Subgroups — C₃₁×#24197 #21480956

Order 7936.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2580	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2012	C₂	1	2	yes	maximal
7	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#556	—	2	4	no	—
15	C₃₁×#556	—	2	4	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#126	—	2	2	no	—
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#117	—	2	1	no	—
25	C₃₁×#461	—	2	4	no	—
26	C₃₁×#442	—	2	4	no	—
27	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#102	C₂²	1	2	yes	—
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₃₁×#399	—	2	1	no	—
37	C₃₁×#393	—	2	1	no	—
38	C₃₁×#390	C₂²	1	2	yes	—
39	C₃₁×#389	C₂²	1	2	yes	—
40	C₃₁×#385	—	2	2	no	—
41	C₃₁×#382	C₂²	1	1	yes	characteristic
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
62	C₃₁×#126	—	2	16	no	—
63	C₃₁×#124	—	2	4	no	—
64	C₃₁×#124	—	2	4	no	—
65	C₃₁×C₂²⋊Q₈	—	2	8	no	—
66	C₄×Q₈×C₃₁	—	2	4	no	—
67	C₄×Q₈×C₃₁	—	2	4	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	D₄	1	4	yes	—
74	C₃₁×#117	—	2	8	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	—	2	2	no	—
91	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#106	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×#103	—	2	4	no	—
96	C₃₁×#103	—	2	4	no	—
97	C₃₁×#103	—	4	2	no	—
98	C₃₁×#102	—	2	2	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×#102	—	4	2	no	—
101	C₃₁×#97	—	2	2	no	—
102	C₃₁×#95	—	2	1	no	—
103	C₃₁×#95	—	2	1	no	—
104	C₃₁×#95	—	2	2	no	—
105	C₂⁴×C₃₁	—	2	1	no	—
106	C₂⁴×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	2	2	no	—
112	C₂×Q₈×C₃₁	—	2	2	no	—
113	C₂×Q₈×C₃₁	—	4	2	no	—
114	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₃₁	#41	1	2	yes	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	4	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	8	2	no	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
140	SD₁₆×C₃₁	—	4	8	no	—
141	D₈×C₃₁	—	2	8	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	#41	1	2	yes	—
151	C₄⋊C₄×C₃₁	—	2	8	no	—
152	C₄⋊C₄×C₃₁	—	2	4	no	—
153	C₄⋊C₄×C₃₁	—	2	4	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	8	2	no	—
162	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	—	2	4	no	—
166	C₂²⋊C₄×C₃₁	—	4	4	no	—
167	C₂²⋊C₄×C₃₁	—	4	2	no	—
168	C₂²⋊C₄×C₃₁	—	8	2	no	—
169	C₄²×C₃₁	—	2	8	no	—
170	C₄²×C₃₁	—	2	2	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	C₂×D₄	1	2	yes	—
173	C₄²×C₃₁	C₂×D₄	1	2	yes	—
174	#24197	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	—	4	1	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₃₁	—	2	4	no	—
187	Q₈×C₃₁	—	2	4	no	—
188	Q₈×C₃₁	—	4	4	no	—
189	Q₈×C₃₁	—	4	2	no	—
190	Q₈×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	2	2	no	—
193	D₄×C₃₁	—	2	2	no	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	4	1	no	—
197	D₄×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	4	8	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	8	4	no	—
219	C₂×C₄×C₃₁	#120	1	2	yes	—
220	C₂×C₄×C₃₁	#120	1	2	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
226	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
227	C₈×C₃₁	—	2	2	no	—
228	C₈×C₃₁	—	4	1	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₂×#556	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂×#442	C₂×C₃₁	1	1	yes	characteristic
233	#2580	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2042	C₂×C₃₁	1	2	yes	complemented
235	#2012	C₂×C₃₁	1	2	yes	—
236	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1960	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1880	C₂×C₃₁	1	2	yes	complemented
239	#1865	C₂×C₃₁	1	2	yes	complemented
240	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	—	4	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	—	8	2	no	—
249	C₂²×C₃₁	—	8	1	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
252	C₂²×C₃₁	#548	1	1	yes	characteristic
253	C₂²×C₃₁	#550	1	1	yes	characteristic
254	C₂²×C₃₁	#575	1	2	yes	—
255	C₂²×C₃₁	#577	1	2	yes	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	4	4	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	—	8	2	no	—
263	C₄×C₃₁	—	8	1	no	—
264	C₄×C₃₁	#525	1	1	yes	characteristic
265	C₄×C₃₁	#537	1	1	yes	characteristic
266	C₄×C₃₁	#546	1	1	yes	characteristic
267	C₄×C₃₁	#547	1	1	yes	characteristic
268	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
269	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
270	#556	—	2	4	no	—
271	#556	—	2	4	no	—
272	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#126	—	2	2	no	—
274	C₂×#124	C₂²×C₃₁	1	2	yes	—
275	C₂×C₂²⋊Q₈	—	2	1	no	—
276	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
277	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#117	—	2	1	no	—
281	#461	—	2	4	no	—
282	#442	—	2	4	no	—
283	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×#103	C₂²×C₃₁	1	2	yes	—
