Subgroups — C₃₁×#60568 #21517327

Order 7936.

2138 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3126	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3122	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3119	C₂	1	8	yes	maximal, complemented
6	C₃₁×#3094	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3080	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3038	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3006	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₃₁×#540	—	2	2	no	—
15	C₃₁×#540	—	2	2	no	—
16	C₃₁×#540	C₂²	1	4	yes	—
17	C₃₁×#540	C₂²	1	4	yes	complemented
18	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃₁×#123	—	2	2	no	—
21	C₂×C₃₁×#121	—	2	2	no	—
22	C₂×C₃₁×#121	C₂²	1	2	yes	—
23	C₃₁×#520	—	2	2	no	—
24	C₃₁×#520	C₂²	1	4	yes	complemented
25	C₃₁×#519	C₂²	1	2	yes	complemented
26	C₃₁×#518	—	2	2	no	—
27	C₃₁×#518	C₂²	1	2	yes	complemented
28	C₄×#41×C₃₁	—	2	2	no	—
29	C₄×#41×C₃₁	C₂²	1	2	yes	—
30	C₄×#41×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
32	C₂×C₃₁×#117	—	2	2	no	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₃₁×#482	C₂²	1	8	yes	complemented
36	C₃₁×#481	C₂²	1	4	yes	complemented
37	C₃₁×#480	C₂²	1	4	yes	complemented
38	C₃₁×#479	C₂²	1	8	yes	complemented
39	C₃₁×#478	C₂²	1	8	yes	complemented
40	C₃₁×#460	C₂²	1	8	yes	complemented
41	C₃₁×#459	C₂²	1	8	yes	complemented
42	C₃₁×#449	C₂²	1	8	yes	complemented
43	C₃₁×#448	C₂²	1	4	yes	complemented
44	C₃₁×#447	C₂²	1	4	yes	complemented
45	C₃₁×#442	C₂²	1	8	yes	complemented
46	C₃₁×#440	C₂²	1	8	yes	complemented
47	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
48	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
49	C₃₁×#433	C₂²	1	4	yes	complemented
50	C₃₁×#433	C₂²	1	4	yes	complemented
51	C₃₁×#428	C₂²	1	4	yes	complemented
52	C₃₁×#423	C₂²	1	2	yes	complemented
53	C₃₁×#418	C₂²	1	4	yes	complemented
54	C₃₁×#417	C₂²	1	4	yes	—
55	C₃₁×#407	C₂²	1	2	yes	—
56	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
57	C₂×#41×C₃₁	—	2	8	no	—
58	C₂×#41×C₃₁	—	2	1	no	—
59	C₂×#41×C₃₁	—	2	1	no	—
60	C₂×#41×C₃₁	C₂³	1	2	yes	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₃₁	—	2	1	no	—
64	C₂²×D₄×C₃₁	—	2	2	no	—
65	C₂²×D₄×C₃₁	—	2	2	no	—
66	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	—	2	1	no	—
69	C₂³×C₄×C₃₁	—	2	2	no	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
72	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
73	C₂×D₈×C₃₁	C₂³	1	8	yes	—
74	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
75	C₃₁×#127	—	2	2	no	—
76	C₃₁×#127	C₂³	1	4	yes	—
77	C₃₁×#125	—	2	2	no	—
78	C₃₁×#125	C₂³	1	4	yes	—
79	C₃₁×#123	—	2	8	no	—
80	C₃₁×#123	—	2	8	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	—	2	2	no	—
83	C₃₁×#121	—	2	8	no	—
84	C₃₁×#121	—	2	8	no	—
85	C₃₁×#121	C₂³	1	4	yes	—
86	C₃₁×#121	C₂³	1	4	yes	—
87	C₄×Q₈×C₃₁	—	2	4	no	—
88	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
89	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
90	C₄×D₄×C₃₁	—	2	4	no	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₄×D₄×C₃₁	—	2	4	no	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	4	no	—
96	C₄×D₄×C₃₁	C₂³	1	4	yes	—
97	C₄×D₄×C₃₁	C₂³	1	8	yes	—
98	C₄×D₄×C₃₁	C₂³	1	4	yes	—
99	C₃₁×#117	—	2	1	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	4	no	—
102	C₃₁×#117	—	2	4	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	D₄	1	2	yes	—
106	C₃₁×#117	D₄	1	2	yes	—
107	C₃₁×#117	C₂³	1	2	yes	—
