Group #15674199 — subgroups

Subgroups — #15674199

Order 2944.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624998	C₂	1	4	yes	maximal
3	#13092287	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	—	2	2	no	—
9	C₂×C₄⋊Dic₂₃	—	2	2	no	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	#10625040	—	2	2	no	—
12	#10625040	C₄	1	4	yes	complemented
13	C₄×C₈×C₂₃	—	2	2	no	—
14	#10624998	—	2	8	no	—
15	C₂²×C₄×C₂₃	D₄	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	2	no	—
18	C₂²×Dic₂₃	—	2	2	no	—
19	C₂×Dic₄₆	—	2	4	no	—
20	C₂×Dic₄₆	—	2	4	no	—
21	C₂×Dic₄₆	—	2	2	no	—
22	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	D₄	1	4	yes	—
26	C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄⋊Dic₂₃	—	2	8	no	—
29	C₄⋊Dic₂₃	—	2	4	no	—
30	C₄⋊Dic₂₃	—	4	4	no	—
31	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
33	C₂×C₄×C₂₃	SD₁₆	1	4	yes	—
34	C₂×C₄×C₂₃	Q₁₆	1	4	yes	—
35	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	#41	1	2	yes	—
39	C₂×Dic₂₃	—	2	4	no	—
40	C₂×Dic₂₃	—	4	2	no	—
41	C₂×Dic₂₃	—	4	2	no	—
42	C₂×Dic₂₃	—	4	2	no	—
43	C₂×Dic₂₃	—	4	4	no	—
44	Dic₄₆	—	2	8	no	—
45	Dic₄₆	—	4	4	no	—
46	C₈×C₂₃	—	2	4	no	—
47	#1956	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₃	#124	1	1	yes	characteristic
50	C₂²×C₂₃	#127	1	1	yes	characteristic
51	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
52	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
53	C₄×C₂₃	#103	1	8	yes	complemented
54	C₄×C₂₃	C₄×D₄	1	2	yes	—
55	C₄×C₂₃	#121	1	2	yes	—
56	Dic₂₃	—	4	4	no	—
57	Dic₂₃	—	8	2	no	—
58	C₂×C₄⋊Q₈	—	23	1	no	—
59	C₂×#103	—	23	4	no	—
60	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
61	#384	—	23	1	no	—
62	C₂×C₂₃	#390	1	1	yes	characteristic
63	C₂×C₂₃	C₂×#103	1	2	yes	—
64	C₂×C₂₃	#487	1	2	yes	—
65	C₂×C₂₃	#492	1	1	yes	characteristic
66	C₂×C₂₃	#494	1	1	yes	characteristic
67	C₂²×Q₈	—	23	2	no	—
68	C₂²×C₈	C₂×D₂₃	1	2	yes	—
69	C₄⋊Q₈	—	23	4	no	—
70	C₄⋊Q₈	—	46	2	no	—
71	C₂×C₄⋊C₄	—	23	2	no	—
72	C₂×C₄⋊C₄	—	46	2	no	—
73	C₂×C₄⋊C₄	—	46	2	no	—
74	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
75	#103	—	46	8	no	—
76	C₄×C₈	—	2	2	no	—
77	C₂₃	#1956	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	23	4	no	—
79	C₂×Q₈	—	46	2	no	—
80	C₂×Q₈	—	46	4	no	—
81	C₂×Q₈	—	46	4	no	—
82	C₂²×C₄	—	46	2	no	—
83	C₂²×C₄	—	46	2	no	—
84	C₂²×C₄	D₉₂	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₉₂	1	4	yes	—
89	C₄⋊C₄	—	46	8	no	—
90	C₄⋊C₄	—	46	4	no	—
91	C₄⋊C₄	—	92	4	no	—
92	C₄²	C₄×D₂₃	1	2	yes	—
93	C₄²	C₂²⋊D₂₃	1	2	yes	—
94	C₂³	C₂×D₉₂	1	1	yes	characteristic
95	Q₈	—	46	8	no	—
96	Q₈	—	92	4	no	—
97	C₂×C₄	—	46	4	no	—
98	C₂×C₄	—	92	2	no	—
99	C₂×C₄	—	92	2	no	—
100	C₂×C₄	—	92	2	no	—
101	C₂×C₄	—	92	4	no	—
102	C₂×C₄	#67263	1	4	yes	—
103	C₂×C₄	Dic₉₂	1	4	yes	—
104	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	C₂×D₉₂	1	2	yes	—
107	C₂×C₄	#67288	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₂²	#10625045	1	1	yes	characteristic
111	C₂²	#10625046	1	1	yes	characteristic
112	C₂²	C₂×#67263	1	2	yes	—
113	C₂²	C₂×Dic₉₂	1	2	yes	—
114	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
115	C₄	—	92	4	no	—
116	C₄	—	184	2	no	—
117	C₄	#10624998	1	8	yes	complemented
118	C₄	C₄×D₉₂	1	2	yes	—
119	C₄	#10625101	1	2	yes	—
120	C₂	#13092319	1	1	yes	characteristic
121	C₂	C₂×#10624998	1	2	yes	—
122	C₂	#13092768	1	2	yes	—
123	C₂	#13092790	1	1	yes	characteristic
124	C₂	#13092797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₂×#10624998

C₂

yes

maximal

#13092287

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

#10625040

—

#10625040

C₄

yes

complemented

C₄×C₈×C₂₃

—

#10624998

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

#1956

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

C₂×SD₁₆

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

#103

yes

complemented

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

D₂₃

yes

characteristic

#384

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

C₂×#103

yes

—

C₂×C₂₃

#487

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

C₄×C₈

—

C₂₃

#1956

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₉₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×D₉₂

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#67263

yes

—

103

C₂×C₄

Dic₉₂

yes

—

104

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

105

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

106

C₂×C₄

C₂×D₉₂

yes

—

107

C₂×C₄

#67288

yes

—

108

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

109

C₈

—

110

C₂²

#10625045

yes

characteristic

111

C₂²

#10625046

yes

characteristic

112

C₂²

C₂×#67263

yes

—

113

C₂²

C₂×Dic₉₂

yes

—

114

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

115

C₄

—

116

C₄

—

184

—

117

C₄

#10624998

yes

complemented

118

C₄

C₄×D₉₂

yes

—

119

C₄

#10625101

yes

—

120

C₂

#13092319

yes

characteristic

121

C₂

C₂×#10624998

yes

—

122

C₂

#13092768

yes

—

123

C₂

#13092790

yes

characteristic

124

C₂

#13092797

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1956
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

#1956

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. #10625040	C₄	semidirect
53. C₄×C₂₃	#103	semidirect
77. C₂₃	#1956	semidirect
117. C₄	#10624998	semidirect

H ≅ G/N

type

12. #10625040

C₄

semidirect

53. C₄×C₂₃

#103

semidirect

77. C₂₃

#1956

semidirect

117. C₄

#10624998

semidirect

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

subgroup	index
C₂×#10625040	2
C₂×#10624998	2
#13092287	2
C₂×C₄×C₈×C₂₃	2
#1956	23

subgroup

index

C₂×C₄×C₈×C₂₃

Subgroups — #15674199

Conjugacy classes of subgroups

Complements

Maximal subgroups