	2	6	no	—
151	C₂²×C₄×C₃₁	—	2	6	no	—
152	C₂²×C₄×C₃₁	—	4	12	no	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₃₁	—	4	6	no	—
157	D₈×C₃₁	—	4	6	no	—
158	D₈×C₃₁	—	4	12	no	—
159	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
160	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
161	C₄⋊C₄×C₃₁	—	2	3	no	—
162	C₄⋊C₄×C₃₁	—	2	3	no	—
163	C₄⋊C₄×C₃₁	—	2	1	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	3	no	—
166	C₄⋊C₄×C₃₁	—	2	3	no	—
167	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
170	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
171	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	2	6	no	—
175	C₂²⋊C₄×C₃₁	—	2	6	no	—
176	C₂²⋊C₄×C₃₁	—	2	3	no	—
177	C₂²⋊C₄×C₃₁	—	2	3	no	—
178	C₂²⋊C₄×C₃₁	—	2	3	no	—
179	C₂²⋊C₄×C₃₁	—	2	3	no	—
180	C₂²⋊C₄×C₃₁	—	2	6	no	—
181	C₄²×C₃₁	—	2	3	no	—
182	C₄²×C₃₁	C₂×D₄	1	3	yes	—
183	C₄²×C₃₁	C₂×D₄	1	3	yes	—
184	C₄²×C₃₁	C₂⁴	1	3	yes	—
185	C₄²×C₃₁	C₂⁴	1	3	yes	—
186	#61907	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₃₁	—	2	3	no	—
188	C₂³×C₃₁	—	2	6	no	—
189	C₂³×C₃₁	—	4	6	no	—
190	C₂³×C₃₁	—	4	6	no	—
191	C₂³×C₃₁	—	4	6	no	—
192	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	Q₈×C₃₁	—	2	6	no	—
194	Q₈×C₃₁	—	2	2	no	—
195	Q₈×C₃₁	—	8	3	no	—
196	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
197	D₄×C₃₁	—	2	6	no	—
198	D₄×C₃₁	—	2	6	no	—
199	D₄×C₃₁	—	2	6	no	—
200	D₄×C₃₁	—	4	6	no	—
201	D₄×C₃₁	—	4	12	no	—
202	D₄×C₃₁	—	4	6	no	—
203	D₄×C₃₁	—	4	12	no	—
204	D₄×C₃₁	—	4	6	no	—
205	D₄×C₃₁	—	8	6	no	—
206	D₄×C₃₁	—	8	6	no	—
207	D₄×C₃₁	—	8	3	no	—
208	C₂×C₄×C₃₁	—	2	6	no	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	3	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	4	12	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	12	no	—
218	C₂×C₄×C₃₁	—	4	6	no	—
219	C₂×C₄×C₃₁	—	8	6	no	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
226	C₈×C₃₁	—	2	2	no	—
227	C₈×C₃₁	—	2	6	no	—
228	#3522	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂×#550	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#3399	C₂×C₃₁	1	6	yes	complemented
231	#3391	C₂×C₃₁	1	2	yes	complemented
232	#3366	C₂×C₃₁	1	6	yes	complemented
233	#3362	C₂×C₃₁	1	2	yes	complemented
234	#3217	C₂×C₃₁	1	3	yes	complemented
235	#3205	C₂×C₃₁	1	3	yes	complemented
236	#3109	C₂×C₃₁	1	3	yes	complemented
237	#3105	C₂×C₃₁	1	3	yes	complemented
238	#2984	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	2	1	no	—
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	4	6	no	—
242	C₂²×C₃₁	—	4	3	no	—
243	C₂²×C₃₁	—	4	6	no	—
244	C₂²×C₃₁	—	4	3	no	—
245	C₂²×C₃₁	—	4	3	no	—
246	C₂²×C₃₁	—	4	6	no	—
247	C₂²×C₃₁	—	8	12	no	—
248	C₂²×C₃₁	#576	1	2	yes	—
249	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₃₁	—	2	1	no	—
251	C₄×C₃₁	—	2	3	no	—
252	C₄×C₃₁	—	2	3	no	—
253	C₄×C₃₁	—	2	3	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	3	no	—
258	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
259	C₄×C₃₁	#585	1	2	yes	—
260	C₂²×#41	—	2	1	no	—
261	#559	C₂²×C₃₁	1	3	yes	complemented
262	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
263	#554	C₂²×C₃₁	1	3	yes	complemented
264	#550	—	2	1	no	—
