Subgroups — C₂×C₃₁×#2091 #21480481

Order 7936.

1102 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#430	C₂	1	3	yes	maximal, complemented
2	C₂×C₃₁×#407	C₂	1	3	yes	maximal
3	C₃₁×#2091	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	3	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	3	yes	—
8	C₃₁×#430	—	2	6	no	—
9	C₃₁×#430	C₂²	1	12	yes	complemented
10	C₂×C₃₁×#108	—	2	6	no	—
11	C₃₁×#407	—	2	6	no	—
12	C₃₁×#407	C₂²	1	12	yes	—
13	C₂×C₃₁×C₂²⋊C₈	—	2	6	no	—
14	C₃₁×#338	C₂²	1	4	yes	complemented
15	C₂⁵×C₃₁	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₃₁	—	2	3	no	—
17	C₂³×C₄×C₃₁	—	2	3	no	—
18	C₂³×C₄×C₃₁	—	2	3	no	—
19	C₂³×C₄×C₃₁	—	2	3	no	—
20	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	6	no	—
25	C₂×M₁₆×C₃₁	—	2	6	no	—
26	C₂×M₁₆×C₃₁	—	2	3	no	—
27	C₂×M₁₆×C₃₁	—	2	6	no	—
28	C₂×M₁₆×C₃₁	—	2	6	no	—
29	C₂×M₁₆×C₃₁	D₄	1	6	yes	—
30	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
31	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
32	C₂²×C₈×C₃₁	—	2	3	no	—
33	C₂²×C₈×C₃₁	D₄	1	6	yes	—
34	C₃₁×#108	—	2	24	no	—
35	C₃₁×C₂²⋊C₈	—	2	24	no	—
36	C₂⁴×C₃₁	—	2	3	no	—
37	C₂⁴×C₃₁	—	2	3	no	—
38	C₂⁴×C₃₁	—	4	4	no	—
39	C₂⁴×C₃₁	C₂×Q₈	1	2	yes	—
40	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	—	2	3	no	—
42	C₂²×C₄×C₃₁	—	2	3	no	—
43	C₂²×C₄×C₃₁	—	2	6	no	—
44	C₂²×C₄×C₃₁	—	2	6	no	—
45	C₂²×C₄×C₃₁	—	2	6	no	—
46	C₂²×C₄×C₃₁	—	2	6	no	—
47	C₂²×C₄×C₃₁	—	4	3	no	—
48	C₂²×C₄×C₃₁	—	4	3	no	—
49	C₂²×C₄×C₃₁	—	4	6	no	—
50	C₂²×C₄×C₃₁	—	4	3	no	—
51	C₂²×C₄×C₃₁	—	4	1	no	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	#41	1	3	yes	—
57	C₂²×C₄×C₃₁	#41	1	3	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₃₁	—	2	12	no	—
60	M₁₆×C₃₁	—	2	12	no	—
61	M₁₆×C₃₁	—	2	12	no	—
62	M₁₆×C₃₁	—	4	6	no	—
63	C₂×C₈×C₃₁	—	2	6	no	—
64	C₂×C₈×C₃₁	—	2	6	no	—
65	C₂×C₈×C₃₁	—	2	6	no	—
66	C₂×C₈×C₃₁	—	4	3	no	—
67	C₂×C₈×C₃₁	—	4	3	no	—
68	C₂×C₈×C₃₁	C₂×D₄	1	12	yes	—
69	C₂×#2091	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	—	2	12	no	—
71	C₂³×C₃₁	—	2	6	no	—
72	C₂³×C₃₁	—	4	2	no	—
73	C₂³×C₃₁	—	4	4	no	—
74	C₂³×C₃₁	—	4	6	no	—
75	C₂³×C₃₁	—	4	12	no	—
76	C₂³×C₃₁	—	4	3	no	—
77	C₂³×C₃₁	—	4	1	no	—
78	C₂³×C₃₁	C₂²⋊Q₈	1	3	yes	—
79	C₂³×C₃₁	C₂²⋊Q₈	1	3	yes	—
80	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₃₁	—	2	6	no	—
82	C₂×C₄×C₃₁	—	4	3	no	—
83	C₂×C₄×C₃₁	—	4	1	no	—
84	C₂×C₄×C₃₁	—	4	6	no	—
85	C₂×C₄×C₃₁	—	4	2	no	—
86	C₂×C₄×C₃₁	—	4	6	no	—
87	C₂×C₄×C₃₁	—	4	2	no	—
88	C₂×C₄×C₃₁	C₂²⋊Q₈	1	3	yes	—
89	C₂×C₄×C₃₁	C₂²⋊Q₈	1	3	yes	—
