Subgroups — C₃₁×#35057 #21491816

Order 7936.

2898 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2849	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2849	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2627	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2618	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2591	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2583	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2579	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2484	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2413	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2406	C₂	1	2	yes	maximal, complemented
13	C₃₁×#2322	C₂	1	4	yes	maximal, complemented
14	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×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	C₃₁×#525	—	2	8	no	—
20	C₂×C₃₁×#123	C₂²	1	4	yes	—
21	C₂×C₃₁×#123	C₂²	1	2	yes	—
22	C₂×C₃₁×#123	C₂²	1	4	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#120	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#117	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₃₁	—	2	8	no	—
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
39	C₃₁×#400	C₂²	1	4	yes	complemented
40	C₃₁×#398	C₂²	1	8	yes	complemented
41	C₃₁×#398	C₂²	1	4	yes	complemented
42	C₃₁×#396	C₂²	1	8	yes	complemented
43	C₃₁×#395	C₂²	1	4	yes	complemented
44	C₃₁×#394	C₂²	1	4	yes	complemented
45	C₃₁×#389	C₂²	1	8	yes	complemented
46	C₃₁×#388	C₂²	1	8	yes	complemented
47	C₃₁×#387	C₂²	1	8	yes	complemented
48	C₃₁×#386	C₂²	1	4	yes	—
49	C₃₁×#386	C₂²	1	4	yes	—
50	C₃₁×#386	C₂²	1	4	yes	—
51	C₃₁×#386	C₂²	1	4	yes	—
52	C₃₁×#385	C₂²	1	4	yes	—
53	C₃₁×#385	C₂²	1	4	yes	—
54	C₃₁×#385	C₂²	1	4	yes	—
55	C₃₁×#385	C₂²	1	4	yes	—
56	C₃₁×#385	C₂²	1	8	yes	complemented
57	C₃₁×#381	C₂²	1	4	yes	—
58	C₃₁×#381	C₂²	1	4	yes	—
59	C₃₁×#381	C₂²	1	2	yes	—
60	C₃₁×#380	C₂²	1	2	yes	—
61	C₃₁×#380	C₂²	1	2	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	—
65	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×#41×C₃₁	—	2	4	no	—
67	C₂×#41×C₃₁	—	4	4	no	—
68	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	—	2	4	no	—
74	C₂³×C₄×C₃₁	—	2	8	no	—
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	2	yes	—
80	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
81	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
82	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
85	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
86	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
87	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
88	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	8	no	—
93	C₃₁×#123	—	2	16	no	—
94	C₃₁×#123	—	4	8	no	—
95	C₃₁×C₂²⋊Q₈	—	2	4	no	—
96	C₃₁×C₂²⋊Q₈	—	2	4	no	—
97	C₃₁×#121	—	2	4	no	—
98	C₃₁×#121	—	2	4	no	—
99	C₃₁×#120	—	2	8	no	—
100	C₄×D₄×C₃₁	—	2	8	no	—
101	C₄×D₄×C₃₁	—	2	8	no	—
102	C₃₁×#117	—	2	4	no	—
103	C₃₁×#117	—	4	4	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
108	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
109	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	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	2	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
130	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	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	4	no	—
140	C₂⁴×C₃₁	—	2	4	no	—
141	C₂⁴×C₃₁	—	2	4	no	—
142	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
143	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
144	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
145	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
146	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
147	#41×C₃₁	—	4	16	no	—
148	C₂×Q₈×C₃₁	—	2	2	no	—
149	C₂×Q₈×C₃₁	—	4	2	no	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	8	no	—
