Subgroups — C₃₁×#36699 #21493458

Order 7936.

1958 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2822	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2821	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2821	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2820	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2817	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2816	C₂	1	2	yes	maximal
9	C₃₁×#2814	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2399	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2398	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2396	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2395	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#533	—	2	8	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃₁×#529	—	2	8	no	—
19	C₂×C₃₁×#126	C₂²	1	2	yes	—
20	C₂×C₃₁×#125	C₂²	1	2	yes	—
21	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#123	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#117	C₂²	1	2	yes	—
34	C₂×C₃₁×#117	C₂²	1	2	yes	—
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#400	C₂²	1	4	yes	—
37	C₃₁×#400	C₂²	1	4	yes	—
38	C₃₁×#400	C₂²	1	4	yes	—
39	C₃₁×#399	C₂²	1	4	yes	—
40	C₃₁×#399	C₂²	1	4	yes	—
41	C₃₁×#398	C₂²	1	4	yes	—
42	C₃₁×#398	C₂²	1	4	yes	—
43	C₃₁×#398	C₂²	1	8	yes	—
44	C₃₁×#398	C₂²	1	4	yes	—
45	C₃₁×#397	C₂²	1	4	yes	—
46	C₃₁×#397	C₂²	1	4	yes	—
47	C₃₁×#397	C₂²	1	8	yes	—
48	C₃₁×#396	C₂²	1	8	yes	—
49	C₃₁×#395	C₂²	1	4	yes	—
50	C₃₁×#395	C₂²	1	4	yes	—
51	C₃₁×#395	C₂²	1	4	yes	—
52	C₃₁×#394	C₂²	1	4	yes	—
53	C₃₁×#394	C₂²	1	8	yes	—
54	C₃₁×#393	C₂²	1	4	yes	—
55	C₃₁×#393	C₂²	1	4	yes	—
56	C₃₁×#383	C₂²	1	2	yes	—
57	C₃₁×#382	C₂²	1	2	yes	—
58	C₃₁×#381	C₂²	1	2	yes	—
59	C₃₁×#380	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	—
61	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
64	C₃₁×#376	C₂²	1	2	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
67	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
71	C₂³×C₄×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	C₃₁×C₄⋊Q₈	—	2	4	no	—
75	C₃₁×C₄⋊Q₈	—	2	4	no	—
76	C₃₁×#126	—	2	8	no	—
77	C₃₁×#125	—	2	4	no	—
78	C₃₁×#125	—	2	8	no	—
79	C₃₁×#124	—	2	4	no	—
80	C₃₁×#123	—	2	8	no	—
81	C₃₁×C₂²⋊Q₈	—	2	4	no	—
82	C₃₁×C₂²⋊Q₈	—	2	8	no	—
83	C₃₁×C₂²⋊Q₈	—	2	8	no	—
84	C₃₁×#121	—	2	4	no	—
85	C₄×Q₈×C₃₁	—	2	4	no	—
86	C₄×Q₈×C₃₁	—	2	8	no	—
87	C₄×Q₈×C₃₁	—	2	8	no	—
88	C₄×D₄×C₃₁	—	2	8	no	—
89	C₄×D₄×C₃₁	—	2	4	no	—
90	C₃₁×#117	—	2	8	no	—
91	C₃₁×#117	—	2	8	no	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	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	8	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	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	4	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	8	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂×C₄²×C₃₁	D₄	1	8	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₃₁×#95	C₂³	1	8	yes	—
125	C₃₁×#95	C₂³	1	8	yes	—
126	C₃₁×#95	C₂³	1	4	yes	—
127	C₃₁×#95	C₂³	1	4	yes	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₃₁×#95	C₂³	1	8	yes	—
130	C₃₁×#95	C₂³	1	4	yes	—
131	C₃₁×#95	C₂³	1	8	yes	—
132	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
133	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂×Q₈×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	—	4	2	no	—
138	C₂×Q₈×C₃₁	—	4	4	no	—
139	C₂×Q₈×C₃₁	—	4	4	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	4	2	no	—
143	C₂×D₄×C₃₁	—	4	4	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	4	2	no	—
148	C₂²×C₄×C₃₁	—	4	2	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	4	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	8	no	—
173	C₄⋊C₄×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	4	no	—
175	C₄⋊C₄×C₃₁	—	2	4	no	—
176	C₄⋊C₄×C₃₁	—	4	4	no	—
177	C₄⋊C₄×C₃₁	—	4	8	no	—
178	C₄⋊C₄×C₃₁	—	4	8	no	—
179	C₂²⋊C₄×C₃₁	—	2	8	no	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	4	no	—
184	C₂²⋊C₄×C₃₁	—	2	4	no	—
185	C₂²⋊C₄×C₃₁	—	4	4	no	—
186	C₂²⋊C₄×C₃₁	—	4	8	no	—
187	C₂²⋊C₄×C₃₁	—	4	8	no	—
188	C₄²×C₃₁	—	2	4	no	—
