Subgroups — C₃₁×#60489 #21517248

Order 7936.

2002 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3084	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3083	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3082	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3081	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3076	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2896	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#544	—	2	2	no	—
17	C₃₁×#544	C₂²	1	4	yes	—
18	C₃₁×#544	C₂²	1	4	yes	complemented
19	C₃₁×#543	—	2	2	no	—
20	C₃₁×#533	—	2	2	no	—
21	C₃₁×#533	—	2	2	no	—
22	C₃₁×#533	C₂²	1	4	yes	complemented
23	C₃₁×#520	—	2	2	no	—
24	C₃₁×#520	C₂²	1	4	yes	—
25	C₃₁×#519	C₂²	1	2	yes	—
26	C₃₁×#518	—	2	2	no	—
27	C₃₁×#518	C₂²	1	2	yes	—
28	C₄×#41×C₃₁	—	2	2	no	—
29	C₄×#41×C₃₁	C₂²	1	4	yes	—
30	C₂×C₃₁×#117	—	2	2	no	—
31	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₃₁×#479	C₂²	1	8	yes	complemented
36	C₃₁×#479	C₂²	1	8	yes	complemented
37	C₃₁×#478	C₂²	1	8	yes	complemented
38	C₃₁×#478	C₂²	1	8	yes	complemented
39	C₃₁×#477	C₂²	1	8	yes	complemented
40	C₃₁×#476	C₂²	1	8	yes	complemented
41	C₃₁×#475	C₂²	1	8	yes	complemented
42	C₃₁×#474	C₂²	1	8	yes	complemented
43	C₃₁×#428	C₂²	1	4	yes	—
44	C₃₁×#428	C₂²	1	4	yes	complemented
45	C₃₁×#427	C₂²	1	4	yes	—
46	C₃₁×#427	C₂²	1	4	yes	complemented
47	C₃₁×#427	C₂²	1	4	yes	complemented
48	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
49	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
50	C₃₁×#424	C₂²	1	4	yes	—
51	C₃₁×#424	C₂²	1	2	yes	complemented
52	C₃₁×#424	C₂²	1	2	yes	complemented
53	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
54	C₃₁×#419	C₂²	1	2	yes	—
55	C₃₁×#418	C₂²	1	2	yes	—
56	C₃₁×#417	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	4	yes	—
58	C₃₁×#416	C₂²	1	4	yes	—
59	C₃₁×#416	C₂²	1	2	yes	—
60	C₃₁×#416	C₂²	1	2	yes	—
61	C₂×C₃₁×#103	C₂²	1	2	yes	—
62	C₂×C₃₁×#102	C₂²	1	2	yes	—
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂×#41×C₃₁	—	2	1	no	—
65	C₂×#41×C₃₁	—	2	4	no	—
66	C₂×#41×C₃₁	Q₈	1	8	yes	complemented
67	C₂×#41×C₃₁	C₂³	1	2	yes	—
68	C₂×#41×C₃₁	C₂³	1	2	yes	—
69	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₃₁	—	2	1	no	—
71	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	—	2	1	no	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₃₁	—	2	4	no	—
75	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₃₁	—	2	4	no	—
78	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
79	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
80	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	—	2	4	no	—
84	C₃₁×C₄⋊Q₈	—	2	4	no	—
85	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
86	C₃₁×#125	—	2	4	no	—
87	C₃₁×#125	—	2	4	no	—
88	C₃₁×#125	C₂³	1	8	yes	—
89	C₃₁×C₂²⋊Q₈	—	2	8	no	—
90	C₃₁×C₂²⋊Q₈	—	2	8	no	—
91	C₃₁×C₂²⋊Q₈	—	2	8	no	—
92	C₄×Q₈×C₃₁	—	2	4	no	—
93	C₄×Q₈×C₃₁	—	2	8	no	—
94	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
95	C₄×D₄×C₃₁	—	2	8	no	—
96	C₄×D₄×C₃₁	—	2	4	no	—
97	C₄×D₄×C₃₁	—	2	8	no	—
98	C₄×D₄×C₃₁	C₂³	1	8	yes	—
99	C₃₁×#117	—	2	1	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	4	no	—
106	C₃₁×#117	—	2	4	no	—
107	C₃₁×#117	—	2	4	no	—
108	C₃₁×#117	D₄	1	2	yes	—
109	C₃₁×#117	D₄	1	2	yes	—
110	C₃₁×#117	C₂³	1	2	yes	—
111	C₃₁×#117	C₂³	1	2	yes	—
112	C₃₁×#117	C₂³	1	2	yes	—
113	C₃₁×#117	C₂³	1	2	yes	—
114	C₃₁×#117	C₂³	1	2	yes	—
115	C₃₁×#117	C₂³	1	2	yes	—
116	C₃₁×#117	C₂³	1	2	yes	—
117	C₃₁×#117	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
127	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
128	C₂×C₄²×C₃₁	—	2	1	no	—
129	C₂×C₄²×C₃₁	—	2	1	no	—
130	C₂×C₄²×C₃₁	—	2	1	no	—
131	C₂×C₄²×C₃₁	—	2	4	no	—
132	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
