Subgroups — C₂×C₃₁×#2364 #21490896

Order 7936.

4262 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2364	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#388	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#387	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#386	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#386	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₃₁×#95	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#121	C₄	1	16	yes	complemented
17	C₂×C₃₁×#121	C₂²	1	8	yes	—
18	C₂×C₄×D₄×C₃₁	—	2	16	no	—
19	C₂×C₄×D₄×C₃₁	C₂²	1	16	yes	complemented
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#388	C₂²	1	16	yes	complemented
31	C₃₁×#387	C₂²	1	16	yes	complemented
32	C₃₁×#386	C₂²	1	16	yes	complemented
33	C₃₁×#386	C₂²	1	16	yes	complemented
34	C₄×C₂²⋊C₄×C₃₁	C₂²	1	16	yes	complemented
35	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
36	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#95	C₂²	1	2	yes	—
38	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#95	C₂²	1	2	yes	—
40	C₂×C₃₁×#95	C₂²	1	2	yes	—
41	C₂×C₃₁×#95	C₂²	1	2	yes	—
42	C₂×C₃₁×#95	C₂²	1	8	yes	complemented
43	C₂⁵×C₃₁	C₂³	1	2	yes	—
44	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	8	no	—
46	C₂²×D₄×C₃₁	—	2	8	no	—
47	C₂²×D₄×C₃₁	—	4	4	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
49	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
50	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
52	C₂³×C₄×C₃₁	—	2	8	no	—
53	C₂³×C₄×C₃₁	—	2	8	no	—
54	C₂³×C₄×C₃₁	—	2	8	no	—
55	C₂³×C₄×C₃₁	—	2	8	no	—
56	C₂³×C₄×C₃₁	—	4	4	no	—
57	C₂³×C₄×C₃₁	—	4	4	no	—
58	C₂³×C₄×C₃₁	—	4	4	no	—
59	C₂³×C₄×C₃₁	C₂×C₄	1	8	yes	complemented
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
70	C₃₁×#121	C₂×C₄	1	64	yes	complemented
71	C₄×D₄×C₃₁	—	2	64	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
80	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
81	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	complemented
82	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄²×C₃₁	—	2	8	no	—
89	C₂×C₄²×C₃₁	—	4	4	no	—
90	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
91	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
92	C₃₁×#95	C₂³	1	4	yes	—
93	C₃₁×#95	C₂³	1	8	yes	—
94	C₃₁×#95	C₂³	1	4	yes	—
95	C₃₁×#95	C₂³	1	8	yes	—
96	C₃₁×#95	C₂³	1	8	yes	—
97	C₃₁×#95	C₂³	1	8	yes	—
98	C₂⁴×C₃₁	—	2	4	no	—
99	C₂⁴×C₃₁	—	2	4	no	—
100	C₂⁴×C₃₁	—	2	4	no	—
101	C₂⁴×C₃₁	—	2	4	no	—
102	C₂⁴×C₃₁	—	2	4	no	—
103	C₂⁴×C₃₁	—	2	4	no	—
104	C₂⁴×C₃₁	—	2	4	no	—
105	C₂⁴×C₃₁	—	2	4	no	—
106	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
107	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
108	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
109	C₂⁴×C₃₁	#41	1	4	yes	—
110	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
111	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
112	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
113	C₂×D₄×C₃₁	—	2	64	no	—
114	C₂×D₄×C₃₁	—	2	16	no	—
115	C₂×D₄×C₃₁	—	2	16	no	—
116	C₂×D₄×C₃₁	—	4	16	no	—
117	C₂×D₄×C₃₁	—	4	8	no	—
118	C₂×D₄×C₃₁	C₂²×C₄	1	32	yes	complemented
119	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	complemented
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	4	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	4	no	—
125	C₂²×C₄×C₃₁	—	2	4	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	16	no	—
130	C₂²×C₄×C₃₁	—	2	4	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	16	no	—
136	C₂²×C₄×C₃₁	—	2	16	no	—
137	C₂²×C₄×C₃₁	—	2	16	no	—
138	C₂²×C₄×C₃₁	—	4	16	no	—
139	C₂²×C₄×C₃₁	—	4	8	no	—
140	C₂²×C₄×C₃₁	—	4	8	no	—
141	C₂²×C₄×C₃₁	—	4	8	no	—
142	C₂²×C₄×C₃₁	—	4	32	no	—
143	C₂²×C₄×C₃₁	—	4	32	no	—
