Subgroups — C₃₁×#24343 #21481102

Order 7936.

1064 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#554	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2059	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2026	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1877	C₂	1	2	yes	maximal
10	C₃₁×#1864	C₂	1	2	yes	maximal
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#554	—	2	4	no	—
14	C₃₁×#554	—	2	4	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
24	C₃₁×#475	—	2	4	no	—
25	C₃₁×#440	—	2	4	no	—
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#106	C₂²	1	2	yes	—
28	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₃₁×#400	—	2	1	no	—
35	C₃₁×#395	—	2	1	no	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#387	—	2	2	no	—
39	C₃₁×#382	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
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	—	2	1	no	—
46	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×Q₈×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	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	Q₈	1	4	yes	complemented
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
59	C₃₁×#125	—	2	4	no	—
60	C₃₁×#125	—	2	4	no	—
61	C₃₁×C₂²⋊Q₈	—	2	8	no	—
62	C₃₁×C₂²⋊Q₈	—	2	16	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×Q₈×C₃₁	—	2	2	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	4	no	—
68	C₄×D₄×C₃₁	—	2	4	no	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	D₄	1	4	yes	—
71	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#107	—	2	2	no	—
89	C₃₁×#106	—	2	4	no	—
90	C₃₁×C₄⋊C₈	—	2	2	no	—
91	C₃₁×#103	—	2	2	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×#97	—	2	2	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₃₁×#95	—	2	2	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₃₁×#95	—	2	2	no	—
100	C₃₁×#95	—	2	1	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	4	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	4	2	no	—
110	C₂×Q₈×C₃₁	—	8	2	no	—
111	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
117	C₂×D₄×C₃₁	#41	1	2	yes	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	4	4	no	—
126	C₂²×C₄×C₃₁	—	8	4	no	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
132	SD₁₆×C₃₁	—	2	8	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
142	C₄⋊C₄×C₃₁	—	2	8	no	—
143	C₄⋊C₄×C₃₁	—	2	8	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	4	no	—
146	C₄⋊C₄×C₃₁	—	2	4	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	4	4	no	—
154	C₄⋊C₄×C₃₁	—	4	2	no	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
156	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₃₁	—	2	8	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	C₂×D₄	1	2	yes	—
163	C₄²×C₃₁	C₂×D₄	1	2	yes	—
164	#24343	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	—	2	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	4	4	no	—
174	Q₈×C₃₁	—	4	1	no	—
175	Q₈×C₃₁	—	4	1	no	—
176	Q₈×C₃₁	—	8	4	no	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	2	2	no	—
180	D₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	2	2	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	4	8	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	4	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	8	8	no	—
202	C₂×C₄×C₃₁	#120	1	2	yes	—
203	C₂×C₄×C₃₁	#120	1	2	yes	—
204	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
209	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	4	1	no	—
212	C₈×C₃₁	—	4	2	no	—
213	C₂×#554	C₂×C₃₁	1	1	yes	characteristic
214	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
216	#2581	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#2059	C₂×C₃₁	1	2	yes	complemented
218	#2026	C₂×C₃₁	1	2	yes	complemented
219	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1877	C₂×C₃₁	1	2	yes	—
222	#1864	C₂×C₃₁	1	2	yes	—
223	#1804	C₂×C₃₁	1	1	yes	characteristic
224	C₂²×C₃₁	—	2	4	no	—
225	C₂²×C₃₁	—	4	4	no	—
226	C₂²×C₃₁	—	4	4	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	—	4	2	no	—
229	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
230	C₂²×C₃₁	#548	1	1	yes	characteristic
231	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
232	C₂²×C₃₁	#575	1	2	yes	—
233	C₂²×C₃₁	#577	1	2	yes	—
234	C₄×C₃₁	—	2	2	no	—
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	4	4	no	—
241	C₄×C₃₁	—	8	2	no	—
242	C₄×C₃₁	—	8	1	no	—
243	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
244	C₄×C₃₁	#536	1	1	yes	characteristic
245	C₄×C₃₁	#548	1	1	yes	characteristic
246	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
247	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
248	#554	—	2	4	no	—
249	#554	—	2	4	no	—
250	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#125	C₂²×C₃₁	1	2	yes	—
252	C₂×C₂²⋊Q₈	—	2	1	no	—
253	C₂×C₂²⋊Q₈	—	2	2	no	—
254	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₂²×C₄⋊C₄	—	2	1	no	—
259	#475	—	2	4	no	—
260	#440	—	2	4	no	—