154	C₂×D₄×C₃₁	—	2	8	no	—
155	C₂×D₄×C₃₁	—	2	8	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	4	2	no	—
158	C₂²×C₄×C₃₁	—	2	32	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	4	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	8	no	—
166	C₂²×C₄×C₃₁	—	2	8	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	4	no	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	—	4	16	no	—
173	C₂²×C₄×C₃₁	—	4	2	no	—
174	C₂²×C₄×C₃₁	—	4	2	no	—
175	C₂²×C₄×C₃₁	—	4	2	no	—
176	C₂²×C₄×C₃₁	—	4	2	no	—
177	C₂²×C₄×C₃₁	—	4	4	no	—
178	C₂²×C₄×C₃₁	—	4	4	no	—
179	C₂²×C₄×C₃₁	—	4	4	no	—
180	C₂²×C₄×C₃₁	—	4	16	no	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₃₁	#41	1	8	yes	complemented
188	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
192	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
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	—	2	8	no	—
202	C₄⋊C₄×C₃₁	—	2	2	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	16	no	—
207	C₄⋊C₄×C₃₁	—	4	4	no	—
208	C₄⋊C₄×C₃₁	—	4	8	no	—
209	C₂²⋊C₄×C₃₁	—	2	4	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	4	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	16	no	—
218	C₂²⋊C₄×C₃₁	—	4	8	no	—
219	C₄²×C₃₁	—	2	4	no	—
220	C₄²×C₃₁	—	4	4	no	—
221	#35057	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₃₁	—	2	32	no	—
223	C₂³×C₃₁	—	2	8	no	—
224	C₂³×C₃₁	—	2	8	no	—
225	C₂³×C₃₁	—	2	8	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	4	no	—
234	C₂³×C₃₁	—	2	4	no	—
235	C₂³×C₃₁	—	2	4	no	—
236	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
237	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
238	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
239	C₂³×C₃₁	C₂×#41	1	4	yes	—
240	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
241	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
242	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
243	Q₈×C₃₁	—	4	4	no	—
244	D₄×C₃₁	—	4	4	no	—
245	D₄×C₃₁	—	4	8	no	—
246	D₄×C₃₁	—	4	8	no	—
247	D₄×C₃₁	—	4	8	no	—
248	C₂×C₄×C₃₁	—	2	16	no	—
249	C₂×C₄×C₃₁	—	2	16	no	—
250	C₂×C₄×C₃₁	—	2	16	no	—
251	C₂×C₄×C₃₁	—	2	8	no	—
252	C₂×C₄×C₃₁	—	2	8	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	2	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	2	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	2	8	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	8	no	—
266	C₂×C₄×C₃₁	—	2	8	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	8	no	—
269	C₂×C₄×C₃₁	—	2	4	no	—
270	C₂×C₄×C₃₁	—	4	4	no	—
271	C₂×C₄×C₃₁	—	4	4	no	—
272	C₂×C₄×C₃₁	—	4	4	no	—
273	C₂×C₄×C₃₁	—	4	4	no	—
274	C₂×C₄×C₃₁	—	4	8	no	—
275	C₂×C₄×C₃₁	—	4	8	no	—
276	C₂×C₄×C₃₁	—	4	8	no	—
277	C₂×C₄×C₃₁	—	4	8	no	—
278	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
283	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
284	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#2849	C₂×C₃₁	1	2	yes	complemented
286	#2849	C₂×C₃₁	1	2	yes	complemented
287	#2627	C₂×C₃₁	1	2	yes	complemented
288	#2618	C₂×C₃₁	1	2	yes	complemented
289	#2591	C₂×C₃₁	1	2	yes	complemented
290	#2583	C₂×C₃₁	1	2	yes	complemented
291	#2579	C₂×C₃₁	1	4	yes	complemented
292	#2484	C₂×C₃₁	1	2	yes	complemented
293	#2413	C₂×C₃₁	1	4	yes	complemented
294	#2406	C₂×C₃₁	1	2	yes	complemented
295	#2322	C₂×C₃₁	1	4	yes	complemented
296	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
297	C₂²×C₃₁	—	2	32	no	—
298	C₂²×C₃₁	—	2	2	no	—
299	C₂²×C₃₁	—	2	2	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	—	2	8	no	—
305	C₂²×C₃₁	—	2	8	no	—
306	C₂²×C₃₁	—	2	8	no	—
307	C₂²×C₃₁	—	4	2	no	—
308	C₂²×C₃₁	—	4	2	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	4	8	no	—
311	C₂²×C₃₁	#525	1	4	yes	—
312	C₂²×C₃₁	#534	1	4	yes	—
313	C₂²×C₃₁	#536	1	4	yes	—
314	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