144	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
146	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
148	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
149	C₂²×C₄×C₃₁	#41	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₄⋊C₄×C₃₁	—	2	16	no	—
158	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
159	C₂²⋊C₄×C₃₁	—	2	8	no	—
160	C₂²⋊C₄×C₃₁	—	2	16	no	—
161	C₂²⋊C₄×C₃₁	—	2	16	no	—
162	C₂²⋊C₄×C₃₁	—	4	8	no	—
163	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
164	C₂²⋊C₄×C₃₁	C₂×D₄	1	16	yes	complemented
165	C₄²×C₃₁	—	2	16	no	—
166	C₄²×C₃₁	—	4	8	no	—
167	C₂×#2364	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₃₁	—	2	8	no	—
169	C₂³×C₃₁	—	2	8	no	—
170	C₂³×C₃₁	—	2	16	no	—
171	C₂³×C₃₁	—	2	16	no	—
172	C₂³×C₃₁	—	2	4	no	—
173	C₂³×C₃₁	—	2	4	no	—
174	C₂³×C₃₁	—	2	16	no	—
175	C₂³×C₃₁	—	2	16	no	—
176	C₂³×C₃₁	—	2	16	no	—
177	C₂³×C₃₁	—	2	4	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	2	4	no	—
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	2	4	no	—
183	C₂³×C₃₁	—	4	8	no	—
184	C₂³×C₃₁	—	4	8	no	—
185	C₂³×C₃₁	—	4	8	no	—
186	C₂³×C₃₁	—	4	8	no	—
187	C₂³×C₃₁	C₄×D₄	1	16	yes	—
188	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
189	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
190	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
191	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
192	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
193	C₂³×C₃₁	C₂×#41	1	4	yes	—
194	C₂³×C₃₁	C₂×#41	1	2	yes	—
195	C₂³×C₃₁	C₂×#41	1	4	yes	—
196	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
197	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
198	D₄×C₃₁	—	2	64	no	—
199	D₄×C₃₁	—	4	16	no	—
200	C₂×C₄×C₃₁	—	2	16	no	—
201	C₂×C₄×C₃₁	—	2	8	no	—
202	C₂×C₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	16	no	—
204	C₂×C₄×C₃₁	—	2	16	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	4	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	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	4	16	no	—
220	C₂×C₄×C₃₁	—	4	32	no	—
221	C₂×C₄×C₃₁	—	4	32	no	—
222	C₂×C₄×C₃₁	—	4	8	no	—
223	C₂×C₄×C₃₁	—	4	8	no	—
224	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
226	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
228	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
229	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic
230	C₂²×C₄×D₄	C₂×C₃₁	1	2	yes	complemented
231	#2364	C₂×C₃₁	1	16	yes	direct product factor
232	C₂×#388	C₂×C₃₁	1	2	yes	complemented
233	C₂×#387	C₂×C₃₁	1	2	yes	complemented
234	C₂×#386	C₂×C₃₁	1	2	yes	complemented
235	C₂×#386	C₂×C₃₁	1	2	yes	complemented
236	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
237	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×#95	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	2	32	no	—
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	8	no	—
243	C₂²×C₃₁	—	2	8	no	—
244	C₂²×C₃₁	—	2	8	no	—
245	C₂²×C₃₁	—	2	8	no	—
246	C₂²×C₃₁	—	2	8	no	—
247	C₂²×C₃₁	—	4	32	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	4	4	no	—
251	C₂²×C₃₁	—	4	4	no	—
252	C₂²×C₃₁	C₂×C₄×D₄	1	8	yes	—
253	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
254	C₂²×C₃₁	C₄×#41	1	4	yes	—
255	C₂²×C₃₁	#518	1	4	yes	—
256	C₂²×C₃₁	#546	1	8	yes	—
257	C₂²×C₃₁	#553	1	8	yes	—
258	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
259	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
260	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
261	C₂²×C₃₁	C₂²×#41	1	2	yes	—
262	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
263	C₄×C₃₁	—	2	8	no	—
264	C₄×C₃₁	—	2	4	no	—
265	C₄×C₃₁	—	2	4	no	—
266	C₄×C₃₁	—	2	8	no	—
267	C₄×C₃₁	—	2	8	no	—
