Group #15674138 — subgroups

Subgroups — #15674138

Order 2944.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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×#10624999	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
3	#13092313	C₂	1	2	yes	maximal
4	#13092179	C₂	1	2	yes	maximal
5	C₂×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
9	C₂×Dic₂₃⋊C₄	—	2	2	no	—
10	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂₃×#107	—	2	2	no	—
13	C₂₃⋊#95	—	2	2	no	—
14	#10624999	C₄	1	4	yes	complemented
15	#10624997	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	2	no	—
19	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
20	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₃	D₄	1	2	yes	—
23	C₂×C₈×C₂₃	Q₈	1	2	yes	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄⋊Dic₂₃	—	2	2	no	—
27	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
28	Dic₂₃⋊C₄	—	4	4	no	—
29	C₄×Dic₂₃	—	2	1	no	—
30	C₄×Dic₂₃	—	2	1	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂×C₂₃⋊C₈	—	2	1	no	—
33	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
34	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	4	2	no	—
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
44	C₂×Dic₂₃	—	2	4	no	—
45	C₂×Dic₂₃	—	2	1	no	—
46	C₂×Dic₂₃	—	2	1	no	—
47	C₂×Dic₂₃	—	2	2	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	4	2	no	—
50	C₂×Dic₂₃	—	4	2	no	—
51	C₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
52	C₂₃⋊C₈	—	2	2	no	—
53	#1944	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
55	C₂²×C₂₃	#121	1	1	yes	characteristic
56	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
58	C₂²×C₂₃	#137	1	2	yes	—
59	C₄×C₂₃	—	8	2	no	—
60	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
62	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
63	C₄×C₂₃	#125	1	1	yes	characteristic
64	Dic₂₃	—	4	4	no	—
65	Dic₂₃	—	4	1	no	—
66	Dic₂₃	—	4	1	no	—
67	C₂×#107	D₂₃	1	1	yes	characteristic
68	C₂×#106	—	23	1	no	—
69	C₂×#97	—	23	1	no	—
70	#389	—	23	2	no	—
71	#340	—	23	2	no	—
72	C₂×C₂₃	#389	1	1	yes	characteristic
73	C₂×C₂₃	#440	1	2	yes	—
74	C₂×C₂₃	#469	1	1	yes	characteristic
75	C₂×C₂₃	#470	1	1	yes	characteristic
76	C₂×C₂₃	#501	1	2	yes	—
77	C₂²×C₈	—	23	1	no	—
78	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
79	C₂×C₄⋊C₄	—	23	2	no	—
80	C₂×C₄⋊C₄	—	46	2	no	—
81	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
82	C₂×C₄²	—	23	1	no	—
83	#107	—	2	2	no	—
84	#106	—	23	4	no	—
85	#97	—	46	2	no	—
86	#95	—	46	2	no	—
87	C₂₃	#1944	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	23	1	no	—
90	C₂²×C₄	—	23	1	no	—
91	C₂²×C₄	—	46	2	no	—
92	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
93	C₂×C₈	—	23	2	no	—
94	C₂×C₈	—	46	1	no	—
95	C₂×C₈	—	46	1	no	—
96	C₂×C₈	Dic₄₆	1	2	yes	—
97	C₂×C₈	C₄×D₂₃	1	2	yes	—
98	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	23	4	no	—
102	C₄⋊C₄	—	46	2	no	—
103	C₄⋊C₄	—	92	4	no	—
104	C₄²	—	46	1	no	—
105	C₄²	—	46	1	no	—
106	C₂³	D₄×D₂₃	1	1	yes	characteristic
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	46	4	no	—
111	C₂×C₄	—	46	2	no	—
112	C₂×C₄	—	46	2	no	—
113	C₂×C₄	—	46	1	no	—
114	C₂×C₄	—	46	1	no	—
115	C₂×C₄	—	92	2	no	—
116	C₂×C₄	—	92	2	no	—
117	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
119	C₂×C₄	#67290	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
121	C₂×C₄	#67292	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
123	C₈	—	46	2	no	—
124	C₈	Dic₂₃⋊C₄	1	4	yes	complemented
125	C₂²	#10625059	1	1	yes	characteristic
126	C₂²	#10625064	1	1	yes	characteristic
127	C₂²	#10625082	1	2	yes	—
128	C₂²	#10625089	1	2	yes	—
129	C₂²	#10625109	1	1	yes	characteristic
130	C₄	—	8	2	no	—
131	C₄	—	92	1	no	—
132	C₄	—	92	1	no	—
133	C₄	—	92	4	no	—
134	C₄	#10625058	1	1	yes	characteristic
135	C₄	#10625061	1	1	yes	characteristic
136	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
137	C₄	#10625117	1	1	yes	characteristic
138	C₂	#13092313	1	1	yes	characteristic
139	C₂	#13092516	1	2	yes	—
140	C₂	#13092667	1	1	yes	characteristic
141	C₂	#13092672	1	1	yes	characteristic
142	C₂	#13092825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624999

C₂

yes

maximal, characteristic

C₂×#10624997

C₂

yes

maximal, characteristic

#13092313

C₂

yes

maximal

#13092179

C₂

yes

maximal

C₂×C₂₃×#107

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#107

—

C₂₃⋊#95

—

#10624999

C₄

yes

complemented

#10624997

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

D₄

yes

characteristic

C₂²×Dic₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

complemented

C₂₃⋊C₈

—

#1944

—

maximal, Sylow

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

#121

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₃

#125

yes

characteristic

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

C₂×#107

D₂₃

yes

characteristic

C₂×#106

—

C₂×#97

—

#389

—

#340

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

#440

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

—

—

—

—

C₂₃

#1944

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₂³

D₄×D₂₃

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×Dic₄₆

yes

characteristic

118

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

119

C₂×C₄

#67290

yes

characteristic

120

C₂×C₄

Q₈×D₂₃

yes

characteristic

121

C₂×C₄

#67292

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

123

C₈

—

124

C₈

Dic₂₃⋊C₄

yes

complemented

125

C₂²

#10625059

yes

characteristic

126

C₂²

#10625064

yes

characteristic

127

C₂²

#10625082

yes

—

128

C₂²

#10625089

yes

—

129

C₂²

#10625109

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#10625058

yes

characteristic

135

C₄

#10625061

yes

characteristic

136

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

137

C₄

#10625117

yes

characteristic

138

C₂

#13092313

yes

characteristic

139

C₂

#13092516

yes

—

140

C₂

#13092667

yes

characteristic

141

C₂

#13092672

yes

characteristic

142

C₂

#13092825

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#107	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#107	D₂₃
hypocenter	C₂₃	#1944
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#107

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
14. #10624999	C₄	semidirect
51. C₈×C₂₃	C₄⋊C₄	semidirect
87. C₂₃	#1944	semidirect
124. C₈	Dic₂₃⋊C₄	semidirect

H ≅ G/N

type

14. #10624999

C₄

semidirect

51. C₈×C₂₃

C₄⋊C₄

semidirect

87. C₂₃

#1944

semidirect

124. C₈

Dic₂₃⋊C₄

semidirect

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

subgroup	index
C₂×#10624999	2
C₂×#10624997	2
#13092313	2
#13092179	2
C₂×C₂₃×#107	2
#1944	23

subgroup

index

C₂×C₂₃×#107

Subgroups — #15674138

Conjugacy classes of subgroups

Complements

Maximal subgroups