133	C₃₁×#107	C₂³	1	4	yes	—
134	C₃₁×#107	C₂³	1	4	yes	—
135	C₃₁×#107	C₂³	1	8	yes	—
136	C₃₁×#106	C₂³	1	4	yes	—
137	C₃₁×#106	C₂³	1	4	yes	—
138	C₃₁×#106	C₂³	1	8	yes	—
139	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
140	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
141	C₃₁×#103	C₂³	1	4	yes	—
142	C₃₁×#103	C₂³	1	4	yes	—
143	C₃₁×#103	C₂³	1	8	yes	—
144	C₃₁×#102	C₂³	1	4	yes	—
145	C₃₁×#102	C₂³	1	4	yes	—
146	C₃₁×#102	C₂³	1	8	yes	—
147	C₂⁴×C₃₁	—	2	1	no	—
148	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
149	#41×C₃₁	—	2	8	no	—
150	#41×C₃₁	—	2	8	no	—
151	#41×C₃₁	—	2	4	no	—
152	#41×C₃₁	—	2	4	no	—
153	#41×C₃₁	C₂×Q₈	1	8	yes	—
154	#41×C₃₁	C₂×Q₈	1	8	yes	—
155	C₂×Q₈×C₃₁	—	2	1	no	—
156	C₂×Q₈×C₃₁	—	2	8	no	—
157	C₂×Q₈×C₃₁	—	2	2	no	—
158	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
159	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
160	C₂×D₄×C₃₁	—	2	8	no	—
161	C₂×D₄×C₃₁	—	2	2	no	—
162	C₂×D₄×C₃₁	—	2	2	no	—
163	C₂×D₄×C₃₁	—	2	1	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
166	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₃₁	—	2	1	no	—
168	C₂²×C₄×C₃₁	—	2	8	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	2	no	—
171	C₂²×C₄×C₃₁	—	2	2	no	—
172	C₂²×C₄×C₃₁	—	2	2	no	—
173	C₂²×C₄×C₃₁	—	2	2	no	—
174	C₂²×C₄×C₃₁	—	2	8	no	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
180	C₂²×C₄×C₃₁	#41	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
184	M₁₆×C₃₁	—	2	4	no	—
185	M₁₆×C₃₁	—	2	4	no	—
186	C₂×C₈×C₃₁	—	2	4	no	—
187	C₂×C₈×C₃₁	—	2	4	no	—
188	C₂×C₈×C₃₁	—	2	2	no	—
189	C₂×C₈×C₃₁	—	2	2	no	—
190	C₂×C₈×C₃₁	—	2	4	no	—
191	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
192	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
193	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	2	8	no	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	8	no	—
199	C₄⋊C₄×C₃₁	—	2	8	no	—
200	C₄⋊C₄×C₃₁	—	2	8	no	—
201	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
202	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
203	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
204	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
205	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	—	2	2	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	—	2	8	no	—
213	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
214	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
215	C₄²×C₃₁	—	2	8	no	—
216	C₄²×C₃₁	—	2	2	no	—
217	C₄²×C₃₁	—	2	2	no	—
218	C₄²×C₃₁	—	2	2	no	—
219	C₄²×C₃₁	—	2	2	no	—
220	C₄²×C₃₁	—	2	2	no	—
221	C₄²×C₃₁	—	2	2	no	—
222	C₄²×C₃₁	C₂×D₄	1	4	yes	—
223	C₄²×C₃₁	C₂⁴	1	4	yes	—
224	#60489	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	4	no	—
227	C₂³×C₃₁	—	4	2	no	—
228	C₂³×C₃₁	—	8	2	no	—
229	C₂³×C₃₁	C₂²⋊Q₈	1	4	yes	—
230	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	Q₈×C₃₁	—	2	4	no	—
234	Q₈×C₃₁	—	2	2	no	—
235	Q₈×C₃₁	—	8	8	no	—
236	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
237	D₄×C₃₁	—	2	8	no	—
238	D₄×C₃₁	—	2	8	no	—
239	D₄×C₃₁	—	2	4	no	—
240	D₄×C₃₁	—	2	2	no	—
241	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	8	no	—
245	C₂×C₄×C₃₁	—	2	8	no	—
246	C₂×C₄×C₃₁	—	2	4	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	4	no	—
249	C₂×C₄×C₃₁	—	4	4	no	—
250	C₂×C₄×C₃₁	—	8	8	no	—
251	C₂×C₄×C₃₁	—	8	2	no	—
252	C₂×C₄×C₃₁	—	8	2	no	—
253	C₂×C₄×C₃₁	—	8	2	no	—
254	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
255	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
256	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
262	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
263	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
264	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
