Subgroups — C₃₁×#35084 #21491843

Order 7936.

2898 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2855	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2848	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2638	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2583	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2561	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2486	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2483	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2415	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2409	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2405	C₂	1	2	yes	maximal
13	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#123	C₂²	1	4	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	—	2	8	no	—
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
26	C₂×C₄×D₄×C₃₁	—	2	8	no	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₃₁×#117	C₂²	1	2	yes	complemented
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#398	C₂²	1	4	yes	complemented
40	C₃₁×#397	C₂²	1	8	yes	complemented
41	C₃₁×#397	C₂²	1	4	yes	complemented
42	C₃₁×#395	C₂²	1	4	yes	complemented
43	C₃₁×#394	C₂²	1	8	yes	—
44	C₃₁×#393	C₂²	1	4	yes	complemented
45	C₃₁×#391	C₂²	1	4	yes	complemented
46	C₃₁×#389	C₂²	1	8	yes	complemented
47	C₃₁×#389	C₂²	1	4	yes	complemented
48	C₃₁×#388	C₂²	1	8	yes	complemented
49	C₃₁×#387	C₂²	1	8	yes	complemented
50	C₃₁×#386	C₂²	1	4	yes	—
51	C₃₁×#385	C₂²	1	4	yes	—
52	C₃₁×#385	C₂²	1	4	yes	—
53	C₃₁×#385	C₂²	1	8	yes	complemented
54	C₃₁×#385	C₂²	1	4	yes	complemented
55	C₃₁×#381	C₂²	1	4	yes	—
56	C₃₁×#381	C₂²	1	4	yes	complemented
57	C₃₁×#381	C₂²	1	2	yes	complemented
58	C₃₁×#380	C₂²	1	2	yes	—
59	C₃₁×#380	C₂²	1	4	yes	complemented
60	C₃₁×#380	C₂²	1	2	yes	complemented
61	C₃₁×#379	C₂²	1	4	yes	—
62	C₃₁×#379	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
65	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
66	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×Q₈×C₃₁	—	2	4	no	—
68	C₂²×Q₈×C₃₁	—	4	4	no	—
69	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
70	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	—	2	4	no	—
72	C₂²×D₄×C₃₁	—	2	8	no	—
73	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
74	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	—	2	4	no	—
76	C₂³×C₄×C₃₁	—	4	4	no	—
77	C₂³×C₄×C₃₁	—	4	4	no	—
78	C₂³×C₄×C₃₁	D₄	1	8	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
87	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
88	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
89	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
91	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#123	—	2	16	no	—
93	C₃₁×C₂²⋊Q₈	—	2	8	no	—
94	C₃₁×C₂²⋊Q₈	—	2	8	no	—
95	C₃₁×C₂²⋊Q₈	—	2	8	no	—
96	C₃₁×C₂²⋊Q₈	—	2	8	no	—
97	C₃₁×C₂²⋊Q₈	—	4	8	no	—
98	C₄×D₄×C₃₁	—	2	8	no	—
99	C₄×D₄×C₃₁	—	2	8	no	—
100	C₄×D₄×C₃₁	—	2	8	no	—
101	C₄×D₄×C₃₁	—	2	8	no	—
102	C₃₁×#117	—	4	4	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
129	C₂×C₄²×C₃₁	—	2	4	no	—
130	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
132	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
133	C₃₁×#95	C₂³	1	4	yes	—
134	C₃₁×#95	C₂³	1	8	yes	—
135	C₃₁×#95	C₂³	1	8	yes	—
136	C₃₁×#95	C₂³	1	8	yes	—
137	C₃₁×#95	C₂³	1	8	yes	—
138	C₃₁×#95	C₂³	1	4	yes	—
139	C₂⁴×C₃₁	—	2	2	no	—
140	C₂⁴×C₃₁	—	2	4	no	—
141	C₂⁴×C₃₁	—	2	4	no	—
142	C₂⁴×C₃₁	—	2	4	no	—
143	C₂⁴×C₃₁	—	2	4	no	—
144	C₂⁴×C₃₁	—	4	2	no	—
145	C₂⁴×C₃₁	—	4	2	no	—
