Subgroups — C₂×C₃₁×#2856 #21493381

Order 7936.

5694 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#121	C₂	1	6	yes	maximal, complemented
3	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2856	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₃₁×#390	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
8	C₂³×D₄×C₃₁	C₂²	1	6	yes	—
9	C₂⁴×C₄×C₃₁	C₂²	1	3	yes	—
10	C₂×C₃₁×#127	—	2	8	no	—
11	C₂×C₃₁×#127	C₂²	1	8	yes	complemented
12	C₂×C₃₁×#121	—	2	48	no	—
13	C₂×C₃₁×#121	C₂²	1	48	yes	complemented
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
15	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	12	yes	—
16	C₂²×C₄²×C₃₁	—	2	8	no	—
17	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₃₁	C₂²	1	8	yes	complemented
20	C₃₁×#390	C₂²	1	48	yes	complemented
21	C₃₁×#384	C₂²	1	8	yes	complemented
22	C₂⁵×C₃₁	C₂³	1	4	yes	—
23	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×D₄×C₃₁	—	2	48	no	—
25	C₂²×D₄×C₃₁	—	2	24	no	—
26	C₂²×D₄×C₃₁	—	2	24	no	—
27	C₂²×D₄×C₃₁	—	2	24	no	—
28	C₂²×D₄×C₃₁	C₂³	1	16	yes	—
29	C₂²×D₄×C₃₁	C₂³	1	24	yes	—
30	C₂³×C₄×C₃₁	—	2	12	no	—
31	C₂³×C₄×C₃₁	—	2	12	no	—
32	C₂³×C₄×C₃₁	D₄	1	24	yes	complemented
33	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
37	C₃₁×#127	—	2	32	no	—
38	C₃₁×#121	—	2	192	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	48	no	—
42	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
43	C₂×C₄²×C₃₁	—	2	12	no	—
44	C₂×C₄²×C₃₁	—	2	32	no	—
45	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
46	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
48	C₂⁴×C₃₁	—	2	2	no	—
49	C₂⁴×C₃₁	—	2	6	no	—
50	C₂⁴×C₃₁	—	2	24	no	—
51	C₂⁴×C₃₁	—	2	8	no	—
52	C₂⁴×C₃₁	—	2	24	no	—
53	C₂⁴×C₃₁	C₂×D₄	1	12	yes	—
54	C₂⁴×C₃₁	C₂⁴	1	8	yes	—
55	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
56	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×D₄×C₃₁	—	2	96	no	—
58	C₂×D₄×C₃₁	—	2	48	no	—
59	C₂×D₄×C₃₁	—	2	48	no	—
60	C₂×D₄×C₃₁	—	2	48	no	—
61	C₂×D₄×C₃₁	—	4	64	no	—
62	C₂×D₄×C₃₁	—	4	96	no	—
63	C₂²×C₄×C₃₁	—	2	6	no	—
64	C₂²×C₄×C₃₁	—	2	6	no	—
65	C₂²×C₄×C₃₁	—	2	96	no	—
66	C₂²×C₄×C₃₁	—	2	24	no	—
67	C₂²×C₄×C₃₁	—	2	24	no	—
68	C₂²×C₄×C₃₁	—	2	24	no	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	48	yes	complemented
73	C₂²×C₄×C₃₁	#41	1	12	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
77	C₄⋊C₄×C₃₁	—	2	48	no	—
78	C₂²⋊C₄×C₃₁	—	2	96	no	—
79	C₄²×C₃₁	—	2	24	no	—
80	C₄²×C₃₁	C₂×D₄	1	16	yes	complemented
81	C₂×#2856	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	—	2	24	no	—
83	C₂³×C₃₁	—	2	24	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	6	no	—
86	C₂³×C₃₁	—	2	24	no	—
87	C₂³×C₃₁	—	2	8	no	—
88	C₂³×C₃₁	—	2	24	no	—
89	C₂³×C₃₁	—	4	48	no	—
90	C₂³×C₃₁	—	4	16	no	—
91	C₂³×C₃₁	—	4	48	no	—
92	C₂³×C₃₁	#127	1	16	yes	complemented
93	C₂³×C₃₁	C₂²×D₄	1	12	yes	—
94	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
95	C₂³×C₃₁	C₂²×D₄	1	6	yes	—
96	C₂³×C₃₁	C₂×#41	1	6	yes	—
97	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
98	D₄×C₃₁	—	4	64	no	—
99	D₄×C₃₁	—	4	96	no	—
100	C₂×C₄×C₃₁	—	2	24	no	—
101	C₂×C₄×C₃₁	—	2	6	no	—
102	C₂×C₄×C₃₁	—	2	6	no	—
103	C₂×C₄×C₃₁	—	2	96	no	—
104	C₂×C₄×C₃₁	—	2	24	no	—
105	C₂×C₄×C₃₁	—	4	16	no	—
106	C₂×C₄×C₃₁	#121	1	48	yes	complemented
107	C₂×C₄×C₃₁	C₂²×D₄	1	12	yes	—
108	C₂×C₄×C₃₁	C₂²×D₄	1	12	yes	—
109	C₂×C₄×C₃₁	C₂²×D₄	1	12	yes	—
110	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
