Group #15676136 — subgroups

Subgroups — #15676136

Order 2944.

444 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 197 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	#13092313	C₂	1	2	yes	maximal
3	#13092312	C₂	1	4	yes	maximal
4	#13092298	C₂	1	4	yes	maximal
5	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal
6	#13092263	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
13	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
16	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
17	#10625041	C₄	1	8	yes	complemented
18	C₂₃⋊#95	C₂²	1	2	yes	—
19	C₂₃⋊#95	C₂²	1	2	yes	—
20	C₂₃⋊#95	C₂²	1	4	yes	—
21	C₂₃⋊#95	C₂²	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₂₃	C₂³	1	4	yes	—
28	C₂²×Dic₂₃	C₂³	1	2	yes	—
29	C₄⋊C₄×C₂₃	—	2	4	no	—
30	C₄⋊C₄×C₂₃	—	4	4	no	—
31	C₄²×C₂₃	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	—
33	Dic₂₃⋊C₄	—	4	4	no	—
34	Dic₂₃⋊C₄	C₂×C₄	1	16	yes	—
35	C₄×Dic₂₃	—	2	2	no	—
36	C₄×Dic₂₃	—	4	4	no	—
37	C₄×Dic₂₃	Q₈	1	4	yes	—
38	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	2	4	no	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₂×Dic₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	2	4	no	—
50	C₂×Dic₂₃	—	2	4	no	—
51	C₂×Dic₂₃	—	2	2	no	—
52	C₂×Dic₂₃	—	2	1	no	—
53	C₂×Dic₂₃	—	2	1	no	—
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	2	2	no	—
56	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
57	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
58	#2330	—	23	1	no	maximal, Sylow
59	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
60	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
62	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
63	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
64	C₄×C₂₃	—	2	4	no	—
65	C₄×C₂₃	—	4	4	no	—
66	C₄×C₂₃	C₄×Q₈	1	4	yes	—
67	Dic₂₃	—	2	8	no	—
68	Dic₂₃	—	2	4	no	—
69	Dic₂₃	—	4	1	no	—
70	Dic₂₃	—	4	1	no	—
71	C₂×#125	—	23	1	no	—
72	#389	—	23	4	no	—
73	#389	—	23	2	no	—
74	#387	—	23	4	no	—
75	#384	D₂₃	1	1	yes	characteristic
76	C₄×C₄⋊C₄	—	23	2	no	—
77	#376	—	23	1	no	—
78	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
79	C₂×C₂₃	#520	1	2	yes	—
80	C₂×C₂₃	#540	1	1	yes	characteristic
81	C₂×C₂₃	#541	1	1	yes	characteristic
82	C₂×C₂₃	#544	1	2	yes	—
83	#125	—	23	8	no	—
84	C₂×C₄⋊C₄	—	23	4	no	—
85	C₂×C₄⋊C₄	—	23	2	no	—
86	C₂×C₄⋊C₄	—	23	4	no	—
87	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
88	C₂×C₄⋊C₄	C₂×D₂₃	1	4	yes	—
89	C₂×C₄²	—	23	1	no	—
90	C₂×C₄²	—	23	1	no	—
91	C₂×C₄²	—	23	4	no	—
92	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
93	#95	—	23	2	no	—
94	#95	—	23	2	no	—
95	#95	—	23	4	no	—
96	#95	—	23	4	no	—
97	C₂₃	#2330	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₄	—	23	2	no	—
99	C₂²×C₄	—	23	4	no	—
100	C₂²×C₄	—	23	1	no	—
101	C₂²×C₄	—	23	1	no	—
102	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
103	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	23	16	no	—
108	C₄⋊C₄	—	23	8	no	—
109	C₄⋊C₄	—	92	4	no	—
110	C₄²	—	23	4	no	—
111	C₄²	—	46	2	no	—
112	C₄²	—	92	4	no	—
113	C₄²	C₄×D₂₃	1	4	yes	—
114	C₂³	C₂³×D₂₃	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	23	8	no	—
121	C₂×C₄	—	23	4	no	—
122	C₂×C₄	—	46	2	no	—
123	C₂×C₄	—	46	4	no	—
124	C₂×C₄	—	46	4	no	—
125	C₂×C₄	—	46	2	no	—
126	C₂×C₄	—	46	1	no	—
127	C₂×C₄	—	46	1	no	—
128	C₂×C₄	—	46	2	no	—
129	C₂×C₄	—	46	2	no	—
130	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
131	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
132	C₂×C₄	#67290	1	2	yes	—
133	C₂×C₄	Q₈×D₂₃	1	2	yes	—
134	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
135	C₂²	C₂×#67290	1	1	yes	characteristic
136	C₂²	#10625155	1	2	yes	—
137	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
138	C₂²	#10625158	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	4	4	no	—
141	C₄	—	46	8	no	—
142	C₄	—	46	4	no	—
143	C₄	—	92	1	no	—
144	C₄	—	92	1	no	—
145	C₄	#10625058	1	4	yes	—
146	C₂	C₂×#10625058	1	1	yes	characteristic
147	C₂	#13092898	1	2	yes	—
148	C₂	#13093021	1	1	yes	characteristic
149	C₂	#13093028	1	1	yes	characteristic
150	C₂	#13093044	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625041

