Group #15673611 — subgroups

Subgroups — #15673611

Order 2944.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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	#13092895	C₂	1	1	yes	maximal, characteristic
2	#13092457	C₂	1	2	yes	maximal
3	C₂×#10625018	C₂	1	1	yes	maximal, characteristic
4	#13092177	C₂	1	2	yes	maximal
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	Q₈×Dic₂₃	—	2	2	no	—
10	Q₈×Dic₂₃	—	2	2	no	—
11	#10625097	—	2	2	no	—
12	#10625097	C₂²	1	2	yes	—
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂₃×#95	—	2	2	no	—
15	#10625018	C₄	1	4	yes	complemented
16	#10624990	—	2	2	no	—
17	#10624989	—	2	2	no	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₂₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₂₃	—	2	1	no	—
23	C₂²×C₄×C₂₃	—	2	1	no	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄²×C₂₃	—	2	2	no	—
28	C₂²⋊Dic₂₃	—	2	2	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	D₄	1	2	yes	—
32	C₄⋊Dic₂₃	D₄	1	2	yes	—
33	C₄×Dic₂₃	—	2	4	no	—
34	C₄×Dic₂₃	—	2	2	no	—
35	#67267	—	2	2	no	—
36	#67267	C₂×C₄	1	4	yes	—
37	C₂×C₂₃⋊C₈	—	2	2	no	—
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₂₃	—	2	4	no	—
40	Q₈×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	4	1	no	—
45	C₂×C₄×C₂₃	—	4	1	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	—	4	1	no	—
48	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	#41	1	2	yes	—
53	C₂×Dic₂₃	—	2	2	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂₃⋊C₈	—	2	4	no	—
56	#1857	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	—	2	1	no	—
58	C₂²×C₂₃	—	2	1	no	—
59	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	#120	1	1	yes	characteristic
61	C₂²×C₂₃	#123	1	1	yes	characteristic
62	C₄×C₂₃	—	2	4	no	—
63	C₄×C₂₃	—	4	2	no	—
64	C₄×C₂₃	—	8	1	no	—
65	C₄×C₂₃	C₄×D₄	1	2	yes	—
66	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
67	Dic₂₃	—	4	2	no	—
68	Dic₂₃	—	4	2	no	—
69	#519	—	23	1	no	—
70	#428	—	23	2	no	—
71	C₂×#101	—	23	1	no	—
72	#391	D₂₃	1	1	yes	characteristic
73	#339	—	23	2	no	—
74	C₂×C₂₃	—	2	1	no	—
75	C₂×C₂₃	—	2	1	no	—
76	C₂×C₂₃	#385	1	1	yes	characteristic
77	C₂×C₂₃	#454	1	1	yes	characteristic
78	C₂×C₂₃	#456	1	1	yes	characteristic
79	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
80	C₂×M₁₆	—	23	2	no	—
81	C₄×Q₈	—	46	2	no	—
82	C₄×Q₈	—	46	2	no	—
83	#117	—	23	2	no	—
84	#117	—	46	2	no	—
85	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
87	#107	—	46	2	no	—
88	#106	—	46	2	no	—
89	#101	—	23	4	no	—
90	#95	—	2	2	no	—
91	C₂₃	#1857	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	Dic₄₆	1	2	yes	—
95	C₂×Q₈	C₄×D₂₃	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
100	M₁₆	—	23	4	no	—
101	M₁₆	—	46	2	no	—
102	C₂×C₈	—	46	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	23	2	no	—
105	C₄⋊C₄	—	23	2	no	—
106	C₄⋊C₄	—	46	2	no	—
107	C₄⋊C₄	—	46	2	no	—
108	C₂²⋊C₄	—	46	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	46	4	no	—
111	C₄²	—	46	2	no	—
112	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	46	2	no	—
123	C₂×C₄	—	46	2	no	—
124	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
125	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₃	1	2	yes	—
128	C₂×C₄	#67290	1	2	yes	—
129	C₈	—	46	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
133	C₂²	#10625106	1	1	yes	characteristic
134	C₂²	#10625108	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	92	2	no	—
139	C₄	—	92	2	no	—
140	C₄	#10625049	1	2	yes	—
141	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#13092289	1	1	yes	characteristic
145	C₂	#13092589	1	1	yes	characteristic
146	C₂	#13092599	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092895

C₂

yes

maximal, characteristic

#13092457

C₂

yes

maximal

C₂×#10625018

C₂

yes

maximal, characteristic

#13092177

C₂

yes

maximal

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

Q₈×Dic₂₃

—

Q₈×Dic₂₃

—

#10625097

—

#10625097

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂₃×#95

—

#10625018

C₄

yes

complemented

#10624990

—

#10624989

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

C₂×C₄

yes

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1857

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

—

—

—

D₂₃

yes

characteristic

#339

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

—

—

—

—

C₂₃

#1857

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₄₆

yes

—

C₂×Q₈

C₄×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

Dic₂₃⋊C₄

yes

—

125

C₂×C₄

C₂×Dic₄₆

yes

characteristic

126

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

127

C₂×C₄

D₄×D₂₃

yes

—

128

C₂×C₄

#67290

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

133

C₂²

#10625106

yes

characteristic

134

C₂²

#10625108

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#10625049

yes

—

141

C₄

(C₂²⋊D₂₃)⋊C₄

yes

—

142

C₂

—

143

C₂

—

144

C₂

#13092289

yes

characteristic

145

C₂

#13092589

yes

characteristic

146

C₂

#13092599

yes

characteristic

	subgroup	G/H
center	C₂²	#10625108
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#391	C₂
socle	C₂²×C₂₃	#120
prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#391	D₂₃
hypocenter	C₂₃	#1857
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#10625108

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#391

C₂

socle

C₂²×C₂₃

#120

prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
15. #10625018	C₄	semidirect
91. C₂₃	#1857	semidirect

H ≅ G/N

type

15. #10625018

C₄

semidirect

91. C₂₃

#1857

semidirect

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

subgroup	index
#13092895	2
#13092457	2
C₂×#10625018	2
#13092177	2
C₂₃×#391	2
#1857	23

subgroup

index

C₂₃×#391

Subgroups — #15673611

Conjugacy classes of subgroups

Complements

Maximal subgroups