90	C₂×C₄×C₃₁	#123	1	3	yes	—
91	C₂×C₄×C₃₁	#123	1	3	yes	—
92	C₂×C₄×C₃₁	#123	1	6	yes	—
93	C₂×C₄×C₃₁	#127	1	3	yes	—
94	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
97	C₈×C₃₁	—	2	12	no	—
98	C₈×C₃₁	—	4	6	no	—
99	C₂×#430	C₂×C₃₁	1	3	yes	complemented
100	C₂×#407	C₂×C₃₁	1	3	yes	—
101	#2091	C₂×C₃₁	1	8	yes	direct product factor
102	C₂×#338	C₂×C₃₁	1	1	yes	characteristic, complemented
103	C₂²×C₃₁	—	2	6	no	—
104	C₂²×C₃₁	—	2	12	no	—
105	C₂²×C₃₁	—	4	3	no	—
106	C₂²×C₃₁	—	4	1	no	—
107	C₂²×C₃₁	—	4	6	no	—
108	C₂²×C₃₁	—	4	12	no	—
109	C₂²×C₃₁	—	4	4	no	—
110	C₂²×C₃₁	—	4	2	no	—
111	C₂²×C₃₁	#399	1	3	yes	—
112	C₂²×C₃₁	#399	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	3	yes	—
114	C₄×C₃₁	—	4	2	no	—
115	C₄×C₃₁	#399	1	3	yes	—
116	C₄×C₃₁	#399	1	1	yes	characteristic
117	C₄×C₃₁	C₂×#123	1	3	yes	—
118	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
119	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
120	C₂²×M₁₆	C₂²×C₃₁	1	3	yes	—
121	C₂²×M₁₆	C₂²×C₃₁	1	3	yes	—
122	#430	—	2	6	no	—
123	#430	C₂²×C₃₁	1	12	yes	complemented
124	C₂×#108	—	2	6	no	—
125	#407	—	2	6	no	—
126	#407	C₂²×C₃₁	1	12	yes	—
127	C₂×C₂²⋊C₈	—	2	6	no	—
128	#338	C₂²×C₃₁	1	4	yes	complemented
129	C₂×C₃₁	—	2	3	no	—
130	C₂×C₃₁	—	2	3	no	—
131	C₂×C₃₁	—	4	4	no	—
132	C₂×C₃₁	#2091	1	2	yes	direct product factor
133	C₂×C₃₁	C₂×#399	1	1	yes	characteristic
134	C₂⁵	Q₈×C₃₁	1	1	yes	characteristic
135	C₂³×C₄	—	2	3	no	—
136	C₂³×C₄	—	2	3	no	—
137	C₂³×C₄	—	2	3	no	—
138	C₂³×C₄	—	2	3	no	—
139	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
140	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
141	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
142	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
143	C₂×M₁₆	—	2	6	no	—
144	C₂×M₁₆	—	2	6	no	—
145	C₂×M₁₆	—	2	3	no	—
146	C₂×M₁₆	—	2	6	no	—
147	C₂×M₁₆	—	2	6	no	—
148	C₂×M₁₆	D₄×C₃₁	1	6	yes	—
149	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
150	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
151	C₂²×C₈	—	2	3	no	—
152	C₂²×C₈	D₄×C₃₁	1	6	yes	—
153	#108	—	2	24	no	—
154	C₂²⋊C₈	—	2	24	no	—
155	C₃₁	C₂×#2091	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	—	2	3	no	—
157	C₂⁴	—	2	3	no	—
158	C₂⁴	—	4	4	no	—
159	C₂⁴	C₂×Q₈×C₃₁	1	2	yes	—
160	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	4	3	no	—
168	C₂²×C₄	—	4	3	no	—
169	C₂²×C₄	—	4	6	no	—
170	C₂²×C₄	—	4	3	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
173	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