146	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
147	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
149	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
151	C₂×Q₈×C₃₁	—	2	4	no	—
152	C₂×Q₈×C₃₁	—	2	4	no	—
153	C₂×Q₈×C₃₁	—	2	8	no	—
154	C₂×Q₈×C₃₁	—	4	4	no	—
155	C₂×Q₈×C₃₁	—	4	16	no	—
156	C₂×D₄×C₃₁	—	2	32	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	4	no	—
159	C₂×D₄×C₃₁	—	2	4	no	—
160	C₂×D₄×C₃₁	—	2	4	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	#41	1	8	yes	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	2	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	—	2	8	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	2	4	no	—
175	C₂²×C₄×C₃₁	—	2	4	no	—
176	C₂²×C₄×C₃₁	—	2	8	no	—
177	C₂²×C₄×C₃₁	—	4	16	no	—
178	C₂²×C₄×C₃₁	—	4	2	no	—
179	C₂²×C₄×C₃₁	—	4	2	no	—
180	C₂²×C₄×C₃₁	—	4	2	no	—
181	C₂²×C₄×C₃₁	—	4	4	no	—
182	C₂²×C₄×C₃₁	—	4	2	no	—
183	C₂²×C₄×C₃₁	—	4	4	no	—
184	C₂²×C₄×C₃₁	—	4	16	no	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
188	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
193	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
198	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	8	no	—
206	C₄⋊C₄×C₃₁	—	2	8	no	—
207	C₄⋊C₄×C₃₁	—	2	16	no	—
208	C₄⋊C₄×C₃₁	—	2	8	no	—
209	C₄⋊C₄×C₃₁	—	4	8	no	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	4	no	—
212	C₂²⋊C₄×C₃₁	—	2	8	no	—
213	C₂²⋊C₄×C₃₁	—	2	8	no	—
214	C₂²⋊C₄×C₃₁	—	2	8	no	—
215	C₂²⋊C₄×C₃₁	—	2	16	no	—
216	C₂²⋊C₄×C₃₁	—	2	8	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	4	no	—
219	C₂²⋊C₄×C₃₁	—	4	4	no	—
220	C₂²⋊C₄×C₃₁	—	4	8	no	—
221	C₄²×C₃₁	—	2	4	no	—
222	C₄²×C₃₁	—	2	4	no	—
223	C₄²×C₃₁	—	4	4	no	—
224	#35084	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	2	8	no	—
230	C₂³×C₃₁	—	2	8	no	—
231	C₂³×C₃₁	—	2	8	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	2	2	no	—
236	C₂³×C₃₁	—	2	4	no	—
237	C₂³×C₃₁	—	2	2	no	—
238	C₂³×C₃₁	—	2	4	no	—
239	C₂³×C₃₁	—	4	4	no	—
240	C₂³×C₃₁	—	4	16	no	—
241	C₂³×C₃₁	—	4	16	no	—
242	C₂³×C₃₁	—	4	4	no	—
243	C₂³×C₃₁	—	4	4	no	—
244	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
246	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
247	C₂³×C₃₁	C₂×#41	1	4	yes	—
248	C₂³×C₃₁	C₂×#41	1	4	yes	—
249	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
250	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
251	Q₈×C₃₁	—	4	8	no	—
252	Q₈×C₃₁	—	4	8	no	—
253	D₄×C₃₁	—	2	16	no	—
254	D₄×C₃₁	—	2	16	no	—
255	C₂×C₄×C₃₁	—	2	8	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	2	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	8	no	—
262	C₂×C₄×C₃₁	—	2	8	no	—
263	C₂×C₄×C₃₁	—	2	8	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	2	2	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	8	no	—
269	C₂×C₄×C₃₁	—	2	8	no	—
270	C₂×C₄×C₃₁	—	2	8	no	—
271	C₂×C₄×C₃₁	—	2	8	no	—
272	C₂×C₄×C₃₁	—	2	4	no	—
273	C₂×C₄×C₃₁	—	2	8	no	—
274	C₂×C₄×C₃₁	—	2	4	no	—
275	C₂×C₄×C₃₁	—	4	4	no	—
276	C₂×C₄×C₃₁	—	4	4	no	—
277	C₂×C₄×C₃₁	—	4	4	no	—
278	C₂×C₄×C₃₁	—	4	8	no	—
279	C₂×C₄×C₃₁	—	4	4	no	—
280	C₂×C₄×C₃₁	—	4	8	no	—
281	C₂×C₄×C₃₁	—	4	8	no	—
282	C₂×C₄×C₃₁	—	4	4	no	—
283	C₂×C₄×C₃₁	—	4	8	no	—
284	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
289	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
290	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
291	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic
292	#2855	C₂×C₃₁	1	2	yes	complemented
293	#2848	C₂×C₃₁	1	2	yes	complemented
294	#2638	C₂×C₃₁	1	4	yes	complemented