265	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
266	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
267	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
268	C₈×C₃₁	—	2	2	no	—
269	C₈×C₃₁	—	2	2	no	—
270	C₈×C₃₁	—	2	4	no	—
271	#3458	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#3428	C₂×C₃₁	1	1	yes	characteristic
273	#3084	C₂×C₃₁	1	4	yes	complemented
274	#3083	C₂×C₃₁	1	4	yes	complemented
275	#3082	C₂×C₃₁	1	4	yes	complemented
276	#3081	C₂×C₃₁	1	4	yes	complemented
277	#3076	C₂×C₃₁	1	8	yes	complemented
278	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#2896	C₂×C₃₁	1	1	yes	characteristic
282	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	2	no	—
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	4	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	8	4	no	—
290	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
291	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
292	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
293	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
294	C₄×C₃₁	—	2	8	no	—
295	C₄×C₃₁	—	2	4	no	—
296	C₄×C₃₁	—	4	4	no	—
297	C₄×C₃₁	—	4	4	no	—
298	C₄×C₃₁	—	4	4	no	—
299	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
300	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
301	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
302	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
303	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
304	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
306	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
307	#544	—	2	2	no	—
308	#544	C₂²×C₃₁	1	4	yes	—
309	#544	C₂²×C₃₁	1	4	yes	complemented
310	#543	—	2	2	no	—
311	#533	—	2	2	no	—
312	#533	—	2	2	no	—
313	#533	C₂²×C₃₁	1	4	yes	complemented
314	#520	—	2	2	no	—
315	#520	C₂²×C₃₁	1	4	yes	—
316	#519	C₂²×C₃₁	1	2	yes	—
317	#518	—	2	2	no	—
318	#518	C₂²×C₃₁	1	2	yes	—
319	C₄×#41	—	2	2	no	—
320	C₄×#41	C₂²×C₃₁	1	4	yes	—
321	C₂×#117	—	2	2	no	—
322	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
326	#479	C₂²×C₃₁	1	8	yes	complemented
327	#479	C₂²×C₃₁	1	8	yes	complemented
328	#478	C₂²×C₃₁	1	8	yes	complemented
329	#478	C₂²×C₃₁	1	8	yes	complemented
330	#477	C₂²×C₃₁	1	8	yes	complemented
331	#476	C₂²×C₃₁	1	8	yes	complemented
332	#475	C₂²×C₃₁	1	8	yes	complemented
333	#474	C₂²×C₃₁	1	8	yes	complemented
334	#428	C₂²×C₃₁	1	4	yes	—
335	#428	C₂²×C₃₁	1	4	yes	complemented
336	#427	C₂²×C₃₁	1	4	yes	—
337	#427	C₂²×C₃₁	1	4	yes	complemented
338	#427	C₂²×C₃₁	1	4	yes	complemented
339	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
340	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
341	#424	C₂²×C₃₁	1	4	yes	—
342	#424	C₂²×C₃₁	1	2	yes	complemented
343	#424	C₂²×C₃₁	1	2	yes	complemented
344	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
345	#419	C₂²×C₃₁	1	2	yes	—
346	#418	C₂²×C₃₁	1	2	yes	—
347	#417	C₂²×C₃₁	1	4	yes	—
348	#416	C₂²×C₃₁	1	4	yes	—
349	#416	C₂²×C₃₁	1	4	yes	—
350	#416	C₂²×C₃₁	1	2	yes	—
351	#416	C₂²×C₃₁	1	2	yes	—
352	C₂×#103	C₂²×C₃₁	1	2	yes	—
353	C₂×#102	C₂²×C₃₁	1	2	yes	—
354	C₂×C₃₁	—	2	2	no	—
355	C₂×C₃₁	—	2	2	no	—
356	C₂×C₃₁	—	2	2	no	—
357	C₂×C₃₁	—	4	4	no	—
358	C₂×C₃₁	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
359	C₂×C₃₁	#3584	1	2	yes	—
360	C₂×#41	—	2	1	no	—
361	C₂×#41	—	2	1	no	—
362	C₂×#41	—	2	4	no	—
363	C₂×#41	Q₈×C₃₁	1	8	yes	complemented
364	C₂×#41	C₂³×C₃₁	1	2	yes	—
365	C₂×#41	C₂³×C₃₁	1	2	yes	—
366	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₂²×D₄	—	2	1	no	—
368	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
371	C₂×M₁₆	—	2	4	no	—
372	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
373	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
374	C₂²×C₈	—	2	4	no	—
375	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