265	#550	—	2	3	no	—
266	#550	C₂²×C₃₁	1	2	yes	—
267	#550	C₂²×C₃₁	1	6	yes	—
268	#550	C₂²×C₃₁	1	1	yes	characteristic, complemented
269	#548	—	2	1	no	—
270	#547	—	2	3	no	—
271	#542	—	2	3	no	—
272	#532	C₂²×C₃₁	1	3	yes	—
273	C₂×#127	C₂²×C₃₁	1	3	yes	—
274	#529	C₂²×C₃₁	1	3	yes	—
275	C₂×#121	—	2	3	no	—
276	#520	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₃₁	1	3	yes	—
279	#496	C₂²×C₃₁	1	6	yes	complemented
280	#493	C₂²×C₃₁	1	6	yes	complemented
281	#490	C₂²×C₃₁	1	6	yes	complemented
282	#486	C₂²×C₃₁	1	6	yes	complemented
283	#484	C₂²×C₃₁	1	1	yes	characteristic
284	#481	C₂²×C₃₁	1	3	yes	—
285	#478	C₂²×C₃₁	1	6	yes	complemented
286	#476	C₂²×C₃₁	1	6	yes	complemented
287	#463	C₂²×C₃₁	1	6	yes	complemented
288	#460	C₂²×C₃₁	1	6	yes	complemented
289	#459	C₂²×C₃₁	1	6	yes	complemented
290	#459	C₂²×C₃₁	1	6	yes	complemented
291	#459	C₂²×C₃₁	1	12	yes	complemented
292	#450	C₂²×C₃₁	1	3	yes	—
293	#447	C₂²×C₃₁	1	3	yes	—
294	#447	C₂²×C₃₁	1	6	yes	—
295	#446	C₂²×C₃₁	1	6	yes	complemented
296	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
297	#442	C₂²×C₃₁	1	12	yes	complemented
298	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
299	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
300	#432	C₂²×C₃₁	1	3	yes	—
301	#431	C₂²×C₃₁	1	3	yes	—
302	C₂×C₃₁	—	2	2	no	—
303	C₂×C₃₁	—	8	3	no	—
304	C₂×C₃₁	—	8	6	no	—
305	C₂×C₃₁	#3487	1	1	yes	characteristic
306	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
308	C₂×#41	—	2	1	no	—
309	C₂×#41	—	2	6	no	—
310	C₂×#41	—	2	2	no	—
311	C₂×#41	—	2	3	no	—
312	C₂×#41	—	4	8	no	—
313	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
314	C₂²×D₄	—	2	3	no	—
315	C₂²×D₄	—	2	3	no	—
316	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
317	C₂³×C₄	—	2	3	no	—
318	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
319	C₂×D₈	C₂³×C₃₁	1	6	yes	—
320	C₂×D₈	C₂³×C₃₁	1	12	yes	—
321	C₄⋊Q₈	—	2	3	no	—
322	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
323	#127	—	2	3	no	—
324	#127	—	2	3	no	—
325	#127	C₂³×C₃₁	1	6	yes	—
326	#127	C₂³×C₃₁	1	6	yes	—
327	#127	C₂³×C₃₁	1	3	yes	—
328	#126	—	2	3	no	—
329	#125	C₂³×C₃₁	1	3	yes	—
330	#124	—	2	3	no	—
331	#124	—	2	3	no	—
332	#124	C₂³×C₃₁	1	3	yes	—
333	#123	—	2	3	no	—
334	#123	—	2	6	no	—
335	C₂²⋊Q₈	—	2	3	no	—
336	C₂²⋊Q₈	—	2	6	no	—
337	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
338	#121	—	2	3	no	—
339	#121	—	2	6	no	—
340	#121	—	2	6	no	—
341	#121	—	2	12	no	—
342	#121	—	2	3	no	—
343	#121	C₂³×C₃₁	1	3	yes	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
346	C₄×Q₈	D₄×C₃₁	1	3	yes	—
347	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
348	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
349	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
350	C₄×D₄	—	2	3	no	—
351	C₄×D₄	—	2	3	no	—
352	C₄×D₄	—	2	3	no	—
353	C₄×D₄	—	2	3	no	—
354	C₄×D₄	—	2	3	no	—
355	C₄×D₄	C₂³×C₃₁	1	6	yes	—
356	C₄×D₄	C₂³×C₃₁	1	6	yes	—
357	C₄×D₄	C₂³×C₃₁	1	3	yes	—
358	C₄×D₄	C₂³×C₃₁	1	6	yes	—
359	#117	—	2	3	no	—
360	C₂×C₄⋊C₄	—	2	3	no	—
361	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×C₂²⋊C₄	—	2	3	no	—
364	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
