Subgroups — C₃₁×#35111 #21491870

Order 7936.

2294 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2855	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2854	C₂	1	2	yes	maximal
5	C₃₁×#2623	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2614	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2604	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2599	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2592	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2584	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2581	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2580	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#529	—	2	8	no	—
17	C₂×C₃₁×#126	C₂²	1	2	yes	—
18	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#123	C₂²	1	4	yes	—
21	C₂×C₃₁×#123	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
25	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#117	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#400	C₂²	1	4	yes	—
38	C₃₁×#399	C₂²	1	4	yes	—
39	C₃₁×#398	C₂²	1	4	yes	—
40	C₃₁×#397	C₂²	1	4	yes	—
41	C₃₁×#395	C₂²	1	4	yes	—
42	C₃₁×#395	C₂²	1	4	yes	—
43	C₃₁×#393	C₂²	1	4	yes	—
44	C₃₁×#393	C₂²	1	4	yes	—
45	C₃₁×#391	C₂²	1	4	yes	—
46	C₃₁×#390	C₂²	1	4	yes	—
47	C₃₁×#389	C₂²	1	8	yes	—
48	C₃₁×#389	C₂²	1	4	yes	—
49	C₃₁×#389	C₂²	1	4	yes	—
50	C₃₁×#388	C₂²	1	8	yes	—
51	C₃₁×#388	C₂²	1	8	yes	—
52	C₃₁×#387	C₂²	1	8	yes	—
53	C₃₁×#387	C₂²	1	8	yes	—
54	C₃₁×#386	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	8	yes	—
56	C₃₁×#385	C₂²	1	4	yes	—
57	C₃₁×#383	C₂²	1	2	yes	—
58	C₃₁×#382	C₂²	1	2	yes	—
59	C₃₁×#381	C₂²	1	4	yes	—
60	C₃₁×#381	C₂²	1	2	yes	—
61	C₃₁×#380	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₃₁×#376	C₂²	1	2	yes	—
65	C₂²×Q₈×C₃₁	—	2	4	no	—
66	C₂²×Q₈×C₃₁	—	2	8	no	—
67	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
69	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#126	—	2	8	no	—
79	C₃₁×#125	—	2	4	no	—
80	C₃₁×#124	—	2	4	no	—
81	C₃₁×#123	—	2	8	no	—
82	C₃₁×#123	—	2	16	no	—
83	C₃₁×C₂²⋊Q₈	—	2	4	no	—
84	C₃₁×#121	—	2	4	no	—
85	C₄×Q₈×C₃₁	—	2	8	no	—
86	C₄×Q₈×C₃₁	—	2	8	no	—
87	C₄×Q₈×C₃₁	—	2	8	no	—
88	C₄×Q₈×C₃₁	—	2	8	no	—
89	C₄×Q₈×C₃₁	—	2	4	no	—
90	C₄×D₄×C₃₁	—	2	8	no	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₃₁×#117	—	2	8	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
116	C₂×C₄²×C₃₁	—	2	8	no	—
117	C₂×C₄²×C₃₁	D₄	1	8	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
126	C₃₁×#95	C₂³	1	4	yes	—
127	C₃₁×#95	C₂³	1	4	yes	—
128	C₃₁×#95	C₂³	1	8	yes	—
129	C₃₁×#95	C₂³	1	8	yes	—
130	C₃₁×#95	C₂³	1	8	yes	—
131	C₃₁×#95	C₂³	1	8	yes	—
132	C₃₁×#95	C₂³	1	4	yes	—
133	C₃₁×#95	C₂³	1	4	yes	—
134	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
135	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
136	C₂×Q₈×C₃₁	—	2	32	no	—
137	C₂×Q₈×C₃₁	—	2	4	no	—
138	C₂×Q₈×C₃₁	—	2	4	no	—
139	C₂×Q₈×C₃₁	—	2	2	no	—
140	C₂×Q₈×C₃₁	—	2	8	no	—
141	C₂×Q₈×C₃₁	—	2	8	no	—
142	C₂×Q₈×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	4	no	—
153	C₂²×C₄×C₃₁	—	4	2	no	—
154	C₂²×C₄×C₃₁	—	4	4	no	—
155	C₂²×C₄×C₃₁	—	4	4	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₃₁	#41	1	8	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₃₁	—	2	8	no	—
173	C₄⋊C₄×C₃₁	—	2	8	no	—
174	C₄⋊C₄×C₃₁	—	2	4	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	16	no	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	4	8	no	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	2	8	no	—
187	C₂²⋊C₄×C₃₁	—	2	4	no	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	4	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	16	no	—
192	C₂²⋊C₄×C₃₁	—	4	8	no	—
193	C₄²×C₃₁	—	2	8	no	—
194	C₄²×C₃₁	—	2	8	no	—
195	C₄²×C₃₁	—	2	4	no	—
196	C₄²×C₃₁	—	4	4	no	—
197	C₄²×C₃₁	C₂×D₄	1	4	yes	—
198	C₄²×C₃₁	C₂×D₄	1	4	yes	—
199	C₄²×C₃₁	C₂×D₄	1	4	yes	—
200	C₄²×C₃₁	C₂×D₄	1	4	yes	—
201	#35111	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	2	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	2	4	no	—
206	C₂³×C₃₁	—	2	4	no	—
207	C₂³×C₃₁	—	2	4	no	—
208	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
209	C₂³×C₃₁	C₂×#41	1	4	yes	—
210	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