376	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
377	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
380	C₄⋊Q₈	—	2	4	no	—
381	C₄⋊Q₈	—	2	4	no	—
382	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
383	#125	—	2	4	no	—
384	#125	—	2	4	no	—
385	#125	C₂³×C₃₁	1	8	yes	—
386	C₂²⋊Q₈	—	2	8	no	—
387	C₂²⋊Q₈	—	2	8	no	—
388	C₂²⋊Q₈	—	2	8	no	—
389	C₄×Q₈	—	2	4	no	—
390	C₄×Q₈	—	2	8	no	—
391	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
392	C₄×D₄	—	2	8	no	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	C₂³×C₃₁	1	8	yes	—
396	#117	—	2	1	no	—
397	#117	—	2	1	no	—
398	#117	—	2	1	no	—
399	#117	—	2	1	no	—
400	#117	—	2	1	no	—
401	#117	—	2	1	no	—
402	#117	—	2	4	no	—
403	#117	—	2	4	no	—
404	#117	—	2	4	no	—
405	#117	D₄×C₃₁	1	2	yes	—
406	#117	D₄×C₃₁	1	2	yes	—
407	#117	C₂³×C₃₁	1	2	yes	—
408	#117	C₂³×C₃₁	1	2	yes	—
409	#117	C₂³×C₃₁	1	2	yes	—
410	#117	C₂³×C₃₁	1	2	yes	—
411	#117	C₂³×C₃₁	1	2	yes	—
412	#117	C₂³×C₃₁	1	2	yes	—
413	#117	C₂³×C₃₁	1	2	yes	—
414	#117	C₂³×C₃₁	1	2	yes	—
415	C₂×C₄⋊C₄	—	2	4	no	—
416	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	4	no	—
424	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	—	2	4	no	—
429	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
430	#107	C₂³×C₃₁	1	4	yes	—
431	#107	C₂³×C₃₁	1	4	yes	—
432	#107	C₂³×C₃₁	1	8	yes	—
433	#106	C₂³×C₃₁	1	4	yes	—
434	#106	C₂³×C₃₁	1	4	yes	—
435	#106	C₂³×C₃₁	1	8	yes	—
436	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
437	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
438	#103	C₂³×C₃₁	1	4	yes	—
439	#103	C₂³×C₃₁	1	4	yes	—
440	#103	C₂³×C₃₁	1	8	yes	—
441	#102	C₂³×C₃₁	1	4	yes	—
442	#102	C₂³×C₃₁	1	4	yes	—
443	#102	C₂³×C₃₁	1	8	yes	—
444	C₃₁	#60489	1	1	yes	characteristic, direct product factor, Sylow
445	C₂⁴	—	2	1	no	—
446	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
447	#41	—	2	8	no	—
448	#41	—	2	8	no	—
449	#41	—	2	4	no	—
450	#41	—	2	4	no	—
451	#41	C₂×Q₈×C₃₁	1	8	yes	—
452	#41	C₂×Q₈×C₃₁	1	8	yes	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	8	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
457	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
458	C₂×D₄	—	2	8	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
464	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
478	C₂²×C₄	#41×C₃₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	M₁₆	—	2	4	no	—
483	M₁₆	—	2	4	no	—
484	C₂×C₈	—	2	4	no	—
485	C₂×C₈	—	2	4	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	4	no	—
489	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
490	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
491	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	8	no	—
497	C₄⋊C₄	—	2	8	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
500	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
501	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
502	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
503	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	8	no	—
511	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
512	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	8	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	C₂×D₄×C₃₁	1	4	yes	—
521	C₄²	C₂⁴×C₃₁	1	4	yes	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	8	2	no	—
526	C₂³	C₃₁×C₂²⋊Q₈	1	4	yes	—
527	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	Q₈	—	2	4	no	—
531	Q₈	—	2	2	no	—
532	Q₈	—	8	8	no	—
533	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
534	D₄	—	2	8	no	—
535	D₄	—	2	8	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	2	no	—
538	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	8	8	no	—
548	C₂×C₄	—	8	2	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	—	8	2	no	—