189	C₄²×C₃₁	—	2	8	no	—
190	C₄²×C₃₁	—	2	8	no	—
191	C₄²×C₃₁	C₂×D₄	1	4	yes	—
192	C₄²×C₃₁	C₂×D₄	1	4	yes	—
193	C₄²×C₃₁	C₂×D₄	1	4	yes	—
194	C₄²×C₃₁	C₂×D₄	1	4	yes	—
195	#36699	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	4	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
204	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
205	Q₈×C₃₁	—	4	4	no	—
206	Q₈×C₃₁	—	4	8	no	—
207	Q₈×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	4	no	—
209	D₄×C₃₁	—	4	8	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	8	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	8	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	8	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	2	8	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	4	8	no	—
233	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
239	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
240	C₂×#533	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
242	#2822	C₂×C₃₁	1	2	yes	complemented
243	#2821	C₂×C₃₁	1	2	yes	complemented
244	#2821	C₂×C₃₁	1	2	yes	complemented
245	#2820	C₂×C₃₁	1	2	yes	complemented
246	#2817	C₂×C₃₁	1	2	yes	complemented
247	#2816	C₂×C₃₁	1	2	yes	—
248	#2814	C₂×C₃₁	1	4	yes	complemented
249	#2399	C₂×C₃₁	1	2	yes	complemented
250	#2398	C₂×C₃₁	1	2	yes	complemented
251	#2396	C₂×C₃₁	1	4	yes	complemented
252	#2395	C₂×C₃₁	1	4	yes	complemented
253	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	2	2	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	—	4	2	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	8	no	—
262	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
263	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
264	C₂²×C₃₁	#585	1	2	yes	—
265	C₂²×C₃₁	#585	1	2	yes	—
266	C₄×C₃₁	—	2	2	no	—
267	C₄×C₃₁	—	2	2	no	—
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	2	2	no	—
270	C₄×C₃₁	—	2	2	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	4	no	—
275	C₄×C₃₁	—	4	4	no	—
276	C₄×C₃₁	—	4	8	no	—
277	C₄×C₃₁	#536	1	4	yes	—
278	#533	—	2	8	no	—
279	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
281	#529	—	2	8	no	—
282	C₂×#126	C₂²×C₃₁	1	2	yes	—
283	C₂×#125	C₂²×C₃₁	1	2	yes	—
284	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
286	C₂×#123	C₂²×C₃₁	1	2	yes	—
287	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
288	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
290	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
292	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
293	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
295	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#117	C₂²×C₃₁	1	2	yes	—
297	C₂×#117	C₂²×C₃₁	1	2	yes	—
298	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
299	#400	C₂²×C₃₁	1	4	yes	—
300	#400	C₂²×C₃₁	1	4	yes	—
301	#400	C₂²×C₃₁	1	4	yes	—
302	#399	C₂²×C₃₁	1	4	yes	—
303	#399	C₂²×C₃₁	1	4	yes	—
304	#398	C₂²×C₃₁	1	4	yes	—
305	#398	C₂²×C₃₁	1	4	yes	—
306	#398	C₂²×C₃₁	1	4	yes	—
307	#398	C₂²×C₃₁	1	8	yes	—
308	#397	C₂²×C₃₁	1	8	yes	—
309	#397	C₂²×C₃₁	1	4	yes	—
310	#397	C₂²×C₃₁	1	4	yes	—
311	#396	C₂²×C₃₁	1	8	yes	—
312	#395	C₂²×C₃₁	1	4	yes	—
313	#395	C₂²×C₃₁	1	4	yes	—
314	#395	C₂²×C₃₁	1	4	yes	—
315	#394	C₂²×C₃₁	1	4	yes	—
316	#394	C₂²×C₃₁	1	8	yes	—
317	#393	C₂²×C₃₁	1	4	yes	—
318	#393	C₂²×C₃₁	1	4	yes	—
319	#383	C₂²×C₃₁	1	2	yes	—
320	#382	C₂²×C₃₁	1	2	yes	—
321	#381	C₂²×C₃₁	1	2	yes	—
322	#380	C₂²×C₃₁	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
325	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
326	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
327	#376	C₂²×C₃₁	1	2	yes	—
328	C₂×C₃₁	—	4	2	no	—
329	C₂×C₃₁	—	8	2	no	—
330	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
331	C₂×C₃₁	#3449	1	1	yes	characteristic
332	C₂×C₃₁	#3449	1	1	yes	characteristic
333	C₂×C₃₁	#3531	1	4	yes	—
334	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
335	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
336	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
338	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂³×C₄	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	C₄⋊Q₈	—	2	4	no	—