295	#2583	C₂×C₃₁	1	2	yes	complemented
296	#2561	C₂×C₃₁	1	2	yes	complemented
297	#2486	C₂×C₃₁	1	2	yes	complemented
298	#2483	C₂×C₃₁	1	2	yes	complemented
299	#2415	C₂×C₃₁	1	4	yes	complemented
300	#2409	C₂×C₃₁	1	2	yes	complemented
301	#2405	C₂×C₃₁	1	2	yes	—
302	#2322	C₂×C₃₁	1	4	yes	complemented
303	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
304	C₂²×C₃₁	—	2	2	no	—
305	C₂²×C₃₁	—	2	2	no	—
306	C₂²×C₃₁	—	2	2	no	—
307	C₂²×C₃₁	—	2	2	no	—
308	C₂²×C₃₁	—	2	8	no	—
309	C₂²×C₃₁	—	2	8	no	—
310	C₂²×C₃₁	—	2	2	no	—
311	C₂²×C₃₁	—	2	4	no	—
312	C₂²×C₃₁	—	2	2	no	—
313	C₂²×C₃₁	—	2	2	no	—
314	C₂²×C₃₁	—	2	8	no	—
315	C₂²×C₃₁	—	4	16	no	—
316	C₂²×C₃₁	—	4	16	no	—
317	C₂²×C₃₁	—	4	2	no	—
318	C₂²×C₃₁	—	4	2	no	—
319	C₂²×C₃₁	—	4	4	no	—
320	C₂²×C₃₁	—	4	4	no	—
321	C₂²×C₃₁	#525	1	4	yes	—
322	C₂²×C₃₁	#536	1	4	yes	—
323	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
324	C₂²×C₃₁	C₂²×#41	1	2	yes	—
325	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
326	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
327	C₄×C₃₁	—	2	4	no	—
328	C₄×C₃₁	—	2	2	no	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	2	2	no	—
331	C₄×C₃₁	—	4	4	no	—
332	C₄×C₃₁	—	4	2	no	—
333	C₄×C₃₁	—	4	4	no	—
334	C₄×C₃₁	—	4	2	no	—
335	C₄×C₃₁	—	4	8	no	—
336	C₄×C₃₁	#534	1	4	yes	—
337	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
338	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
339	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
340	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×#123	C₂²×C₃₁	1	4	yes	—
342	C₂×C₂²⋊Q₈	—	2	8	no	—
343	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
344	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
345	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
346	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
347	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
348	C₂×C₄×D₄	—	2	8	no	—
349	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
350	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
351	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
352	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
353	C₂×#117	C₂²×C₃₁	1	2	yes	complemented
354	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
355	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
356	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
357	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
358	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
359	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
360	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
361	#398	C₂²×C₃₁	1	4	yes	complemented
362	#397	C₂²×C₃₁	1	4	yes	complemented
363	#397	C₂²×C₃₁	1	8	yes	complemented
364	#395	C₂²×C₃₁	1	4	yes	complemented
365	#394	C₂²×C₃₁	1	8	yes	—
366	#393	C₂²×C₃₁	1	4	yes	complemented
367	#391	C₂²×C₃₁	1	4	yes	complemented
368	#389	C₂²×C₃₁	1	8	yes	complemented
369	#389	C₂²×C₃₁	1	4	yes	complemented
370	#388	C₂²×C₃₁	1	8	yes	complemented
371	#387	C₂²×C₃₁	1	8	yes	complemented
372	#386	C₂²×C₃₁	1	4	yes	—
373	#385	C₂²×C₃₁	1	4	yes	—
374	#385	C₂²×C₃₁	1	4	yes	—
375	#385	C₂²×C₃₁	1	8	yes	complemented
376	#385	C₂²×C₃₁	1	4	yes	complemented
377	#381	C₂²×C₃₁	1	4	yes	—
378	#381	C₂²×C₃₁	1	4	yes	complemented
379	#381	C₂²×C₃₁	1	2	yes	complemented
380	#380	C₂²×C₃₁	1	2	yes	—
381	#380	C₂²×C₃₁	1	4	yes	complemented
382	#380	C₂²×C₃₁	1	2	yes	complemented
383	#379	C₂²×C₃₁	1	4	yes	—
384	#379	C₂²×C₃₁	1	2	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