211	Q₈×C₃₁	—	2	16	no	—
212	Q₈×C₃₁	—	2	16	no	—
213	Q₈×C₃₁	—	4	4	no	—
214	D₄×C₃₁	—	4	8	no	—
215	D₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	2	8	no	—
217	C₂×C₄×C₃₁	—	2	16	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	8	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	8	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	8	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	4	8	no	—
240	C₂×C₄×C₃₁	—	4	8	no	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
248	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
249	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
250	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#2855	C₂×C₃₁	1	2	yes	complemented
252	#2854	C₂×C₃₁	1	2	yes	—
253	#2623	C₂×C₃₁	1	2	yes	complemented
254	#2614	C₂×C₃₁	1	2	yes	complemented
255	#2604	C₂×C₃₁	1	4	yes	complemented
256	#2599	C₂×C₃₁	1	4	yes	complemented
257	#2592	C₂×C₃₁	1	2	yes	complemented
258	#2584	C₂×C₃₁	1	2	yes	complemented
259	#2581	C₂×C₃₁	1	2	yes	complemented
260	#2580	C₂×C₃₁	1	2	yes	complemented
261	#2322	C₂×C₃₁	1	4	yes	complemented
262	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	2	no	—
265	C₂²×C₃₁	—	2	8	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	—	4	2	no	—
269	C₂²×C₃₁	—	4	8	no	—
270	C₂²×C₃₁	#525	1	4	yes	—
271	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
272	C₂²×C₃₁	C₂²×#41	1	2	yes	—
273	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
274	C₂²×C₃₁	#585	1	2	yes	—
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	8	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	—	4	2	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	2	no	—
285	C₄×C₃₁	—	4	8	no	—
286	C₄×C₃₁	#536	1	4	yes	—
287	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	#529	—	2	8	no	—
289	C₂×#126	C₂²×C₃₁	1	2	yes	—
290	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#123	C₂²×C₃₁	1	4	yes	—
293	C₂×#123	C₂²×C₃₁	1	2	yes	—
294	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×Q₈	—	2	8	no	—
297	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
300	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
301	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
302	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
303	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#117	C₂²×C₃₁	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
307	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
308	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
309	#400	C₂²×C₃₁	1	4	yes	—
310	#399	C₂²×C₃₁	1	4	yes	—
311	#398	C₂²×C₃₁	1	4	yes	—
312	#397	C₂²×C₃₁	1	4	yes	—
313	#395	C₂²×C₃₁	1	4	yes	—
314	#395	C₂²×C₃₁	1	4	yes	—
315	#393	C₂²×C₃₁	1	4	yes	—
316	#393	C₂²×C₃₁	1	4	yes	—
317	#391	C₂²×C₃₁	1	4	yes	—
318	#390	C₂²×C₃₁	1	4	yes	—
319	#389	C₂²×C₃₁	1	8	yes	—
320	#389	C₂²×C₃₁	1	4	yes	—
321	#389	C₂²×C₃₁	1	4	yes	—
322	#388	C₂²×C₃₁	1	8	yes	—
323	#388	C₂²×C₃₁	1	8	yes	—
324	#387	C₂²×C₃₁	1	8	yes	—
325	#387	C₂²×C₃₁	1	8	yes	—
326	#386	C₂²×C₃₁	1	4	yes	—
327	#385	C₂²×C₃₁	1	8	yes	—
328	#385	C₂²×C₃₁	1	4	yes	—
329	#383	C₂²×C₃₁	1	2	yes	—
330	#382	C₂²×C₃₁	1	2	yes	—
331	#381	C₂²×C₃₁	1	4	yes	—
332	#381	C₂²×C₃₁	1	2	yes	—
333	#380	C₂²×C₃₁	1	2	yes	—
334	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
335	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
336	#376	C₂²×C₃₁	1	2	yes	—
337	C₂×C₃₁	—	2	4	no	—
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
341	C₂×C₃₁	#3449	1	1	yes	characteristic
342	C₂×C₃₁	#3507	1	4	yes	—
343	C₂²×Q₈	—	2	4	no	—
344	C₂²×Q₈	—	2	8	no	—
345	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
346	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
347	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
349	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
353	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
354	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	#126	—	2	8	no	—
357	#125	—	2	4	no	—
358	#124	—	2	4	no	—
359	#123	—	2	8	no	—
360	#123	—	2	16	no	—
361	C₂²⋊Q₈	—	2	4	no	—
362	#121	—	2	4	no	—
363	C₄×Q₈	—	2	8	no	—
364	C₄×Q₈	—	2	8	no	—
365	C₄×Q₈	—	2	8	no	—
366	C₄×Q₈	—	2	8	no	—