268	C₄×C₃₁	—	4	4	no	—
269	C₄×C₃₁	—	4	4	no	—
270	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
271	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
272	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
273	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
274	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#121	C₄×C₃₁	1	16	yes	complemented
276	C₂×#121	C₂²×C₃₁	1	8	yes	—
277	C₂×C₄×D₄	—	2	16	no	—
278	C₂×C₄×D₄	C₂²×C₃₁	1	16	yes	complemented
279	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
280	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
282	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
284	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
285	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
286	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
287	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
288	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
289	#388	C₂²×C₃₁	1	16	yes	complemented
290	#387	C₂²×C₃₁	1	16	yes	complemented
291	#386	C₂²×C₃₁	1	16	yes	complemented
292	#386	C₂²×C₃₁	1	16	yes	complemented
293	C₄×C₂²⋊C₄	C₂²×C₃₁	1	16	yes	complemented
294	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
295	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#95	C₂²×C₃₁	1	2	yes	—
297	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×#95	C₂²×C₃₁	1	2	yes	—
299	C₂×#95	C₂²×C₃₁	1	2	yes	—
300	C₂×#95	C₂²×C₃₁	1	2	yes	—
301	C₂×#95	C₂²×C₃₁	1	8	yes	complemented
302	C₂×C₃₁	—	2	8	no	—
303	C₂×C₃₁	—	4	8	no	—
304	C₂×C₃₁	#2364	1	8	yes	direct product factor
305	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
306	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
307	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
308	C₂×C₃₁	C₂×#546	1	2	yes	—
309	C₂×C₃₁	C₂×#553	1	2	yes	—
310	C₂⁵	C₂³×C₃₁	1	2	yes	—
311	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×D₄	—	2	8	no	—
313	C₂²×D₄	—	2	8	no	—
314	C₂²×D₄	—	4	4	no	—
315	C₂²×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
316	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	complemented
317	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
318	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
319	C₂³×C₄	—	2	8	no	—
320	C₂³×C₄	—	2	8	no	—
321	C₂³×C₄	—	2	8	no	—
322	C₂³×C₄	—	2	8	no	—
323	C₂³×C₄	—	4	4	no	—
324	C₂³×C₄	—	4	4	no	—
325	C₂³×C₄	—	4	4	no	—
326	C₂³×C₄	C₂×C₄×C₃₁	1	8	yes	complemented
327	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
328	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
329	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
330	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
331	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
333	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
334	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
335	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
336	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
337	#121	C₂×C₄×C₃₁	1	64	yes	complemented
338	C₄×D₄	—	2	64	no	—
339	C₂×C₄⋊C₄	—	2	8	no	—
340	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
343	C₂×C₂²⋊C₄	—	2	4	no	—
344	C₂×C₂²⋊C₄	—	2	8	no	—
345	C₂×C₂²⋊C₄	—	2	8	no	—
346	C₂×C₂²⋊C₄	—	4	4	no	—
347	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
348	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	complemented
349	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
355	C₂×C₄²	—	2	8	no	—
356	C₂×C₄²	—	4	4	no	—
357	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
358	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
359	#95	C₂³×C₃₁	1	4	yes	—
360	#95	C₂³×C₃₁	1	8	yes	—
361	#95	C₂³×C₃₁	1	4	yes	—
362	#95	C₂³×C₃₁	1	8	yes	—
363	#95	C₂³×C₃₁	1	8	yes	—