365	#107	C₂³×C₃₁	1	2	yes	—
366	#106	C₂³×C₃₁	1	6	yes	—
367	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
368	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
369	#103	C₂³×C₃₁	1	2	yes	—
370	#102	C₂³×C₃₁	1	6	yes	—
371	#102	C₂³×C₃₁	1	6	yes	—
372	#102	C₂³×C₃₁	1	6	yes	—
373	#102	C₂³×C₃₁	1	12	yes	—
374	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
376	#97	C₂³×C₃₁	1	6	yes	—
377	C₄×C₈	C₂³×C₃₁	1	6	yes	—
378	C₃₁	#61907	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	—	2	3	no	—
380	#41	—	4	24	no	—
381	#41	—	4	4	no	—
382	#41	—	4	8	no	—
383	#41	—	8	6	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	2	3	no	—
386	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
387	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	—	2	3	no	—
391	C₂×D₄	—	2	3	no	—
392	C₂×D₄	—	2	6	no	—
393	C₂×D₄	—	2	6	no	—
394	C₂×D₄	—	2	6	no	—
395	C₂×D₄	—	2	6	no	—
396	C₂×D₄	—	2	3	no	—
397	C₂×D₄	—	4	6	no	—
398	C₂×D₄	—	4	12	no	—
399	C₂×D₄	—	8	6	no	—
400	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
401	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	3	no	—
404	C₂²×C₄	—	2	6	no	—
405	C₂²×C₄	—	2	6	no	—
406	C₂²×C₄	—	2	6	no	—
407	C₂²×C₄	—	4	12	no	—
408	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
409	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
411	SD₁₆	—	4	6	no	—
412	D₈	—	4	6	no	—
413	D₈	—	4	12	no	—
414	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
415	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
416	C₄⋊C₄	—	2	3	no	—
417	C₄⋊C₄	—	2	3	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	3	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
423	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
424	C₄⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
425	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
426	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
428	C₂²⋊C₄	—	2	1	no	—
429	C₂²⋊C₄	—	2	6	no	—
430	C₂²⋊C₄	—	2	6	no	—
431	C₂²⋊C₄	—	2	3	no	—
432	C₂²⋊C₄	—	2	3	no	—
433	C₂²⋊C₄	—	2	3	no	—
434	C₂²⋊C₄	—	2	3	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₄²	—	2	3	no	—
437	C₄²	C₂×D₄×C₃₁	1	3	yes	—
438	C₄²	C₂×D₄×C₃₁	1	3	yes	—
439	C₄²	C₂⁴×C₃₁	1	3	yes	—
440	C₄²	C₂⁴×C₃₁	1	3	yes	—
441	C₂³	—	2	3	no	—
442	C₂³	—	2	6	no	—
443	C₂³	—	4	6	no	—
444	C₂³	—	4	6	no	—
445	C₂³	—	4	6	no	—
446	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
447	Q₈	—	2	6	no	—
448	Q₈	—	2	2	no	—
449	Q₈	—	8	3	no	—
450	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
451	D₄	—	2	6	no	—
452	D₄	—	2	6	no	—
453	D₄	—	2	6	no	—
454	D₄	—	4	6	no	—
455	D₄	—	4	12	no	—
456	D₄	—	4	6	no	—
457	D₄	—	4	12	no	—
458	D₄	—	4	6	no	—
459	D₄	—	8	6	no	—
460	D₄	—	8	6	no	—
461	D₄	—	8	3	no	—
462	C₂×C₄	—	2	6	no	—
463	C₂×C₄	—	2	6	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	3	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	4	12	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	12	no	—
472	C₂×C₄	—	4	6	no	—
473	C₂×C₄	—	8	6	no	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
476	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
477	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
