Group #15676238 — subgroups

Subgroups — #15676238

Order 2944.

872 conjugacy classes of subgroups; only the 313 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 313 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×Q₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
2	#13092320	C₂	1	6	yes	maximal, complemented
3	#13092277	C₂	1	3	yes	maximal, complemented
4	C₄×C₂²⋊Dic₂₃	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	—	2	8	no	—
7	C₂×#41×C₂₃	C₄	1	8	yes	complemented
8	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₂₃	C₂²	1	3	yes	—
10	C₂³×C₄×C₂₃	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₂₃	C₂²	1	6	yes	—
12	C₂×C₂²⋊Dic₂₃	C₂²	1	6	yes	—
13	Q₈×Dic₂₃	—	2	8	no	—
14	C₂×C₄⋊Dic₂₃	C₂²	1	6	yes	—
15	C₂×C₄×Dic₂₃	C₂²	1	6	yes	—
16	C₂₃⋊#95	C₂²	1	1	yes	characteristic
17	C₂₃⋊#95	C₂²	1	3	yes	—
18	C₂⁴×C₂₃	C₂³	1	3	yes	—
19	#41×C₂₃	—	2	32	no	—
20	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
21	C₂×Q₈×C₂₃	D₄	1	8	yes	—
22	C₂×D₄×C₂₃	—	2	12	no	—
23	C₂×D₄×C₂₃	C₂×C₄	1	12	yes	—
24	C₂²×C₄×C₂₃	—	2	12	no	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	12	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂³	1	3	yes	—
28	C₂²×Dic₂₃	C₂³	1	6	yes	—
29	C₂²×Dic₂₃	C₂³	1	2	yes	—
30	C₂²⋊Dic₂₃	—	2	12	no	—
31	C₂²⋊Dic₂₃	—	4	6	no	—
32	C₄⋊Dic₂₃	—	2	12	no	—
33	C₄×Dic₂₃	—	2	12	no	—
34	C₂³×C₂₃	—	2	6	no	—
35	C₂³×C₂₃	—	2	6	no	—
36	C₂³×C₂₃	—	2	3	no	—
37	C₂³×C₂₃	—	2	3	no	—
38	C₂³×C₂₃	C₂²×C₄	1	6	yes	—
39	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
40	Q₈×C₂₃	C₂²⋊C₄	1	16	yes	complemented
41	D₄×C₂₃	—	2	24	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	24	no	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	12	yes	—
47	C₂×Dic₂₃	—	2	2	no	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	2	6	no	—
50	C₂×Dic₂₃	—	2	6	no	—
51	C₂×Dic₂₃	—	2	6	no	—
52	C₂×Dic₂₃	#41	1	4	yes	complemented
53	#2341	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	—	2	6	no	—
55	C₂²×C₂₃	—	2	6	no	—
56	C₂²×C₂₃	—	2	6	no	—
57	C₂²×C₂₃	—	4	6	no	—
58	C₂²×C₂₃	—	4	6	no	—
59	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
61	C₂²×C₂₃	C₂×#41	1	2	yes	—
62	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
64	C₄×C₂₃	—	2	2	no	—
65	C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
66	Dic₂₃	—	2	4	no	—
67	Dic₂₃	—	4	6	no	—
68	C₂²×#41	D₂₃	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	23	2	no	—
70	#390	—	23	6	no	—
71	#380	—	23	3	no	—
72	C₄×C₂²⋊C₄	—	23	3	no	—
73	C₂×C₂₃	—	4	6	no	—
74	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
76	C₂×C₂₃	#520	1	1	yes	characteristic
77	C₂×C₂₃	#548	1	2	yes	—
78	C₂×C₂₃	#550	1	2	yes	—
79	C₂×#41	—	2	8	no	—
80	C₂×#41	Dic₂₃	1	8	yes	complemented
81	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₂₃	1	3	yes	—
83	C₂³×C₄	C₂×D₂₃	1	3	yes	—
84	C₄×Q₈	—	46	8	no	—
85	C₂×C₄⋊C₄	—	23	6	no	—
86	C₂×C₂²⋊C₄	—	23	6	no	—
87	C₂×C₂²⋊C₄	—	23	6	no	—
88	C₂×C₄²	—	23	6	no	—
89	#95	—	23	1	no	—
90	#95	—	23	3	no	—
91	C₂₃	#2341	1	1	yes	characteristic, complemented, Sylow
92	C₂⁴	C₂²×D₂₃	1	3	yes	—
93	#41	—	2	32	no	—
94	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
95	C₂×Q₈	C₂²⋊D₂₃	1	8	yes	—
96	C₂×D₄	—	2	12	no	—
97	C₂×D₄	C₂×Dic₂₃	1	12	yes	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	—	23	2	no	—
100	C₂²×C₄	—	23	6	no	—
101	C₂²×C₄	C₂×Dic₂₃	1	12	yes	—
102	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₃	1	3	yes	—
104	C₄⋊C₄	—	46	12	no	—
105	C₂²⋊C₄	—	46	12	no	—
106	C₂²⋊C₄	—	92	6	no	—
107	C₄²	—	46	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	—	2	3	no	—
110	C₂³	—	2	3	no	—
111	C₂³	—	2	6	no	—
112	C₂³	C₂²×Dic₂₃	1	6	yes	—
113	C₂³	C₂³×D₂₃	1	1	yes	characteristic
114	Q₈	C₂²⋊Dic₂₃	1	16	yes	complemented
115	D₄	—	2	24	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	23	4	no	—
119	C₂×C₄	—	46	2	no	—
120	C₂×C₄	—	46	2	no	—
121	C₂×C₄	—	46	6	no	—
122	C₂×C₄	—	46	6	no	—
123	C₂×C₄	—	46	6	no	—
124	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
125	C₂×C₄	C₂²×Dic₂₃	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₂₃	1	12	yes	—
127	C₂²	—	2	6	no	—
128	C₂²	—	2	6	no	—
129	C₂²	—	2	6	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	4	6	no	—
132	C₂²	#41×D₂₃	1	2	yes	—
133	C₂²	#10625160	1	1	yes	characteristic
134	C₂²	#10625161	1	1	yes	characteristic
135	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
136	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
137	C₄	—	2	2	no	—
138	C₄	—	46	4	no	—
139	C₄	—	92	6	no	—
140	C₄	C₂×C₂²⋊Dic₂₃	1	12	yes	—
141	C₂	—	4	6	no	—
142	C₂	C₂²×C₂²⋊Dic₂₃	1	1	yes	characteristic
143	C₂	#41×Dic₂₃	1	1	yes	characteristic
144	C₂	#13092900	1	1	yes	characteristic
145	C₂	#13093074	1	2	yes	—
146	C₂	#13093086	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₂₃

