Subgroups — C₃₁×#60676 #21517435

Order 7936.

1998 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3172	C₂	1	8	yes	maximal, complemented
3	C₃₁×#3171	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3135	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3054	C₂	1	4	yes	maximal, complemented
8	C₄×C₃₁×#135	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#533	C₂²	1	4	yes	—
13	C₃₁×#533	C₂²	1	2	yes	—
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₃₁×#532	C₂²	1	4	yes	—
16	C₃₁×#529	—	2	4	no	—
17	C₃₁×#529	—	2	4	no	—
18	C₃₁×#529	C₂²	1	8	yes	—
19	C₃₁×#525	—	2	4	no	—
20	C₃₁×#525	C₂²	1	4	yes	—
21	C₄×#41×C₃₁	—	2	4	no	—
22	C₄×#41×C₃₁	C₂²	1	4	yes	—
23	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#501	C₂²	1	8	yes	complemented
25	C₃₁×#497	C₂²	1	4	yes	complemented
26	C₃₁×#496	C₂²	1	4	yes	complemented
27	C₃₁×#491	C₂²	1	4	yes	complemented
28	C₃₁×#490	C₂²	1	4	yes	complemented
29	C₃₁×#489	C₂²	1	8	yes	complemented
30	C₃₁×#470	C₂²	1	8	yes	—
31	C₃₁×#469	C₂²	1	8	yes	—
32	C₃₁×#468	C₂²	1	16	yes	—
33	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	complemented
34	C₄×SD₁₆×C₃₁	C₂²	1	16	yes	complemented
35	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
36	C₃₁×#427	C₂²	1	4	yes	—
37	C₃₁×#416	C₂²	1	8	yes	—
38	C₄×M₁₆×C₃₁	C₂²	1	8	yes	—
39	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
42	C₂×#41×C₃₁	—	2	4	no	—
43	C₂×#41×C₃₁	C₂³	1	4	yes	—
44	C₂³×C₄×C₃₁	D₄	1	2	yes	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#135	—	2	16	no	—
47	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
48	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
49	C₂×D₈×C₃₁	C₂³	1	4	yes	—
50	C₂×M₁₆×C₃₁	—	2	8	no	—
51	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
56	C₃₁×#127	C₂³	1	2	yes	—
57	C₃₁×#126	—	2	8	no	—
58	C₃₁×#125	—	2	4	no	—
59	C₃₁×#125	C₂³	1	4	yes	—
60	C₃₁×#124	—	2	4	no	—
61	C₃₁×#124	C₂³	1	4	yes	—
62	C₃₁×#123	—	2	8	no	—
63	C₃₁×#123	—	2	4	no	—
64	C₃₁×C₂²⋊Q₈	—	2	4	no	—
65	C₃₁×C₂²⋊Q₈	—	2	4	no	—
66	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
67	C₃₁×#121	—	2	4	no	—
68	C₃₁×#121	—	2	4	no	—
69	C₃₁×#121	C₂³	1	8	yes	—
70	C₃₁×#120	—	2	4	no	—
71	C₄×Q₈×C₃₁	—	2	4	no	—
72	C₄×Q₈×C₃₁	—	2	4	no	—
73	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
74	C₄×D₄×C₃₁	—	2	8	no	—
75	C₄×D₄×C₃₁	—	2	8	no	—
76	C₄×D₄×C₃₁	—	2	8	no	—
77	C₄×D₄×C₃₁	—	2	4	no	—
78	C₄×D₄×C₃₁	—	2	4	no	—
79	C₄×D₄×C₃₁	C₂³	1	8	yes	—
80	C₃₁×#117	—	2	4	no	—
81	C₃₁×#117	—	2	8	no	—
82	C₃₁×#117	—	2	4	no	—
83	C₃₁×#117	C₂³	1	4	yes	—
84	C₂×C₄²×C₃₁	—	2	4	no	—
85	C₂×C₄²×C₃₁	D₄	1	2	yes	—
86	C₂×C₄²×C₃₁	D₄	1	2	yes	—
87	C₂×C₄²×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×#107	C₂³	1	8	yes	—
92	C₃₁×#106	C₂³	1	8	yes	—
93	C₃₁×#103	C₂³	1	16	yes	—
94	C₃₁×#102	C₂³	1	16	yes	—
95	C₃₁×#97	C₂³	1	4	yes	—
96	C₃₁×#97	C₂³	1	4	yes	—
97	C₄×C₈×C₃₁	C₂³	1	8	yes	—
98	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
99	#41×C₃₁	—	2	16	no	—
100	#41×C₃₁	—	4	16	no	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
103	C₂×D₄×C₃₁	—	2	4	no	—
104	C₂×D₄×C₃₁	—	2	8	no	—
105	C₂×D₄×C₃₁	—	2	4	no	—
106	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	8	no	—
