Subgroups — C₃₁×#61411 #21518170

Order 7936.

1838 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3398	C₂	1	8	yes	maximal, complemented
3	C₃₁×#3395	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3393	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3156	C₂	1	2	yes	maximal
6	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3044	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3039	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2950	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#137	C₂²	1	4	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#557	C₂²	1	8	yes	—
17	C₃₁×#556	C₂²	1	8	yes	—
18	C₃₁×#552	—	2	8	no	—
19	C₃₁×#541	—	2	4	no	—
20	C₃₁×#536	C₂²	1	4	yes	—
21	C₃₁×#533	—	2	4	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
24	C₃₁×#519	C₂²	1	4	yes	—
25	C₄×#41×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#497	C₂²	1	8	yes	—
27	C₃₁×#495	C₂²	1	4	yes	—
28	C₃₁×#494	C₂²	1	4	yes	—
29	C₃₁×#494	C₂²	1	4	yes	—
30	C₃₁×#494	C₂²	1	4	yes	—
31	C₃₁×#464	C₂²	1	16	yes	—
32	C₃₁×#462	C₂²	1	8	yes	—
33	C₃₁×#462	C₂²	1	16	yes	—
34	C₃₁×#461	C₂²	1	8	yes	—
35	C₃₁×#446	C₂²	1	4	yes	—
36	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
37	C₃₁×#441	C₂²	1	8	yes	—
38	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
39	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
40	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
41	C₄×D₈×C₃₁	C₂²	1	2	yes	—
42	C₃₁×#427	C₂²	1	1	yes	characteristic
43	C₃₁×#424	C₂²	1	2	yes	—
44	C₃₁×#423	C₂²	1	1	yes	characteristic
45	C₃₁×#419	C₂²	1	4	yes	—
46	C₃₁×#416	C₂²	1	2	yes	—
47	C₃₁×#405	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₂×#41×C₃₁	C₂³	1	2	yes	—
50	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
51	C₃₁×#137	—	4	8	no	—
52	C₃₁×#135	—	2	4	no	—
53	C₂×Q₁₆×C₃₁	—	2	8	no	—
54	C₂×Q₁₆×C₃₁	—	4	4	no	—
55	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
57	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
59	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
60	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
62	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	—	2	2	no	—
67	C₃₁×C₄⋊Q₈	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
71	C₃₁×#126	—	2	16	no	—
72	C₃₁×#125	—	2	8	no	—
73	C₃₁×#125	—	2	16	no	—
74	C₃₁×#125	D₄	1	2	yes	—
75	C₃₁×#124	C₂³	1	8	yes	—
76	C₃₁×#123	—	2	8	no	—
77	C₃₁×C₂²⋊Q₈	—	2	8	no	—
78	C₃₁×C₂²⋊Q₈	—	2	8	no	—
79	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
80	C₄×Q₈×C₃₁	—	2	8	no	—
81	C₄×Q₈×C₃₁	—	2	8	no	—
82	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
83	C₄×Q₈×C₃₁	C₂³	1	16	yes	—
84	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
85	C₄×D₄×C₃₁	—	2	4	no	—
86	C₄×D₄×C₃₁	C₂³	1	4	yes	—
87	C₃₁×#117	—	2	8	no	—
88	C₃₁×#117	—	2	8	no	—
89	C₃₁×#117	—	2	2	no	—
90	C₃₁×#117	C₂³	1	2	yes	—
91	C₃₁×#117	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₂×C₄²×C₃₁	—	2	2	no	—
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×#107	C₂³	1	2	yes	—
96	C₃₁×#106	C₂³	1	2	yes	—
97	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
99	C₃₁×#103	C₂³	1	4	yes	—
100	C₃₁×#103	C₂³	1	16	yes	—
101	C₃₁×#102	C₂³	1	4	yes	—
102	C₃₁×#97	C₂³	1	4	yes	—
103	C₄×C₈×C₃₁	C₂³	1	2	yes	—
104	C₄×C₈×C₃₁	C₂³	1	2	yes	—
105	C₄×C₈×C₃₁	C₂³	1	4	yes	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	C₂×Q₈×C₃₁	—	2	4	no	—
109	C₂×Q₈×C₃₁	—	2	8	no	—
110	C₂×Q₈×C₃₁	—	2	8	no	—
111	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
112	C₂×Q₈×C₃₁	C₂⁴	1	8	yes	—
113	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	Q₁₆×C₃₁	—	2	8	no	—
122	Q₁₆×C₃₁	—	2	8	no	—
123	Q₁₆×C₃₁	—	2	2	no	—
124	Q₁₆×C₃₁	—	4	8	no	—
125	Q₁₆×C₃₁	—	4	8	no	—
126	SD₁₆×C₃₁	—	2	4	no	—
127	SD₁₆×C₃₁	—	4	8	no	—
128	D₈×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	2	4	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	#41	1	4	yes	—
132	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
133	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
134	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
135	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₃₁	—	2	8	no	—
137	C₄⋊C₄×C₃₁	—	2	8	no	—
138	C₄⋊C₄×C₃₁	—	2	4	no	—
139	C₄⋊C₄×C₃₁	—	2	16	no	—
140	C₄⋊C₄×C₃₁	—	2	16	no	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
142	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
143	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
144	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
