Subgroups — C₃₁×#23774 #21480533

Order 7936.

820 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2099	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2093	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2086	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2067	C₂	1	2	yes	maximal
7	C₃₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2023	C₂	1	1	yes	maximal, characteristic
9	C₄×C₃₁×#107	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#102	C₂	1	2	yes	maximal
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#391	—	2	1	no	—
23	C₃₁×#390	C₂²	1	1	yes	characteristic
24	C₃₁×#390	C₂²	1	1	yes	characteristic
25	C₃₁×#389	—	2	1	no	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#389	C₂²	1	1	yes	characteristic
28	C₃₁×#389	C₂²	1	1	yes	characteristic
29	C₃₁×#388	—	2	1	no	—
30	C₃₁×#388	—	2	1	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₃₁×C₄⋊Q₈	—	2	4	no	—
49	C₄×Q₈×C₃₁	—	4	4	no	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	Q₈	1	4	yes	complemented
52	C₃₁×#117	—	4	4	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#107	—	2	4	no	—
79	C₃₁×#107	—	2	4	no	—
80	C₃₁×C₄⋊C₈	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₄×C₈×C₃₁	—	2	4	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	4	no	—
91	C₂×Q₈×C₃₁	—	4	2	no	—
92	C₂×Q₈×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	4	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	4	2	no	—
104	C₂²×C₄×C₃₁	—	4	2	no	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	#41	1	4	yes	—
118	C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	4	2	no	—
126	C₄⋊C₄×C₃₁	—	4	2	no	—
127	C₄⋊C₄×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
131	C₄⋊C₄×C₃₁	#41	1	4	yes	—
132	C₂²⋊C₄×C₃₁	—	2	4	no	—
133	C₂²⋊C₄×C₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	4	1	no	—
142	C₄²×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	—	4	4	no	—
144	C₄²×C₃₁	D₈	1	4	yes	—
145	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
146	C₄²×C₃₁	#41	1	2	yes	—
147	#23774	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	—	4	1	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	—	4	1	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₃₁	—	4	8	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	8	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
187	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
189	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
190	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
191	C₂×C₄×C₃₁	#135	1	2	yes	—
192	C₂×C₄×C₃₁	#137	1	2	yes	—
193	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
196	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
197	C₈×C₃₁	—	2	4	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#2293	C₂×C₃₁	1	1	yes	characteristic
201	#2099	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2093	C₂×C₃₁	1	1	yes	characteristic
203	#2086	C₂×C₃₁	1	2	yes	complemented
204	#2067	C₂×C₃₁	1	2	yes	—
205	#2057	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#2023	C₂×C₃₁	1	1	yes	characteristic
207	C₄×#107	C₂×C₃₁	1	2	yes	complemented
208	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₄×#102	C₂×C₃₁	1	2	yes	—
210	C₂²×C₃₁	—	4	2	no	—
211	C₂²×C₃₁	—	4	2	no	—
212	C₂²×C₃₁	—	4	2	no	—
213	C₂²×C₃₁	—	8	1	no	—
214	C₂²×C₃₁	—	8	1	no	—
215	C₂²×C₃₁	—	8	1	no	—
216	C₂²×C₃₁	—	8	1	no	—
217	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
218	C₂²×C₃₁	#525	1	1	yes	characteristic
219	C₂²×C₃₁	#532	1	1	yes	characteristic
220	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
221	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
222	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
223	C₂²×C₃₁	#575	1	1	yes	characteristic
224	C₄×C₃₁	—	2	2	no	—
225	C₄×C₃₁	—	2	2	no	—
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	4	4	no	—
229	C₄×C₃₁	—	4	1	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	8	1	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	#474	1	4	yes	—
235	C₄×C₃₁	#529	1	2	yes	—
236	C₄×C₃₁	#533	1	2	yes	—
237	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₄×Q₈	—	2	1	no	—
239	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#117	—	2	1	no	—
241	C₂×#107	C₂²×C₃₁	1	2	yes	—
242	C₂×#107	C₂²×C₃₁	1	2	yes	—
243	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
244	C₂×#102	C₂²×C₃₁	1	2	yes	—
245	C₂×#102	C₂²×C₃₁	1	2	yes	—
246	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
247	#391	—	2	1	no	—
248	#390	C₂²×C₃₁	1	1	yes	characteristic
249	#390	C₂²×C₃₁	1	1	yes	characteristic
250	#389	—	2	1	no	—
251	#389	C₂²×C₃₁	1	2	yes	—
252	#389	C₂²×C₃₁	1	1	yes	characteristic