386	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
387	C₂×C₃₁	—	2	4	no	—
388	C₂×C₃₁	—	2	4	no	—
389	C₂×C₃₁	—	4	2	no	—
390	C₂×C₃₁	—	4	4	no	—
391	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
392	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
393	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
394	C₂×C₃₁	#3520	1	4	yes	—
395	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
396	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
397	C₂²×Q₈	—	2	4	no	—
398	C₂²×Q₈	—	4	4	no	—
399	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
400	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
401	C₂²×D₄	—	2	4	no	—
402	C₂²×D₄	—	2	8	no	—
403	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
404	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
405	C₂³×C₄	—	2	4	no	—
406	C₂³×C₄	—	4	4	no	—
407	C₂³×C₄	—	4	4	no	—
408	C₂³×C₄	D₄×C₃₁	1	8	yes	—
409	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
411	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
412	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
413	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
415	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
416	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
417	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
418	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
419	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
420	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
421	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
422	#123	—	2	16	no	—
423	C₂²⋊Q₈	—	2	8	no	—
424	C₂²⋊Q₈	—	2	8	no	—
425	C₂²⋊Q₈	—	2	8	no	—
426	C₂²⋊Q₈	—	2	8	no	—
427	C₂²⋊Q₈	—	4	8	no	—
428	C₄×D₄	—	2	8	no	—
429	C₄×D₄	—	2	8	no	—
430	C₄×D₄	—	2	8	no	—
431	C₄×D₄	—	2	8	no	—
432	#117	—	4	4	no	—
433	C₂×C₄⋊C₄	—	2	4	no	—
434	C₂×C₄⋊C₄	—	2	4	no	—
435	C₂×C₄⋊C₄	—	2	8	no	—
436	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
438	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
439	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
440	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
441	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
443	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
444	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
445	C₂×C₂²⋊C₄	—	2	4	no	—
446	C₂×C₂²⋊C₄	—	2	8	no	—
447	C₂×C₂²⋊C₄	—	2	4	no	—
448	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
459	C₂×C₄²	—	2	4	no	—
460	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
461	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
462	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
463	#95	C₂³×C₃₁	1	4	yes	—
464	#95	C₂³×C₃₁	1	4	yes	—
465	#95	C₂³×C₃₁	1	8	yes	—
466	#95	C₂³×C₃₁	1	8	yes	—
467	#95	C₂³×C₃₁	1	8	yes	—
468	#95	C₂³×C₃₁	1	8	yes	—
469	C₃₁	#35084	1	1	yes	characteristic, direct product factor, Sylow
470	C₂⁴	—	2	2	no	—
471	C₂⁴	—	2	4	no	—
472	C₂⁴	—	2	4	no	—
473	C₂⁴	—	2	4	no	—
474	C₂⁴	—	2	4	no	—
475	C₂⁴	—	4	2	no	—
476	C₂⁴	—	4	2	no	—
477	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
478	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
479	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
480	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	8	no	—
485	C₂×Q₈	—	4	4	no	—
486	C₂×Q₈	—	4	16	no	—
487	C₂×D₄	—	2	32	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	2	4	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	#41×C₃₁	1	8	yes	—
494	C₂²×C₄	—	2	2	no	—
