Group #23757927 — subgroups

Subgroups — #23757927

Order 10048.

292 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 115 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×#14578953	C₂	1	2	yes	maximal, complemented
2	#19097771	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097755	C₂	1	4	yes	maximal, complemented
4	#19097751	C₂	1	2	yes	maximal, complemented
5	#19097745	C₂	1	2	yes	maximal, complemented
6	C₄×D₆₂₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Dic₁₅₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#14578953	—	2	8	no	—
10	C₂×Dic₃₁₄	C₂²	1	2	yes	—
11	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₅₇	C₂²	1	4	yes	—
13	C₂²×Dic₁₅₇	C₂²	1	4	yes	—
14	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
17	C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
18	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
19	#14578931	C₂²	1	2	yes	—
20	#14578931	C₂²	1	4	yes	—
21	C₂×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₅₇	C₂²	1	2	yes	—
25	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₅₇	C₂³	1	2	yes	—
27	Dic₃₁₄	—	2	4	no	—
28	D₆₂₈	—	2	2	no	—
29	C₂²⋊D₁₅₇	—	2	8	no	—
30	C₂×Dic₁₅₇	—	2	8	no	—
31	C₂×Dic₁₅₇	C₂³	1	2	yes	—
32	C₂×Dic₁₅₇	C₂³	1	4	yes	—
33	C₄×D₁₅₇	—	2	4	no	—
34	C₂³×C₁₅₇	C₂³	1	2	yes	—
35	D₄×C₁₅₇	D₄	1	4	yes	complemented
36	C₂×C₄×C₁₅₇	—	2	4	no	—
37	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
41	C₂²×C₁₅₇	—	2	2	no	—
42	C₂²×C₁₅₇	—	2	2	no	—
43	C₂²×C₁₅₇	C₂×D₄	1	4	yes	—
44	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₅₇	—	2	2	no	—
46	C₂×D₁₅₇	—	2	2	no	—
47	C₂×D₁₅₇	—	2	2	no	—
48	C₄×C₁₅₇	—	2	2	no	—
49	C₄×C₁₅₇	—	2	1	no	—
50	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
51	C₄×C₁₅₇	#41	1	2	yes	complemented
52	Dic₁₅₇	—	2	2	no	—
53	Dic₁₅₇	—	2	4	no	—
54	C₂×C₁₅₇	—	2	4	no	—
55	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₁₅₇	C₂×#41	1	1	yes	characteristic
57	C₂×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₅₇	—	4	2	no	—
59	C₁₅₇	#547	1	1	yes	characteristic, complemented, Sylow
60	#547	—	157	1	no	maximal, Sylow
61	C₂×#41	—	157	2	no	—
62	C₄⋊Q₈	—	157	1	no	—
63	#123	—	157	4	no	—
64	C₂²⋊Q₈	—	157	2	no	—
65	#121	—	157	2	no	—
66	C₄×D₄	—	157	1	no	—
67	C₄×D₄	D₁₅₇	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄	—	157	2	no	—
69	#41	—	314	8	no	—
70	C₂×Q₈	—	157	2	no	—
71	C₂×D₄	—	157	4	no	—
72	C₂×D₄	—	157	1	no	—
73	C₂×D₄	C₂×D₁₅₇	1	1	yes	characteristic
74	C₂²×C₄	—	157	4	no	—
75	C₂²×C₄	—	157	2	no	—
76	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
77	C₄⋊C₄	—	157	1	no	—
78	C₄⋊C₄	—	157	2	no	—
79	C₄⋊C₄	—	157	4	no	—
80	C₄⋊C₄	—	157	2	no	—
81	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
82	C₂²⋊C₄	—	157	2	no	—
83	C₂²⋊C₄	—	157	4	no	—
84	C₂²⋊C₄	C₂×D₁₅₇	1	2	yes	—
85	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
86	C₂³	—	157	2	no	—
87	C₂³	C₂²×D₁₅₇	1	2	yes	—
88	Q₈	—	314	4	no	—
89	D₄	—	314	8	no	—
90	D₄	—	314	2	no	—
91	D₄	D₆₂₈	1	4	yes	complemented
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	314	8	no	—
94	C₂×C₄	—	314	4	no	—
95	C₂×C₄	—	157	2	no	—
96	C₂×C₄	—	157	4	no	—
97	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
99	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
101	C₂²	—	2	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	314	2	no	—
104	C₂²	—	314	2	no	—
105	C₂²	—	314	2	no	—
106	C₂²	C₂×D₆₂₈	1	4	yes	—
107	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	314	2	no	—
111	C₄	—	314	4	no	—
112	C₄	C₂×D₆₂₈	1	2	yes	—
113	C₄	#14578953	1	2	yes	complemented
114	C₂	—	2	4	no	—
115	C₂	—	628	2	no	—
116	C₂	C₂²×D₆₂₈	1	1	yes	characteristic
117	C₂	C₂×#14578953	1	1	yes	characteristic
118	C₂	#19097818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578953