110	C₂²×C₄×C₃₁	—	4	4	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	Q₁₆×C₃₁	—	2	8	no	—
120	SD₁₆×C₃₁	—	2	16	no	—
121	D₈×C₃₁	—	2	8	no	—
122	M₁₆×C₃₁	—	2	16	no	—
123	C₂×C₈×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	#41	1	8	yes	complemented
125	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
126	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
127	C₄⋊C₄×C₃₁	—	2	4	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	8	no	—
130	C₄⋊C₄×C₃₁	—	2	8	no	—
131	C₄⋊C₄×C₃₁	—	2	8	no	—
132	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
133	C₂²⋊C₄×C₃₁	—	2	8	no	—
134	C₂²⋊C₄×C₃₁	—	2	4	no	—
135	C₂²⋊C₄×C₃₁	—	2	8	no	—
136	C₂²⋊C₄×C₃₁	—	2	8	no	—
137	C₂²⋊C₄×C₃₁	—	2	8	no	—
138	C₄²×C₃₁	—	2	8	no	—
139	C₄²×C₃₁	—	2	4	no	—
140	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₃₁	C₂×D₄	1	4	yes	—
142	C₄²×C₃₁	C₂×D₄	1	2	yes	—
143	C₄²×C₃₁	C₂×D₄	1	4	yes	—
144	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₃₁	C₂⁴	1	2	yes	—
146	C₄²×C₃₁	C₂⁴	1	2	yes	—
147	#60676	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	2	4	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
152	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₃₁	—	2	8	no	—
154	Q₈×C₃₁	—	4	8	no	—
155	D₄×C₃₁	—	2	8	no	—
156	D₄×C₃₁	—	4	8	no	—
157	D₄×C₃₁	—	4	16	no	—
158	D₄×C₃₁	—	4	8	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	8	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	8	no	—
166	C₂×C₄×C₃₁	—	4	4	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	8	no	—
169	C₂×C₄×C₃₁	—	4	16	no	—
170	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
171	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
176	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
177	C₈×C₃₁	—	2	4	no	—
178	C₈×C₃₁	C₂×#41	1	8	yes	—
179	#3444	C₂×C₃₁	1	2	yes	complemented
180	#3172	C₂×C₃₁	1	8	yes	complemented
181	#3171	C₂×C₃₁	1	8	yes	complemented
182	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#3150	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#3135	C₂×C₃₁	1	2	yes	complemented
185	#3054	C₂×C₃₁	1	4	yes	complemented
186	C₄×#135	C₂×C₃₁	1	4	yes	complemented
187	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂²×C₃₁	—	2	2	no	—
189	C₂²×C₃₁	—	2	2	no	—
190	C₂²×C₃₁	—	2	4	no	—
191	C₂²×C₃₁	—	4	4	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	4	8	no	—
194	C₂²×C₃₁	#532	1	2	yes	—
195	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
196	C₄×C₃₁	—	2	2	no	—
197	C₄×C₃₁	—	2	2	no	—
198	C₄×C₃₁	—	2	4	no	—
199	C₄×C₃₁	—	4	4	no	—
200	C₄×C₃₁	—	4	8	no	—
201	C₄×C₃₁	#532	1	2	yes	—
202	C₄×C₃₁	#575	1	4	yes	—
203	C₄×C₃₁	C₂²×#41	1	2	yes	—
204	C₂×#135	C₂²×C₃₁	1	4	yes	—
205	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
206	#533	C₂²×C₃₁	1	4	yes	—
207	#533	C₂²×C₃₁	1	2	yes	—
208	#532	C₂²×C₃₁	1	2	yes	—
209	#532	C₂²×C₃₁	1	4	yes	—
210	#529	—	2	4	no	—
211	#529	—	2	4	no	—
212	#529	C₂²×C₃₁	1	8	yes	—
213	#525	—	2	4	no	—
214	#525	C₂²×C₃₁	1	4	yes	—
215	C₄×#41	—	2	4	no	—
216	C₄×#41	C₂²×C₃₁	1	4	yes	—
217	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
218	#501	C₂²×C₃₁	1	8	yes	complemented
219	#497	C₂²×C₃₁	1	4	yes	complemented
