Subgroups — C₃₁×#23757 #21480516

Order 7936.

796 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2083	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2076	C₂	1	2	yes	maximal
7	C₃₁×#2072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2036	C₂	1	2	yes	maximal
9	C₄×C₃₁×#107	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#103	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	—	2	1	no	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#391	C₂²	1	1	yes	characteristic
23	C₃₁×#389	C₂²	1	1	yes	characteristic
24	C₃₁×#387	—	2	1	no	—
25	C₃₁×#385	—	2	1	no	—
26	C₃₁×#384	C₂²	1	1	yes	characteristic
27	C₃₁×#384	C₂²	1	1	yes	characteristic
28	C₃₁×#382	—	2	1	no	—
29	C₃₁×#380	—	2	1	no	—
30	C₃₁×#380	C₂²	1	1	yes	characteristic
31	C₃₁×#380	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₄×Q₈×C₃₁	—	2	4	no	—
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	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	4	no	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	—	2	4	no	—
77	C₃₁×#107	—	2	4	no	—
78	C₃₁×#103	—	2	4	no	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×C₂²⋊C₈	—	2	4	no	—
81	C₄×C₈×C₃₁	—	2	4	no	—
82	C₃₁×#95	—	2	1	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×Q₈×C₃₁	—	2	1	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	#41	1	2	yes	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
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	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	4	1	no	—
109	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	#41	1	4	yes	—
122	C₄⋊C₄×C₃₁	—	2	4	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	#41	1	2	yes	—
137	C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	4	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	4	2	no	—
145	C₄²×C₃₁	—	4	4	no	—
146	C₄²×C₃₁	#41	1	4	yes	—
147	#23757	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	2	1	no	—
149	C₂³×C₃₁	—	2	1	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
155	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	Q₈×C₃₁	—	2	4	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	2	1	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	8	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	4	4	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	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	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	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	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	#123	1	2	yes	—
190	C₂×C₄×C₃₁	#123	1	2	yes	—
191	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
192	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
193	C₂×C₄×C₃₁	#135	1	2	yes	—
194	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
199	C₈×C₃₁	—	2	4	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₄×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
202	#2285	C₂×C₃₁	1	1	yes	characteristic
203	#2100	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#2095	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#2083	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#2076	C₂×C₃₁	1	2	yes	—
207	#2072	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#2036	C₂×C₃₁	1	2	yes	—
209	C₄×#107	C₂×C₃₁	1	2	yes	complemented
210	C₄×#103	C₂×C₃₁	1	2	yes	complemented
211	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	4	1	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
220	C₂²×C₃₁	#525	1	1	yes	characteristic
221	C₂²×C₃₁	#532	1	1	yes	characteristic
222	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
223	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂²×C₃₁	#575	1	1	yes	characteristic
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	4	1	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	8	1	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	—	8	1	no	—
236	C₄×C₃₁	#484	1	4	yes	—
237	C₄×C₃₁	#529	1	2	yes	—
238	C₄×C₃₁	#529	1	2	yes	—
239	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#117	—	2	1	no	—
242	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#107	C₂²×C₃₁	1	2	yes	—
244	C₂×#107	C₂²×C₃₁	1	2	yes	—
245	C₂×#103	C₂²×C₃₁	1	2	yes	—
246	C₂×#103	C₂²×C₃₁	1	2	yes	—
247	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
248	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
249	#391	C₂²×C₃₁	1	1	yes	characteristic
