Group #15674665 — subgroups

Subgroups — #15674665

Order 2944.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₉₂	C₂	1	1	yes	maximal, characteristic
2	#13092425	C₂	1	2	yes	maximal
3	C₂×#10625006	C₂	1	2	yes	maximal
4	#13092182	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	C₂²	1	2	yes	—
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	2	yes	—
11	C₂×Dic₉₂	—	2	4	no	—
12	C₂×Dic₉₂	—	2	2	no	—
13	C₂₃×C₄⋊C₈	—	2	2	no	—
14	#10625006	—	2	4	no	—
15	#10624992	—	2	4	no	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	2	no	—
18	C₂×Dic₄₆	—	2	2	no	—
19	C₂×Dic₄₆	—	2	2	no	—
20	C₂×Dic₄₆	—	2	2	no	—
21	C₂×Dic₄₆	—	2	2	no	—
22	C₂×Dic₄₆	—	2	2	no	—
23	C₂×Dic₄₆	D₄	1	4	yes	—
24	M₁₆×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₂₃	—	2	1	no	—
30	C₄²×C₂₃	—	2	1	no	—
31	#67267	—	2	2	no	—
32	#67267	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	Dic₉₂	—	2	8	no	—
35	C₂³×C₂₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	#41	1	2	yes	—
41	C₂×Dic₂₃	—	2	4	no	—
42	C₂×Dic₂₃	—	4	2	no	—
43	C₂×Dic₂₃	—	4	2	no	—
44	Dic₄₆	—	2	4	no	—
45	Dic₄₆	—	2	4	no	—
46	Dic₄₆	—	4	2	no	—
47	Dic₄₆	—	4	2	no	—
48	C₈×C₂₃	—	2	2	no	—
49	C₈×C₂₃	—	4	1	no	—
50	C₂₃⋊C₈	—	4	2	no	—
51	#2047	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	#121	1	1	yes	characteristic
54	C₂²×C₂₃	#123	1	2	yes	—
55	C₄×C₂₃	—	4	2	no	—
56	C₄×C₂₃	#120	1	1	yes	characteristic
57	C₄×C₂₃	#121	1	2	yes	—
58	C₄×C₂₃	#124	1	1	yes	characteristic
59	Dic₂₃	—	4	4	no	—
60	C₂²×Q₁₆	—	23	1	no	—
61	#424	D₂₃	1	1	yes	characteristic
62	#419	—	23	2	no	—
63	C₂×#101	—	23	2	no	—
64	#341	—	23	1	no	—
65	C₂×C₂₃	—	2	2	no	—
66	C₂×C₂₃	#394	1	1	yes	characteristic
67	C₂×C₂₃	#473	1	2	yes	—
68	C₂²×Q₈	—	23	2	no	—
69	C₂×Q₁₆	—	46	4	no	—
70	C₂×Q₁₆	—	46	2	no	—
71	C₂×M₁₆	—	23	2	no	—
72	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
74	#117	C₂×D₂₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#103	—	46	4	no	—
77	#101	—	46	4	no	—
78	C₂₃	#2047	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	23	4	no	—
80	C₂×Q₈	—	46	2	no	—
81	C₂×Q₈	—	46	2	no	—
82	C₂×Q₈	—	46	2	no	—
83	C₂×Q₈	—	46	2	no	—
84	C₂×Q₈	—	46	2	no	—
85	C₂²×C₄	—	46	2	no	—
86	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
87	Q₁₆	—	46	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	46	2	no	—
90	M₁₆	—	46	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	46	2	no	—
94	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
95	C₄⋊C₄	D₉₂	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#67288	1	1	yes	characteristic
99	Q₈	—	46	4	no	—
100	Q₈	—	46	4	no	—
101	Q₈	—	92	2	no	—
102	Q₈	—	92	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	46	4	no	—
105	C₂×C₄	—	92	2	no	—
106	C₂×C₄	—	92	2	no	—
107	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
108	C₂×C₄	D₄×D₂₃	1	2	yes	—
109	C₂×C₄	#67292	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	92	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#10625065	1	2	yes	—
116	C₂²	#10625101	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	92	4	no	—
119	C₄	#10625066	1	2	yes	—
120	C₄	#10625108	1	1	yes	characteristic
121	C₄	#10625118	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#13092334	1	1	yes	characteristic
124	C₂	#13092692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₉₂

C₂

yes

maximal, characteristic

#13092425

C₂

yes

maximal

C₂×#10625006

C₂

yes

maximal

#13092182

C₂

yes

maximal, characteristic

C₂₃×#424

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×Dic₉₂

—

C₂×Dic₉₂

—

C₂₃×C₄⋊C₈

—

#10625006

—

#10624992

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

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

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#2047

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#121

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

#121

yes

—

C₄×C₂₃

#124

yes

characteristic

Dic₂₃

—

C₂²×Q₁₆

—

#424

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

#394

yes

characteristic

C₂×C₂₃

#473

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₂₃

#2047

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

D₉₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#67288

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₉₂

yes

characteristic

108

C₂×C₄

D₄×D₂₃

yes

—

109

C₂×C₄

#67292

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

#10625065

yes

—

116

C₂²

#10625101

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#10625066

yes

—

120

C₄

#10625108

yes

characteristic

121

C₄

#10625118

yes

characteristic

122

C₂

—

123

C₂

#13092334

yes

characteristic

124

C₂

#13092692

yes

—

	subgroup	G/H
center	C₂²	#10625101
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#424	C₂
socle	C₂²×C₂₃	#121
prime radical	C₂³×C₂₃	#41
hypercenter	#424	D₂₃
hypocenter	C₂₃	#2047
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#10625101

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#424

C₂

C₂²×C₂₃

C₂³×C₂₃

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
78. C₂₃	#2047	semidirect

H ≅ G/N

type

78. C₂₃

#2047

semidirect

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

subgroup	index
C₂²×Dic₉₂	2
#13092425	2
C₂×#10625006	2
#13092182	2
C₂₃×#424	2
#2047	23

subgroup

index

C₂²×Dic₉₂

C₂₃×#424

Subgroups — #15674665

Conjugacy classes of subgroups

Complements

Maximal subgroups