315	C₂²×C₃₁	C₂²×#41	1	2	yes	—
316	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
317	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
318	C₄×C₃₁	—	2	8	no	—
319	C₄×C₃₁	—	2	2	no	—
320	C₄×C₃₁	—	2	2	no	—
321	C₄×C₃₁	—	2	2	no	—
322	C₄×C₃₁	—	2	2	no	—
323	C₄×C₃₁	—	4	2	no	—
324	C₄×C₃₁	—	4	4	no	—
325	C₄×C₃₁	—	4	2	no	—
326	C₄×C₃₁	—	4	2	no	—
327	C₄×C₃₁	—	4	8	no	—
328	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
329	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
330	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
331	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
332	#525	—	2	8	no	—
333	C₂×#123	C₂²×C₃₁	1	4	yes	—
334	C₂×#123	C₂²×C₃₁	1	2	yes	—
335	C₂×#123	C₂²×C₃₁	1	4	yes	—
336	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
339	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×#120	C₂²×C₃₁	1	2	yes	—
341	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
342	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
343	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×#117	C₂²×C₃₁	1	2	yes	—
345	C₂²×C₄⋊C₄	—	2	8	no	—
346	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
347	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
348	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
349	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
350	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
351	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
352	#400	C₂²×C₃₁	1	4	yes	complemented
353	#398	C₂²×C₃₁	1	8	yes	complemented
354	#398	C₂²×C₃₁	1	4	yes	complemented
355	#396	C₂²×C₃₁	1	8	yes	complemented
356	#395	C₂²×C₃₁	1	4	yes	complemented
357	#394	C₂²×C₃₁	1	4	yes	complemented
358	#389	C₂²×C₃₁	1	8	yes	complemented
359	#388	C₂²×C₃₁	1	8	yes	complemented
360	#387	C₂²×C₃₁	1	8	yes	complemented
361	#386	C₂²×C₃₁	1	4	yes	—
362	#386	C₂²×C₃₁	1	4	yes	—
363	#386	C₂²×C₃₁	1	4	yes	—
364	#386	C₂²×C₃₁	1	4	yes	—
365	#385	C₂²×C₃₁	1	4	yes	—
366	#385	C₂²×C₃₁	1	4	yes	—
367	#385	C₂²×C₃₁	1	4	yes	—
368	#385	C₂²×C₃₁	1	4	yes	—
369	#385	C₂²×C₃₁	1	8	yes	complemented
370	#381	C₂²×C₃₁	1	4	yes	—
371	#381	C₂²×C₃₁	1	4	yes	—
372	#381	C₂²×C₃₁	1	2	yes	—
373	#380	C₂²×C₃₁	1	2	yes	—
374	#380	C₂²×C₃₁	1	2	yes	—
375	#379	C₂²×C₃₁	1	2	yes	—
376	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
377	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
378	C₂×C₃₁	—	2	4	no	—
379	C₂×C₃₁	—	2	4	no	—
380	C₂×C₃₁	—	2	4	no	—
381	C₂×C₃₁	—	8	2	no	—
382	C₂×C₃₁	C₂×#525	1	1	yes	characteristic
383	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
384	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
385	C₂×C₃₁	#3492	1	4	yes	—
386	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×#41	—	2	4	no	—
388	C₂×#41	—	4	4	no	—
389	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
391	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
392	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
393	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂³×C₄	—	2	4	no	—
395	C₂³×C₄	—	2	8	no	—
396	C₂³×C₄	—	2	4	no	—
397	C₂³×C₄	—	4	4	no	—
398	C₂³×C₄	—	4	4	no	—
399	C₂³×C₄	D₄×C₃₁	1	8	yes	—
400	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
401	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
402	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
403	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
404	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
405	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
406	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
407	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
408	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
409	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
411	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