345	#126	—	2	8	no	—
346	#125	—	2	4	no	—
347	#125	—	2	8	no	—
348	#124	—	2	4	no	—
349	#123	—	2	8	no	—
350	C₂²⋊Q₈	—	2	4	no	—
351	C₂²⋊Q₈	—	2	8	no	—
352	C₂²⋊Q₈	—	2	8	no	—
353	#121	—	2	4	no	—
354	C₄×Q₈	—	2	4	no	—
355	C₄×Q₈	—	2	8	no	—
356	C₄×Q₈	—	2	8	no	—
357	C₄×D₄	—	2	8	no	—
358	C₄×D₄	—	2	4	no	—
359	#117	—	2	8	no	—
360	#117	—	2	8	no	—
361	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
362	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
385	C₂×C₄²	D₄×C₃₁	1	8	yes	—
386	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
387	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
388	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
389	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
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	8	yes	—
397	#95	C₂³×C₃₁	1	8	yes	—
398	#95	C₂³×C₃₁	1	4	yes	—
399	#95	C₂³×C₃₁	1	8	yes	—
400	#95	C₂³×C₃₁	1	8	yes	—
401	C₃₁	#36699	1	1	yes	characteristic, direct product factor, Sylow
402	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
403	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	4	2	no	—
408	C₂×Q₈	—	4	4	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	—	4	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
424	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
428	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	4	4	no	—
447	C₄⋊C₄	—	4	8	no	—
448	C₄⋊C₄	—	4	8	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	—	4	8	no	—
457	C₂²⋊C₄	—	4	8	no	—
458	C₄²	—	2	4	no	—
459	C₄²	—	2	8	no	—
460	C₄²	—	2	8	no	—
461	C₄²	C₂×D₄×C₃₁	1	4	yes	—
462	C₄²	C₂×D₄×C₃₁	1	4	yes	—
463	C₄²	C₂×D₄×C₃₁	1	4	yes	—
464	C₄²	C₂×D₄×C₃₁	1	4	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	4	no	—
472	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
473	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
474	Q₈	—	4	4	no	—
475	Q₈	—	4	8	no	—
476	Q₈	—	4	8	no	—
477	D₄	—	4	4	no	—
478	D₄	—	4	8	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	8	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	8	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
507	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
508	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
509	C₂²	—	2	2	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	8	no	—
517	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
518	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
519	C₂²	C₃₁×#585	1	2	yes	—
520	C₂²	C₃₁×#585	1	2	yes	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	4	no	—
530	C₄	—	4	4	no	—
531	C₄	—	4	8	no	—
532	C₄	C₃₁×#536	1	4	yes	—
533	C₂	—	4	2	no	—
534	C₂	—	8	2	no	—
535	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
536	C₂	C₃₁×#3449	1	1	yes	characteristic
537	C₂	C₃₁×#3449	1	1	yes	characteristic
538	C₂	C₃₁×#3531	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂×C₃₁×#121	C₂²
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. C₂×C₃₁×#533	C₂	semidirect
3. C₃₁×#2822	C₂	semidirect
4. C₃₁×#2821	C₂	semidirect
5. C₃₁×#2821	C₂	semidirect
6. C₃₁×#2820	C₂	semidirect
7. C₃₁×#2817	C₂	semidirect
9. C₃₁×#2814	C₂	semidirect
10. C₃₁×#2399	C₂	semidirect
11. C₃₁×#2398	C₂	semidirect
12. C₃₁×#2396	C₂	semidirect
13. C₃₁×#2395	C₂	semidirect
14. C₃₁×#2283	C₂	semidirect
195. #36699	C₃₁	direct (N × H)
240. C₂×#533	C₂×C₃₁	semidirect
242. #2822	C₂×C₃₁	semidirect
243. #2821	C₂×C₃₁	semidirect
244. #2821	C₂×C₃₁	semidirect
245. #2820	C₂×C₃₁	semidirect
246. #2817	C₂×C₃₁	semidirect
248. #2814	C₂×C₃₁	semidirect
249. #2399	C₂×C₃₁	semidirect
250. #2398	C₂×C₃₁	semidirect
251. #2396	C₂×C₃₁	semidirect
252. #2395	C₂×C₃₁	semidirect
253. #2283	C₂×C₃₁	semidirect
401. C₃₁	#36699	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×#533	2
C₂×C₃₁×#529	2
C₃₁×#2822	2
C₃₁×#2821	2
C₃₁×#2820	2
C₃₁×#2817	2
C₃₁×#2816	2
C₃₁×#2814	2
C₃₁×#2399	2
C₃₁×#2398	2
C₃₁×#2396	2
C₃₁×#2395	2
C₃₁×#2283	2
#36699	31

Subgroups — C31×#36699 #21493458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36699 #21493458