Group #15674548 — subgroups

Subgroups — #15674548

Order 2944.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#10625041	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625017	C₂	1	2	yes	maximal
3	#13092179	C₂	1	2	yes	maximal
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	C₂×Dic₂₃⋊C₄	—	2	2	no	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	#10625041	—	2	4	no	—
13	C₂₃×#95	—	2	2	no	—
14	#10625017	—	2	4	no	—
15	C₂×Q₈×C₂₃	—	2	1	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	Q₈	1	2	yes	—
19	C₂²×C₄×C₂₃	—	2	1	no	—
20	C₂²×C₄×C₂₃	—	2	1	no	—
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	—	2	2	no	—
26	C₄⋊Dic₂₃	—	2	2	no	—
27	C₄⋊Dic₂₃	D₄	1	4	yes	—
28	Dic₂₃⋊C₄	—	2	8	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	Q₈×C₂₃	—	2	4	no	—
34	Q₈×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	4	1	no	—
40	C₂×C₄×C₂₃	—	4	1	no	—
41	C₂×C₄×C₂₃	—	4	1	no	—
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×Dic₂₃	—	2	4	no	—
48	C₂×Dic₂₃	—	4	2	no	—
49	C₂×Dic₂₃	—	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	2	no	—
59	C₄×C₂₃	—	8	1	no	—
60	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₃	#124	1	2	yes	—
62	Dic₂₃	—	4	4	no	—
63	C₂×#125	—	23	1	no	—
64	C₂×#103	—	23	2	no	—
65	#391	D₂₃	1	1	yes	characteristic
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	C₂×D₂₃	1	1	yes	characteristic
74	C₂²×C₈	—	23	2	no	—
75	#125	—	46	4	no	—
76	C₂×C₄⋊C₄	—	23	2	no	—
77	C₂×C₄⋊C₄	—	46	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
80	#103	—	46	4	no	—
81	#95	—	2	2	no	—
82	C₂₃	#2026	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	Dic₄₆	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	46	2	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	2	no	—
96	C₄⋊C₄	—	23	4	no	—
97	C₄⋊C₄	—	46	8	no	—
98	C₄⋊C₄	—	46	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	4	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	46	4	no	—
112	C₂×C₄	—	92	2	no	—
113	C₂×C₄	—	92	2	no	—
114	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
115	C₂×C₄	#67288	1	1	yes	characteristic
116	C₂×C₄	D₄×D₂₃	1	2	yes	—
117	C₂×C₄	#67290	1	2	yes	—
118	C₈	—	92	2	no	—
119	C₂²	#10625095	1	1	yes	characteristic
120	C₂²	#10625107	1	1	yes	characteristic
121	C₂²	#10625108	1	1	yes	characteristic
122	C₂²	#10625113	1	2	yes	—
123	C₂²	#10625121	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	—	8	1	no	—
127	C₄	—	92	4	no	—
128	C₄	#10625049	1	2	yes	—
129	C₄	#10625056	1	2	yes	—
130	C₂	#13092329	1	1	yes	characteristic
131	C₂	#13092577	1	2	yes	—
132	C₂	#13092724	1	2	yes	—
133	C₂	#13092771	1	1	yes	characteristic
134	C₂	#13092785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625041

C₂

yes

maximal, characteristic

C₂×#10625017

C₂

yes

maximal

#13092179

C₂

yes

maximal

#13092172

C₂

yes

maximal, characteristic

C₂₃×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625041

—

C₂₃×#95

—

#10625017

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

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

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

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

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#124

yes

—

Dic₂₃

—

C₂×#125

—

C₂×#103

—

#391

D₂₃

yes

characteristic

#340

—

#336

—

C₂×C₂₃

#393

yes

characteristic

C₂×C₂₃

#452

yes

—

C₂×C₂₃

#479

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

#95

—

C₂₃

#2026

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₄₆

yes

—

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

—

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

C₂×Dic₄₆

yes

characteristic

115

C₂×C₄

#67288

yes

characteristic

116

C₂×C₄

D₄×D₂₃

yes

—

117

C₂×C₄

#67290

yes

—

118

C₈

—

119

C₂²

#10625095

yes

characteristic

120

C₂²

#10625107

yes

characteristic

121

C₂²

#10625108

yes

characteristic

122

C₂²

#10625113

yes

—

123

C₂²

#10625121

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#10625049

yes

—

129

C₄

#10625056

yes

—

130

C₂

#13092329

yes

characteristic

131

C₂

#13092577

yes

—

132

C₂

#13092724

yes

—

133

C₂

#13092771

yes

characteristic

134

C₂

#13092785

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#391	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#391	D₂₃
hypocenter	C₂₃	#2026
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#391

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

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×#10625041	2
C₂×#10625017	2
#13092179	2
#13092172	2
C₂₃×#391	2
#2026	23

subgroup

index

C₂₃×#391

Subgroups — #15674548

Conjugacy classes of subgroups

Complements

Maximal subgroups