286	C₂×#103	C₂²×C₃₁	1	2	yes	—
287	C₂×#103	C₂²×C₃₁	1	2	yes	—
288	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×#102	C₂²×C₃₁	1	2	yes	—
291	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
292	#399	—	2	1	no	—
293	#393	—	2	1	no	—
294	#390	C₂²×C₃₁	1	2	yes	—
295	#389	C₂²×C₃₁	1	2	yes	—
296	#385	—	2	2	no	—
297	#382	C₂²×C₃₁	1	1	yes	characteristic
298	#340	C₂²×C₃₁	1	2	yes	—
299	#336	C₂²×C₃₁	1	2	yes	—
300	#336	C₂²×C₃₁	1	2	yes	—
301	C₂×C₃₁	—	8	2	no	—
302	C₂×C₃₁	—	8	2	no	—
303	C₂×C₃₁	#2407	1	1	yes	characteristic
304	C₂×C₃₁	#3005	1	1	yes	characteristic
305	C₂×C₃₁	#3010	1	1	yes	characteristic
306	C₂×C₃₁	#3376	1	2	yes	—
307	C₂×C₃₁	#3401	1	2	yes	—
308	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
310	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂³×C₄	—	2	1	no	—
313	C₂³×C₄	—	2	1	no	—
314	C₂×SD₁₆	—	2	4	no	—
315	C₂×SD₁₆	—	2	4	no	—
316	C₂×SD₁₆	—	4	2	no	—
317	C₂×D₈	—	2	2	no	—
318	C₂×D₈	—	2	2	no	—
319	C₂×D₈	—	2	2	no	—
320	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
321	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
325	#126	—	2	16	no	—
326	#124	—	2	4	no	—
327	#124	—	2	4	no	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₄×Q₈	—	2	4	no	—
330	C₄×Q₈	—	2	4	no	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	D₄×C₃₁	1	4	yes	—
337	#117	—	2	8	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₄²	—	2	2	no	—
354	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
355	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
356	#106	—	2	2	no	—
357	C₄⋊C₈	—	2	2	no	—
358	#103	—	2	4	no	—
359	#103	—	2	4	no	—
360	#103	—	4	2	no	—
361	#102	—	2	2	no	—
362	#102	—	2	2	no	—
363	#102	—	4	2	no	—
364	#97	—	2	2	no	—
365	#95	—	2	1	no	—
366	#95	—	2	1	no	—
367	#95	—	2	2	no	—
368	C₃₁	#24197	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁴	—	2	1	no	—
370	C₂⁴	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	4	2	no	—
378	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
390	C₂×D₄	#41×C₃₁	1	2	yes	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	8	2	no	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	SD₁₆	—	4	8	no	—
405	D₈	—	2	8	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	#41×C₃₁	1	2	yes	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	8	2	no	—
426	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	4	2	no	—
432	C₂²⋊C₄	—	8	2	no	—
433	C₄²	—	2	8	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	2	no	—
436	C₄²	C₂×D₄×C₃₁	1	2	yes	—
437	C₄²	C₂×D₄×C₃₁	1	2	yes	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	1	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	4	no	—
451	Q₈	—	4	4	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	8	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	8	4	no	—
482	C₂×C₄	C₃₁×#120	1	2	yes	—
483	C₂×C₄	C₃₁×#120	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
489	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
490	C₈	—	2	2	no	—
491	C₈	—	4	1	no	—
492	C₈	—	4	2	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	8	2	no	—
499	C₂²	—	8	2	no	—
500	C₂²	—	8	2	no	—
501	C₂²	—	8	1	no	—
502	C₂²	—	8	1	no	—
503	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
504	C₂²	C₃₁×#548	1	1	yes	characteristic
505	C₂²	C₃₁×#550	1	1	yes	characteristic
506	C₂²	C₃₁×#575	1	2	yes	—
507	C₂²	C₃₁×#577	1	2	yes	—
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	4	4	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	2	no	—
514	C₄	—	8	2	no	—
515	C₄	—	8	1	no	—
516	C₄	C₃₁×#525	1	1	yes	characteristic
517	C₄	C₃₁×#537	1	1	yes	characteristic
518	C₄	C₃₁×#546	1	1	yes	characteristic
519	C₄	C₃₁×#547	1	1	yes	characteristic
520	C₂	—	8	2	no	—
521	C₂	—	8	2	no	—
522	C₂	C₃₁×#2407	1	1	yes	characteristic
523	C₂	C₃₁×#3005	1	1	yes	characteristic
524	C₂	C₃₁×#3010	1	1	yes	characteristic
525	C₂	C₃₁×#3376	1	2	yes	—
526	C₂	C₃₁×#3401	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#556	C₂	semidirect
2. C₂×C₃₁×#461	C₂	semidirect
4. C₃₁×#2580	C₂	semidirect
5. C₃₁×#2042	C₂	semidirect
7. C₃₁×#1985	C₂	semidirect
8. C₃₁×#1960	C₂	semidirect
9. C₃₁×#1880	C₂	semidirect
10. C₃₁×#1865	C₂	semidirect
11. C₃₁×#1804	C₂	semidirect
174. #24197	C₃₁	direct (N × H)
230. C₂×#556	C₂×C₃₁	semidirect
231. C₂×#461	C₂×C₃₁	semidirect
233. #2580	C₂×C₃₁	semidirect
234. #2042	C₂×C₃₁	semidirect
236. #1985	C₂×C₃₁	semidirect
237. #1960	C₂×C₃₁	semidirect
238. #1880	C₂×C₃₁	semidirect
239. #1865	C₂×C₃₁	semidirect
240. #1804	C₂×C₃₁	semidirect
368. C₃₁	#24197	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#556	2
C₂×C₃₁×#461	2
C₂×C₃₁×#442	2
C₃₁×#2580	2
C₃₁×#2042	2
C₃₁×#2012	2
C₃₁×#1985	2
C₃₁×#1960	2
C₃₁×#1880	2
C₃₁×#1865	2
C₃₁×#1804	2
#24197	31

Subgroups — C31×#24197 #21480956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24197 #21480956