Subgroups — C₃₁×#34629 #21491388

Order 7936.

1558 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2655	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2652	C₂	1	4	yes	maximal
3	C₃₁×#2647	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2645	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2642	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2629	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#2625	C₂	1	2	yes	maximal
8	C₃₁×#2621	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2402	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2401	C₂	1	4	yes	maximal
11	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
14	C₃₁×#2295	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#401	C₂²	1	4	yes	—
24	C₃₁×#400	C₂²	1	4	yes	—
25	C₃₁×#400	C₂²	1	4	yes	—
26	C₃₁×#400	C₂²	1	4	yes	—
27	C₃₁×#400	C₂²	1	4	yes	—
28	C₃₁×#400	C₂²	1	4	yes	—
29	C₃₁×#400	C₂²	1	4	yes	—
30	C₃₁×#400	C₂²	1	4	yes	—
31	C₃₁×#398	C₂²	1	4	yes	—
32	C₃₁×#398	C₂²	1	4	yes	—
33	C₃₁×#398	C₂²	1	4	yes	—
34	C₃₁×#398	C₂²	1	4	yes	—
35	C₃₁×#397	C₂²	1	4	yes	—
36	C₃₁×#397	C₂²	1	4	yes	—
37	C₃₁×#397	C₂²	1	4	yes	—
38	C₃₁×#396	C₂²	1	4	yes	—
39	C₃₁×#391	C₂²	1	2	yes	—
40	C₃₁×#391	C₂²	1	2	yes	—
41	C₃₁×#391	C₂²	1	2	yes	—
42	C₃₁×#391	C₂²	1	2	yes	—
43	C₃₁×#389	C₂²	1	4	yes	—
44	C₃₁×#389	C₂²	1	2	yes	—
45	C₃₁×#389	C₂²	1	2	yes	—
46	C₃₁×#389	C₂²	1	2	yes	—
47	C₃₁×#389	C₂²	1	2	yes	—
48	C₃₁×#387	C₂²	1	4	yes	—
49	C₃₁×#387	C₂²	1	4	yes	—
50	C₃₁×#387	C₂²	1	4	yes	—
51	C₃₁×#387	C₂²	1	4	yes	—
52	C₃₁×#385	C₂²	1	4	yes	—
53	C₃₁×#384	C₂²	1	1	yes	characteristic
54	C₃₁×#382	C₂²	1	1	yes	characteristic
55	C₃₁×#380	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
66	C₃₁×#376	C₂²	1	1	yes	characteristic
67	C₃₁×#376	C₂²	1	1	yes	characteristic
68	C₃₁×#376	C₂²	1	4	yes	—
69	C₄³×C₃₁	C₂²	1	1	yes	characteristic
70	C₄³×C₃₁	C₂²	1	1	yes	characteristic
71	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	—	2	4	no	—
76	C₃₁×#125	—	2	4	no	—
77	C₃₁×C₂²⋊Q₈	—	2	8	no	—
78	C₄×Q₈×C₃₁	—	2	8	no	—
79	C₃₁×#117	—	2	4	no	—
80	C₃₁×#117	—	2	4	no	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄²×C₃₁	—	2	4	no	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₃₁×#95	C₂³	1	4	yes	—
115	C₃₁×#95	C₂³	1	4	yes	—
116	C₃₁×#95	C₂³	1	4	yes	—
117	C₃₁×#95	C₂³	1	4	yes	—
118	C₃₁×#95	C₂³	1	4	yes	—
119	C₃₁×#95	C₂³	1	4	yes	—
120	C₃₁×#95	C₂³	1	4	yes	—
121	C₃₁×#95	C₂³	1	4	yes	—
122	C₃₁×#95	C₂³	1	4	yes	—
123	C₃₁×#95	C₂³	1	4	yes	—
124	C₃₁×#95	C₂³	1	4	yes	—
125	C₃₁×#95	C₂³	1	2	yes	—
126	C₃₁×#95	C₂³	1	2	yes	—
127	C₃₁×#95	C₂³	1	2	yes	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₃₁×#95	C₂³	1	4	yes	—
130	C₃₁×#95	C₂³	1	4	yes	—
131	C₃₁×#95	C₂³	1	2	yes	—
132	C₃₁×#95	C₂³	1	4	yes	—
133	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	4	2	no	—
143	C₂²×C₄×C₃₁	#41	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₄⋊C₄×C₃₁	—	2	4	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	8	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	8	no	—
164	C₄⋊C₄×C₃₁	—	2	8	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	4	4	no	—
168	C₄⋊C₄×C₃₁	—	4	4	no	—
169	C₄⋊C₄×C₃₁	—	4	4	no	—
170	C₄⋊C₄×C₃₁	—	4	4	no	—
171	C₄⋊C₄×C₃₁	—	4	4	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	—	2	8	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	4	4	no	—
176	C₄²×C₃₁	—	2	2	no	—
177	C₄²×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	—	2	2	no	—
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	2	8	no	—
183	C₄²×C₃₁	—	2	4	no	—
184	C₄²×C₃₁	—	2	4	no	—
185	C₄²×C₃₁	—	2	8	no	—
186	C₄²×C₃₁	—	4	2	no	—
187	C₄²×C₃₁	—	4	2	no	—
188	C₄²×C₃₁	#41	1	4	yes	—
189	C₄²×C₃₁	#41	1	4	yes	—
190	#34629	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂³×C₃₁	—	2	1	no	—
192	C₂³×C₃₁	—	2	1	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	C₂×#41	1	2	yes	—
196	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
197	Q₈×C₃₁	—	4	8	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	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	8	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	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	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