145	C₂²⋊C₄×C₃₁	—	2	2	no	—
146	C₂²⋊C₄×C₃₁	—	2	4	no	—
147	C₂²⋊C₄×C₃₁	—	2	16	no	—
148	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	4	no	—
151	C₄²×C₃₁	—	2	16	no	—
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂⁴	1	2	yes	—
155	C₄²×C₃₁	C₂⁴	1	4	yes	—
156	#61411	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	8	no	—
160	Q₈×C₃₁	—	2	2	no	—
161	Q₈×C₃₁	—	2	8	no	—
162	Q₈×C₃₁	—	2	8	no	—
163	Q₈×C₃₁	—	2	2	no	—
164	Q₈×C₃₁	—	4	8	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	8	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	4	8	no	—
173	C₂×C₄×C₃₁	—	4	4	no	—
174	C₂×C₄×C₃₁	—	4	8	no	—
175	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
176	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
179	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
180	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
181	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
182	C₈×C₃₁	—	2	2	no	—
183	C₈×C₃₁	—	2	4	no	—
184	C₈×C₃₁	C₂×#41	1	2	yes	—
185	C₈×C₃₁	C₂×#41	1	2	yes	—
186	#3509	C₂×C₃₁	1	2	yes	complemented
187	#3398	C₂×C₃₁	1	8	yes	complemented
188	#3395	C₂×C₃₁	1	4	yes	complemented
189	#3393	C₂×C₃₁	1	4	yes	complemented
190	#3156	C₂×C₃₁	1	2	yes	—
191	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#3044	C₂×C₃₁	1	4	yes	complemented
193	#3039	C₂×C₃₁	1	2	yes	complemented
194	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#2950	C₂×C₃₁	1	2	yes	complemented
196	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
197	C₂²×C₃₁	—	2	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	4	no	—
200	C₂²×C₃₁	#550	1	2	yes	—
201	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	8	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	4	8	no	—
206	C₄×C₃₁	—	4	4	no	—
207	C₄×C₃₁	#550	1	2	yes	—
208	C₄×C₃₁	#578	1	4	yes	—
209	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
210	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×#137	C₂²×C₃₁	1	4	yes	—
212	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
213	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
214	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
215	#557	C₂²×C₃₁	1	8	yes	—
216	#556	C₂²×C₃₁	1	8	yes	—
217	#552	—	2	8	no	—
218	#541	—	2	4	no	—
219	#536	C₂²×C₃₁	1	4	yes	—
220	#533	—	2	4	no	—
221	#533	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
223	#519	C₂²×C₃₁	1	4	yes	—
224	C₄×#41	C₂²×C₃₁	1	2	yes	—
225	#497	C₂²×C₃₁	1	8	yes	—
226	#495	C₂²×C₃₁	1	4	yes	—
227	#494	C₂²×C₃₁	1	4	yes	—
228	#494	C₂²×C₃₁	1	4	yes	—
229	#494	C₂²×C₃₁	1	4	yes	—
230	#464	C₂²×C₃₁	1	16	yes	—
231	#462	C₂²×C₃₁	1	8	yes	—
232	#462	C₂²×C₃₁	1	16	yes	—
233	#461	C₂²×C₃₁	1	8	yes	—
234	#446	C₂²×C₃₁	1	4	yes	—
235	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
236	#441	C₂²×C₃₁	1	8	yes	—
237	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
238	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
239	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
240	C₄×D₈	C₂²×C₃₁	1	2	yes	—
241	#427	C₂²×C₃₁	1	1	yes	characteristic
242	#424	C₂²×C₃₁	1	2	yes	—
243	#423	C₂²×C₃₁	1	1	yes	characteristic
244	#419	C₂²×C₃₁	1	4	yes	—
245	#416	C₂²×C₃₁	1	2	yes	—
246	#405	C₂²×C₃₁	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₃₁	—	2	2	no	—
249	C₂×C₃₁	—	8	2	no	—
250	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
251	C₂×C₃₁	C₂×#578	1	2	yes	—
252	C₂×#41	C₂³×C₃₁	1	2	yes	—
253	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
254	#137	—	4	8	no	—
255	#135	—	2	4	no	—
256	C₂×Q₁₆	—	2	8	no	—
257	C₂×Q₁₆	—	4	4	no	—
258	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
259	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
260	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
261	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
262	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
263	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
264	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
265	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₄⋊Q₈	—	2	2	no	—
270	C₄⋊Q₈	—	2	2	no	—
271	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
272	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
273	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
274	#126	—	2	16	no	—
275	#125	—	2	8	no	—
276	#125	—	2	16	no	—
277	#125	D₄×C₃₁	1	2	yes	—
278	#124	C₂³×C₃₁	1	8	yes	—
279	#123	—	2	8	no	—
280	C₂²⋊Q₈	—	2	8	no	—
281	C₂²⋊Q₈	—	2	8	no	—
282	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