250	#389	C₂²×C₃₁	1	1	yes	characteristic
251	#387	—	2	1	no	—
252	#385	—	2	1	no	—
253	#384	C₂²×C₃₁	1	1	yes	characteristic
254	#384	C₂²×C₃₁	1	1	yes	characteristic
255	#382	—	2	1	no	—
256	#380	—	2	1	no	—
257	#380	C₂²×C₃₁	1	1	yes	characteristic
258	#380	C₂²×C₃₁	1	2	yes	—
259	C₄×C₄⋊C₄	—	2	1	no	—
260	C₄×C₄⋊C₄	—	2	1	no	—
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
262	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₄×C₂²⋊C₄	—	2	1	no	—
264	#340	C₂²×C₃₁	1	2	yes	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	2	yes	—
268	#336	C₂²×C₃₁	1	1	yes	characteristic
269	#336	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₃₁	—	4	2	no	—
271	C₂×C₃₁	#2395	1	1	yes	characteristic
272	C₂×C₃₁	C₂×#484	1	1	yes	characteristic
273	C₂×C₃₁	#3091	1	1	yes	characteristic
274	C₂×C₃₁	#3104	1	1	yes	characteristic
275	C₂×C₃₁	#3106	1	1	yes	characteristic
276	C₂×C₃₁	#3170	1	1	yes	characteristic
277	C₂×C₃₁	#3171	1	1	yes	characteristic
278	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
283	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
284	C₂²⋊Q₈	—	2	2	no	—
285	C₂²⋊Q₈	—	2	2	no	—
286	C₄×Q₈	—	2	4	no	—
287	#117	—	4	4	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₄²	—	2	4	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄²	D₄×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	#103	—	2	4	no	—
314	#103	—	2	4	no	—
315	C₂²⋊C₈	—	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	1	no	—
322	#95	—	2	2	no	—
323	#95	—	2	1	no	—
324	C₃₁	#23757	1	1	yes	characteristic, direct product factor, Sylow
325	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	#41×C₃₁	1	2	yes	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	4	1	no	—
345	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	#41×C₃₁	1	4	yes	—
358	C₄⋊C₄	—	2	4	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	#41×C₃₁	1	2	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	4	2	no	—
380	C₄²	—	4	2	no	—
381	C₄²	—	4	4	no	—
382	C₄²	#41×C₃₁	1	4	yes	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
390	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
391	Q₈	—	2	4	no	—
392	Q₈	—	4	2	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	8	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	4	4	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	C₃₁×#123	1	2	yes	—
425	C₂×C₄	C₃₁×#123	1	2	yes	—
426	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
427	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
428	C₂×C₄	C₃₁×#135	1	2	yes	—
429	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
430	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
434	C₈	—	2	4	no	—
435	C₈	—	4	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	—	4	1	no	—
442	C₂²	—	4	1	no	—
443	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
444	C₂²	C₃₁×#525	1	1	yes	characteristic
445	C₂²	C₃₁×#532	1	1	yes	characteristic
446	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
447	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
448	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
449	C₂²	C₃₁×#575	1	1	yes	characteristic
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	1	no	—
460	C₄	C₃₁×#484	1	4	yes	—
461	C₄	C₃₁×#529	1	2	yes	—
462	C₄	C₃₁×#529	1	2	yes	—
463	C₂	—	4	2	no	—
464	C₂	C₃₁×#2395	1	1	yes	characteristic
465	C₂	C₂×C₃₁×#484	1	1	yes	characteristic
466	C₂	C₃₁×#3091	1	1	yes	characteristic
467	C₂	C₃₁×#3104	1	1	yes	characteristic
468	C₂	C₃₁×#3106	1	1	yes	characteristic
469	C₂	C₃₁×#3170	1	1	yes	characteristic
470	C₂	C₃₁×#3171	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	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2100	C₂	semidirect
4. C₃₁×#2095	C₂	semidirect
5. C₃₁×#2083	C₂	semidirect
7. C₃₁×#2072	C₂	semidirect
9. C₄×C₃₁×#107	C₂	semidirect
10. C₄×C₃₁×#103	C₂	semidirect
147. #23757	C₃₁	direct (N × H)
203. #2100	C₂×C₃₁	semidirect
204. #2095	C₂×C₃₁	semidirect
205. #2083	C₂×C₃₁	semidirect
207. #2072	C₂×C₃₁	semidirect
209. C₄×#107	C₂×C₃₁	semidirect
210. C₄×#103	C₂×C₃₁	semidirect
324. C₃₁	#23757	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×C₂²⋊Q₈	2
C₃₁×#2285	2
C₃₁×#2100	2
C₃₁×#2095	2
C₃₁×#2083	2
C₃₁×#2076	2
C₃₁×#2072	2
C₃₁×#2036	2
C₄×C₃₁×#107	2
C₄×C₃₁×#103	2
C₄×C₃₁×C₂²⋊C₈	2
#23757	31

Subgroups — C31×#23757 #21480516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23757 #21480516