Group #16856050 — subgroups

Subgroups — C₂×#14521417 #16856050

Order 3968.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758447	C₂	1	1	yes	maximal, characteristic
2	#14521417	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758423	C₂	1	4	yes	maximal
4	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₃₁×C₄⋊Q₈	—	2	2	no	—
8	C₃₁×C₄⋊Q₈	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₂×Dic₃₁⋊C₄	—	2	2	no	—
13	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
14	#11758447	—	2	2	no	—
15	#11758447	C₂²	1	4	yes	—
16	#11758423	C₂²	1	16	yes	—
17	#11758391	C₂²	1	4	yes	—
18	C₂×Q₈×C₃₁	—	2	2	no	—
19	C₂×Q₈×C₃₁	—	2	8	no	—
20	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
21	C₂²×C₄×C₃₁	—	2	2	no	—
22	C₂²×C₄×C₃₁	D₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₁	—	2	2	no	—
25	C₄⋊C₄×C₃₁	—	2	8	no	—
26	C₄²×C₃₁	D₄	1	2	yes	—
27	C₄²×C₃₁	C₂³	1	2	yes	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₄⋊Dic₃₁	C₂³	1	4	yes	—
30	Dic₃₁⋊C₄	—	2	8	no	—
31	C₂×C₃₁⋊C₈	—	2	4	no	—
32	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
33	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₃₁	—	2	8	no	—
35	Q₈×C₃₁	—	4	4	no	—
36	C₂×C₄×C₃₁	—	2	4	no	—
37	C₂×C₄×C₃₁	—	2	4	no	—
38	C₂×C₄×C₃₁	—	4	4	no	—
39	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	4	yes	—
42	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	4	no	—
44	C₂×Dic₃₁	—	4	4	no	—
45	C₃₁⋊C₈	—	2	4	no	—
46	C₂×#491	—	31	1	no	maximal, Sylow
47	C₂²×C₃₁	#124	1	2	yes	—
48	C₂²×C₃₁	#137	1	4	yes	—
49	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
50	C₄×C₃₁	—	2	4	no	—
51	C₄×C₃₁	—	4	4	no	—
52	C₄×C₃₁	#124	1	2	yes	—
53	C₄×C₃₁	C₂×#41	1	2	yes	—
54	Dic₃₁	—	4	4	no	—
55	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
56	C₂×#125	—	31	1	no	—
57	#491	—	31	8	no	—
58	C₂×#103	—	31	4	no	—
59	C₂×#97	—	31	1	no	—
60	C₂×C₃₁	#491	1	4	yes	complemented
61	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
62	C₂×C₃₁	C₂×#137	1	2	yes	—
63	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
64	C₂²×C₈	—	31	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	C₂×D₃₁	1	4	yes	—
67	#125	—	31	4	no	—
68	#125	—	62	2	no	—
69	C₂×C₄⋊C₄	—	2	2	no	—
70	C₂×C₄⋊C₄	—	31	2	no	—
71	C₂×C₄⋊C₄	—	62	2	no	—
72	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
73	#103	—	31	16	no	—
74	#97	—	31	4	no	—
75	C₃₁	C₂×#491	1	1	yes	characteristic, complemented, Sylow
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	62	2	no	—
81	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
82	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
83	C₂×C₈	—	31	4	no	—
84	C₂×C₈	—	62	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	31	4	no	—
87	C₄⋊C₄	—	62	8	no	—
88	C₄⋊C₄	—	62	2	no	—
89	C₄²	C₂²⋊D₃₁	1	2	yes	—
90	C₄²	C₂²×D₃₁	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	62	4	no	—
98	C₂×C₄	—	124	4	no	—
99	C₂×C₄	#92497	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
102	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
103	C₈	—	62	4	no	—
104	C₂²	#11758513	1	2	yes	—
105	C₂²	#11758519	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	124	4	no	—
110	C₄	#11758513	1	2	yes	—
111	C₄	C₂×#92497	1	2	yes	—
112	C₂	#14521417	1	4	yes	direct product factor
113	C₂	C₂×#11758513	1	1	yes	characteristic
114	C₂	C₂×#11758519	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal, characteristic

#14521417

C₂

yes

maximal, direct product factor

C₂×#11758423

C₂

yes

maximal

C₂×#11758391

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758447

—

#11758447

C₂²

yes

—

#11758423

C₂²

yes

—

#11758391

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

C₂×#491

—

maximal, Sylow

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

#137

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

D₃₁

yes

—

—

—

C₂×#97

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

#97

—

C₃₁

C₂×#491

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

#92497

yes

—

100

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

102

C₂×C₄

C₂³×D₃₁

yes

characteristic

103

C₈

—

104

C₂²

#11758513

yes

—

105

C₂²

#11758519

yes

—

106

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

124

—

110

C₄

#11758513

yes

—

111

C₄

C₂×#92497

yes

—

112

C₂

#14521417

yes

direct product factor

113

C₂

C₂×#11758513

yes

characteristic

114

C₂

C₂×#11758519

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	C₂×#491
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₃₁

hypocenter

C₃₁

C₂×#491

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #14521417	C₂	direct (N × H)
60. C₂×C₃₁	#491	semidirect
75. C₃₁	C₂×#491	semidirect
112. C₂	#14521417	direct (N × H)

H ≅ G/N

type

2. #14521417

C₂

direct (N × H)

60. C₂×C₃₁

#491

semidirect

75. C₃₁

C₂×#491

semidirect

112. C₂

#14521417

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11758447	2
#14521417	2
C₂×#11758423	2
C₂×#11758391	2
C₂×C₃₁×C₄⋊Q₈	2
C₂×#491	31

subgroup

index

C₂×C₃₁×C₄⋊Q₈

Subgroups — C₂×#14521417 #16856050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521417 #16856050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521417 #16856050