220	#496	C₂²×C₃₁	1	4	yes	complemented
221	#491	C₂²×C₃₁	1	4	yes	complemented
222	#490	C₂²×C₃₁	1	4	yes	complemented
223	#489	C₂²×C₃₁	1	8	yes	complemented
224	#470	C₂²×C₃₁	1	8	yes	—
225	#469	C₂²×C₃₁	1	8	yes	—
226	#468	C₂²×C₃₁	1	16	yes	—
227	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	complemented
228	C₄×SD₁₆	C₂²×C₃₁	1	16	yes	complemented
229	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
230	#427	C₂²×C₃₁	1	4	yes	—
231	#416	C₂²×C₃₁	1	8	yes	—
232	C₄×M₁₆	C₂²×C₃₁	1	8	yes	—
233	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
236	C₂×C₃₁	—	2	2	no	—
237	C₂×C₃₁	—	4	2	no	—
238	C₂×C₃₁	—	8	4	no	—
239	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#575	1	2	yes	—
241	C₂×#41	—	2	4	no	—
242	C₂×#41	C₂³×C₃₁	1	4	yes	—
243	C₂³×C₄	D₄×C₃₁	1	2	yes	—
244	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
245	#135	—	2	16	no	—
246	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
247	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
248	C₂×D₈	C₂³×C₃₁	1	4	yes	—
249	C₂×M₁₆	—	2	8	no	—
250	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
254	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
255	#127	C₂³×C₃₁	1	2	yes	—
256	#126	—	2	8	no	—
257	#125	—	2	4	no	—
258	#125	C₂³×C₃₁	1	4	yes	—
259	#124	—	2	4	no	—
260	#124	C₂³×C₃₁	1	4	yes	—
261	#123	—	2	8	no	—
262	#123	—	2	4	no	—
263	C₂²⋊Q₈	—	2	4	no	—
264	C₂²⋊Q₈	—	2	4	no	—
265	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
266	#121	—	2	4	no	—
267	#121	—	2	4	no	—
268	#121	C₂³×C₃₁	1	8	yes	—
269	#120	—	2	4	no	—
270	C₄×Q₈	—	2	4	no	—
271	C₄×Q₈	—	2	4	no	—
272	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
273	C₄×D₄	—	2	8	no	—
274	C₄×D₄	—	2	8	no	—
275	C₄×D₄	—	2	8	no	—
276	C₄×D₄	—	2	4	no	—
277	C₄×D₄	—	2	4	no	—
278	C₄×D₄	C₂³×C₃₁	1	8	yes	—
279	#117	—	2	4	no	—
280	#117	—	2	8	no	—
281	#117	—	2	4	no	—
282	#117	C₂³×C₃₁	1	4	yes	—
283	C₂×C₄²	—	2	4	no	—
284	C₂×C₄²	D₄×C₃₁	1	2	yes	—
285	C₂×C₄²	D₄×C₃₁	1	2	yes	—
286	C₂×C₄²	D₄×C₃₁	1	2	yes	—
287	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	#107	C₂³×C₃₁	1	8	yes	—
291	#106	C₂³×C₃₁	1	8	yes	—
292	#103	C₂³×C₃₁	1	16	yes	—
293	#102	C₂³×C₃₁	1	16	yes	—
294	#97	C₂³×C₃₁	1	4	yes	—
295	#97	C₂³×C₃₁	1	4	yes	—
296	C₄×C₈	C₂³×C₃₁	1	8	yes	—
297	C₃₁	#60676	1	1	yes	characteristic, direct product factor, Sylow
298	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
299	#41	—	2	16	no	—
300	#41	—	4	16	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	8	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	8	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
317	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
318	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
319	Q₁₆	—	2	8	no	—
320	SD₁₆	—	2	16	no	—
321	D₈	—	2	8	no	—
322	M₁₆	—	2	16	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	#41×C₃₁	1	8	yes	complemented
325	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
326	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
333	C₂²⋊C₄	—	2	8	no	—
334	C₂²⋊C₄	—	2	4	no	—
335	C₂²⋊C₄	—	2	8	no	—
336	C₂²⋊C₄	—	2	8	no	—
337	C₂²⋊C₄	—	2	8	no	—
338	C₄²	—	2	8	no	—