111	C₂²×#127	C₂×C₃₁	1	1	yes	characteristic, complemented
112	C₂²×#121	C₂×C₃₁	1	6	yes	complemented
113	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
114	#2856	C₂×C₃₁	1	16	yes	direct product factor
115	C₂×#390	C₂×C₃₁	1	6	yes	complemented
116	C₂×#384	C₂×C₃₁	1	1	yes	characteristic, complemented
117	C₂²×C₃₁	—	2	32	no	—
118	C₂²×C₃₁	—	2	12	no	—
119	C₂²×C₃₁	—	2	12	no	—
120	C₂²×C₃₁	—	4	24	no	—
121	C₂²×C₃₁	—	4	8	no	—
122	C₂²×C₃₁	—	4	24	no	—
123	C₂²×C₃₁	—	8	32	no	—
124	C₂²×C₃₁	C₂×#121	1	12	yes	—
125	C₂²×C₃₁	C₂×#127	1	4	yes	—
126	C₂²×C₃₁	C₂×#127	1	8	yes	complemented
127	C₂²×C₃₁	#532	1	12	yes	—
128	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
129	C₂²×C₃₁	C₂³×D₄	1	3	yes	—
130	C₂²×C₃₁	C₂²×#41	1	3	yes	—
131	C₄×C₃₁	—	2	12	no	—
132	C₄×C₃₁	—	4	8	no	—
133	C₄×C₃₁	C₂×#121	1	24	yes	complemented
134	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
135	C₂³×D₄	C₂²×C₃₁	1	6	yes	—
136	C₂⁴×C₄	C₂²×C₃₁	1	3	yes	—
137	C₂×#127	—	2	8	no	—
138	C₂×#127	C₂²×C₃₁	1	8	yes	complemented
139	C₂×#121	—	2	48	no	—
140	C₂×#121	C₂²×C₃₁	1	48	yes	complemented
141	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
142	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	12	yes	—
143	C₂²×C₄²	—	2	8	no	—
144	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
145	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₃₁	1	8	yes	complemented
147	#390	C₂²×C₃₁	1	48	yes	complemented
148	#384	C₂²×C₃₁	1	8	yes	complemented
149	C₂×C₃₁	—	2	8	no	—
150	C₂×C₃₁	—	8	8	no	—
151	C₂×C₃₁	#2856	1	8	yes	direct product factor
152	C₂×C₃₁	C₂²×#121	1	3	yes	—
153	C₂×C₃₁	C₂²×#127	1	1	yes	characteristic
154	C₂×C₃₁	C₂×#532	1	3	yes	—
155	C₂⁵	C₂³×C₃₁	1	4	yes	—
156	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
157	C₂²×D₄	—	2	48	no	—
158	C₂²×D₄	—	2	24	no	—
159	C₂²×D₄	—	2	24	no	—
160	C₂²×D₄	—	2	24	no	—
161	C₂²×D₄	C₂³×C₃₁	1	16	yes	—
162	C₂²×D₄	C₂³×C₃₁	1	24	yes	—
163	C₂³×C₄	—	2	12	no	—
164	C₂³×C₄	—	2	12	no	—
165	C₂³×C₄	D₄×C₃₁	1	24	yes	complemented
166	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
167	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
168	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
169	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
170	#127	—	2	32	no	—
171	#121	—	2	192	no	—
172	C₂×C₄⋊C₄	—	2	24	no	—
173	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
174	C₂×C₂²⋊C₄	—	2	48	no	—
175	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
176	C₂×C₄²	—	2	12	no	—
177	C₂×C₄²	—	2	32	no	—
178	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
179	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
180	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
181	C₃₁	C₂×#2856	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	—	2	2	no	—
183	C₂⁴	—	2	6	no	—
184	C₂⁴	—	2	24	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	—	2	24	no	—
187	C₂⁴	C₂×D₄×C₃₁	1	12	yes	—
188	C₂⁴	C₂⁴×C₃₁	1	8	yes	—
189	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
190	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
191	C₂×D₄	—	2	96	no	—
192	C₂×D₄	—	2	48	no	—
193	C₂×D₄	—	2	48	no	—
194	C₂×D₄	—	2	48	no	—
195	C₂×D₄	—	4	64	no	—
196	C₂×D₄	—	4	96	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	2	96	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	2	24	no	—
202	C₂²×C₄	—	2	24	no	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
206	C₂²×C₄	C₂×D₄×C₃₁	1	48	yes	complemented
207	C₂²×C₄	#41×C₃₁	1	12	yes	—