232	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
233	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
234	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
235	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
236	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
237	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
238	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
239	#2655	C₂×C₃₁	1	4	yes	complemented
240	#2652	C₂×C₃₁	1	4	yes	—
241	#2647	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#2645	C₂×C₃₁	1	2	yes	complemented
243	#2642	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#2629	C₂×C₃₁	1	1	yes	characteristic
245	#2625	C₂×C₃₁	1	2	yes	—
246	#2621	C₂×C₃₁	1	1	yes	characteristic
247	#2402	C₂×C₃₁	1	4	yes	complemented
248	#2401	C₂×C₃₁	1	4	yes	—
249	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₄×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
252	#2295	C₂×C₃₁	1	2	yes	complemented
253	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	4	1	no	—
257	C₂²×C₃₁	—	4	1	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
260	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
261	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
262	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
263	C₂²×C₃₁	#585	1	2	yes	—
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	2	2	no	—
266	C₄×C₃₁	—	2	4	no	—
267	C₄×C₃₁	—	4	4	no	—
268	C₄×C₃₁	—	4	1	no	—
269	C₄×C₃₁	—	4	4	no	—
270	C₄×C₃₁	—	4	1	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	4	4	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	2	no	—
275	C₄×C₃₁	—	4	4	no	—
276	C₄×C₃₁	#529	1	4	yes	—
277	C₄×C₃₁	#541	1	4	yes	—
278	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
282	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
284	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
285	#401	C₂²×C₃₁	1	4	yes	—
286	#400	C₂²×C₃₁	1	4	yes	—
287	#400	C₂²×C₃₁	1	4	yes	—
288	#400	C₂²×C₃₁	1	4	yes	—
289	#400	C₂²×C₃₁	1	4	yes	—
290	#400	C₂²×C₃₁	1	4	yes	—
291	#400	C₂²×C₃₁	1	4	yes	—
292	#400	C₂²×C₃₁	1	4	yes	—
293	#398	C₂²×C₃₁	1	4	yes	—
294	#398	C₂²×C₃₁	1	4	yes	—
295	#398	C₂²×C₃₁	1	4	yes	—
296	#398	C₂²×C₃₁	1	4	yes	—
297	#397	C₂²×C₃₁	1	4	yes	—
298	#397	C₂²×C₃₁	1	4	yes	—
299	#397	C₂²×C₃₁	1	4	yes	—
300	#396	C₂²×C₃₁	1	4	yes	—
301	#391	C₂²×C₃₁	1	2	yes	—
302	#391	C₂²×C₃₁	1	2	yes	—
303	#391	C₂²×C₃₁	1	2	yes	—
304	#391	C₂²×C₃₁	1	2	yes	—
305	#389	C₂²×C₃₁	1	2	yes	—
306	#389	C₂²×C₃₁	1	4	yes	—
307	#389	C₂²×C₃₁	1	2	yes	—
308	#389	C₂²×C₃₁	1	2	yes	—
309	#389	C₂²×C₃₁	1	2	yes	—
310	#387	C₂²×C₃₁	1	4	yes	—
311	#387	C₂²×C₃₁	1	4	yes	—
312	#387	C₂²×C₃₁	1	4	yes	—
313	#387	C₂²×C₃₁	1	4	yes	—
314	#385	C₂²×C₃₁	1	4	yes	—
315	#384	C₂²×C₃₁	1	1	yes	characteristic
316	#382	C₂²×C₃₁	1	1	yes	characteristic
317	#380	C₂²×C₃₁	1	2	yes	—
318	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
319	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
320	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
321	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
325	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
326	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
327	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
328	#376	C₂²×C₃₁	1	1	yes	characteristic
329	#376	C₂²×C₃₁	1	1	yes	characteristic
330	#376	C₂²×C₃₁	1	4	yes	—
331	C₄³	C₂²×C₃₁	1	1	yes	characteristic
332	C₄³	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×C₃₁	—	4	2	no	—
334	C₂×C₃₁	C₂×#529	1	1	yes	characteristic
335	C₂×C₃₁	C₂×#541	1	1	yes	characteristic
336	C₂×C₃₁	#3455	1	1	yes	characteristic
337	C₂×C₃₁	#3461	1	2	yes	—
338	C₂×C₃₁	#3461	1	2	yes	—
339	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
340	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₄⋊Q₈	—	2	4	no	—
344	#125	—	2	4	no	—
345	C₂²⋊Q₈	—	2	8	no	—
346	C₄×Q₈	—	2	8	no	—
347	#117	—	2	4	no	—
348	#117	—	2	4	no	—
349	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
355	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
357	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
359	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
360	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
