Group #15674545 — subgroups

Subgroups — #15674545

Order 2944.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#10624992	C₂	1	2	yes	maximal
2	#13092323	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	2	yes	maximal
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
7	C₂₃×#125	—	2	4	no	—
8	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	C₂₃⋊#95	—	2	2	no	—
14	#10624992	—	2	4	no	—
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	D₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	1	no	—
19	C₂²×Dic₂₃	—	2	1	no	—
20	C₂×Dic₄₆	—	2	2	no	—
21	C₂×Dic₄₆	—	2	2	no	—
22	C₂×Dic₄₆	—	2	1	no	—
23	C₂×Dic₄₆	Q₈	1	2	yes	—
24	C₄⋊C₄×C₂₃	—	2	8	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
27	C₄²×C₂₃	—	2	2	no	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	#41	1	2	yes	—
42	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×Dic₂₃	—	2	2	no	—
44	C₂×Dic₂₃	—	4	1	no	—
45	C₂×Dic₂₃	—	4	1	no	—
46	C₂×Dic₂₃	—	4	1	no	—
47	C₂×Dic₂₃	—	4	2	no	—
48	Dic₄₆	—	2	4	no	—
49	Dic₄₆	—	4	2	no	—
50	C₂₃⋊C₈	—	4	2	no	—
51	#2026	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	#120	1	1	yes	characteristic
53	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₂₃	#124	1	1	yes	characteristic
55	C₂²×C₂₃	#135	1	2	yes	—
56	C₂²×C₂₃	#137	1	2	yes	—
57	C₄×C₂₃	—	4	2	no	—
58	C₄×C₂₃	—	4	4	no	—
59	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₂₃	#124	1	2	yes	—
61	Dic₂₃	—	4	2	no	—
62	Dic₂₃	—	8	1	no	—
63	C₂×#125	D₂₃	1	1	yes	characteristic
64	C₂×#103	—	23	2	no	—
65	#391	—	23	1	no	—
66	#340	—	23	2	no	—
67	#336	—	23	1	no	—
68	C₂×C₂₃	#393	1	1	yes	characteristic
69	C₂×C₂₃	#452	1	2	yes	—
70	C₂×C₂₃	#479	1	2	yes	—
71	C₂×C₂₃	#488	1	1	yes	characteristic
72	C₂×C₂₃	#491	1	1	yes	characteristic
73	C₂²×Q₈	—	23	1	no	—
74	C₂²×C₈	—	23	2	no	—
75	#125	—	2	4	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	23	1	no	—
78	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
79	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
80	#103	—	46	4	no	—
81	#95	—	46	2	no	—
82	C₂₃	#2026	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	23	2	no	—
84	C₂×Q₈	—	46	1	no	—
85	C₂×Q₈	—	46	2	no	—
86	C₂×Q₈	—	46	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	46	1	no	—
89	C₂²×C₄	—	46	1	no	—
90	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
92	C₂×C₈	—	46	2	no	—
93	C₂×C₈	—	46	2	no	—
94	C₂×C₈	—	46	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	46	2	no	—
98	C₄⋊C₄	C₂²⋊D₂₃	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
101	Q₈	—	46	4	no	—
102	Q₈	—	92	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	46	2	no	—
110	C₂×C₄	—	92	1	no	—
111	C₂×C₄	—	92	1	no	—
112	C₂×C₄	—	92	1	no	—
113	C₂×C₄	—	92	2	no	—
114	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
115	C₂×C₄	#67292	1	2	yes	—
116	C₂×C₄	#67292	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
118	C₈	—	92	2	no	—
119	C₂²	#10625117	1	1	yes	characteristic
120	C₂²	#10625118	1	1	yes	characteristic
121	C₂²	#10625121	1	2	yes	—
122	C₂²	#10625122	1	2	yes	—
123	C₂²	#10625124	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	92	2	no	—
127	C₄	—	184	1	no	—
128	C₄	#10625117	1	2	yes	—
129	C₄	#10625118	1	2	yes	—
130	C₂	#13092328	1	1	yes	characteristic
131	C₂	#13092579	1	2	yes	—
132	C₂	#13092721	1	2	yes	—
133	C₂	#13092774	1	1	yes	characteristic
134	C₂	#13092787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624992

C₂

yes

maximal

#13092323

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

#13092172

C₂

yes

maximal, characteristic

C₂×C₂₃×#125

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

characteristic

C₂₃×#125

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

—

#10624992

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

Q₈

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂²×C₂₃

C₂²⋊Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#125

D₂₃

yes

—

—

—

—

C₂×C₂₃

#393

yes

characteristic

C₂×C₂₃

#452

yes

—

C₂×C₂₃

#479

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

#95

—

C₂₃

#2026

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₃

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

Q₈×D₂₃

yes

characteristic

115

C₂×C₄

#67292

yes

—

116

C₂×C₄

#67292

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

118

C₈

—

119

C₂²

#10625117

yes

characteristic

120

C₂²

#10625118

yes

characteristic

121

C₂²

#10625121

yes

—

122

C₂²

#10625122

yes

—

123

C₂²

#10625124

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

184

—

128

C₄

#10625117

yes

—

129

C₄

#10625118

yes

—

130

C₂

#13092328

yes

characteristic

131

C₂

#13092579

yes

—

132

C₂

#13092721

yes

—

133

C₂

#13092774

yes

characteristic

134

C₂

#13092787

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	#2026
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#125

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
82. C₂₃	#2026	semidirect

H ≅ G/N

type

82. C₂₃

#2026

semidirect

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

subgroup	index
C₂×#10624992	2
#13092323	2
#13092179	2
#13092172	2
C₂×C₂₃×#125	2
#2026	23

subgroup

index

C₂×C₂₃×#125

Subgroups — #15674545

Conjugacy classes of subgroups

Complements

Maximal subgroups