283	C₄×Q₈	—	2	8	no	—
284	C₄×Q₈	—	2	8	no	—
285	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
286	C₄×Q₈	C₂³×C₃₁	1	16	yes	—
287	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
288	C₄×D₄	—	2	4	no	—
289	C₄×D₄	C₂³×C₃₁	1	4	yes	—
290	#117	—	2	8	no	—
291	#117	—	2	8	no	—
292	#117	—	2	2	no	—
293	#117	C₂³×C₃₁	1	2	yes	—
294	#117	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	—	2	4	no	—
296	C₂×C₄²	—	2	2	no	—
297	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
298	#107	C₂³×C₃₁	1	2	yes	—
299	#106	C₂³×C₃₁	1	2	yes	—
300	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
301	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
302	#103	C₂³×C₃₁	1	4	yes	—
303	#103	C₂³×C₃₁	1	16	yes	—
304	#102	C₂³×C₃₁	1	4	yes	—
305	#97	C₂³×C₃₁	1	4	yes	—
306	C₄×C₈	C₂³×C₃₁	1	2	yes	—
307	C₄×C₈	C₂³×C₃₁	1	2	yes	—
308	C₄×C₈	C₂³×C₃₁	1	4	yes	—
309	C₃₁	#61411	1	1	yes	characteristic, direct product factor, Sylow
310	#41	—	2	4	no	—
311	#41	—	2	4	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	—	2	8	no	—
314	C₂×Q₈	—	2	8	no	—
315	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
316	C₂×Q₈	C₂⁴×C₃₁	1	8	yes	—
317	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
325	Q₁₆	—	2	8	no	—
326	Q₁₆	—	2	8	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	8	no	—
329	Q₁₆	—	4	8	no	—
330	SD₁₆	—	2	4	no	—
331	SD₁₆	—	4	8	no	—
332	D₈	—	2	2	no	—
333	M₁₆	—	2	4	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	#41×C₃₁	1	4	yes	—
336	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
337	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
338	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	16	no	—
344	C₄⋊C₄	—	2	16	no	—
345	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
346	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
347	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
348	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	4	no	—
351	C₂²⋊C₄	—	2	16	no	—
352	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	4	no	—
355	C₄²	—	2	16	no	—
356	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₄²	C₂⁴×C₃₁	1	2	yes	—
359	C₄²	C₂⁴×C₃₁	1	4	yes	—
360	C₂³	—	2	2	no	—
361	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
362	Q₈	—	2	8	no	—
363	Q₈	—	2	2	no	—
364	Q₈	—	2	8	no	—
365	Q₈	—	2	8	no	—
366	Q₈	—	2	2	no	—
367	Q₈	—	4	8	no	—
368	D₄	—	2	2	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	4	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	4	8	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	4	8	no	—
378	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
379	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
380	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
382	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
383	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
384	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
385	C₈	—	2	2	no	—
386	C₈	—	2	4	no	—
387	C₈	C₂×#41×C₃₁	1	2	yes	—
388	C₈	C₂×#41×C₃₁	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	4	no	—
392	C₂²	C₃₁×#550	1	2	yes	—
393	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
394	C₄	—	2	2	no	—
395	C₄	—	2	8	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	8	no	—
398	C₄	—	4	4	no	—
399	C₄	C₃₁×#550	1	2	yes	—
400	C₄	C₃₁×#578	1	4	yes	—
401	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
402	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
403	C₂	—	2	2	no	—
404	C₂	—	8	2	no	—
405	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
406	C₂	C₂×C₃₁×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3509	C₂	semidirect
2. C₃₁×#3398	C₂	semidirect
3. C₃₁×#3395	C₂	semidirect
4. C₃₁×#3393	C₂	semidirect
6. C₂×C₃₁×#494	C₂	semidirect
7. C₃₁×#3044	C₂	semidirect
8. C₃₁×#3039	C₂	semidirect
9. C₃₁×#2960	C₂	semidirect
10. C₃₁×#2950	C₂	semidirect
156. #61411	C₃₁	direct (N × H)
186. #3509	C₂×C₃₁	semidirect
187. #3398	C₂×C₃₁	semidirect
188. #3395	C₂×C₃₁	semidirect
189. #3393	C₂×C₃₁	semidirect
191. C₂×#494	C₂×C₃₁	semidirect
192. #3044	C₂×C₃₁	semidirect
193. #3039	C₂×C₃₁	semidirect
194. #2960	C₂×C₃₁	semidirect
195. #2950	C₂×C₃₁	semidirect
309. C₃₁	#61411	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×#3509	2
C₃₁×#3398	2
C₃₁×#3395	2
C₃₁×#3393	2
C₃₁×#3156	2
C₂×C₃₁×#494	2
C₃₁×#3044	2
C₃₁×#3039	2
C₃₁×#2960	2
C₃₁×#2950	2
C₄×C₃₁×#135	2
#61411	31

Subgroups — C31×#61411 #21518170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61411 #21518170