C₂

yes

maximal, complemented

#19097771

C₂

yes

maximal, characteristic, complemented

#19097755

C₂

yes

maximal, complemented

#19097751

C₂

yes

maximal, complemented

#19097745

C₂

yes

maximal, complemented

C₄×D₆₂₈

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, complemented

C₄×D₄×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578953

—

C₂×Dic₃₁₄

C₂²

yes

—

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578931

C₂²

yes

—

#14578931

C₂²

yes

—

C₂×D₄×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₇

C₂²

yes

—

C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂³

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₄×D₁₅₇

—

C₂³×C₁₅₇

C₂³

yes

—

D₄×C₁₅₇

D₄

yes

complemented

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

#41

yes

complemented

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

characteristic

D₁₅₇

—

C₁₅₇

#547

yes

characteristic, complemented, Sylow

#547

—

157

maximal, Sylow

C₂×#41

—

157

—

C₄⋊Q₈

—

157

—

#123

—

157

—

C₂²⋊Q₈

—

157

—

#121

—

157

—

C₄×D₄

—

157

—

C₄×D₄

D₁₅₇

yes

characteristic, complemented

C₂×C₄⋊C₄

—

157

—

#41

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

C₂×D₁₅₇

yes

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

—

Q₈

—

314

—

D₄

—

314

—

D₄

—

314

—

D₄

D₆₂₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

100

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

314

—

104

C₂²

—

314

—

105

C₂²

—

314

—

106

C₂²

C₂×D₆₂₈

yes

—

107

C₂²

C₂³×D₁₅₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

314

—

111

C₄

—

314

—

112

C₄

C₂×D₆₂₈

yes

—

113

C₄

#14578953

yes

complemented

114

C₂

—

115

C₂

—

628

—

116

C₂

C₂²×D₆₂₈

yes

characteristic

117

C₂

C₂×#14578953

yes

characteristic

118

C₂

#19097818

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₄×D₄×C₁₅₇	C₂
hypercenter	C₄×D₄	D₁₅₇
hypocenter	C₁₅₇	#547
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₄×D₄×C₁₅₇

C₂

hypercenter

C₄×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#14578953	C₂	semidirect
2. #19097771	C₂	semidirect
3. #19097755	C₂	semidirect
4. #19097751	C₂	semidirect
5. #19097745	C₂	semidirect
6. C₄×D₆₂₈	C₂	semidirect
7. C₂×C₄⋊Dic₁₅₇	C₂	semidirect
8. C₄×D₄×C₁₅₇	C₂	semidirect
35. D₄×C₁₅₇	D₄	semidirect
51. C₄×C₁₅₇	#41	semidirect
59. C₁₅₇	#547	semidirect
67. C₄×D₄	D₁₅₇	semidirect
91. D₄	D₆₂₈	semidirect
113. C₄	#14578953	semidirect

H ≅ G/N

type

1. C₂×#14578953

C₂

semidirect

2. #19097771

C₂

semidirect

3. #19097755

C₂

semidirect

4. #19097751

C₂

semidirect

5. #19097745

C₂

semidirect

6. C₄×D₆₂₈

C₂

semidirect

7. C₂×C₄⋊Dic₁₅₇

C₂

semidirect

8. C₄×D₄×C₁₅₇

C₂

semidirect

35. D₄×C₁₅₇

D₄

semidirect

51. C₄×C₁₅₇

#41

semidirect

59. C₁₅₇

#547

semidirect

67. C₄×D₄

D₁₅₇

semidirect

91. D₄

D₆₂₈

semidirect

113. C₄

#14578953

semidirect

Subgroups — #23757927

Conjugacy classes of subgroups

Complements