208	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
209	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
210	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
211	C₄⋊C₄	—	2	48	no	—
212	C₂²⋊C₄	—	2	96	no	—
213	C₄²	—	2	24	no	—
214	C₄²	C₂×D₄×C₃₁	1	16	yes	complemented
215	C₂³	—	2	24	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	6	no	—
219	C₂³	—	2	24	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	2	24	no	—
222	C₂³	—	4	48	no	—
223	C₂³	—	4	16	no	—
224	C₂³	—	4	48	no	—
225	C₂³	C₃₁×#127	1	16	yes	complemented
226	C₂³	C₂²×D₄×C₃₁	1	12	yes	—
227	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
228	C₂³	C₂²×D₄×C₃₁	1	6	yes	—
229	C₂³	C₂×#41×C₃₁	1	6	yes	—
230	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
231	D₄	—	4	64	no	—
232	D₄	—	4	96	no	—
233	C₂×C₄	—	2	24	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	96	no	—
237	C₂×C₄	—	2	24	no	—
238	C₂×C₄	—	4	16	no	—
239	C₂×C₄	C₃₁×#121	1	48	yes	complemented
240	C₂×C₄	C₂²×D₄×C₃₁	1	12	yes	—
241	C₂×C₄	C₂²×D₄×C₃₁	1	12	yes	—
242	C₂×C₄	C₂²×D₄×C₃₁	1	12	yes	—
243	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
244	C₂²	—	2	32	no	—
245	C₂²	—	2	12	no	—
246	C₂²	—	2	12	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	4	8	no	—
249	C₂²	—	4	24	no	—
250	C₂²	—	8	32	no	—
251	C₂²	C₂×C₃₁×#121	1	12	yes	—
252	C₂²	C₂×C₃₁×#127	1	4	yes	—
253	C₂²	C₂×C₃₁×#127	1	8	yes	complemented
254	C₂²	C₃₁×#532	1	12	yes	—
255	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
256	C₂²	C₂³×D₄×C₃₁	1	3	yes	—
257	C₂²	C₂²×#41×C₃₁	1	3	yes	—
258	C₄	—	2	12	no	—
259	C₄	—	4	8	no	—
260	C₄	C₂×C₃₁×#121	1	24	yes	complemented
261	C₂	—	2	8	no	—
262	C₂	—	8	8	no	—
263	C₂	C₃₁×#2856	1	8	yes	direct product factor
264	C₂	C₂²×C₃₁×#121	1	3	yes	—
265	C₂	C₂²×C₃₁×#127	1	1	yes	characteristic
266	C₂	C₂×C₃₁×#532	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#127	C₂	semidirect
2. C₂²×C₃₁×#121	C₂	semidirect
3. C₂³×C₄²×C₃₁	C₂	semidirect
4. C₃₁×#2856	C₂	direct (N × H)
5. C₂×C₃₁×#390	C₂	semidirect
6. C₂×C₃₁×#384	C₂	semidirect
11. C₂×C₃₁×#127	C₂²	semidirect
13. C₂×C₃₁×#121	C₂²	semidirect
19. C₂²×C₄²×C₃₁	C₂²	semidirect
20. C₃₁×#390	C₂²	semidirect
21. C₃₁×#384	C₂²	semidirect
32. C₂³×C₄×C₃₁	D₄	semidirect
45. C₂×C₄²×C₃₁	D₄	semidirect
72. C₂²×C₄×C₃₁	C₂×D₄	semidirect
80. C₄²×C₃₁	C₂×D₄	semidirect
81. C₂×#2856	C₃₁	direct (N × H)
92. C₂³×C₃₁	#127	semidirect
106. C₂×C₄×C₃₁	#121	semidirect
111. C₂²×#127	C₂×C₃₁	semidirect
112. C₂²×#121	C₂×C₃₁	semidirect
113. C₂³×C₄²	C₂×C₃₁	semidirect
114. #2856	C₂×C₃₁	direct (N × H)
115. C₂×#390	C₂×C₃₁	semidirect
116. C₂×#384	C₂×C₃₁	semidirect
126. C₂²×C₃₁	C₂×#127	semidirect
133. C₄×C₃₁	C₂×#121	semidirect
138. C₂×#127	C₂²×C₃₁	semidirect
140. C₂×#121	C₂²×C₃₁	semidirect
146. C₂²×C₄²	C₂²×C₃₁	semidirect
147. #390	C₂²×C₃₁	semidirect
148. #384	C₂²×C₃₁	semidirect
151. C₂×C₃₁	#2856	direct (N × H)
165. C₂³×C₄	D₄×C₃₁	semidirect
178. C₂×C₄²	D₄×C₃₁	semidirect
181. C₃₁	C₂×#2856	direct (N × H)
206. C₂²×C₄	C₂×D₄×C₃₁	semidirect
214. C₄²	C₂×D₄×C₃₁	semidirect
225. C₂³	C₃₁×#127	semidirect
239. C₂×C₄	C₃₁×#121	semidirect
253. C₂²	C₂×C₃₁×#127	semidirect
260. C₄	C₂×C₃₁×#121	semidirect
263. C₂	C₃₁×#2856	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×#127	2
C₂²×C₃₁×#121	2
C₂³×C₄²×C₃₁	2
C₃₁×#2856	2
C₂×C₃₁×#390	2
C₂×C₃₁×#384	2
C₂×#2856	31

Subgroups — C2×C31×#2856 #21493381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2856 #21493381