495	C₂²×C₄	—	2	4	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	8	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	4	no	—
507	C₂²×C₄	—	2	8	no	—
508	C₂²×C₄	—	4	16	no	—
509	C₂²×C₄	—	4	2	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	4	2	no	—
512	C₂²×C₄	—	4	4	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	—	4	4	no	—
515	C₂²×C₄	—	4	16	no	—
516	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
517	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
518	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
519	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
520	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
521	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
522	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
524	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
525	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
526	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
527	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
530	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	8	no	—
537	C₄⋊C₄	—	2	8	no	—
538	C₄⋊C₄	—	2	16	no	—
539	C₄⋊C₄	—	2	8	no	—
540	C₄⋊C₄	—	4	8	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	—	2	4	no	—
543	C₂²⋊C₄	—	2	8	no	—
544	C₂²⋊C₄	—	2	8	no	—
545	C₂²⋊C₄	—	2	8	no	—
546	C₂²⋊C₄	—	2	16	no	—
547	C₂²⋊C₄	—	2	8	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	4	4	no	—
551	C₂²⋊C₄	—	4	8	no	—
552	C₄²	—	2	4	no	—
553	C₄²	—	2	4	no	—
554	C₄²	—	4	4	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	8	no	—
560	C₂³	—	2	8	no	—
561	C₂³	—	2	8	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	2	no	—
566	C₂³	—	2	4	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	4	no	—
569	C₂³	—	4	4	no	—
570	C₂³	—	4	16	no	—
571	C₂³	—	4	16	no	—
572	C₂³	—	4	4	no	—
573	C₂³	—	4	4	no	—
574	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
575	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
576	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
577	C₂³	C₂×#41×C₃₁	1	4	yes	—
578	C₂³	C₂×#41×C₃₁	1	4	yes	—
579	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
580	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
581	Q₈	—	4	8	no	—
582	Q₈	—	4	8	no	—
583	D₄	—	2	16	no	—
584	D₄	—	2	16	no	—
585	C₂×C₄	—	2	8	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	8	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	2	8	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	8	no	—
599	C₂×C₄	—	2	8	no	—
600	C₂×C₄	—	2	8	no	—
601	C₂×C₄	—	2	8	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	8	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	4	4	no	—
606	C₂×C₄	—	4	4	no	—
607	C₂×C₄	—	4	4	no	—
608	C₂×C₄	—	4	8	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	8	no	—
611	C₂×C₄	—	4	8	no	—
612	C₂×C₄	—	4	4	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
616	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
617	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
618	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
619	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	8	no	—
625	C₂²	—	2	8	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	4	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	8	no	—
631	C₂²	—	4	16	no	—
632	C₂²	—	4	16	no	—
633	C₂²	—	4	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	4	no	—
636	C₂²	—	4	4	no	—
637	C₂²	C₃₁×#525	1	4	yes	—
638	C₂²	C₃₁×#536	1	4	yes	—
639	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
640	C₂²	C₂²×#41×C₃₁	1	2	yes	—