C₂

yes

maximal, complemented

#13092320

C₂

yes

maximal, complemented

#13092277

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₂₃

C₂

yes

maximal, complemented

C₂²×#41×C₂₃

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

—

C₂×#41×C₂₃

C₄

yes

complemented

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₃

C₂²

yes

—

C₂³×C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

Q₈×Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

characteristic

C₂₃⋊#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

Q₈×C₂₃

C₂²⋊C₄

yes

complemented

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×D₄

yes

—

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₂₃

2₋¹⁺⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

—

Dic₂₃

—

Dic₂₃

—

C₂²×#41

D₂₃

yes

characteristic

C₂×C₄×Q₈

—

#390

—

#380

—

C₄×C₂²⋊C₄

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₃

C₄×#41

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#550

yes

—

C₂×#41

—

C₂×#41

Dic₂₃

yes

complemented

C₂²×Q₈

C₂×D₂₃

yes

characteristic

C₂²×D₄

C₂×D₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

C₂₃

#2341

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₂₃

yes

—

#41

—

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×Dic₂₃

yes

—

102

C₂²×C₄

C₂²×D₂₃

yes

characteristic

103

C₂²×C₄

C₂²×D₂₃

yes

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

C₂²×Dic₂₃

yes

—

113

C₂³

C₂³×D₂₃

yes

characteristic

114

Q₈

C₂²⋊Dic₂₃

yes

complemented

115

D₄

—

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

C₂²×Dic₂₃

yes

—

125

C₂×C₄

C₂²×Dic₂₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#41×D₂₃

yes

—

133

C₂²

#10625160

yes

characteristic

134

C₂²

#10625161

yes

characteristic

135

C₂²

C₂³×Dic₂₃

yes

characteristic

136

C₂²

C₂²×C₂²⋊D₂₃

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₂×C₂²⋊Dic₂₃

yes

—

141

C₂

—

142

C₂

C₂²×C₂²⋊Dic₂₃

yes

characteristic

143

C₂

#41×Dic₂₃

yes

characteristic

144

C₂

#13092900

yes

characteristic

145

C₂

#13093074

yes

—

146

C₂

#13093086

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting, prime radical	C₂²×#41×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂²×#41	D₂₃
hypocenter	C₂₃	#2341
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₃

commutator

C₂²×C₂₃

C₂³×C₄

Fitting, prime radical

C₂²×#41×C₂₃

C₂

socle, cosocle

C₂³×C₂₃

C₂⁴

hypercenter

C₂²×#41

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂³ ◁ C₂²×#41

N	H ≅ G/N	type
1. C₂×Q₈×Dic₂₃	C₂	semidirect
2. #13092320	C₂	semidirect
3. #13092277	C₂	semidirect
4. C₄×C₂²⋊Dic₂₃	C₂	semidirect
7. C₂×#41×C₂₃	C₄	semidirect
40. Q₈×C₂₃	C₂²⋊C₄	semidirect
52. C₂×Dic₂₃	#41	semidirect
80. C₂×#41	Dic₂₃	semidirect
91. C₂₃	#2341	semidirect
114. Q₈	C₂²⋊Dic₂₃	semidirect

H ≅ G/N

type

1. C₂×Q₈×Dic₂₃

C₂

semidirect

2. #13092320

C₂

semidirect

3. #13092277

C₂

semidirect

4. C₄×C₂²⋊Dic₂₃

C₂

semidirect

7. C₂×#41×C₂₃

C₄

semidirect

40. Q₈×C₂₃

C₂²⋊C₄

semidirect

52. C₂×Dic₂₃

#41

semidirect

80. C₂×#41

Dic₂₃

semidirect

91. C₂₃

#2341

semidirect

114. Q₈

C₂²⋊Dic₂₃

semidirect

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

subgroup	index
C₂×Q₈×Dic₂₃	2
#13092320	2
#13092277	2
C₄×C₂²⋊Dic₂₃	2
C₂²×#41×C₂₃	2
#2341	23

subgroup

index

C₂×Q₈×Dic₂₃

#13092320

#13092277

C₄×C₂²⋊Dic₂₃

C₂²×#41×C₂₃

#2341

Subgroups — #15676238

Conjugacy classes of subgroups

Complements

Maximal subgroups