C₂

yes

maximal, characteristic

#13092313

C₂

yes

maximal

#13092312

C₂

yes

maximal

#13092298

C₂

yes

maximal

C₄⋊C₄×Dic₂₃

C₂

yes

maximal

#13092263

C₂

yes

maximal, characteristic

C₂₃×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

C₂²

yes

—

C₂×Dic₂₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

#10625041

C₄

yes

complemented

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂₃⋊#95

C₂²

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

C₂×C₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

Q₈

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂×Q₈

yes

—

#2330

—

maximal, Sylow

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×Q₈

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

characteristic

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₂²×C₂₃

2₋¹⁺⁴

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic

C₄×C₄⋊C₄

—

#376

—

C₂×C₂₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₂₃

#520

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

—

C₂₃

#2330

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₂₃

yes

—

103

C₂²×C₄

C₂²×D₂₃

yes

—

104

C₂²×C₄

C₂²×D₂₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₄×D₂₃

yes

—

114

C₂³

C₂³×D₂₃

yes

characteristic

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

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×C₄×D₂₃

yes

—

131

C₂×C₄

C₂×C₄×D₂₃

yes

—

132

C₂×C₄

#67290

yes

—

133

C₂×C₄

Q₈×D₂₃

yes

—

134

C₂²

C₂²×C₄×D₂₃

yes

characteristic

135

C₂²

C₂×#67290

yes

characteristic

136

C₂²

#10625155

yes

—

137

C₂²

C₂×Q₈×D₂₃

yes

characteristic

138

C₂²

#10625158

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#10625058

yes

—

146

C₂

C₂×#10625058

yes

characteristic

147

C₂

#13092898

yes

—

148

C₂

#13093021

yes

characteristic

149

C₂

#13093028

yes

characteristic

150

C₂

#13093044

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#384	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	#384	D₂₃
hypocenter	C₂₃	#2330
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₃

commutator

C₂²×C₂₃

C₂³×C₄

Fitting

C₂₃×#384

C₂

socle, prime radical, cosocle

C₂³×C₂₃

C₂⁴

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
17. #10625041	C₄	semidirect
97. C₂₃	#2330	semidirect

H ≅ G/N

type

17. #10625041

C₄

semidirect

97. C₂₃

#2330

semidirect

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

subgroup	index
C₂×#10625041	2
#13092313	2
#13092312	2
#13092298	2
C₄⋊C₄×Dic₂₃	2
#13092263	2
C₂₃×#384	2
#2330	23

subgroup

index

C₄⋊C₄×Dic₂₃

C₂₃×#384

Subgroups — #15676136

Conjugacy classes of subgroups

Complements

Maximal subgroups