364	#95	C₂³×C₃₁	1	8	yes	—
365	C₃₁	C₂×#2364	1	1	yes	characteristic, direct product factor, Sylow
366	C₂⁴	—	2	4	no	—
367	C₂⁴	—	2	4	no	—
368	C₂⁴	—	2	4	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	—	2	4	no	—
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	2	4	no	—
373	C₂⁴	—	2	4	no	—
374	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
375	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
376	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
377	C₂⁴	#41×C₃₁	1	4	yes	—
378	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
379	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
380	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
381	C₂×D₄	—	2	64	no	—
382	C₂×D₄	—	2	16	no	—
383	C₂×D₄	—	2	16	no	—
384	C₂×D₄	—	4	16	no	—
385	C₂×D₄	—	4	8	no	—
386	C₂×D₄	C₂²×C₄×C₃₁	1	32	yes	complemented
387	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	complemented
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	16	no	—
404	C₂²×C₄	—	2	16	no	—
405	C₂²×C₄	—	2	16	no	—
406	C₂²×C₄	—	4	16	no	—
407	C₂²×C₄	—	4	8	no	—
408	C₂²×C₄	—	4	8	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	32	no	—
411	C₂²×C₄	—	4	32	no	—
412	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
413	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
414	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
415	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	complemented
416	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
417	C₂²×C₄	#41×C₃₁	1	8	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
420	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
421	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
425	C₄⋊C₄	—	2	16	no	—
426	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	complemented
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	16	no	—
429	C₂²⋊C₄	—	2	16	no	—
430	C₂²⋊C₄	—	4	8	no	—
431	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
432	C₂²⋊C₄	C₂×D₄×C₃₁	1	16	yes	complemented
433	C₄²	—	2	16	no	—
434	C₄²	—	4	8	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	8	no	—
437	C₂³	—	2	16	no	—
438	C₂³	—	2	16	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	16	no	—
442	C₂³	—	2	16	no	—
443	C₂³	—	2	16	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	4	8	no	—
451	C₂³	—	4	8	no	—
452	C₂³	—	4	8	no	—
453	C₂³	—	4	8	no	—
454	C₂³	C₄×D₄×C₃₁	1	16	yes	—
455	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
456	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
457	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
458	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
459	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
460	C₂³	C₂×#41×C₃₁	1	4	yes	—
461	C₂³	C₂×#41×C₃₁	1	2	yes	—
462	C₂³	C₂×#41×C₃₁	1	4	yes	—
463	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
464	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
465	D₄	—	2	64	no	—
466	D₄	—	4	16	no	—
467	C₂×C₄	—	2	16	no	—
468	C₂×C₄	—	2	8	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	16	no	—
471	C₂×C₄	—	2	16	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	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	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	4	16	no	—
487	C₂×C₄	—	4	32	no	—
488	C₂×C₄	—	4	32	no	—
489	C₂×C₄	—	4	8	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
492	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
493	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
494	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
495	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
496	C₂²	—	2	32	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	8	no	—
