Subgroups — C₃₁×#24341 #21481100

Order 7936.

1068 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2580	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2057	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2023	C₂	1	2	yes	maximal
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal
10	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	Q₈²×C₃₁	—	2	4	no	—
14	Q₈²×C₃₁	—	2	4	no	—
15	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
16	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₃₁	—	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×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#474	—	2	4	no	—
25	C₃₁×#440	—	2	4	no	—
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#107	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×#399	—	2	1	no	—
35	C₃₁×#393	—	2	1	no	—
36	C₃₁×#390	C₂²	1	2	yes	—
37	C₃₁×#389	C₂²	1	2	yes	—
38	C₃₁×#385	—	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	2	no	—
46	C₂²×Q₈×C₃₁	—	2	1	no	—
47	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	—	2	1	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	16	no	—
58	C₃₁×C₄⋊Q₈	—	2	8	no	—
59	C₃₁×C₄⋊Q₈	—	2	4	no	—
60	C₃₁×C₄⋊Q₈	—	2	4	no	—
61	C₃₁×C₄⋊Q₈	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
63	C₄×Q₈×C₃₁	—	2	8	no	—
64	C₄×Q₈×C₃₁	—	2	4	no	—
65	C₄×Q₈×C₃₁	—	2	4	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	D₄	1	4	yes	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	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	1	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₄²×C₃₁	—	2	2	no	—
87	C₂×C₄²×C₃₁	—	2	1	no	—
88	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#107	—	2	2	no	—
91	C₃₁×#107	—	2	4	no	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×#103	—	2	2	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	4	no	—
97	C₃₁×#97	—	2	2	no	—
98	C₃₁×#95	—	2	1	no	—
99	C₃₁×#95	—	2	1	no	—
100	C₃₁×#95	—	2	2	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	8	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	4	no	—
105	C₂×Q₈×C₃₁	—	2	4	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	—	8	2	no	—
110	C₂×Q₈×C₃₁	—	8	2	no	—
111	C₂×Q₈×C₃₁	—	8	2	no	—
112	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
114	C₂×Q₈×C₃₁	#41	1	2	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	C₂×Q₈	1	2	yes	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	—	8	2	no	—
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	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	SD₁₆×C₃₁	—	2	8	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
143	C₄⋊C₄×C₃₁	—	2	8	no	—
144	C₄⋊C₄×C₃₁	—	2	8	no	—
145	C₄⋊C₄×C₃₁	—	2	4	no	—
146	C₄⋊C₄×C₃₁	—	2	4	no	—
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	8	2	no	—
156	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	4	2	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₂²⋊C₄×C₃₁	—	8	2	no	—
161	C₄²×C₃₁	—	2	8	no	—
162	C₄²×C₃₁	—	2	4	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	C₂×D₄	1	2	yes	—
166	C₄²×C₃₁	C₂×D₄	1	2	yes	—
167	#24341	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	2	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	Q₈×C₃₁	—	2	8	no	—
175	Q₈×C₃₁	—	2	2	no	—
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	Q₈×C₃₁	—	8	4	no	—
179	Q₈×C₃₁	—	8	8	no	—
180	Q₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
181	D₄×C₃₁	—	2	2	no	—
182	D₄×C₃₁	—	2	2	no	—
183	D₄×C₃₁	—	4	1	no	—
184	D₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	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	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	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	1	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	8	4	no	—