551	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
552	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
553	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
554	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
555	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
557	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
559	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
560	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
561	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
562	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
563	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
564	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	2	no	—
567	C₈	—	2	4	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	8	4	no	—
575	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
576	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
577	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
578	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
579	C₄	—	2	4	no	—
580	C₄	—	2	8	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	4	no	—
584	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
585	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
586	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
587	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
588	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
589	C₂	—	2	2	no	—
590	C₂	—	2	2	no	—
591	C₂	—	2	2	no	—
592	C₂	—	4	4	no	—
593	C₂	C₂²×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
594	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3458	C₂	semidirect
3. C₃₁×#3084	C₂	semidirect
4. C₃₁×#3083	C₂	semidirect
5. C₃₁×#3082	C₂	semidirect
6. C₃₁×#3081	C₂	semidirect
7. C₃₁×#3076	C₂	semidirect
8. C₃₁×#2911	C₂	semidirect
9. C₂×C₃₁×#427	C₂	semidirect
10. C₂×C₃₁×#424	C₂	semidirect
18. C₃₁×#544	C₂²	semidirect
22. C₃₁×#533	C₂²	semidirect
35. C₃₁×#479	C₂²	semidirect
36. C₃₁×#479	C₂²	semidirect
37. C₃₁×#478	C₂²	semidirect
38. C₃₁×#478	C₂²	semidirect
39. C₃₁×#477	C₂²	semidirect
40. C₃₁×#476	C₂²	semidirect
41. C₃₁×#475	C₂²	semidirect
42. C₃₁×#474	C₂²	semidirect
44. C₃₁×#428	C₂²	semidirect
46. C₃₁×#427	C₂²	semidirect
47. C₃₁×#427	C₂²	semidirect
48. C₂×C₃₁×#107	C₂²	semidirect
49. C₂×C₃₁×#106	C₂²	semidirect
51. C₃₁×#424	C₂²	semidirect
52. C₃₁×#424	C₂²	semidirect
53. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
66. C₂×#41×C₃₁	Q₈	semidirect
224. #60489	C₃₁	direct (N × H)
271. #3458	C₂×C₃₁	semidirect
273. #3084	C₂×C₃₁	semidirect
274. #3083	C₂×C₃₁	semidirect
275. #3082	C₂×C₃₁	semidirect
276. #3081	C₂×C₃₁	semidirect
277. #3076	C₂×C₃₁	semidirect
278. #2911	C₂×C₃₁	semidirect
279. C₂×#427	C₂×C₃₁	semidirect
280. C₂×#424	C₂×C₃₁	semidirect
309. #544	C₂²×C₃₁	semidirect
313. #533	C₂²×C₃₁	semidirect
326. #479	C₂²×C₃₁	semidirect
327. #479	C₂²×C₃₁	semidirect
328. #478	C₂²×C₃₁	semidirect
329. #478	C₂²×C₃₁	semidirect
330. #477	C₂²×C₃₁	semidirect
331. #476	C₂²×C₃₁	semidirect
332. #475	C₂²×C₃₁	semidirect
333. #474	C₂²×C₃₁	semidirect
335. #428	C₂²×C₃₁	semidirect
337. #427	C₂²×C₃₁	semidirect
338. #427	C₂²×C₃₁	semidirect
339. C₂×#107	C₂²×C₃₁	semidirect
340. C₂×#106	C₂²×C₃₁	semidirect
342. #424	C₂²×C₃₁	semidirect
343. #424	C₂²×C₃₁	semidirect
344. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
363. C₂×#41	Q₈×C₃₁	semidirect
444. C₃₁	#60489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3458	2
C₃₁×#3428	2
C₃₁×#3084	2
C₃₁×#3083	2
C₃₁×#3082	2
C₃₁×#3081	2
C₃₁×#3076	2
C₃₁×#2911	2
C₂×C₃₁×#427	2
C₂×C₃₁×#424	2
C₃₁×#2896	2
C₂×C₃₁×#416	2
#60489	31

Subgroups — C31×#60489 #21517248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60489 #21517248