339	C₄²	—	2	4	no	—
340	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₄²	C₂×D₄×C₃₁	1	4	yes	—
342	C₄²	C₂×D₄×C₃₁	1	2	yes	—
343	C₄²	C₂×D₄×C₃₁	1	4	yes	—
344	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₄²	C₂⁴×C₃₁	1	2	yes	—
346	C₄²	C₂⁴×C₃₁	1	2	yes	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	4	2	no	—
350	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
351	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
352	Q₈	—	2	8	no	—
353	Q₈	—	4	8	no	—
354	D₄	—	2	8	no	—
355	D₄	—	4	8	no	—
356	D₄	—	4	16	no	—
357	D₄	—	4	8	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	—	4	16	no	—
369	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
370	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
371	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
372	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
374	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
375	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
376	C₈	—	2	4	no	—
377	C₈	C₂×#41×C₃₁	1	8	yes	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	4	no	—
381	C₂²	—	4	4	no	—
382	C₂²	—	4	4	no	—
383	C₂²	—	4	8	no	—
384	C₂²	C₃₁×#532	1	2	yes	—
385	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
386	C₄	—	2	2	no	—
387	C₄	—	2	4	no	—
388	C₄	—	2	2	no	—
389	C₄	—	4	4	no	—
390	C₄	—	4	8	no	—
391	C₄	C₃₁×#532	1	2	yes	—
392	C₄	C₃₁×#575	1	4	yes	—
393	C₄	C₂²×#41×C₃₁	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	4	2	no	—
396	C₂	—	8	4	no	—
397	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
398	C₂	C₂×C₃₁×#575	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3444	C₂	semidirect
2. C₃₁×#3172	C₂	semidirect
3. C₃₁×#3171	C₂	semidirect
4. C₃₁×#3162	C₂	semidirect
5. C₃₁×#3150	C₂	semidirect
6. C₃₁×#3135	C₂	semidirect
7. C₃₁×#3054	C₂	semidirect
8. C₄×C₃₁×#135	C₂	semidirect
9. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
24. C₃₁×#501	C₂²	semidirect
25. C₃₁×#497	C₂²	semidirect
26. C₃₁×#496	C₂²	semidirect
27. C₃₁×#491	C₂²	semidirect
28. C₃₁×#490	C₂²	semidirect
29. C₃₁×#489	C₂²	semidirect
33. C₄×Q₁₆×C₃₁	C₂²	semidirect
34. C₄×SD₁₆×C₃₁	C₂²	semidirect
35. C₄×D₈×C₃₁	C₂²	semidirect
124. C₂×C₈×C₃₁	#41	semidirect
147. #60676	C₃₁	direct (N × H)
179. #3444	C₂×C₃₁	semidirect
180. #3172	C₂×C₃₁	semidirect
181. #3171	C₂×C₃₁	semidirect
182. #3162	C₂×C₃₁	semidirect
183. #3150	C₂×C₃₁	semidirect
184. #3135	C₂×C₃₁	semidirect
185. #3054	C₂×C₃₁	semidirect
186. C₄×#135	C₂×C₃₁	semidirect
187. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
218. #501	C₂²×C₃₁	semidirect
219. #497	C₂²×C₃₁	semidirect
220. #496	C₂²×C₃₁	semidirect
221. #491	C₂²×C₃₁	semidirect
222. #490	C₂²×C₃₁	semidirect
223. #489	C₂²×C₃₁	semidirect
227. C₄×Q₁₆	C₂²×C₃₁	semidirect
228. C₄×SD₁₆	C₂²×C₃₁	semidirect
229. C₄×D₈	C₂²×C₃₁	semidirect
297. C₃₁	#60676	direct (N × H)
324. C₂×C₈	#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×#3444	2
C₃₁×#3172	2
C₃₁×#3171	2
C₃₁×#3162	2
C₃₁×#3150	2
C₃₁×#3135	2
C₃₁×#3054	2
C₄×C₃₁×#135	2
C₂×C₄×M₁₆×C₃₁	2
#60676	31

Subgroups — C31×#60676 #21517435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60676 #21517435