Group #34101112 — subgroups

Subgroups — #34101112

Order 20096.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#23757830	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
3	#23757034	C₂	1	4	yes	maximal
4	#423×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	#19097771	C₄	1	2	yes	complemented
6	#19097763	C₄	1	2	yes	complemented
7	#19097750	—	2	2	no	—
8	C₄×Dic₃₁₄	—	2	2	no	—
9	#19097730	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₅₇	—	2	2	no	—
11	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₅₇	—	2	2	no	—
15	C₂×Dic₃₁₄	—	2	2	no	—
16	C₂²×Dic₁₅₇	—	2	2	no	—
17	C₄⋊Dic₁₅₇	—	2	2	no	—
18	C₄⋊Dic₁₅₇	C₂×C₄	1	4	yes	—
19	#14578933	—	2	4	no	—
20	C₂²⋊Dic₁₅₇	—	2	2	no	—
21	C₄×Dic₁₅₇	—	2	8	no	—
22	C₄×Dic₁₅₇	—	2	2	no	—
23	C₄×Dic₁₅₇	—	2	2	no	—
24	C₄×Dic₁₅₇	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₅₇	—	2	4	no	—
28	C₂×C₈×C₁₅₇	—	2	2	no	—
29	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₅₇	D₄	1	2	yes	—
32	Dic₃₁₄	—	4	4	no	—
33	C₂×Dic₁₅₇	—	2	8	no	—
34	C₂×Dic₁₅₇	—	2	4	no	—
35	C₂×Dic₁₅₇	—	2	4	no	—
36	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	—	2	4	no	—
38	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₅₇	#41	1	2	yes	—
44	C₈×C₁₅₇	—	4	2	no	—
45	C₂²×C₁₅₇	—	2	2	no	—
46	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₅₇	—	2	2	no	—
49	C₄×C₁₅₇	#104	1	4	yes	—
50	C₄×C₁₅₇	C₄×Q₈	1	2	yes	—
51	C₄×C₁₅₇	#124	1	1	yes	characteristic
52	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₅₇	—	2	8	no	—
54	Dic₁₅₇	—	4	4	no	—
55	C₂×C₁₅₇	—	2	2	no	—
56	C₂×C₁₅₇	#391	1	1	yes	characteristic
57	C₂×C₁₅₇	C₂×#104	1	2	yes	—
58	C₁₅₇	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	157	1	no	maximal, Sylow
60	#533	—	157	1	no	—
61	#423	D₁₅₇	1	1	yes	characteristic
62	C₄³	—	157	1	no	—
63	#339	—	157	4	no	—
64	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
65	C₄⋊Q₈	—	157	2	no	—
66	#125	—	157	2	no	—
67	C₂²⋊Q₈	—	314	2	no	—
68	C₄×Q₈	—	314	2	no	—
69	#117	—	157	2	no	—
70	C₂×C₄²	—	314	2	no	—
71	C₂×C₄²	—	157	2	no	—
72	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	314	2	no	—
75	C₂²×C₄	—	314	2	no	—
76	C₂²×C₄	D₆₂₈	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	314	4	no	—
81	C₄⋊C₄	—	314	2	no	—
82	C₄⋊C₄	—	157	4	no	—
83	C₂²⋊C₄	—	314	2	no	—
84	C₄²	—	314	8	no	—
85	C₄²	—	314	2	no	—
86	C₄²	—	314	2	no	—
87	C₄²	—	157	4	no	—
88	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
89	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
91	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
92	Q₈	—	628	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	314	8	no	—
95	C₂×C₄	—	314	4	no	—
96	C₂×C₄	—	314	4	no	—
97	C₂×C₄	#14578931	1	2	yes	—
98	C₂×C₄	#14578931	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
102	C₂×C₄	#14578953	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#14578931	1	1	yes	characteristic
106	C₂²	#19097751	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	314	8	no	—
109	C₄	—	628	4	no	—
110	C₄	#19097697	1	4	yes	—
111	C₄	#19097736	1	2	yes	—
112	C₄	#19097758	1	1	yes	characteristic
113	C₄	#19097769	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#23757178	1	1	yes	characteristic
116	C₂	C₂×#19097697	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23757830

C₂

yes

maximal, characteristic

C₄²×Dic₁₅₇

C₂

yes

maximal, characteristic

#23757034

C₂

yes

maximal

#423×C₁₅₇

C₂

yes

maximal, characteristic

#19097771

C₄

yes

complemented

#19097763

C₄

yes

complemented

#19097750

—

C₄×Dic₃₁₄

—

#19097730

C₂²

yes

—

C₂×C₄×Dic₁₅₇

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₂×M₁₆×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂×C₄

yes

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

Q₈

yes

complemented

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

M₁₆×C₁₅₇

—

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₅₇

C₂²⋊Q₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#104

yes

—

C₄×C₁₅₇

C₄×Q₈

yes

—

C₄×C₁₅₇

#124

yes

characteristic

C₄×C₁₅₇

C₄⋊Q₈

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#391

yes

characteristic

C₂×C₁₅₇

C₂×#104

yes

—

C₁₅₇

#1994

yes

characteristic, complemented, Sylow

#1994

—

157

maximal, Sylow

#533

—

157

—

#423

D₁₅₇

yes

characteristic

C₄³

—

157

—

#339

—

157

—

C₂×M₁₆

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

157

—

#125

—

157

—

C₂²⋊Q₈

—

314

—

C₄×Q₈

—

314

—

#117

—

157

—

C₂×C₄²

—

314

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

314

—

C₂²×C₄

—

314

—

C₂²×C₄

D₆₂₈

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

314

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

314

—

C₄²

—

157

—

C₄²

C₄×D₁₅₇

yes

characteristic

C₄²

C₄×D₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

—

C₂³

C₂×D₆₂₈

yes

characteristic

Q₈

—

628

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

#14578931

yes

—

C₂×C₄

#14578931

yes

—

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

101

C₂×C₄

Q₈×D₁₅₇

yes

—

102

C₂×C₄

#14578953

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×#14578931

yes

characteristic

106

C₂²

#19097751

yes

—

107

C₄

—

108

C₄

—

314

—

109

C₄

—

628

—

110

C₄

#19097697

yes

—

111

C₄

#19097736

yes

—

112

C₄

#19097758

yes

characteristic

113

C₄

#19097769

yes

characteristic

114

C₂

—

115

C₂

#23757178

yes

characteristic

116

C₂

C₂×#19097697

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#423×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#423	D₁₅₇
hypocenter	C₁₅₇	#1994
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#423×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #19097771	C₄	semidirect
6. #19097763	C₄	semidirect
24. C₄×Dic₁₅₇	Q₈	semidirect
58. C₁₅₇	#1994	semidirect

H ≅ G/N

type

5. #19097771

C₄

semidirect

6. #19097763

C₄

semidirect

24. C₄×Dic₁₅₇

Q₈

semidirect

58. C₁₅₇

#1994

semidirect

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

subgroup	index
#23757830	2
C₄²×Dic₁₅₇	2
#23757034	2
#423×C₁₅₇	2
#1994	157

subgroup

index

#23757830

C₄²×Dic₁₅₇

#23757034

#423×C₁₅₇

#1994

157

Subgroups — #34101112

Conjugacy classes of subgroups

Complements

Maximal subgroups