Subgroups — C₃₁×#21901 #21478660

Order 7936.

1058 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3455	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1986	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1980	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1897	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1896	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1806	C₂	1	4	yes	maximal, complemented
9	C₃₁×#542	C₄	1	4	yes	complemented
10	C₃₁×#541	C₄	1	2	yes	complemented
11	C₃₁×#540	C₄	1	2	yes	complemented
12	C₃₁×#539	—	2	2	no	—
13	C₃₁×#538	—	2	2	no	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₃₁×#529	—	2	4	no	—
17	C₃₁×#525	C₂²	1	2	yes	—
18	C₄×#41×C₃₁	C₂²	1	2	yes	—
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#433	—	2	4	no	—
23	C₃₁×#423	C₂²	1	2	yes	—
24	C₂×C₃₁×#104	C₂²	1	4	yes	—
25	C₂×C₃₁×#104	C₂²	1	4	yes	—
26	C₃₁×#407	C₂²	1	2	yes	—
27	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#389	—	2	2	no	—
29	C₃₁×#385	—	2	2	no	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₃₁×#339	C₂²	1	4	yes	—
33	C₃₁×#339	C₂²	1	4	yes	—
34	C₂×#41×C₃₁	C₂³	1	2	yes	—
35	C₂³×C₄×C₃₁	—	2	1	no	—
36	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
38	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
40	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
41	C₃₁×#126	—	2	4	no	—
42	C₃₁×#126	—	2	4	no	—
43	C₃₁×#125	—	2	4	no	—
44	C₃₁×#125	C₂×C₄	1	2	yes	complemented
45	C₃₁×#124	—	2	4	no	—
46	C₃₁×#124	C₂×C₄	1	2	yes	complemented
47	C₃₁×#123	—	2	2	no	—
48	C₃₁×#123	—	2	4	no	—
49	C₃₁×#123	—	2	4	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	2	4	no	—
52	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
53	C₃₁×#121	—	2	2	no	—
54	C₃₁×#121	—	2	4	no	—
55	C₃₁×#121	C₂×C₄	1	4	yes	—
56	C₃₁×#120	—	2	2	no	—
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	2	4	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	—	4	2	no	—
65	C₃₁×#117	C₂×C₄	1	2	yes	—
66	C₃₁×#117	C₂×C₄	1	2	yes	—
67	C₃₁×#117	D₄	1	4	yes	—
68	C₃₁×#117	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₂×C₄²×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	—	4	2	no	—
78	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊C₈	—	2	4	no	—
83	C₃₁×#104	—	4	4	no	—
84	C₃₁×#104	—	4	4	no	—
85	C₃₁×C₂²⋊C₈	—	2	4	no	—
86	C₃₁×#97	—	2	2	no	—
87	C₄×C₈×C₃₁	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
91	#41×C₃₁	—	2	4	no	—
92	#41×C₃₁	—	2	4	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	4	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₃₁	—	2	4	no	—
114	M₁₆×C₃₁	—	2	4	no	—
115	C₂×C₈×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	4	4	no	—
122	C₄⋊C₄×C₃₁	—	4	4	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
125	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
126	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
127	C₂²⋊C₄×C₃₁	—	2	4	no	—
128	C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂²⋊C₄×C₃₁	—	2	4	no	—
130	C₂²⋊C₄×C₃₁	—	2	4	no	—
131	C₂²⋊C₄×C₃₁	—	2	4	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
134	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
135	C₄²×C₃₁	—	2	4	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	4	no	—
138	C₄²×C₃₁	—	2	4	no	—
139	C₄²×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
141	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₃₁	C₂×D₄	1	2	yes	—
144	C₄²×C₃₁	C₂×D₄	1	2	yes	—
145	#21901	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	4	no	—
153	Q₈×C₃₁	—	8	2	no	—
154	D₄×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	8	2	no	—
158	D₄×C₃₁	—	8	4	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	8	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	8	4	no	—
176	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
177	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
183	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
184	C₈×C₃₁	—	4	4	no	—
185	#3455	C₂×C₃₁	1	1	yes	characteristic
186	#2921	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1986	C₂×C₃₁	1	2	yes	complemented
189	#1980	C₂×C₃₁	1	2	yes	complemented
190	#1897	C₂×C₃₁	1	2	yes	complemented
191	#1896	C₂×C₃₁	1	2	yes	complemented
192	#1806	C₂×C₃₁	1	4	yes	complemented
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	4	no	—
200	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₃₁	#535	1	1	yes	characteristic
202	C₂²×C₃₁	#536	1	1	yes	characteristic
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	4	no	—
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	4	4	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	4	4	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	#520	1	2	yes	—
212	C₄×C₃₁	#537	1	2	yes	—
213	#542	C₄×C₃₁	1	4	yes	complemented
214	#541	C₄×C₃₁	1	2	yes	complemented
215	#540	C₄×C₃₁	1	2	yes	complemented
216	#539	—	2	2	no	—
217	#538	—	2	2	no	—
218	#533	C₂²×C₃₁	1	1	yes	characteristic
219	#532	C₂²×C₃₁	1	1	yes	characteristic
220	#529	—	2	4	no	—
221	#525	C₂²×C₃₁	1	2	yes	—
222	C₄×#41	C₂²×C₃₁	1	2	yes	—
223	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
225	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
226	#433	—	2	4	no	—
227	#423	C₂²×C₃₁	1	2	yes	—
228	C₂×#104	C₂²×C₃₁	1	4	yes	—
229	C₂×#104	C₂²×C₃₁	1	4	yes	—
230	#407	C₂²×C₃₁	1	2	yes	—
231	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
232	#389	—	2	2	no	—
233	#385	—	2	2	no	—