206	C₂×C₄×C₃₁	#120	1	2	yes	—
207	C₂×C₄×C₃₁	#120	1	2	yes	—
208	C₂×C₄×C₃₁	#137	1	4	yes	—
209	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
214	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
215	C₈×C₃₁	—	2	2	no	—
216	C₈×C₃₁	—	4	1	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₂×Q₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂×#474	C₂×C₃₁	1	1	yes	characteristic
220	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
221	#2580	C₂×C₃₁	1	1	yes	characteristic
222	#2057	C₂×C₃₁	1	2	yes	complemented
223	#2023	C₂×C₃₁	1	2	yes	—
224	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1880	C₂×C₃₁	1	2	yes	—
227	#1863	C₂×C₃₁	1	2	yes	complemented
228	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	4	2	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	—	8	1	no	—
233	C₂²×C₃₁	—	8	1	no	—
234	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
235	C₂²×C₃₁	#548	1	1	yes	characteristic
236	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×#137	1	2	yes	—
238	C₂²×C₃₁	#576	1	2	yes	—
239	C₄×C₃₁	—	2	4	no	—
240	C₄×C₃₁	—	2	2	no	—
241	C₄×C₃₁	—	2	4	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	4	2	no	—
244	C₄×C₃₁	—	4	4	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	8	2	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
249	C₄×C₃₁	#536	1	1	yes	characteristic
250	C₄×C₃₁	#548	1	1	yes	characteristic
251	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
252	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
253	Q₈²	—	2	4	no	—
254	Q₈²	—	2	4	no	—
255	C₂×C₄⋊Q₈	—	2	2	no	—
256	C₂×C₄⋊Q₈	—	2	1	no	—
257	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
258	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×Q₈	—	2	1	no	—
260	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
261	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
264	#474	—	2	4	no	—
265	#440	—	2	4	no	—
266	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#107	C₂²×C₃₁	1	2	yes	—
268	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#103	C₂²×C₃₁	1	2	yes	—
270	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#102	C₂²×C₃₁	1	2	yes	—
272	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
274	#399	—	2	1	no	—
275	#393	—	2	1	no	—
276	#390	C₂²×C₃₁	1	2	yes	—
277	#389	C₂²×C₃₁	1	2	yes	—
278	#385	—	2	2	no	—
279	#382	C₂²×C₃₁	1	1	yes	characteristic
280	#340	C₂²×C₃₁	1	2	yes	—
281	#340	C₂²×C₃₁	1	2	yes	—
282	#340	C₂²×C₃₁	1	2	yes	—
283	#336	C₂²×C₃₁	1	2	yes	—
284	#336	C₂²×C₃₁	1	2	yes	—
285	C₂×C₃₁	—	8	2	no	—
286	C₂×C₃₁	#2409	1	1	yes	characteristic
287	C₂×C₃₁	#3004	1	1	yes	characteristic
288	C₂×C₃₁	#3008	1	1	yes	characteristic
289	C₂×C₃₁	#3375	1	2	yes	—
290	C₂×C₃₁	#3385	1	2	yes	—
291	C₂²×Q₈	—	2	2	no	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
294	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂³×C₄	—	2	1	no	—
297	C₂×SD₁₆	—	2	2	no	—
298	C₂×SD₁₆	—	2	2	no	—
299	C₂×SD₁₆	Q₈×C₃₁	1	4	yes	complemented
300	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
302	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄⋊Q₈	—	2	16	no	—
304	C₄⋊Q₈	—	2	8	no	—
305	C₄⋊Q₈	—	2	4	no	—
306	C₄⋊Q₈	—	2	4	no	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
309	C₄×Q₈	—	2	8	no	—
310	C₄×Q₈	—	2	4	no	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	—	2	2	no	—
313	C₄×Q₈	—	2	2	no	—
314	C₄×Q₈	—	2	2	no	—
315	C₄×Q₈	D₄×C₃₁	1	4	yes	—
316	C₄×D₄	—	2	2	no	—
317	C₄×D₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
324	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
335	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
336	#107	—	2	2	no	—
337	#107	—	2	4	no	—
338	C₄⋊C₈	—	2	2	no	—
339	#103	—	2	2	no	—
