Group #15674949 — subgroups

Subgroups — #15674949

Order 2944.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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	#13092445	C₂	1	2	yes	maximal, complemented
2	#13092433	C₂	1	2	yes	maximal
3	#13092385	C₂	1	1	yes	maximal, characteristic
4	#13092174	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67288	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	#10625097	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂₃×#108	—	2	2	no	—
12	#10625007	—	2	4	no	—
13	(C₂×D₂₃)⋊C₈	—	2	2	no	—
14	Dic₂₃⋊C₈	—	2	4	no	—
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₂₃	—	2	1	no	—
17	#67288	—	2	2	no	—
18	#67288	—	2	2	no	—
19	C₂×D₉₂	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₂₃	—	2	1	no	—
21	C₂×Dic₄₆	Q₈	1	1	yes	characteristic
22	M₁₆×C₂₃	—	2	2	no	—
23	M₁₆×C₂₃	—	2	2	no	—
24	M₁₆×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	D₄	1	2	yes	—
27	C₂²⋊Dic₂₃	—	2	2	no	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄×Dic₂₃	D₄	1	4	yes	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂³×C₂₃	#41	1	1	yes	characteristic
33	C₂²×D₂₃	—	2	1	no	—
34	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₂₃	#41	1	2	yes	—
38	C₂²⋊D₂₃	—	4	2	no	—
39	C₂×Dic₂₃	—	2	1	no	—
40	C₂×Dic₂₃	—	2	4	no	—
41	D₉₂	—	2	2	no	—
42	C₄×D₂₃	—	4	1	no	—
43	C₄×D₂₃	—	4	1	no	—
44	Dic₄₆	—	2	2	no	—
45	C₈×C₂₃	—	2	2	no	—
46	C₈×C₂₃	—	4	2	no	—
47	C₂₃⋊C₈	—	4	1	no	—
48	#2098	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	—	4	1	no	—
50	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₃	#123	1	2	yes	—
52	C₂×D₂₃	—	4	2	no	—
53	C₂×D₂₃	—	4	1	no	—
54	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₂₃	#123	1	1	yes	characteristic
56	C₄×C₂₃	#127	1	1	yes	characteristic
57	Dic₂₃	—	4	4	no	—
58	Dic₂₃	—	4	1	no	—
59	#430	D₂₃	1	1	yes	characteristic, complemented
60	#424	—	23	2	no	—
61	#421	—	23	2	no	—
62	#408	—	23	1	no	—
63	H(3,ℤ/4)	—	23	1	no	—
64	C₂×C₂₃	—	2	2	no	—
65	C₂×C₂₃	—	2	1	no	—
66	C₂×C₂₃	#399	1	1	yes	characteristic
67	D₂₃	—	8	1	no	—
68	C₂×#41	—	23	1	no	—
69	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
70	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
71	C₂²×C₈	—	23	1	no	—
72	#117	—	23	2	no	—
73	#108	—	2	2	no	—
74	C₄⋊C₈	—	46	4	no	—
75	#104	—	46	4	no	—
76	C₂²⋊C₈	—	46	2	no	—
77	C₂₃	#2098	1	1	yes	characteristic, complemented, Sylow
78	#41	—	46	2	no	—
79	#41	—	46	2	no	—
80	C₂×Q₈	—	23	1	no	—
81	C₂×D₄	—	23	1	no	—
82	C₂×D₄	—	46	1	no	—
83	C₂²×C₄	—	46	1	no	—
84	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	46	2	no	—
90	C₂×C₈	—	46	1	no	—
91	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
92	C₄⋊C₄	—	46	2	no	—
93	C₂²⋊C₄	—	46	2	no	—
94	C₄²	—	23	4	no	—
95	C₂³	—	46	1	no	—
96	C₂³	#67292	1	1	yes	characteristic
97	Q₈	—	46	2	no	—
98	D₄	—	46	2	no	—
99	D₄	—	92	2	no	—
100	C₂×C₄	—	46	4	no	—
101	C₂×C₄	—	46	1	no	—
102	C₂×C₄	—	92	1	no	—
103	C₂×C₄	—	92	1	no	—
104	C₂×C₄	#67288	1	2	yes	—
105	C₂×C₄	D₄×D₂₃	1	2	yes	—
106	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	92	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	92	2	no	—
113	C₂²	—	92	1	no	—
114	C₂²	#10625065	1	2	yes	—
115	C₂²	#10625117	1	1	yes	characteristic
116	C₄	—	92	4	no	—
117	C₄	—	92	1	no	—
118	C₄	#10625067	1	2	yes	—
119	C₄	#10625100	1	1	yes	characteristic
120	C₄	#10625111	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	184	1	no	—
124	C₂	#13092352	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092445

C₂

yes

maximal, complemented

#13092433

C₂

yes

maximal

#13092385

C₂

yes

maximal, characteristic

#13092174

C₂

yes

maximal, characteristic, complemented

C₂₃×#430

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#108

—

#10625007

—

(C₂×D₂₃)⋊C₈

—

Dic₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

—

#67288

—

#67288

—

C₂×D₉₂

Q₈

yes

characteristic

C₂×C₄×D₂₃

—

C₂×Dic₄₆

Q₈

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

#41

yes

characteristic

C₂²×D₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#2098

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

#430

D₂₃

yes

characteristic, complemented

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#399

yes

characteristic

D₂₃

—

C₂×#41

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#117

—

#108

—

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

C₂₃

#2098

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

#67292

yes

characteristic

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#67288

yes

—

105

C₂×C₄

D₄×D₂₃

yes

—

106

C₂×C₄

Q₈×D₂₃

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#10625065

yes

—

115

C₂²

#10625117

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#10625067

yes

—

119

C₄

#10625100

yes

characteristic

120

C₄

#10625111

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

184

—

124

C₂

#13092352

yes

characteristic

	subgroup	G/H
center	C₄	#10625111
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#430	C₂
socle	C₂×C₂₃	#399
prime radical	C₂×#67288	C₂²
hypercenter	#430	D₂₃
hypocenter	C₂₃	#2098
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#10625111

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#430

C₂

C₂×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #13092445	C₂	semidirect
4. #13092174	C₂	semidirect
5. C₂₃×#430	C₂	semidirect
59. #430	D₂₃	semidirect
77. C₂₃	#2098	semidirect

H ≅ G/N

type

1. #13092445

C₂

semidirect

4. #13092174

C₂

semidirect

5. C₂₃×#430

C₂

semidirect

59. #430

D₂₃

semidirect

77. C₂₃

#2098

semidirect

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

subgroup	index
#13092445	2
#13092433	2
#13092385	2
#13092174	2
C₂₃×#430	2
#2098	23

subgroup

index

C₂₃×#430

Subgroups — #15674949

Conjugacy classes of subgroups

Complements

Maximal subgroups