Subgroups — C₂×C₃₁×#2862 #21493391

Order 7936.

2814 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃₁×#2862	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#391	C₂	1	8	yes	maximal
4	C₂×C₃₁×#382	C₂	1	4	yes	maximal
5	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₃₁	C₂²	1	4	yes	—
7	C₂×C₃₁×C₄⋊Q₈	—	2	16	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	16	yes	—
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
11	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#391	C₂²	1	64	yes	—
15	C₃₁×#382	C₂²	1	32	yes	—
16	C₂×C₃₁×#95	C₂²	1	16	yes	—
17	C₄³×C₃₁	C₂²	1	8	yes	—
18	C₂²×Q₈×C₃₁	—	2	16	no	—
19	C₂²×Q₈×C₃₁	—	2	32	no	—
20	C₂²×Q₈×C₃₁	C₂³	1	16	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
25	C₃₁×C₄⋊Q₈	—	2	64	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
29	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
30	C₂×C₄²×C₃₁	—	2	16	no	—
31	C₂×C₄²×C₃₁	D₄	1	8	yes	—
32	C₂×C₄²×C₃₁	Q₈	1	16	yes	complemented
33	C₂×C₄²×C₃₁	C₂³	1	16	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
36	C₃₁×#95	C₂³	1	64	yes	—
37	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
38	C₂×Q₈×C₃₁	—	2	32	no	—
39	C₂×Q₈×C₃₁	—	2	64	no	—
40	C₂×Q₈×C₃₁	—	4	64	no	—
41	C₂²×C₄×C₃₁	—	2	16	no	—
42	C₂²×C₄×C₃₁	—	2	16	no	—
43	C₂²×C₄×C₃₁	—	2	32	no	—
44	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
45	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₃₁	C₂×Q₈	1	16	yes	—
47	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	—
48	C₂²×C₄×C₃₁	#41	1	16	yes	—
49	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
50	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
51	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
53	C₄⋊C₄×C₃₁	—	2	32	no	—
54	C₄⋊C₄×C₃₁	—	2	32	no	—
55	C₄²×C₃₁	—	2	32	no	—
56	C₄²×C₃₁	C₂×D₄	1	16	yes	—
57	C₄²×C₃₁	C₂×Q₈	1	32	yes	complemented
58	C₂×#2862	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
60	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
61	C₂³×C₃₁	C₂×#41	1	8	yes	—
62	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
63	Q₈×C₃₁	—	4	64	no	—
64	C₂×C₄×C₃₁	—	2	32	no	—
65	C₂×C₄×C₃₁	—	2	16	no	—
66	C₂×C₄×C₃₁	—	2	16	no	—
67	C₂×C₄×C₃₁	—	2	32	no	—
68	C₂×C₄×C₃₁	—	4	32	no	—
69	C₂×C₄×C₃₁	#124	1	32	yes	—
70	C₂×C₄×C₃₁	C₄⋊Q₈	1	16	yes	—
71	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
72	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
73	C₂×C₄×C₃₁	C₂²×Q₈	1	16	yes	—
74	C₂×C₄×C₃₁	C₂²×Q₈	1	8	yes	—
75	C₂×C₄×C₃₁	C₂×#41	1	16	yes	—
76	C₂²×C₄⋊Q₈	C₂×C₃₁	1	2	yes	—
77	#2862	C₂×C₃₁	1	16	yes	direct product factor
78	C₂×#391	C₂×C₃₁	1	8	yes	—
79	C₂×#382	C₂×C₃₁	1	4	yes	—
80	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
81	C₂²×C₃₁	C₂×#124	1	8	yes	—
82	C₂²×C₃₁	C₂×C₄⋊Q₈	1	4	yes	—
83	C₂²×C₃₁	#533	1	16	yes	—
84	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
85	C₂²×C₃₁	C₂³×Q₈	1	2	yes	—
86	C₂²×C₃₁	C₂²×#41	1	4	yes	—
87	C₄×C₃₁	—	2	16	no	—
88	C₄×C₃₁	—	4	16	no	—
89	C₄×C₃₁	C₂×#124	1	16	yes	—
90	C₄×C₃₁	C₂×C₄⋊Q₈	1	8	yes	—
91	C₂³×Q₈	C₂²×C₃₁	1	4	yes	—
92	C₂×C₄⋊Q₈	—	2	16	no	—
93	C₂×C₄⋊Q₈	C₂²×C₃₁	1	16	yes	—
94	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
95	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
96	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
97	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
98	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
99	#391	C₂²×C₃₁	1	64	yes	—
100	#382	C₂²×C₃₁	1	32	yes	—
101	C₂×#95	C₂²×C₃₁	1	16	yes	—
102	C₄³	C₂²×C₃₁	1	8	yes	—
103	C₂×C₃₁	#2862	1	8	yes	direct product factor