412	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
413	#123	—	2	8	no	—
414	#123	—	2	16	no	—
415	#123	—	4	8	no	—
416	C₂²⋊Q₈	—	2	4	no	—
417	C₂²⋊Q₈	—	2	4	no	—
418	#121	—	2	4	no	—
419	#121	—	2	4	no	—
420	#120	—	2	8	no	—
421	C₄×D₄	—	2	8	no	—
422	C₄×D₄	—	2	8	no	—
423	#117	—	2	4	no	—
424	#117	—	4	4	no	—
425	C₂×C₄⋊C₄	—	2	8	no	—
426	C₂×C₄⋊C₄	—	2	8	no	—
427	C₂×C₄⋊C₄	—	2	8	no	—
428	C₂×C₄⋊C₄	—	2	8	no	—
429	C₂×C₄⋊C₄	—	2	8	no	—
430	C₂×C₄⋊C₄	—	2	8	no	—
431	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
432	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
433	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
434	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
435	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
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	8	yes	—
440	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
441	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
448	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
451	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
452	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
454	#95	C₂³×C₃₁	1	4	yes	—
455	#95	C₂³×C₃₁	1	4	yes	—
456	#95	C₂³×C₃₁	1	8	yes	—
457	#95	C₂³×C₃₁	1	8	yes	—
458	#95	C₂³×C₃₁	1	8	yes	—
459	#95	C₂³×C₃₁	1	8	yes	—
460	C₃₁	#35057	1	1	yes	characteristic, direct product factor, Sylow
461	C₂⁴	—	2	4	no	—
462	C₂⁴	—	2	4	no	—
463	C₂⁴	—	2	4	no	—
464	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
465	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
466	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
467	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
468	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
469	#41	—	4	16	no	—
470	C₂×Q₈	—	2	2	no	—
471	C₂×Q₈	—	4	2	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	2	4	no	—
475	C₂×D₄	—	2	8	no	—
476	C₂×D₄	—	2	8	no	—
477	C₂×D₄	—	2	8	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	4	2	no	—
480	C₂²×C₄	—	2	32	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	8	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	4	16	no	—
495	C₂²×C₄	—	4	2	no	—
496	C₂²×C₄	—	4	2	no	—
497	C₂²×C₄	—	4	2	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	—	4	4	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	—	4	4	no	—
502	C₂²×C₄	—	4	16	no	—
503	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
504	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
505	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
506	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
507	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
509	C₂²×C₄	#41×C₃₁	1	8	yes	complemented
510	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
511	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
512	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
520	C₄⋊C₄	—	2	8	no	—
521	C₄⋊C₄	—	2	8	no	—
522	C₄⋊C₄	—	2	8	no	—
523	C₄⋊C₄	—	2	8	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	16	no	—
529	C₄⋊C₄	—	4	4	no	—
530	C₄⋊C₄	—	4	8	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	8	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	16	no	—
540	C₂²⋊C₄	—	4	8	no	—
541	C₄²	—	2	4	no	—
542	C₄²	—	4	4	no	—
543	C₂³	—	2	32	no	—
544	C₂³	—	2	8	no	—
545	C₂³	—	2	8	no	—
546	C₂³	—	2	8	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	2	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	2	4	no	—
556	C₂³	—	2	4	no	—
557	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
558	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
559	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
560	C₂³	C₂×#41×C₃₁	1	4	yes	—
561	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
562	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
563	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
