Subgroups — C₂×C₃₁×#1276 #21464624

Order 7936.

974 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	2	yes	maximal
2	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1276	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄²×C₃₁	—	2	2	no	—
5	C₂²×C₄²×C₃₁	C₄	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#431	—	2	2	no	—
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	C₄	1	4	yes	complemented
10	C₃₁×#431	C₂²	1	4	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	complemented
12	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
13	C₂×C₄×C₈×C₃₁	—	2	2	no	—
14	C₄³×C₃₁	—	2	2	no	—
15	C₄³×C₃₁	C₄	1	2	yes	—
16	C₄³×C₃₁	C₂²	1	2	yes	—
17	C₂³×C₄×C₃₁	—	2	2	no	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	D₄	1	2	yes	—
20	C₂²×C₈×C₃₁	—	2	4	no	—
21	C₂×C₄²×C₃₁	—	2	16	no	—
22	C₂×C₄²×C₃₁	—	2	4	no	—
23	C₂×C₄²×C₃₁	—	2	2	no	—
24	C₂×C₄²×C₃₁	—	2	4	no	—
25	C₂×C₄²×C₃₁	—	2	4	no	—
26	C₂×C₄²×C₃₁	C₈	1	8	yes	complemented
27	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
39	C₃₁×C₂²⋊C₈	—	2	4	no	—
40	C₃₁×C₂²⋊C₈	—	2	4	no	—
41	C₄×C₈×C₃₁	—	2	4	no	—
42	C₄×C₈×C₃₁	—	2	4	no	—
43	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	—	2	16	no	—
45	C₂²×C₄×C₃₁	—	2	8	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	—	2	8	no	—
48	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₃₁	—	2	8	no	—
63	C₂×C₈×C₃₁	—	4	8	no	—
64	C₄²×C₃₁	—	2	16	no	—
65	C₄²×C₃₁	—	2	16	no	—
66	C₄²×C₃₁	—	2	8	no	—
67	C₄²×C₃₁	—	2	4	no	—
68	C₄²×C₃₁	C₄²	1	2	yes	—
69	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
73	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
75	C₄²×C₃₁	C₂×C₈	1	8	yes	complemented
76	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
77	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
79	C₂×#1276	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	2	4	no	—
81	C₂³×C₃₁	#95	1	1	yes	characteristic
82	C₂³×C₃₁	#95	1	1	yes	characteristic
83	C₂³×C₃₁	C₂²⋊C₈	1	4	yes	—
84	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	—	2	16	no	—
86	C₂×C₄×C₃₁	—	2	32	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	8	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	#95	1	2	yes	—
94	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
96	C₂×C₄×C₃₁	#95	1	2	yes	—
97	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₃₁	#97	1	2	yes	—
100	C₂×C₄×C₃₁	#97	1	2	yes	—
101	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
102	C₂×C₄×C₃₁	#104	1	8	yes	—
103	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
104	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
105	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
111	C₈×C₃₁	—	4	8	no	—
112	C₂×#431	C₂×C₃₁	1	2	yes	—
113	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
114	#1276	C₂×C₃₁	1	4	yes	direct product factor
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	8	no	—
118	C₂²×C₃₁	#336	1	2	yes	—
119	C₂²×C₃₁	#336	1	2	yes	—
120	C₂²×C₃₁	#339	1	4	yes	—
121	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
122	C₂²×C₃₁	C₂×C₂²⋊C₈	1	2	yes	—
123	C₄×C₃₁	—	2	16	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	#336	1	2	yes	—
127	C₄×C₃₁	#336	1	2	yes	—
128	C₄×C₃₁	#339	1	4	yes	—
129	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
130	C₄×C₃₁	C₂×#97	1	1	yes	characteristic
131	C₄×C₃₁	C₂×#104	1	4	yes	—
132	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
133	C₂²×C₄²	—	2	2	no	—
134	C₂²×C₄²	C₄×C₃₁	1	2	yes	—
135	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
136	#431	—	2	2	no	—
137	#431	—	2	2	no	—
138	#431	C₄×C₃₁	1	4	yes	complemented
139	#431	C₂²×C₃₁	1	4	yes	—
140	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	complemented
141	C₂×C₂²⋊C₈	—	2	2	no	—
142	C₂×C₄×C₈	—	2	2	no	—
143	C₄³	—	2	2	no	—
144	C₄³	C₄×C₃₁	1	2	yes	—
145	C₄³	C₂²×C₃₁	1	2	yes	—
146	C₂×C₃₁	—	2	2	no	—
147	C₂×C₃₁	—	2	2	no	—
148	C₂×C₃₁	#1276	1	4	yes	direct product factor
149	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
150	C₂×C₃₁	C₂×#339	1	2	yes	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₃₁	1	2	yes	—
154	C₂²×C₈	—	2	4	no	—