108	C₃₁×#117	C₂³	1	2	yes	—
109	C₃₁×#117	C₂³	1	2	yes	—
110	C₃₁×#117	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
121	C₂×C₄²×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	—	2	2	no	—
123	C₂×C₄²×C₃₁	—	2	2	no	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₃₁×#107	C₂³	1	8	yes	—
127	C₃₁×#106	C₂³	1	8	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
129	C₃₁×#103	C₂³	1	8	yes	—
130	C₃₁×#102	C₂³	1	8	yes	—
131	C₃₁×#102	C₂³	1	8	yes	—
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
135	C₃₁×#97	C₂³	1	4	yes	—
136	C₄×C₈×C₃₁	C₂³	1	4	yes	—
137	C₂⁴×C₃₁	—	2	1	no	—
138	C₂⁴×C₃₁	—	2	2	no	—
139	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
140	#41×C₃₁	—	2	8	no	—
141	#41×C₃₁	—	2	8	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	2	4	no	—
144	#41×C₃₁	—	2	4	no	—
145	#41×C₃₁	#41	1	8	yes	—
146	C₂×Q₈×C₃₁	—	2	4	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
149	C₂×D₄×C₃₁	—	2	8	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	2	4	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	2	8	no	—
158	C₂×D₄×C₃₁	—	2	8	no	—
159	C₂×D₄×C₃₁	—	8	4	no	—
160	C₂×D₄×C₃₁	—	8	4	no	—
161	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
162	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	2	1	no	—
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	2	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	4	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	—	2	8	no	—
173	C₂²×C₄×C₃₁	—	8	4	no	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	SD₁₆×C₃₁	—	2	16	no	—
183	D₈×C₃₁	—	2	16	no	—
184	M₁₆×C₃₁	—	2	8	no	—
185	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	4	no	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
203	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
204	C₄²×C₃₁	—	2	2	no	—
205	C₄²×C₃₁	—	2	4	no	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	2	no	—
208	C₄²×C₃₁	C₂×D₄	1	2	yes	—
209	C₄²×C₃₁	C₂×D₄	1	2	yes	—
210	C₄²×C₃₁	C₂⁴	1	2	yes	—
211	C₄²×C₃₁	C₂⁴	1	2	yes	—
212	#60568	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂³×C₃₁	—	2	4	no	—
214	C₂³×C₃₁	—	2	2	no	—
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	4	2	no	—
218	C₂³×C₃₁	—	4	4	no	—
219	C₂³×C₃₁	—	4	4	no	—
220	C₂³×C₃₁	—	4	4	no	—
221	C₂³×C₃₁	—	8	2	no	—
222	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₃₁	—	2	4	no	—
226	Q₈×C₃₁	—	2	2	no	—
227	Q₈×C₃₁	C₂×#41	1	4	yes	—
228	D₄×C₃₁	—	2	8	no	—
229	D₄×C₃₁	—	2	4	no	—
230	D₄×C₃₁	—	2	8	no	—
231	D₄×C₃₁	—	2	2	no	—
232	D₄×C₃₁	—	2	8	no	—
233	D₄×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	8	8	no	—
235	D₄×C₃₁	—	8	4	no	—
236	D₄×C₃₁	—	8	4	no	—
237	D₄×C₃₁	C₂×#41	1	4	yes	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	8	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	8	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	4	2	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	4	no	—
252	C₂×C₄×C₃₁	—	8	4	no	—
253	C₂×C₄×C₃₁	—	8	4	no	—
254	C₂×C₄×C₃₁	—	8	8	no	—
255	C₂×C₄×C₃₁	—	8	2	no	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
259	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
262	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
263	C₈×C₃₁	—	2	8	no	—
264	C₂×#540	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#3428	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#3126	C₂×C₃₁	1	4	yes	complemented
267	#3122	C₂×C₃₁	1	4	yes	complemented
268	#3119	C₂×C₃₁	1	8	yes	complemented