253	#389	C₂²×C₃₁	1	1	yes	characteristic
254	#388	—	2	1	no	—
255	#388	—	2	1	no	—
256	#384	C₂²×C₃₁	1	1	yes	characteristic
257	C₄×C₄⋊C₄	—	2	1	no	—
258	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
259	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₄×C₂²⋊C₄	—	2	1	no	—
261	C₄³	C₂²×C₃₁	1	1	yes	characteristic
262	#340	C₂²×C₃₁	1	2	yes	—
263	#340	C₂²×C₃₁	1	2	yes	—
264	#340	C₂²×C₃₁	1	2	yes	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#336	C₂²×C₃₁	1	1	yes	characteristic
267	#336	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₃₁	—	8	2	no	—
269	C₂×C₃₁	#2396	1	1	yes	characteristic
270	C₂×C₃₁	C₂×#474	1	1	yes	characteristic
271	C₂×C₃₁	#3076	1	1	yes	characteristic
272	C₂×C₃₁	#3100	1	1	yes	characteristic
273	C₂×C₃₁	#3108	1	1	yes	characteristic
274	C₂×C₃₁	#3169	1	1	yes	characteristic
275	C₂×C₃₁	#3172	1	1	yes	characteristic
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
280	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
281	C₄⋊Q₈	—	2	4	no	—
282	C₄×Q₈	—	4	4	no	—
283	C₄×D₄	—	2	2	no	—
284	C₄×D₄	Q₈×C₃₁	1	4	yes	complemented
285	#117	—	4	4	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	#107	—	2	4	no	—
312	#107	—	2	4	no	—
313	C₄⋊C₈	—	2	4	no	—
314	#102	—	2	4	no	—
315	#102	—	2	4	no	—
316	C₄×C₈	—	2	4	no	—
317	#95	—	2	1	no	—
318	#95	—	2	1	no	—
319	#95	—	2	2	no	—
320	#95	—	2	1	no	—
321	#95	—	2	2	no	—
322	C₃₁	#23774	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	—	2	1	no	—
324	C₂×Q₈	—	2	4	no	—
325	C₂×Q₈	—	4	2	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	2	no	—
338	C₂²×C₄	—	4	2	no	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	#41×C₃₁	1	4	yes	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	—	4	2	no	—
361	C₄⋊C₄	—	4	4	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
365	C₄⋊C₄	#41×C₃₁	1	4	yes	—
366	C₂²⋊C₄	—	2	4	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	4	2	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	4	1	no	—
376	C₄²	—	4	2	no	—
377	C₄²	—	4	4	no	—
378	C₄²	D₈×C₃₁	1	4	yes	—
379	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
380	C₄²	#41×C₃₁	1	2	yes	—
381	C₂³	—	2	2	no	—
382	C₂³	—	4	1	no	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
389	Q₈	—	4	8	no	—
390	D₄	—	2	4	no	—
391	D₄	—	4	2	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	8	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
420	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
421	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
422	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
423	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
424	C₂×C₄	C₃₁×#135	1	2	yes	—
425	C₂×C₄	C₃₁×#137	1	2	yes	—
426	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
429	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
430	C₈	—	2	4	no	—
431	C₈	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	8	1	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	1	no	—
439	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
440	C₂²	C₃₁×#525	1	1	yes	characteristic
441	C₂²	C₃₁×#532	1	1	yes	characteristic
442	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
443	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
444	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
445	C₂²	C₃₁×#575	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	1	no	—
452	C₄	—	4	2	no	—
453	C₄	—	8	1	no	—
454	C₄	—	8	1	no	—
455	C₄	—	8	2	no	—
456	C₄	C₃₁×#474	1	4	yes	—
457	C₄	C₃₁×#529	1	2	yes	—
458	C₄	C₃₁×#533	1	2	yes	—
459	C₂	—	8	2	no	—
460	C₂	C₃₁×#2396	1	1	yes	characteristic
461	C₂	C₂×C₃₁×#474	1	1	yes	characteristic
462	C₂	C₃₁×#3076	1	1	yes	characteristic
463	C₂	C₃₁×#3100	1	1	yes	characteristic
464	C₂	C₃₁×#3108	1	1	yes	characteristic
465	C₂	C₃₁×#3169	1	1	yes	characteristic
466	C₂	C₃₁×#3172	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₃₁×#2099	C₂	semidirect
5. C₃₁×#2086	C₂	semidirect
7. C₃₁×#2057	C₂	semidirect
9. C₄×C₃₁×#107	C₂	semidirect
10. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
51. C₄×D₄×C₃₁	Q₈	semidirect
147. #23774	C₃₁	direct (N × H)
199. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
201. #2099	C₂×C₃₁	semidirect
203. #2086	C₂×C₃₁	semidirect
205. #2057	C₂×C₃₁	semidirect
207. C₄×#107	C₂×C₃₁	semidirect
208. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
284. C₄×D₄	Q₈×C₃₁	semidirect
322. C₃₁	#23774	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2293	2
C₃₁×#2099	2
C₃₁×#2093	2
C₃₁×#2086	2
C₃₁×#2067	2
C₃₁×#2057	2
C₃₁×#2023	2
C₄×C₃₁×#107	2
C₄×C₃₁×C₄⋊C₈	2
C₄×C₃₁×#102	2
#23774	31

Subgroups — C31×#23774 #21480533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23774 #21480533