504	C₂²	—	4	32	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	4	no	—
508	C₂²	—	4	4	no	—
509	C₂²	C₂×C₄×D₄×C₃₁	1	8	yes	—
510	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
511	C₂²	C₄×#41×C₃₁	1	4	yes	—
512	C₂²	C₃₁×#518	1	4	yes	—
513	C₂²	C₃₁×#546	1	8	yes	—
514	C₂²	C₃₁×#553	1	8	yes	—
515	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
516	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
517	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
518	C₂²	C₂²×#41×C₃₁	1	2	yes	—
519	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
520	C₄	—	2	8	no	—
521	C₄	—	2	4	no	—
522	C₄	—	2	4	no	—
523	C₄	—	2	8	no	—
524	C₄	—	2	8	no	—
525	C₄	—	4	4	no	—
526	C₄	—	4	4	no	—
527	C₂	—	2	8	no	—
528	C₂	—	4	8	no	—
529	C₂	C₃₁×#2364	1	8	yes	direct product factor
530	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
531	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
532	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#546	1	2	yes	—
534	C₂	C₂×C₃₁×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₄×C₃₁	C₂	semidirect
3. C₃₁×#2364	C₂	direct (N × H)
4. C₂×C₃₁×#388	C₂	semidirect
5. C₂×C₃₁×#387	C₂	semidirect
6. C₂×C₃₁×#386	C₂	semidirect
7. C₂×C₃₁×#386	C₂	semidirect
8. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
10. C₂²×C₃₁×#95	C₂	semidirect
16. C₂×C₃₁×#121	C₄	semidirect
19. C₂×C₄×D₄×C₃₁	C₂²	semidirect
30. C₃₁×#388	C₂²	semidirect
31. C₃₁×#387	C₂²	semidirect
32. C₃₁×#386	C₂²	semidirect
33. C₃₁×#386	C₂²	semidirect
34. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
42. C₂×C₃₁×#95	C₂²	semidirect
48. C₂²×D₄×C₃₁	C₂×C₄	semidirect
49. C₂²×D₄×C₃₁	C₂×C₄	semidirect
59. C₂³×C₄×C₃₁	C₂×C₄	semidirect
70. C₃₁×#121	C₂×C₄	semidirect
73. C₂×C₄⋊C₄×C₃₁	C₂×C₄	semidirect
80. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
81. C₂×C₂²⋊C₄×C₃₁	D₄	semidirect
118. C₂×D₄×C₃₁	C₂²×C₄	semidirect
119. C₂×D₄×C₃₁	C₂²×C₄	semidirect
147. C₂²×C₄×C₃₁	C₂²×C₄	semidirect
158. C₄⋊C₄×C₃₁	C₂²×C₄	semidirect
163. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
164. C₂²⋊C₄×C₃₁	C₂×D₄	semidirect
167. C₂×#2364	C₃₁	direct (N × H)
230. C₂²×C₄×D₄	C₂×C₃₁	semidirect
231. #2364	C₂×C₃₁	direct (N × H)
232. C₂×#388	C₂×C₃₁	semidirect
233. C₂×#387	C₂×C₃₁	semidirect
234. C₂×#386	C₂×C₃₁	semidirect
235. C₂×#386	C₂×C₃₁	semidirect
236. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
237. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
238. C₂²×#95	C₂×C₃₁	semidirect
275. C₂×#121	C₄×C₃₁	semidirect
278. C₂×C₄×D₄	C₂²×C₃₁	semidirect
289. #388	C₂²×C₃₁	semidirect
290. #387	C₂²×C₃₁	semidirect
291. #386	C₂²×C₃₁	semidirect
292. #386	C₂²×C₃₁	semidirect
293. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
294. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
301. C₂×#95	C₂²×C₃₁	semidirect
304. C₂×C₃₁	#2364	direct (N × H)
315. C₂²×D₄	C₂×C₄×C₃₁	semidirect
316. C₂²×D₄	C₂×C₄×C₃₁	semidirect
326. C₂³×C₄	C₂×C₄×C₃₁	semidirect
337. #121	C₂×C₄×C₃₁	semidirect
340. C₂×C₄⋊C₄	C₂×C₄×C₃₁	semidirect
347. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
348. C₂×C₂²⋊C₄	D₄×C₃₁	semidirect
365. C₃₁	C₂×#2364	direct (N × H)
386. C₂×D₄	C₂²×C₄×C₃₁	semidirect
387. C₂×D₄	C₂²×C₄×C₃₁	semidirect
415. C₂²×C₄	C₂²×C₄×C₃₁	semidirect
426. C₄⋊C₄	C₂²×C₄×C₃₁	semidirect
431. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
432. C₂²⋊C₄	C₂×D₄×C₃₁	semidirect
529. C₂	C₃₁×#2364	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×#121	2
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2364	2
C₂×C₃₁×#388	2
C₂×C₃₁×#387	2
C₂×C₃₁×#386	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂²×C₃₁×#95	2
C₂×#2364	31

Subgroups — C2×C31×#2364 #21490896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2364 #21490896