478	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
479	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	6	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	4	6	no	—
485	C₂²	—	4	3	no	—
486	C₂²	—	4	6	no	—
487	C₂²	—	4	3	no	—
488	C₂²	—	4	3	no	—
489	C₂²	—	4	6	no	—
490	C₂²	—	8	12	no	—
491	C₂²	C₃₁×#576	1	2	yes	—
492	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
493	C₄	—	2	1	no	—
494	C₄	—	2	3	no	—
495	C₄	—	2	3	no	—
496	C₄	—	2	3	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	1	no	—
500	C₄	—	4	3	no	—
501	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
502	C₄	C₃₁×#585	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	8	6	no	—
505	C₂	—	8	3	no	—
506	C₂	C₃₁×#3487	1	1	yes	characteristic
507	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
508	C₂	C₂×C₃₁×#576	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 ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3522	C₂	semidirect
2. C₂×C₃₁×#550	C₂	semidirect
3. C₃₁×#3399	C₂	semidirect
4. C₃₁×#3391	C₂	semidirect
5. C₃₁×#3366	C₂	semidirect
6. C₃₁×#3362	C₂	semidirect
7. C₃₁×#3217	C₂	semidirect
8. C₃₁×#3205	C₂	semidirect
9. C₃₁×#3109	C₂	semidirect
10. C₃₁×#3105	C₂	semidirect
11. C₃₁×#2984	C₂	semidirect
13. C₃₁×#559	C₂²	semidirect
14. Q₈²×C₃₁	C₂²	semidirect
15. C₃₁×#554	C₂²	semidirect
20. C₃₁×#550	C₂²	semidirect
31. C₃₁×#496	C₂²	semidirect
32. C₃₁×#493	C₂²	semidirect
33. C₃₁×#490	C₂²	semidirect
34. C₃₁×#486	C₂²	semidirect
37. C₃₁×#478	C₂²	semidirect
38. C₃₁×#476	C₂²	semidirect
39. C₃₁×#463	C₂²	semidirect
40. C₃₁×#460	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#459	C₂²	semidirect
43. C₃₁×#459	C₂²	semidirect
47. C₃₁×#446	C₂²	semidirect
48. C₈×Q₈×C₃₁	C₂²	semidirect
49. C₃₁×#442	C₂²	semidirect
50. C₄×SD₁₆×C₃₁	C₂²	semidirect
51. C₄×D₈×C₃₁	C₂²	semidirect
186. #61907	C₃₁	direct (N × H)
228. #3522	C₂×C₃₁	semidirect
229. C₂×#550	C₂×C₃₁	semidirect
230. #3399	C₂×C₃₁	semidirect
231. #3391	C₂×C₃₁	semidirect
232. #3366	C₂×C₃₁	semidirect
233. #3362	C₂×C₃₁	semidirect
234. #3217	C₂×C₃₁	semidirect
235. #3205	C₂×C₃₁	semidirect
236. #3109	C₂×C₃₁	semidirect
237. #3105	C₂×C₃₁	semidirect
238. #2984	C₂×C₃₁	semidirect
261. #559	C₂²×C₃₁	semidirect
262. Q₈²	C₂²×C₃₁	semidirect
263. #554	C₂²×C₃₁	semidirect
268. #550	C₂²×C₃₁	semidirect
279. #496	C₂²×C₃₁	semidirect
280. #493	C₂²×C₃₁	semidirect
281. #490	C₂²×C₃₁	semidirect
282. #486	C₂²×C₃₁	semidirect
285. #478	C₂²×C₃₁	semidirect
286. #476	C₂²×C₃₁	semidirect
287. #463	C₂²×C₃₁	semidirect
288. #460	C₂²×C₃₁	semidirect
289. #459	C₂²×C₃₁	semidirect
290. #459	C₂²×C₃₁	semidirect
291. #459	C₂²×C₃₁	semidirect
295. #446	C₂²×C₃₁	semidirect
296. C₈×Q₈	C₂²×C₃₁	semidirect
297. #442	C₂²×C₃₁	semidirect
298. C₄×SD₁₆	C₂²×C₃₁	semidirect
299. C₄×D₈	C₂²×C₃₁	semidirect
378. C₃₁	#61907	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3522	2
C₂×C₃₁×#550	2
C₃₁×#3399	2
C₃₁×#3391	2
C₃₁×#3366	2
C₃₁×#3362	2
C₃₁×#3217	2
C₃₁×#3205	2
C₃₁×#3109	2
C₃₁×#3105	2
C₃₁×#2984	2
#61907	31

Subgroups — C31×#61907 #21518666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61907 #21518666