104	C₂×C₃₁	C₂²×#124	1	2	yes	—
105	C₂×C₃₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₃₁	C₂×#533	1	4	yes	—
107	C₂²×Q₈	—	2	16	no	—
108	C₂²×Q₈	—	2	32	no	—
109	C₂²×Q₈	C₂³×C₃₁	1	16	yes	—
110	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
111	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
112	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
113	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
114	C₄⋊Q₈	—	2	64	no	—
115	C₂×C₄⋊C₄	—	2	16	no	—
116	C₂×C₄⋊C₄	—	2	16	no	—
117	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
118	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
119	C₂×C₄²	—	2	16	no	—
120	C₂×C₄²	D₄×C₃₁	1	8	yes	—
121	C₂×C₄²	Q₈×C₃₁	1	16	yes	complemented
122	C₂×C₄²	C₂³×C₃₁	1	16	yes	—
123	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
124	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
125	#95	C₂³×C₃₁	1	64	yes	—
126	C₃₁	C₂×#2862	1	1	yes	characteristic, direct product factor, Sylow
127	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
128	C₂×Q₈	—	2	32	no	—
129	C₂×Q₈	—	2	64	no	—
130	C₂×Q₈	—	4	64	no	—
131	C₂²×C₄	—	2	16	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	32	no	—
134	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
135	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
136	C₂²×C₄	C₂×Q₈×C₃₁	1	16	yes	—
137	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	—
138	C₂²×C₄	#41×C₃₁	1	16	yes	—
139	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
140	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
141	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
142	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
143	C₄⋊C₄	—	2	32	no	—
144	C₄⋊C₄	—	2	32	no	—
145	C₄²	—	2	32	no	—
146	C₄²	C₂×D₄×C₃₁	1	16	yes	—
147	C₄²	C₂×Q₈×C₃₁	1	32	yes	complemented
148	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
149	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
150	C₂³	C₂×#41×C₃₁	1	8	yes	—
151	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
152	Q₈	—	4	64	no	—
153	C₂×C₄	—	2	32	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	32	no	—
157	C₂×C₄	—	4	32	no	—
158	C₂×C₄	C₃₁×#124	1	32	yes	—
159	C₂×C₄	C₃₁×C₄⋊Q₈	1	16	yes	—
160	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
161	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
162	C₂×C₄	C₂²×Q₈×C₃₁	1	16	yes	—
163	C₂×C₄	C₂²×Q₈×C₃₁	1	8	yes	—
164	C₂×C₄	C₂×#41×C₃₁	1	16	yes	—
165	C₂²	C₂×C₃₁×#124	1	8	yes	—
166	C₂²	C₂×C₃₁×C₄⋊Q₈	1	4	yes	—
167	C₂²	C₃₁×#533	1	16	yes	—
168	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
169	C₂²	C₂³×Q₈×C₃₁	1	2	yes	—
170	C₂²	C₂²×#41×C₃₁	1	4	yes	—
171	C₄	—	2	16	no	—
172	C₄	—	4	16	no	—
173	C₄	C₂×C₃₁×#124	1	16	yes	—
174	C₄	C₂×C₃₁×C₄⋊Q₈	1	8	yes	—
175	C₂	C₃₁×#2862	1	8	yes	direct product factor
176	C₂	C₂²×C₃₁×#124	1	2	yes	—
177	C₂	C₂²×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
178	C₂	C₂×C₃₁×#533	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	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
2. C₃₁×#2862	C₂	direct (N × H)
32. C₂×C₄²×C₃₁	Q₈	semidirect
57. C₄²×C₃₁	C₂×Q₈	semidirect
58. C₂×#2862	C₃₁	direct (N × H)
77. #2862	C₂×C₃₁	direct (N × H)
103. C₂×C₃₁	#2862	direct (N × H)
121. C₂×C₄²	Q₈×C₃₁	semidirect
126. C₃₁	C₂×#2862	direct (N × H)
147. C₄²	C₂×Q₈×C₃₁	semidirect
175. C₂	C₃₁×#2862	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×C₄⋊Q₈	2
C₃₁×#2862	2
C₂×C₃₁×#391	2
C₂×C₃₁×#382	2
C₂×C₄³×C₃₁	2
C₂×#2862	31

Subgroups — C2×C31×#2862 #21493391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2862 #21493391