Group #34110475 — subgroups

Subgroups — C₂×#23757641 #34110475

Order 20096.

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×#19097763	C₂	1	1	yes	maximal, characteristic
2	#23757641	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19097695	C₂	1	4	yes	maximal
4	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929434	C₂²	1	2	yes	—
7	#19097763	—	2	2	no	—
8	#19097763	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
10	C₂×#14578933	—	2	2	no	—
11	#19097695	C₂²	1	16	yes	—
12	#19097682	C₂²	1	4	yes	—
13	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
15	C₄⋊Q₈×C₁₅₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
17	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₅₇	—	2	2	no	—
19	C₂×#11929434	—	2	4	no	—
20	C₂×#11929434	C₂³	1	4	yes	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
23	#14578933	—	2	8	no	—
24	C₂×Q₈×C₁₅₇	—	2	2	no	—
25	C₂×Q₈×C₁₅₇	—	2	8	no	—
26	C₂×Q₈×C₁₅₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₅₇	—	2	2	no	—
28	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
29	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₅₇	—	2	8	no	—
31	C₄²×C₁₅₇	D₄	1	2	yes	—
32	C₄²×C₁₅₇	C₂³	1	2	yes	—
33	C₂×Dic₁₅₇	—	2	4	no	—
34	C₂×Dic₁₅₇	—	4	4	no	—
35	#11929434	—	2	4	no	—
36	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₅₇	—	2	8	no	—
38	Q₈×C₁₅₇	—	4	4	no	—
39	C₂×C₄×C₁₅₇	—	2	4	no	—
40	C₂×C₄×C₁₅₇	—	2	4	no	—
41	C₂×C₄×C₁₅₇	—	4	4	no	—
42	C₂×C₄×C₁₅₇	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	#41	1	4	yes	—
45	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
46	C₂²×C₁₅₇	#124	1	2	yes	—
47	C₂²×C₁₅₇	#137	1	4	yes	—
48	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₅₇	—	2	4	no	—
50	C₄×C₁₅₇	—	4	4	no	—
51	C₄×C₁₅₇	#124	1	2	yes	—
52	C₄×C₁₅₇	C₂×#41	1	2	yes	—
53	Dic₁₅₇	—	4	4	no	—
54	C₂×C₁₅₇	#491	1	4	yes	complemented
55	C₂×C₁₅₇	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₅₇	C₂×#137	1	2	yes	—
57	C₁₅₇	C₂×#491	1	1	yes	characteristic, complemented, Sylow
58	C₂×#491	—	157	1	no	maximal, Sylow
59	C₂×C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
60	C₂×#125	—	157	1	no	—
61	#491	—	157	8	no	—
62	C₂×#103	—	157	4	no	—
63	C₂×#97	—	157	1	no	—
64	C₂²×Q₈	C₂×D₁₅₇	1	2	yes	—
65	C₂²×C₈	—	157	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₅₇	1	4	yes	—
68	#125	—	314	2	no	—
69	#125	—	157	4	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	314	2	no	—
72	C₂×C₄⋊C₄	—	157	2	no	—
73	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
74	#103	—	157	16	no	—
75	#97	—	157	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂²×D₁₅₇	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	314	2	no	—
81	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
83	C₂×C₈	—	314	4	no	—
84	C₂×C₈	—	157	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	314	8	no	—
87	C₄⋊C₄	—	314	2	no	—
88	C₄⋊C₄	—	157	4	no	—
89	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
90	C₄²	C₂²×D₁₅₇	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₅₇	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	—	314	4	no	—
98	C₂×C₄	—	628	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
101	C₂×C₄	#14578953	1	4	yes	—
102	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
103	C₈	—	314	4	no	—
104	C₂²	#19097758	1	2	yes	—
105	C₂²	#19097802	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	628	4	no	—
110	C₄	#19097758	1	2	yes	—
111	C₄	C₂×#14578953	1	2	yes	—
112	C₂	#23757641	1	4	yes	direct product factor
113	C₂	C₂×#19097758	1	1	yes	characteristic
114	C₂	C₂×#19097802	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097763

C₂

yes

maximal, characteristic

#23757641

C₂

yes

maximal, direct product factor

C₂×#19097695

C₂

yes

maximal

C₂×#19097682

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

#19097763

—

#19097763

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×#14578933

—

#19097695

C₂²

yes

—

#19097682

C₂²

yes

—

C₂²×Q₈×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

—

C₂×#11929434

—

C₂×#11929434

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

#14578933

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₂²×C₁₅₇

#124

yes

—

C₂²×C₁₅₇

#137

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#124

yes

—

C₄×C₁₅₇

C₂×#41

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

—

C₁₅₇

C₂×#491

yes

characteristic, complemented, Sylow

C₂×#491

—

157

maximal, Sylow

C₂×C₄⋊Q₈

D₁₅₇

yes

characteristic

C₂×#125

—

157

—

#491

—

157

—

C₂×#103

—

157

—

C₂×#97

—

157

—

C₂²×Q₈

C₂×D₁₅₇

yes

—

C₂²×C₈

—

157

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₅₇

yes

—

#125

—

314

—

#125

—

157

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#103

—

157

—

#97

—

157

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₅₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₄²

C₂²×D₁₅₇

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

628

—

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

101

C₂×C₄

#14578953

yes

—

102

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

103

C₈

—

314

—

104

C₂²

#19097758

yes

—

105

C₂²

#19097802

yes

—

106

C₂²

C₂²×C₂²⋊D₁₅₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

628

—

110

C₄

#19097758

yes

—

111

C₄

C₂×#14578953

yes

—

112

C₂

#23757641

yes

direct product factor

113

C₂

C₂×#19097758

yes

characteristic

114

C₂

C₂×#19097802

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×C₄⋊Q₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	C₂×#491
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×C₄⋊Q₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₅₇

hypocenter

C₁₅₇

C₂×#491

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #23757641	C₂	direct (N × H)
54. C₂×C₁₅₇	#491	semidirect
57. C₁₅₇	C₂×#491	semidirect
112. C₂	#23757641	direct (N × H)

H ≅ G/N

type

2. #23757641

C₂

direct (N × H)

54. C₂×C₁₅₇

#491

semidirect

57. C₁₅₇

C₂×#491

semidirect

112. C₂

#23757641

direct (N × H)

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

subgroup	index
C₂×#19097763	2
#23757641	2
C₂×#19097695	2
C₂×#19097682	2
C₂×C₄⋊Q₈×C₁₅₇	2
C₂×#491	157

subgroup

index

C₂×C₄⋊Q₈×C₁₅₇

157

Subgroups — C₂×#23757641 #34110475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23757641 #34110475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757641 #34110475