234	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
235	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
236	#339	C₂²×C₃₁	1	4	yes	—
237	#339	C₂²×C₃₁	1	4	yes	—
238	C₂×C₃₁	—	2	1	no	—
239	C₂×C₃₁	—	2	1	no	—
240	C₂×C₃₁	—	4	2	no	—
241	C₂×C₃₁	—	8	2	no	—
242	C₂×C₃₁	#2311	1	1	yes	characteristic
243	C₂×C₃₁	#2901	1	2	yes	—
244	C₂×#41	C₂³×C₃₁	1	2	yes	—
245	C₂³×C₄	—	2	1	no	—
246	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
248	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
249	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
250	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
251	#126	—	2	4	no	—
252	#126	—	2	4	no	—
253	#125	—	2	4	no	—
254	#125	C₂×C₄×C₃₁	1	2	yes	complemented
255	#124	—	2	4	no	—
256	#124	C₂×C₄×C₃₁	1	2	yes	complemented
257	#123	—	2	2	no	—
258	#123	—	2	4	no	—
259	#123	—	2	4	no	—
260	C₂²⋊Q₈	—	2	2	no	—
261	C₂²⋊Q₈	—	2	4	no	—
262	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
263	#121	—	2	2	no	—
264	#121	—	2	4	no	—
265	#121	C₂×C₄×C₃₁	1	4	yes	—
266	#120	—	2	2	no	—
267	C₄×Q₈	—	2	2	no	—
268	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
269	C₄×D₄	—	2	4	no	—
270	C₄×D₄	—	2	4	no	—
271	C₄×D₄	—	2	2	no	—
272	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
273	#117	—	2	2	no	—
274	#117	—	4	2	no	—
275	#117	C₂×C₄×C₃₁	1	2	yes	—
276	#117	C₂×C₄×C₃₁	1	2	yes	—
277	#117	D₄×C₃₁	1	4	yes	—
278	#117	C₂³×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
283	C₂×C₂²⋊C₄	—	2	1	no	—
284	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	—	4	2	no	—
288	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
289	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	—	2	4	no	—
293	#104	—	4	4	no	—
294	#104	—	4	4	no	—
295	C₂²⋊C₈	—	2	4	no	—
296	#97	—	2	2	no	—
297	C₄×C₈	—	2	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	C₃₁	#21901	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
302	#41	—	2	4	no	—
303	#41	—	2	4	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	4	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
323	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	C₂×C₈	—	2	4	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	4	no	—
333	C₄⋊C₄	—	4	4	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
336	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
337	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
338	C₂²⋊C₄	—	2	4	no	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	4	no	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	4	2	no	—
344	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
345	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
346	C₄²	—	2	4	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	4	no	—
349	C₄²	—	2	4	no	—
350	C₄²	—	4	2	no	—
351	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
352	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₃₁	1	2	yes	—
355	C₄²	C₂×D₄×C₃₁	1	2	yes	—
356	C₂³	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
361	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	8	2	no	—
368	D₄	—	8	4	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	8	4	no	—
386	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
387	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
388	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
393	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
394	C₈	—	4	4	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	4	no	—
402	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
403	C₂²	C₃₁×#535	1	1	yes	characteristic
404	C₂²	C₃₁×#536	1	1	yes	characteristic
405	C₄	—	2	2	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	4	no	—
412	C₄	—	8	2	no	—
413	C₄	C₃₁×#520	1	2	yes	—
414	C₄	C₃₁×#537	1	2	yes	—
415	C₂	—	2	1	no	—
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	8	2	no	—
419	C₂	C₃₁×#2311	1	1	yes	characteristic
420	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2921	C₂	semidirect
3. C₃₁×#2296	C₂	semidirect
4. C₃₁×#1986	C₂	semidirect
5. C₃₁×#1980	C₂	semidirect
6. C₃₁×#1897	C₂	semidirect
7. C₃₁×#1896	C₂	semidirect
8. C₃₁×#1806	C₂	semidirect
9. C₃₁×#542	C₄	semidirect
10. C₃₁×#541	C₄	semidirect
11. C₃₁×#540	C₄	semidirect
38. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
39. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
40. C₃₁×#127	C₂×C₄	semidirect
44. C₃₁×#125	C₂×C₄	semidirect
46. C₃₁×#124	C₂×C₄	semidirect
58. C₄×Q₈×C₃₁	C₂×C₄	semidirect
62. C₄×D₄×C₃₁	C₂×C₄	semidirect
145. #21901	C₃₁	direct (N × H)
186. #2921	C₂×C₃₁	semidirect
187. #2296	C₂×C₃₁	semidirect
188. #1986	C₂×C₃₁	semidirect
189. #1980	C₂×C₃₁	semidirect
190. #1897	C₂×C₃₁	semidirect
191. #1896	C₂×C₃₁	semidirect
192. #1806	C₂×C₃₁	semidirect
213. #542	C₄×C₃₁	semidirect
214. #541	C₄×C₃₁	semidirect
215. #540	C₄×C₃₁	semidirect
248. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
249. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
250. #127	C₂×C₄×C₃₁	semidirect
254. #125	C₂×C₄×C₃₁	semidirect
256. #124	C₂×C₄×C₃₁	semidirect
268. C₄×Q₈	C₂×C₄×C₃₁	semidirect
272. C₄×D₄	C₂×C₄×C₃₁	semidirect
300. C₃₁	#21901	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×#3455	2
C₃₁×#2921	2
C₃₁×#2296	2
C₃₁×#1986	2
C₃₁×#1980	2
C₃₁×#1897	2
C₃₁×#1896	2
C₃₁×#1806	2
#21901	31

Subgroups — C31×#21901 #21478660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21901 #21478660