340	#103	—	2	4	no	—
341	#102	—	2	2	no	—
342	#102	—	2	4	no	—
343	#97	—	2	2	no	—
344	#95	—	2	1	no	—
345	#95	—	2	1	no	—
346	#95	—	2	2	no	—
347	C₃₁	#24341	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	—	2	1	no	—
349	C₂×Q₈	—	2	8	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	4	no	—
352	C₂×Q₈	—	2	4	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	8	2	no	—
357	C₂×Q₈	—	8	2	no	—
358	C₂×Q₈	—	8	2	no	—
359	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
360	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
361	C₂×Q₈	#41×C₃₁	1	2	yes	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	8	2	no	—
375	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
380	SD₁₆	—	2	8	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
390	C₄⋊C₄	—	2	8	no	—
391	C₄⋊C₄	—	2	8	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	8	2	no	—
403	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
404	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
405	C₂²⋊C₄	—	4	2	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₂²⋊C₄	—	8	2	no	—
408	C₄²	—	2	8	no	—
409	C₄²	—	2	4	no	—
410	C₄²	—	2	2	no	—
411	C₄²	—	2	2	no	—
412	C₄²	C₂×D₄×C₃₁	1	2	yes	—
413	C₄²	C₂×D₄×C₃₁	1	2	yes	—
414	C₂³	—	2	2	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	Q₈	—	2	8	no	—
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	—	2	2	no	—
424	Q₈	—	8	4	no	—
425	Q₈	—	8	8	no	—
426	Q₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
427	D₄	—	2	2	no	—
428	D₄	—	2	2	no	—
429	D₄	—	4	1	no	—
430	D₄	—	4	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	8	4	no	—
452	C₂×C₄	C₃₁×#120	1	2	yes	—
453	C₂×C₄	C₃₁×#120	1	2	yes	—
454	C₂×C₄	C₃₁×#137	1	4	yes	—
455	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
458	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
460	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
461	C₈	—	2	2	no	—
462	C₈	—	4	1	no	—
463	C₈	—	4	2	no	—
464	C₂²	—	4	4	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	8	2	no	—
467	C₂²	—	8	1	no	—
468	C₂²	—	8	1	no	—
469	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
470	C₂²	C₃₁×#548	1	1	yes	characteristic
471	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
472	C₂²	C₂×C₃₁×#137	1	2	yes	—
473	C₂²	C₃₁×#576	1	2	yes	—
474	C₄	—	2	4	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	4	no	—
477	C₄	—	2	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	4	no	—
480	C₄	—	4	2	no	—
481	C₄	—	8	2	no	—
482	C₄	—	8	1	no	—
483	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
484	C₄	C₃₁×#536	1	1	yes	characteristic
485	C₄	C₃₁×#548	1	1	yes	characteristic
486	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
487	C₂	—	8	2	no	—
488	C₂	C₃₁×#2409	1	1	yes	characteristic
489	C₂	C₃₁×#3004	1	1	yes	characteristic
490	C₂	C₃₁×#3008	1	1	yes	characteristic
491	C₂	C₃₁×#3375	1	2	yes	—
492	C₂	C₃₁×#3385	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
1. C₂×Q₈²×C₃₁	C₂	semidirect
5. C₃₁×#2057	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1985	C₂	semidirect
10. C₃₁×#1863	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
53. C₂×SD₁₆×C₃₁	Q₈	semidirect
167. #24341	C₃₁	direct (N × H)
218. C₂×Q₈²	C₂×C₃₁	semidirect
222. #2057	C₂×C₃₁	semidirect
224. #1993	C₂×C₃₁	semidirect
225. #1985	C₂×C₃₁	semidirect
227. #1863	C₂×C₃₁	semidirect
228. #1805	C₂×C₃₁	semidirect
299. C₂×SD₁₆	Q₈×C₃₁	semidirect
347. C₃₁	#24341	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×Q₈²×C₃₁	2
C₂×C₃₁×#474	2
C₂×C₃₁×#440	2
C₃₁×#2580	2
C₃₁×#2057	2
C₃₁×#2023	2
C₃₁×#1993	2
C₃₁×#1985	2
C₃₁×#1880	2
C₃₁×#1863	2
C₃₁×#1805	2
#24341	31

Subgroups — C31×#24341 #21481100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24341 #21481100