269	#3094	C₂×C₃₁	1	2	yes	complemented
270	#3080	C₂×C₃₁	1	2	yes	complemented
271	#3038	C₂×C₃₁	1	2	yes	complemented
272	#3006	C₂×C₃₁	1	2	yes	complemented
273	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	4	yes	complemented
274	#2921	C₂×C₃₁	1	1	yes	characteristic, complemented
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	2	no	—
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	2	1	no	—
280	C₂²×C₃₁	—	2	4	no	—
281	C₂²×C₃₁	—	4	4	no	—
282	C₂²×C₃₁	—	4	4	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	8	2	no	—
286	C₂²×C₃₁	—	8	8	no	—
287	C₂²×C₃₁	—	8	4	no	—
288	C₂²×C₃₁	#525	1	2	yes	—
289	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₃₁	—	2	4	no	—
291	C₄×C₃₁	—	2	4	no	—
292	C₄×C₃₁	—	2	1	no	—
293	C₄×C₃₁	—	2	4	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	4	4	no	—
296	C₄×C₃₁	—	4	4	no	—
297	C₄×C₃₁	#525	1	2	yes	—
298	C₄×C₃₁	C₂²×#41	1	2	yes	—
299	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
301	#540	—	2	2	no	—
302	#540	—	2	2	no	—
303	#540	C₂²×C₃₁	1	4	yes	—
304	#540	C₂²×C₃₁	1	4	yes	complemented
305	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
306	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	C₂×#123	—	2	2	no	—
308	C₂×#121	—	2	2	no	—
309	C₂×#121	C₂²×C₃₁	1	2	yes	—
310	#520	—	2	2	no	—
311	#520	C₂²×C₃₁	1	4	yes	complemented
312	#519	C₂²×C₃₁	1	2	yes	complemented
313	#518	—	2	2	no	—
314	#518	C₂²×C₃₁	1	2	yes	complemented
315	C₄×#41	—	2	2	no	—
316	C₄×#41	C₂²×C₃₁	1	2	yes	—
317	C₄×#41	C₂²×C₃₁	1	2	yes	—
318	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
319	C₂×#117	—	2	2	no	—
320	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
322	#482	C₂²×C₃₁	1	8	yes	complemented
323	#481	C₂²×C₃₁	1	4	yes	complemented
324	#480	C₂²×C₃₁	1	4	yes	complemented
325	#479	C₂²×C₃₁	1	8	yes	complemented
326	#478	C₂²×C₃₁	1	8	yes	complemented
327	#460	C₂²×C₃₁	1	8	yes	complemented
328	#459	C₂²×C₃₁	1	8	yes	complemented
329	#449	C₂²×C₃₁	1	8	yes	complemented
330	#448	C₂²×C₃₁	1	4	yes	complemented
331	#447	C₂²×C₃₁	1	4	yes	complemented
332	#442	C₂²×C₃₁	1	8	yes	complemented
333	#440	C₂²×C₃₁	1	8	yes	complemented
334	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
335	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
336	#433	C₂²×C₃₁	1	4	yes	complemented
337	#433	C₂²×C₃₁	1	4	yes	complemented
338	#428	C₂²×C₃₁	1	4	yes	complemented
339	#423	C₂²×C₃₁	1	2	yes	complemented
340	#418	C₂²×C₃₁	1	4	yes	complemented
341	#417	C₂²×C₃₁	1	4	yes	—
342	#407	C₂²×C₃₁	1	2	yes	—
343	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
344	C₂×C₃₁	—	2	2	no	—
345	C₂×C₃₁	—	4	1	no	—
346	C₂×C₃₁	—	4	1	no	—
347	C₂×C₃₁	—	4	4	no	—
348	C₂×C₃₁	—	8	4	no	—
349	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
350	C₂×C₃₁	#3585	1	2	yes	—
351	C₂×#41	—	2	8	no	—
352	C₂×#41	—	2	1	no	—
353	C₂×#41	—	2	1	no	—
354	C₂×#41	C₂³×C₃₁	1	2	yes	—
355	C₂×#41	C₂³×C₃₁	1	2	yes	—
356	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	—	2	2	no	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
361	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂³×C₄	—	2	1	no	—
363	C₂³×C₄	—	2	2	no	—
364	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
365	AGL(1,ℤ/8)	—	2	16	no	—
366	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
367	C₂×D₈	C₂³×C₃₁	1	8	yes	—
368	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
369	#127	—	2	2	no	—
370	#127	C₂³×C₃₁	1	4	yes	—
371	#125	—	2	2	no	—
372	#125	C₂³×C₃₁	1	4	yes	—