641	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
642	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
643	C₄	—	2	4	no	—
644	C₄	—	2	2	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	2	no	—
647	C₄	—	4	4	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	8	no	—
652	C₄	C₃₁×#534	1	4	yes	—
653	C₂	—	2	4	no	—
654	C₂	—	2	4	no	—
655	C₂	—	4	2	no	—
656	C₂	—	4	4	no	—
657	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
658	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
659	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
660	C₂	C₃₁×#3520	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	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
1. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#2855	C₂	semidirect
4. C₃₁×#2848	C₂	semidirect
5. C₃₁×#2638	C₂	semidirect
6. C₃₁×#2583	C₂	semidirect
7. C₃₁×#2561	C₂	semidirect
8. C₃₁×#2486	C₂	semidirect
9. C₃₁×#2483	C₂	semidirect
10. C₃₁×#2415	C₂	semidirect
11. C₃₁×#2409	C₂	semidirect
13. C₃₁×#2322	C₂	semidirect
14. C₃₁×#2280	C₂	semidirect
15. C₂³×Q₈×C₃₁	C₂²	semidirect
23. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
24. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
25. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
31. C₂×C₃₁×#117	C₂²	semidirect
33. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
39. C₃₁×#398	C₂²	semidirect
40. C₃₁×#397	C₂²	semidirect
41. C₃₁×#397	C₂²	semidirect
42. C₃₁×#395	C₂²	semidirect
44. C₃₁×#393	C₂²	semidirect
45. C₃₁×#391	C₂²	semidirect
46. C₃₁×#389	C₂²	semidirect
47. C₃₁×#389	C₂²	semidirect
48. C₃₁×#388	C₂²	semidirect
49. C₃₁×#387	C₂²	semidirect
53. C₃₁×#385	C₂²	semidirect
54. C₃₁×#385	C₂²	semidirect
56. C₃₁×#381	C₂²	semidirect
57. C₃₁×#381	C₂²	semidirect
59. C₃₁×#380	C₂²	semidirect
60. C₃₁×#380	C₂²	semidirect
64. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
224. #35084	C₃₁	direct (N × H)
290. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
292. #2855	C₂×C₃₁	semidirect
293. #2848	C₂×C₃₁	semidirect
294. #2638	C₂×C₃₁	semidirect
295. #2583	C₂×C₃₁	semidirect
296. #2561	C₂×C₃₁	semidirect
297. #2486	C₂×C₃₁	semidirect
298. #2483	C₂×C₃₁	semidirect
299. #2415	C₂×C₃₁	semidirect
300. #2409	C₂×C₃₁	semidirect
302. #2322	C₂×C₃₁	semidirect
303. #2280	C₂×C₃₁	semidirect
337. C₂³×Q₈	C₂²×C₃₁	semidirect
345. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
346. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
347. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
353. C₂×#117	C₂²×C₃₁	semidirect
355. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
356. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
361. #398	C₂²×C₃₁	semidirect
362. #397	C₂²×C₃₁	semidirect
363. #397	C₂²×C₃₁	semidirect
364. #395	C₂²×C₃₁	semidirect
366. #393	C₂²×C₃₁	semidirect
367. #391	C₂²×C₃₁	semidirect
368. #389	C₂²×C₃₁	semidirect
369. #389	C₂²×C₃₁	semidirect
370. #388	C₂²×C₃₁	semidirect
371. #387	C₂²×C₃₁	semidirect
375. #385	C₂²×C₃₁	semidirect
376. #385	C₂²×C₃₁	semidirect
378. #381	C₂²×C₃₁	semidirect
379. #381	C₂²×C₃₁	semidirect
381. #380	C₂²×C₃₁	semidirect
382. #380	C₂²×C₃₁	semidirect
386. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
469. C₃₁	#35084	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2855	2
C₃₁×#2848	2
C₃₁×#2638	2
C₃₁×#2583	2
C₃₁×#2561	2
C₃₁×#2486	2
C₃₁×#2483	2
C₃₁×#2415	2
C₃₁×#2409	2
C₃₁×#2405	2
C₃₁×#2322	2
C₃₁×#2280	2
#35084	31

Subgroups — C31×#35084 #21491843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35084 #21491843