174	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
176	C₂²×C₄	#41×C₃₁	1	3	yes	—
177	C₂²×C₄	#41×C₃₁	1	3	yes	—
178	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
179	M₁₆	—	2	12	no	—
180	M₁₆	—	2	12	no	—
181	M₁₆	—	2	12	no	—
182	M₁₆	—	4	6	no	—
183	C₂×C₈	—	2	6	no	—
184	C₂×C₈	—	2	6	no	—
185	C₂×C₈	—	2	6	no	—
186	C₂×C₈	—	4	3	no	—
187	C₂×C₈	—	4	3	no	—
188	C₂×C₈	C₂×D₄×C₃₁	1	12	yes	—
189	C₂³	—	2	12	no	—
190	C₂³	—	2	6	no	—
191	C₂³	—	4	2	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	4	6	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	4	3	no	—
196	C₂³	—	4	1	no	—
197	C₂³	C₃₁×C₂²⋊Q₈	1	3	yes	—
198	C₂³	C₃₁×C₂²⋊Q₈	1	3	yes	—
199	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	6	no	—
205	C₂×C₄	—	4	6	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₃₁×C₂²⋊Q₈	1	3	yes	—
208	C₂×C₄	C₃₁×C₂²⋊Q₈	1	3	yes	—
209	C₂×C₄	C₃₁×#123	1	3	yes	—
210	C₂×C₄	C₃₁×#123	1	3	yes	—
211	C₂×C₄	C₃₁×#123	1	6	yes	—
212	C₂×C₄	C₃₁×#127	1	3	yes	—
213	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
215	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
216	C₈	—	2	12	no	—
217	C₈	—	4	6	no	—
218	C₂²	—	2	6	no	—
219	C₂²	—	2	12	no	—
220	C₂²	—	4	3	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	6	no	—
223	C₂²	—	4	12	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₃₁×#399	1	3	yes	—
227	C₂²	C₃₁×#399	1	1	yes	characteristic
228	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	3	yes	—
229	C₄	—	4	2	no	—
230	C₄	C₃₁×#399	1	1	yes	characteristic
231	C₄	C₃₁×#399	1	3	yes	—
232	C₄	C₂×C₃₁×#123	1	3	yes	—
233	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
234	C₂	—	2	3	no	—
235	C₂	—	2	3	no	—
236	C₂	—	4	4	no	—
237	C₂	C₃₁×#2091	1	2	yes	direct product factor
238	C₂	C₂×C₃₁×#399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	#399
prime radical	C₂⁵×C₃₁	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#430	C₂	semidirect
3. C₃₁×#2091	C₂	direct (N × H)
4. C₂×C₃₁×#338	C₂	semidirect
9. C₃₁×#430	C₂²	semidirect
14. C₃₁×#338	C₂²	semidirect
69. C₂×#2091	C₃₁	direct (N × H)
99. C₂×#430	C₂×C₃₁	semidirect
101. #2091	C₂×C₃₁	direct (N × H)
102. C₂×#338	C₂×C₃₁	semidirect
123. #430	C₂²×C₃₁	semidirect
128. #338	C₂²×C₃₁	semidirect
132. C₂×C₃₁	#2091	direct (N × H)
155. C₃₁	C₂×#2091	direct (N × H)
237. C₂	C₃₁×#2091	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×#430	2
C₂×C₃₁×#407	2
C₃₁×#2091	2
C₂×C₃₁×#338	2
C₂×#2091	31

Subgroups — C2×C31×#2091 #21480481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2091 #21480481