261	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#106	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×#103	C₂²×C₃₁	1	2	yes	—
265	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#102	C₂²×C₃₁	1	2	yes	—
267	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
269	#400	—	2	1	no	—
270	#395	—	2	1	no	—
271	#391	C₂²×C₃₁	1	2	yes	—
272	#389	C₂²×C₃₁	1	2	yes	—
273	#387	—	2	2	no	—
274	#382	C₂²×C₃₁	1	1	yes	characteristic
275	#340	C₂²×C₃₁	1	2	yes	—
276	#340	C₂²×C₃₁	1	2	yes	—
277	#340	C₂²×C₃₁	1	2	yes	—
278	#336	C₂²×C₃₁	1	2	yes	—
279	#336	C₂²×C₃₁	1	2	yes	—
280	C₂×C₃₁	—	4	4	no	—
281	C₂×C₃₁	#2409	1	1	yes	characteristic
282	C₂×C₃₁	#3005	1	1	yes	characteristic
283	C₂×C₃₁	#3009	1	1	yes	characteristic
284	C₂×C₃₁	#3376	1	2	yes	—
285	C₂×C₃₁	#3388	1	2	yes	—
286	C₂²×Q₈	—	2	1	no	—
287	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
288	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂³×C₄	—	2	1	no	—
291	C₂³×C₄	—	2	2	no	—
292	C₂×SD₁₆	—	2	2	no	—
293	C₂×SD₁₆	—	2	2	no	—
294	C₂×SD₁₆	Q₈×C₃₁	1	4	yes	complemented
295	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄⋊Q₈	—	2	2	no	—
299	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
300	#125	—	2	4	no	—
301	#125	—	2	4	no	—
302	C₂²⋊Q₈	—	2	8	no	—
303	C₂²⋊Q₈	—	2	16	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	2	no	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	2	4	no	—
310	C₄×D₄	—	2	2	no	—
311	C₄×D₄	D₄×C₃₁	1	4	yes	—
312	C₂×C₄⋊C₄	—	2	8	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
328	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
329	#107	—	2	2	no	—
330	#106	—	2	4	no	—
331	C₄⋊C₈	—	2	2	no	—
332	#103	—	2	2	no	—
333	#103	—	2	4	no	—
334	#102	—	2	2	no	—
335	#102	—	2	4	no	—
336	#97	—	2	2	no	—
337	#95	—	2	1	no	—
338	#95	—	2	1	no	—
339	#95	—	2	2	no	—
340	#95	—	2	2	no	—
341	#95	—	2	2	no	—
342	C₃₁	#24343	1	1	yes	characteristic, direct product factor, Sylow
343	C₂⁴	—	2	1	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	4	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	4	2	no	—
352	C₂×Q₈	—	8	2	no	—
353	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
358	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
359	C₂×D₄	#41×C₃₁	1	2	yes	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	—	8	4	no	—
369	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
374	SD₁₆	—	2	8	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	8	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	4	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	4	4	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
398	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	4	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	4	2	no	—
404	C₄²	C₂×D₄×C₃₁	1	2	yes	—
405	C₄²	C₂×D₄×C₃₁	1	2	yes	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
412	Q₈	—	2	2	no	—
413	Q₈	—	2	2	no	—
414	Q₈	—	4	4	no	—
415	Q₈	—	4	1	no	—
416	Q₈	—	4	1	no	—
417	Q₈	—	8	4	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	2	2	no	—
421	D₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	4	8	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	8	8	no	—
443	C₂×C₄	C₃₁×#120	1	2	yes	—
444	C₂×C₄	C₃₁×#120	1	2	yes	—
445	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
450	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	4	1	no	—
453	C₈	—	4	2	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	4	4	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
460	C₂²	C₃₁×#548	1	1	yes	characteristic
461	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
462	C₂²	C₃₁×#575	1	2	yes	—
463	C₂²	C₃₁×#577	1	2	yes	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	4	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	1	no	—
473	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
474	C₄	C₃₁×#536	1	1	yes	characteristic
475	C₄	C₃₁×#548	1	1	yes	characteristic
476	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
477	C₂	—	4	4	no	—
478	C₂	C₃₁×#2409	1	1	yes	characteristic
479	C₂	C₃₁×#3005	1	1	yes	characteristic
480	C₂	C₃₁×#3009	1	1	yes	characteristic
481	C₂	C₃₁×#3376	1	2	yes	—
482	C₂	C₃₁×#3388	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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
2. C₂×C₃₁×#475	C₂	semidirect
4. C₃₁×#2581	C₂	semidirect
5. C₃₁×#2059	C₂	semidirect
6. C₃₁×#2026	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1985	C₂	semidirect
53. C₂×SD₁₆×C₃₁	Q₈	semidirect
164. #24343	C₃₁	direct (N × H)
214. C₂×#475	C₂×C₃₁	semidirect
216. #2581	C₂×C₃₁	semidirect
217. #2059	C₂×C₃₁	semidirect
218. #2026	C₂×C₃₁	semidirect
219. #1993	C₂×C₃₁	semidirect
220. #1985	C₂×C₃₁	semidirect
294. C₂×SD₁₆	Q₈×C₃₁	semidirect
342. C₃₁	#24343	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×#554	2
C₂×C₃₁×#475	2
C₂×C₃₁×#440	2
C₃₁×#2581	2
C₃₁×#2059	2
C₃₁×#2026	2
C₃₁×#1993	2
C₃₁×#1985	2
C₃₁×#1877	2
C₃₁×#1864	2
C₃₁×#1804	2
#24343	31

Subgroups — C31×#24343 #21481102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24343 #21481102