373	#123	—	2	8	no	—
374	#123	—	2	8	no	—
375	#121	—	2	2	no	—
376	#121	—	2	2	no	—
377	#121	—	2	8	no	—
378	#121	—	2	8	no	—
379	#121	C₂³×C₃₁	1	4	yes	—
380	#121	C₂³×C₃₁	1	4	yes	—
381	C₄×Q₈	—	2	4	no	—
382	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
383	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	4	no	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	C₂³×C₃₁	1	8	yes	—
391	C₄×D₄	C₂³×C₃₁	1	4	yes	—
392	C₄×D₄	C₂³×C₃₁	1	4	yes	—
393	#117	—	2	1	no	—
394	#117	—	2	1	no	—
395	#117	—	2	4	no	—
396	#117	—	2	4	no	—
397	#117	—	2	2	no	—
398	#117	—	2	2	no	—
399	#117	D₄×C₃₁	1	2	yes	—
400	#117	D₄×C₃₁	1	2	yes	—
401	#117	C₂³×C₃₁	1	2	yes	—
402	#117	C₂³×C₃₁	1	2	yes	—
403	#117	C₂³×C₃₁	1	2	yes	—
404	#117	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	—	2	4	no	—
407	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	4	no	—
414	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
415	C₂×C₄²	—	2	1	no	—
416	C₂×C₄²	—	2	2	no	—
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
420	#107	C₂³×C₃₁	1	8	yes	—
421	#106	C₂³×C₃₁	1	8	yes	—
422	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
423	#103	C₂³×C₃₁	1	8	yes	—
424	#102	C₂³×C₃₁	1	8	yes	—
425	#102	C₂³×C₃₁	1	8	yes	—
426	#102	C₂³×C₃₁	1	8	yes	—
427	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
428	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
429	#97	C₂³×C₃₁	1	4	yes	—
430	C₄×C₈	C₂³×C₃₁	1	4	yes	—
431	C₃₁	#60568	1	1	yes	characteristic, direct product factor, Sylow
432	C₂⁴	—	2	1	no	—
433	C₂⁴	—	2	2	no	—
434	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	2	8	no	—
437	#41	—	2	4	no	—
438	#41	—	2	4	no	—
439	#41	—	2	4	no	—
440	#41	#41×C₃₁	1	8	yes	—
441	C₂×Q₈	—	2	4	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	8	4	no	—
455	C₂×D₄	—	8	4	no	—
456	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
457	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
458	C₂²×C₄	—	2	8	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	8	no	—
468	C₂²×C₄	—	8	4	no	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
477	SD₁₆	—	2	16	no	—
478	D₈	—	2	16	no	—
479	M₁₆	—	2	8	no	—
480	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	8	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
487	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
488	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
489	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
498	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
499	C₄²	—	2	2	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	4	no	—
502	C₄²	—	2	2	no	—
503	C₄²	C₂×D₄×C₃₁	1	2	yes	—
504	C₄²	C₂×D₄×C₃₁	1	2	yes	—
505	C₄²	C₂⁴×C₃₁	1	2	yes	—
506	C₄²	C₂⁴×C₃₁	1	2	yes	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	4	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	4	no	—
515	C₂³	—	8	2	no	—
516	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
517	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
519	Q₈	—	2	4	no	—
520	Q₈	—	2	2	no	—
521	Q₈	C₂×#41×C₃₁	1	4	yes	—
522	D₄	—	2	8	no	—
523	D₄	—	2	4	no	—
524	D₄	—	2	8	no	—
525	D₄	—	2	2	no	—
526	D₄	—	2	8	no	—
527	D₄	—	4	4	no	—
528	D₄	—	8	8	no	—
529	D₄	—	8	4	no	—
530	D₄	—	8	4	no	—
531	D₄	C₂×#41×C₃₁	1	4	yes	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	8	4	no	—
547	C₂×C₄	—	8	4	no	—
548	C₂×C₄	—	8	8	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
552	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