367	C₂×C₄²	—	2	4	no	—
368	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
372	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
376	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
379	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
380	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
382	#95	C₂³×C₃₁	1	2	yes	—
383	#95	C₂³×C₃₁	1	2	yes	—
384	#95	C₂³×C₃₁	1	2	yes	—
385	#95	C₂³×C₃₁	1	2	yes	—
386	#95	C₂³×C₃₁	1	4	yes	—
387	#95	C₂³×C₃₁	1	4	yes	—
388	#95	C₂³×C₃₁	1	4	yes	—
389	#95	C₂³×C₃₁	1	4	yes	—
390	#95	C₂³×C₃₁	1	4	yes	—
391	#95	C₂³×C₃₁	1	4	yes	—
392	#95	C₂³×C₃₁	1	4	yes	—
393	#95	C₂³×C₃₁	1	4	yes	—
394	#95	C₂³×C₃₁	1	4	yes	—
395	#95	C₂³×C₃₁	1	4	yes	—
396	#95	C₂³×C₃₁	1	4	yes	—
397	#95	C₂³×C₃₁	1	4	yes	—
398	#95	C₂³×C₃₁	1	4	yes	—
399	#95	C₂³×C₃₁	1	4	yes	—
400	#95	C₂³×C₃₁	1	4	yes	—
401	C₃₁	#34629	1	1	yes	characteristic, direct product factor, Sylow
402	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	4	2	no	—
412	C₂²×C₄	#41×C₃₁	1	4	yes	—
413	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
420	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
424	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	8	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	4	4	no	—
437	C₄⋊C₄	—	4	4	no	—
438	C₄⋊C₄	—	4	4	no	—
439	C₄⋊C₄	—	4	4	no	—
440	C₄⋊C₄	—	4	4	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	4	4	no	—
445	C₄²	—	2	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	2	no	—
451	C₄²	—	2	8	no	—
452	C₄²	—	2	4	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	8	no	—
455	C₄²	—	4	2	no	—
456	C₄²	—	4	2	no	—
457	C₄²	#41×C₃₁	1	4	yes	—
458	C₄²	#41×C₃₁	1	4	yes	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	C₂×#41×C₃₁	1	2	yes	—
464	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
465	Q₈	—	4	8	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
500	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
501	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
502	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
503	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
504	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
505	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
506	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	4	1	no	—
510	C₂²	—	4	1	no	—
511	C₂²	—	4	2	no	—
512	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
513	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
514	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
515	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
516	C₂²	C₃₁×#585	1	2	yes	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	4	no	—
520	C₄	—	4	4	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	1	no	—
523	C₄	—	4	4	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	4	no	—
529	C₄	C₃₁×#529	1	4	yes	—
530	C₄	C₃₁×#541	1	4	yes	—
531	C₂	—	4	2	no	—
532	C₂	C₂×C₃₁×#529	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#541	1	1	yes	characteristic
534	C₂	C₃₁×#3455	1	1	yes	characteristic
535	C₂	C₃₁×#3461	1	2	yes	—
536	C₂	C₃₁×#3461	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂⁴×C₃₁	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2655	C₂	semidirect
3. C₃₁×#2647	C₂	semidirect
4. C₃₁×#2645	C₂	semidirect
5. C₃₁×#2642	C₂	semidirect
9. C₃₁×#2402	C₂	semidirect
11. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
12. C₄×C₃₁×#125	C₂	semidirect
14. C₃₁×#2295	C₂	semidirect
190. #34629	C₃₁	direct (N × H)
239. #2655	C₂×C₃₁	semidirect
241. #2647	C₂×C₃₁	semidirect
242. #2645	C₂×C₃₁	semidirect
243. #2642	C₂×C₃₁	semidirect
247. #2402	C₂×C₃₁	semidirect
249. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
250. C₄×#125	C₂×C₃₁	semidirect
252. #2295	C₂×C₃₁	semidirect
401. C₃₁	#34629	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×#2655	2
C₃₁×#2652	2
C₃₁×#2647	2
C₃₁×#2645	2
C₃₁×#2642	2
C₃₁×#2629	2
C₃₁×#2625	2
C₃₁×#2621	2
C₃₁×#2402	2
C₃₁×#2401	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#125	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#2295	2
C₄×C₃₁×#117	2
#34629	31

Subgroups — C31×#34629 #21491388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34629 #21491388