Group #15683919 — subgroups

Subgroups — #15683919

Order 2944.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#13092974	C₂	1	1	yes	maximal, characteristic
2	#13092827	C₂	1	4	yes	maximal, complemented
3	#13092823	C₂	1	4	yes	maximal, complemented
4	#13092453	C₂	1	4	yes	maximal
5	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
7	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂₃×#125	C₂²	1	2	yes	—
9	C₂₃×C₂²⋊Q₈	—	2	2	no	—
10	C₄×Q₈×C₂₃	—	2	2	no	—
11	C₂₃×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	#10625097	C₂²	1	2	yes	—
14	#10625095	—	2	2	no	—
15	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
16	#10625041	C₂²	1	2	yes	—
17	#10625040	C₂²	1	2	yes	—
18	C₄×Dic₄₆	—	2	2	no	—
19	#10624990	C₂²	1	8	yes	—
20	#10624989	C₂²	1	8	yes	—
21	#10624985	C₂²	1	2	yes	—
22	#10624985	C₂²	1	2	yes	—
23	C₂×Q₈×C₂₃	—	2	2	no	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₄₆	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₂²⋊C₄×C₂₃	—	2	2	no	—
31	C₄²×C₂₃	—	2	2	no	—
32	C₄²×C₂₃	D₄	1	2	yes	—
33	C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₂₃	—	2	2	no	—
36	C₄⋊Dic₂₃	—	2	2	no	—
37	C₄⋊Dic₂₃	C₂³	1	4	yes	—
38	Dic₂₃⋊C₄	—	2	4	no	—
39	C₄×Dic₂₃	—	2	2	no	—
40	C₂×C₂₃⋊C₈	Q₈	1	8	yes	complemented
41	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
42	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₂₃	—	4	4	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₂₃	—	2	4	no	—
52	Dic₄₆	—	4	4	no	—
53	C₂₃⋊C₈	C₂×Q₈	1	8	yes	—
54	#3162	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
57	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	—	4	4	no	—
61	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
62	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
63	Dic₂₃	—	4	4	no	—
64	#533	—	23	1	no	—
65	#533	D₂₃	1	1	yes	characteristic
66	#501	—	23	4	no	—
67	#501	—	23	4	no	—
68	#427	—	23	4	no	—
69	C₂×#97	—	23	1	no	—
70	C₂×C₂₃	—	2	2	no	—
71	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₂₃	#575	1	2	yes	—
73	C₂²×C₈	—	23	2	no	—
74	C₄⋊Q₈	—	23	2	no	—
75	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
76	#125	—	23	2	no	—
77	#125	C₂×D₂₃	1	2	yes	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₂²⋊Q₈	—	46	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	46	2	no	—
82	#117	—	23	2	no	—
83	#117	C₂×D₂₃	1	2	yes	—
84	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
85	#107	—	23	8	no	—
86	#106	—	23	8	no	—
87	#97	—	23	2	no	—
88	#97	—	23	2	no	—
89	C₂₃	#3162	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	46	2	no	—
92	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
94	C₂×C₈	—	23	8	no	—
95	C₂×C₈	—	23	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	23	4	no	—
99	C₄⋊C₄	—	46	4	no	—
100	C₄⋊C₄	—	46	2	no	—
101	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	46	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	46	2	no	—
106	C₄²	C₂²⋊D₂₃	1	2	yes	—
107	C₄²	C₂²×D₂₃	1	1	yes	characteristic
108	C₄²	C₂²×D₂₃	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	92	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	46	4	no	—
115	C₂×C₄	Q₈×D₂₃	1	4	yes	—
116	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
119	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
120	C₈	—	23	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#10625115	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	92	4	no	—
128	C₄	#10625115	1	2	yes	—
129	C₄	C₂×Q₈×D₂₃	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#10625115	1	1	yes	characteristic
132	C₂	#13093215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

#13092827

C₂

yes

maximal, complemented

#13092823

C₂

yes

maximal, complemented

#13092453

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic

C₂₃×#533

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#125

C₂²

yes

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625041

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

—

#10624990

C₂²

yes

—

#10624989

C₂²

yes

—

#10624985

C₂²

yes

—

#10624985

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#3162

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

—

Dic₂₃

—

#533

—

#533

D₂₃

yes

—

—

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

#575

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#125

—

#125

C₂×D₂₃

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

—

C₂₃

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₂₃

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²⋊D₂₃

yes

—

107

C₄²

C₂²×D₂₃

yes

characteristic

108

C₄²

C₂²×D₂₃

yes

characteristic

109

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

Q₈×D₂₃

yes

—

116

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

117

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

119

C₂×C₄

C₂³×D₂₃

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

#10625115

yes

—

123

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#10625115

yes

—

129

C₄

C₂×Q₈×D₂₃

yes

—

130

C₂

—

131

C₂

C₂×#10625115

yes

characteristic

132

C₂

#13093215

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#533	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#533	D₂₃
hypocenter	C₂₃	#3162
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#533

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #13092827	C₂	semidirect
3. #13092823	C₂	semidirect
40. C₂×C₂₃⋊C₈	Q₈	semidirect
89. C₂₃	#3162	semidirect

H ≅ G/N

type

2. #13092827

C₂

semidirect

3. #13092823

C₂

semidirect

40. C₂×C₂₃⋊C₈

Q₈

semidirect

89. C₂₃

#3162

semidirect

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

subgroup	index
#13092974	2
#13092827	2
#13092823	2
#13092453	2
C₂×#10624985	2
C₂₃×#533	2
#3162	23

subgroup

index

C₂₃×#533

Subgroups — #15683919

Conjugacy classes of subgroups

Complements

Maximal subgroups