367	C₄×Q₈	—	2	4	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	#117	—	2	8	no	—
371	C₂×C₄⋊C₄	—	2	8	no	—
372	C₂×C₄⋊C₄	—	2	4	no	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
384	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
394	C₂×C₄²	—	2	8	no	—
395	C₂×C₄²	D₄×C₃₁	1	8	yes	—
396	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
397	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
398	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
399	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
400	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
404	#95	C₂³×C₃₁	1	4	yes	—
405	#95	C₂³×C₃₁	1	4	yes	—
406	#95	C₂³×C₃₁	1	4	yes	—
407	#95	C₂³×C₃₁	1	4	yes	—
408	#95	C₂³×C₃₁	1	8	yes	—
409	#95	C₂³×C₃₁	1	8	yes	—
410	#95	C₂³×C₃₁	1	8	yes	—
411	#95	C₂³×C₃₁	1	8	yes	—
412	C₃₁	#35111	1	1	yes	characteristic, direct product factor, Sylow
413	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
414	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
415	C₂×Q₈	—	2	32	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	8	no	—
420	C₂×Q₈	—	2	8	no	—
421	C₂×Q₈	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
437	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
438	C₂²×C₄	#41×C₃₁	1	8	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₄⋊C₄	—	2	8	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	16	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	4	8	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	16	no	—
471	C₂²⋊C₄	—	4	8	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	2	8	no	—
474	C₄²	—	2	4	no	—
475	C₄²	—	4	4	no	—
476	C₄²	C₂×D₄×C₃₁	1	4	yes	—
477	C₄²	C₂×D₄×C₃₁	1	4	yes	—
478	C₄²	C₂×D₄×C₃₁	1	4	yes	—
479	C₄²	C₂×D₄×C₃₁	1	4	yes	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	4	no	—
486	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
487	C₂³	C₂×#41×C₃₁	1	4	yes	—
488	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
489	Q₈	—	2	16	no	—
490	Q₈	—	2	16	no	—
491	Q₈	—	4	4	no	—
492	D₄	—	4	8	no	—
493	D₄	—	4	4	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	16	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
520	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
525	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
526	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	2	8	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	8	no	—
534	C₂²	C₃₁×#525	1	4	yes	—
535	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
536	C₂²	C₂²×#41×C₃₁	1	2	yes	—
537	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
538	C₂²	C₃₁×#585	1	2	yes	—
539	C₄	—	2	2	no	—
540	C₄	—	2	2	no	—
541	C₄	—	2	8	no	—
542	C₄	—	2	2	no	—
543	C₄	—	2	2	no	—
544	C₄	—	2	2	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	4	no	—
548	C₄	—	4	2	no	—
549	C₄	—	4	8	no	—
550	C₄	C₃₁×#536	1	4	yes	—
551	C₂	—	2	4	no	—
552	C₂	—	8	2	no	—
553	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
554	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
555	C₂	C₃₁×#3449	1	1	yes	characteristic
556	C₂	C₃₁×#3507	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#2855	C₂	semidirect
5. C₃₁×#2623	C₂	semidirect
6. C₃₁×#2614	C₂	semidirect
7. C₃₁×#2604	C₂	semidirect
8. C₃₁×#2599	C₂	semidirect
9. C₃₁×#2592	C₂	semidirect
10. C₃₁×#2584	C₂	semidirect
11. C₃₁×#2581	C₂	semidirect
12. C₃₁×#2580	C₂	semidirect
13. C₃₁×#2322	C₂	semidirect
14. C₃₁×#2283	C₂	semidirect
201. #35111	C₃₁	direct (N × H)
250. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
251. #2855	C₂×C₃₁	semidirect
253. #2623	C₂×C₃₁	semidirect
254. #2614	C₂×C₃₁	semidirect
255. #2604	C₂×C₃₁	semidirect
256. #2599	C₂×C₃₁	semidirect
257. #2592	C₂×C₃₁	semidirect
258. #2584	C₂×C₃₁	semidirect
259. #2581	C₂×C₃₁	semidirect
260. #2580	C₂×C₃₁	semidirect
261. #2322	C₂×C₃₁	semidirect
262. #2283	C₂×C₃₁	semidirect
412. C₃₁	#35111	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×#529	2
C₂²×C₄×Q₈×C₃₁	2
C₃₁×#2855	2
C₃₁×#2854	2
C₃₁×#2623	2
C₃₁×#2614	2
C₃₁×#2604	2
C₃₁×#2599	2
C₃₁×#2592	2
C₃₁×#2584	2
C₃₁×#2581	2
C₃₁×#2580	2
C₃₁×#2322	2
C₃₁×#2283	2
#35111	31

Subgroups — C31×#35111 #21491870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35111 #21491870