155	C₂×C₄²	—	2	16	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	4	no	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	—	2	4	no	—
160	C₂×C₄²	C₈×C₃₁	1	8	yes	complemented
161	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
162	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
165	C₂×C₄²	D₄×C₃₁	1	2	yes	—
166	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
173	C₂²⋊C₈	—	2	4	no	—
174	C₂²⋊C₈	—	2	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	C₃₁	C₂×#1276	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
186	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
187	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
189	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
190	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
191	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
194	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	8	no	—
199	C₄²	—	2	16	no	—
200	C₄²	—	2	16	no	—
201	C₄²	—	2	8	no	—
202	C₄²	—	2	4	no	—
203	C₄²	C₄²×C₃₁	1	2	yes	—
204	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
205	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
206	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
207	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
208	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
209	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
210	C₄²	C₂×C₈×C₃₁	1	8	yes	complemented
211	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
212	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
214	C₂³	—	2	4	no	—
215	C₂³	C₃₁×#95	1	1	yes	characteristic
216	C₂³	C₃₁×#95	1	1	yes	characteristic
217	C₂³	C₃₁×C₂²⋊C₈	1	4	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	2	32	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	C₃₁×#95	1	2	yes	—
228	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
229	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
230	C₂×C₄	C₃₁×#95	1	2	yes	—
231	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
232	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
233	C₂×C₄	C₃₁×#97	1	2	yes	—
234	C₂×C₄	C₃₁×#97	1	2	yes	—
235	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
236	C₂×C₄	C₃₁×#104	1	8	yes	—
237	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
238	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
239	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
244	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
245	C₈	—	4	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	8	no	—
249	C₂²	C₃₁×#336	1	2	yes	—
250	C₂²	C₃₁×#336	1	2	yes	—
251	C₂²	C₃₁×#339	1	4	yes	—
252	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
253	C₂²	C₂×C₃₁×C₂²⋊C₈	1	2	yes	—
254	C₄	—	2	16	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	C₃₁×#336	1	2	yes	—
258	C₄	C₃₁×#336	1	2	yes	—
259	C₄	C₃₁×#339	1	4	yes	—
260	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
261	C₄	C₂×C₃₁×#97	1	1	yes	characteristic
262	C₄	C₂×C₃₁×#104	1	4	yes	—
263	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
264	C₂	—	2	2	no	—
265	C₂	—	2	2	no	—
266	C₂	C₃₁×#1276	1	4	yes	direct product factor
267	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
268	C₂	C₂×C₃₁×#339	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₁	D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	#95
prime radical	C₂⁴×C₃₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1276	C₂	direct (N × H)
9. C₃₁×#431	C₄	semidirect
11. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
26. C₂×C₄²×C₃₁	C₈	semidirect
38. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
75. C₄²×C₃₁	C₂×C₈	semidirect
79. C₂×#1276	C₃₁	direct (N × H)
114. #1276	C₂×C₃₁	direct (N × H)
138. #431	C₄×C₃₁	semidirect
140. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
148. C₂×C₃₁	#1276	direct (N × H)
160. C₂×C₄²	C₈×C₃₁	semidirect
172. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
177. C₃₁	C₂×#1276	direct (N × H)
210. C₄²	C₂×C₈×C₃₁	semidirect
266. C₂	C₃₁×#1276	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×#431	2
C₂×C₄³×C₃₁	2
C₃₁×#1276	2
C₂×#1276	31

Subgroups — C2×C31×#1276 #21464624

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1276 #21464624