553	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
556	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
557	C₈	—	2	8	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	8	2	no	—
569	C₂²	—	8	8	no	—
570	C₂²	—	8	4	no	—
571	C₂²	C₃₁×#525	1	2	yes	—
572	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
573	C₄	—	2	1	no	—
574	C₄	—	2	4	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	4	no	—
577	C₄	—	2	4	no	—
578	C₄	—	4	4	no	—
579	C₄	—	4	4	no	—
580	C₄	C₃₁×#525	1	2	yes	—
581	C₄	C₂²×#41×C₃₁	1	2	yes	—
582	C₂	—	2	2	no	—
583	C₂	—	4	1	no	—
584	C₂	—	4	1	no	—
585	C₂	—	4	4	no	—
586	C₂	—	8	4	no	—
587	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
588	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#540	C₂	semidirect
2. C₃₁×#3428	C₂	semidirect
3. C₃₁×#3126	C₂	semidirect
4. C₃₁×#3122	C₂	semidirect
5. C₃₁×#3119	C₂	semidirect
6. C₃₁×#3094	C₂	semidirect
7. C₃₁×#3080	C₂	semidirect
8. C₃₁×#3038	C₂	semidirect
9. C₃₁×#3006	C₂	semidirect
10. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
11. C₃₁×#2921	C₂	semidirect
17. C₃₁×#540	C₂²	semidirect
18. C₂×C₃₁×#127	C₂²	semidirect
19. C₂×C₃₁×#125	C₂²	semidirect
24. C₃₁×#520	C₂²	semidirect
25. C₃₁×#519	C₂²	semidirect
27. C₃₁×#518	C₂²	semidirect
31. C₂×C₄×D₄×C₃₁	C₂²	semidirect
35. C₃₁×#482	C₂²	semidirect
36. C₃₁×#481	C₂²	semidirect
37. C₃₁×#480	C₂²	semidirect
38. C₃₁×#479	C₂²	semidirect
39. C₃₁×#478	C₂²	semidirect
40. C₃₁×#460	C₂²	semidirect
41. C₃₁×#459	C₂²	semidirect
42. C₃₁×#449	C₂²	semidirect
43. C₃₁×#448	C₂²	semidirect
44. C₃₁×#447	C₂²	semidirect
45. C₃₁×#442	C₂²	semidirect
46. C₃₁×#440	C₂²	semidirect
47. C₄×SD₁₆×C₃₁	C₂²	semidirect
48. C₄×D₈×C₃₁	C₂²	semidirect
49. C₃₁×#433	C₂²	semidirect
50. C₃₁×#433	C₂²	semidirect
51. C₃₁×#428	C₂²	semidirect
52. C₃₁×#423	C₂²	semidirect
53. C₃₁×#418	C₂²	semidirect
56. C₄×M₁₆×C₃₁	C₂²	semidirect
212. #60568	C₃₁	direct (N × H)
264. C₂×#540	C₂×C₃₁	semidirect
265. #3428	C₂×C₃₁	semidirect
266. #3126	C₂×C₃₁	semidirect
267. #3122	C₂×C₃₁	semidirect
268. #3119	C₂×C₃₁	semidirect
269. #3094	C₂×C₃₁	semidirect
270. #3080	C₂×C₃₁	semidirect
271. #3038	C₂×C₃₁	semidirect
272. #3006	C₂×C₃₁	semidirect
273. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
274. #2921	C₂×C₃₁	semidirect
304. #540	C₂²×C₃₁	semidirect
305. C₂×#127	C₂²×C₃₁	semidirect
306. C₂×#125	C₂²×C₃₁	semidirect
311. #520	C₂²×C₃₁	semidirect
312. #519	C₂²×C₃₁	semidirect
314. #518	C₂²×C₃₁	semidirect
318. C₂×C₄×D₄	C₂²×C₃₁	semidirect
322. #482	C₂²×C₃₁	semidirect
323. #481	C₂²×C₃₁	semidirect
324. #480	C₂²×C₃₁	semidirect
325. #479	C₂²×C₃₁	semidirect
326. #478	C₂²×C₃₁	semidirect
327. #460	C₂²×C₃₁	semidirect
328. #459	C₂²×C₃₁	semidirect
329. #449	C₂²×C₃₁	semidirect
330. #448	C₂²×C₃₁	semidirect
331. #447	C₂²×C₃₁	semidirect
332. #442	C₂²×C₃₁	semidirect
333. #440	C₂²×C₃₁	semidirect
334. C₄×SD₁₆	C₂²×C₃₁	semidirect
335. C₄×D₈	C₂²×C₃₁	semidirect
336. #433	C₂²×C₃₁	semidirect
337. #433	C₂²×C₃₁	semidirect
338. #428	C₂²×C₃₁	semidirect
339. #423	C₂²×C₃₁	semidirect
340. #418	C₂²×C₃₁	semidirect
343. C₄×M₁₆	C₂²×C₃₁	semidirect
431. C₃₁	#60568	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×#540	2
C₃₁×#3428	2
C₃₁×#3126	2
C₃₁×#3122	2
C₃₁×#3119	2
C₃₁×#3094	2
C₃₁×#3080	2
C₃₁×#3038	2
C₃₁×#3006	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2921	2
#60568	31

Subgroups — C31×#60568 #21517327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60568 #21517327