564	Q₈	—	4	4	no	—
565	D₄	—	4	4	no	—
566	D₄	—	4	8	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	8	no	—
569	C₂×C₄	—	2	16	no	—
570	C₂×C₄	—	2	16	no	—
571	C₂×C₄	—	2	16	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	8	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	8	no	—
587	C₂×C₄	—	2	8	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	8	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	4	4	no	—
592	C₂×C₄	—	4	4	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	8	no	—
596	C₂×C₄	—	4	8	no	—
597	C₂×C₄	—	4	8	no	—
598	C₂×C₄	—	4	8	no	—
599	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
600	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
601	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
602	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
603	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
604	C₂²	—	2	32	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	8	no	—
612	C₂²	—	2	8	no	—
613	C₂²	—	2	8	no	—
614	C₂²	—	4	2	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	8	no	—
618	C₂²	C₃₁×#525	1	4	yes	—
619	C₂²	C₃₁×#534	1	4	yes	—
620	C₂²	C₃₁×#536	1	4	yes	—
621	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
622	C₂²	C₂²×#41×C₃₁	1	2	yes	—
623	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
624	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
625	C₄	—	2	8	no	—
626	C₄	—	2	2	no	—
627	C₄	—	2	2	no	—
628	C₄	—	2	2	no	—
629	C₄	—	2	2	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	4	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	2	no	—
634	C₄	—	4	8	no	—
635	C₂	—	2	4	no	—
636	C₂	—	2	4	no	—
637	C₂	—	2	4	no	—
638	C₂	—	8	2	no	—
639	C₂	C₂×C₃₁×#525	1	1	yes	characteristic
640	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
641	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
642	C₂	C₃₁×#3492	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#525	C₂
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. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
3. C₃₁×#2849	C₂	semidirect
4. C₃₁×#2849	C₂	semidirect
5. C₃₁×#2627	C₂	semidirect
6. C₃₁×#2618	C₂	semidirect
7. C₃₁×#2591	C₂	semidirect
8. C₃₁×#2583	C₂	semidirect
9. C₃₁×#2579	C₂	semidirect
10. C₃₁×#2484	C₂	semidirect
11. C₃₁×#2413	C₂	semidirect
12. C₃₁×#2406	C₂	semidirect
13. C₃₁×#2322	C₂	semidirect
14. C₃₁×#2280	C₂	semidirect
39. C₃₁×#400	C₂²	semidirect
40. C₃₁×#398	C₂²	semidirect
41. C₃₁×#398	C₂²	semidirect
42. C₃₁×#396	C₂²	semidirect
43. C₃₁×#395	C₂²	semidirect
44. C₃₁×#394	C₂²	semidirect
45. C₃₁×#389	C₂²	semidirect
46. C₃₁×#388	C₂²	semidirect
47. C₃₁×#387	C₂²	semidirect
56. C₃₁×#385	C₂²	semidirect
187. C₂²×C₄×C₃₁	#41	semidirect
221. #35057	C₃₁	direct (N × H)
284. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
285. #2849	C₂×C₃₁	semidirect
286. #2849	C₂×C₃₁	semidirect
287. #2627	C₂×C₃₁	semidirect
288. #2618	C₂×C₃₁	semidirect
289. #2591	C₂×C₃₁	semidirect
290. #2583	C₂×C₃₁	semidirect
291. #2579	C₂×C₃₁	semidirect
292. #2484	C₂×C₃₁	semidirect
293. #2413	C₂×C₃₁	semidirect
294. #2406	C₂×C₃₁	semidirect
295. #2322	C₂×C₃₁	semidirect
296. #2280	C₂×C₃₁	semidirect
352. #400	C₂²×C₃₁	semidirect
353. #398	C₂²×C₃₁	semidirect
354. #398	C₂²×C₃₁	semidirect
355. #396	C₂²×C₃₁	semidirect
356. #395	C₂²×C₃₁	semidirect
357. #394	C₂²×C₃₁	semidirect
358. #389	C₂²×C₃₁	semidirect
359. #388	C₂²×C₃₁	semidirect
360. #387	C₂²×C₃₁	semidirect
369. #385	C₂²×C₃₁	semidirect
460. C₃₁	#35057	direct (N × H)
509. C₂²×C₄	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₂³×C₄⋊C₄×C₃₁	2
C₃₁×#2849	2
C₃₁×#2627	2
C₃₁×#2618	2
C₃₁×#2591	2
C₃₁×#2583	2
C₃₁×#2579	2
C₃₁×#2484	2
C₃₁×#2413	2
C₃₁×#2406	2
C₃₁×#2322	2
C₃₁×#2280	2
#35057	31

Subgroups — C31×#35057 #21491816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35057 #21491816