Group #15675979 — subgroups

Subgroups — #15675979

Order 2944.

1104 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 309 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×#10625097	C₂	1	2	yes	maximal, complemented
2	#13092320	C₂	1	8	yes	maximal, complemented
3	#13092273	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁵×C₂₃	C₂²	1	2	yes	—
6	C₂²×D₄×C₂₃	—	2	8	no	—
7	C₂²×D₄×C₂₃	C₄	1	8	yes	complemented
8	C₂²×D₄×C₂₃	C₂²	1	4	yes	—
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₂₃	C₂²	1	16	yes	complemented
12	#10625097	—	2	8	no	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	complemented
14	C₂×C₄×Dic₂₃	C₂²	1	4	yes	complemented
15	C₂⁴×C₂₃	—	2	8	no	—
16	C₂⁴×C₂₃	—	4	8	no	—
17	C₂⁴×C₂₃	C₂×C₄	1	8	yes	—
18	C₂⁴×C₂₃	C₂³	1	4	yes	—
19	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₂₃	—	2	32	no	—
21	C₂×D₄×C₂₃	—	2	8	no	—
22	C₂×D₄×C₂₃	—	2	8	no	—
23	C₂×D₄×C₂₃	C₂×C₄	1	16	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₃	D₄	1	8	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×Dic₂₃	C₂³	1	8	yes	—
28	C₂²⋊Dic₂₃	—	2	8	no	—
29	C₂²⋊Dic₂₃	—	4	16	no	—
30	C₄⋊Dic₂₃	—	2	8	no	—
31	C₄×Dic₂₃	—	2	8	no	—
32	C₂³×C₂₃	—	2	4	no	—
33	C₂³×C₂₃	—	2	16	no	—
34	C₂³×C₂₃	—	2	8	no	—
35	C₂³×C₂₃	—	2	16	no	—
36	C₂³×C₂₃	—	4	16	no	—
37	C₂³×C₂₃	—	4	32	no	—
38	C₂³×C₂₃	C₂²×C₄	1	8	yes	—
39	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
40	C₂³×C₂₃	C₂×D₄	1	4	yes	—
41	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
42	D₄×C₂₃	—	2	32	no	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	16	yes	complemented
44	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
47	C₂×Dic₂₃	—	2	8	no	—
48	C₂×Dic₂₃	—	2	16	no	—
49	#2309	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	—	2	4	no	—
51	C₂²×C₂₃	—	2	8	no	—
52	C₂²×C₂₃	—	2	8	no	—
53	C₂²×C₂₃	—	2	16	no	—
54	C₂²×C₂₃	—	4	32	no	—
55	C₂²×C₂₃	—	4	16	no	—
56	C₂²×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
57	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
58	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
59	C₂²×C₂₃	H(5,𝔽₂)	1	4	yes	—
60	C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
61	Dic₂₃	—	4	8	no	—
62	C₂³×D₄	D₂₃	1	1	yes	characteristic
63	C₂×#117	—	23	2	no	—
64	#390	—	23	8	no	—
65	#379	—	23	4	no	—
66	C₂×C₂₃	—	2	4	no	—
67	C₂×C₂₃	—	4	8	no	—
68	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₂₃	#518	1	2	yes	—
70	C₂×C₂₃	#535	1	4	yes	—
71	C₂⁵	C₂×D₂₃	1	2	yes	—
72	C₂²×D₄	—	2	8	no	—
73	C₂²×D₄	Dic₂₃	1	8	yes	complemented
74	C₂²×D₄	C₂×D₂₃	1	4	yes	—
75	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
76	#117	—	46	8	no	—
77	C₂×C₄⋊C₄	—	23	4	no	—
78	C₂×C₂²⋊C₄	—	23	4	no	—
79	C₂×C₂²⋊C₄	—	23	16	no	—
80	C₂×C₄²	—	23	4	no	—
81	C₂₃	#2309	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	2	8	no	—
83	C₂⁴	—	4	8	no	—
84	C₂⁴	C₂×Dic₂₃	1	8	yes	—
85	C₂⁴	C₂²×D₂₃	1	4	yes	—
86	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
87	C₂×D₄	—	2	32	no	—
88	C₂×D₄	—	2	8	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	C₂×Dic₂₃	1	16	yes	—
91	C₂²×C₄	—	23	8	no	—
92	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
93	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	—
94	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
95	C₄⋊C₄	—	46	8	no	—
96	C₂²⋊C₄	—	46	8	no	—
97	C₂²⋊C₄	—	92	16	no	—
98	C₄²	—	46	8	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	16	no	—
101	C₂³	—	2	16	no	—
102	C₂³	—	2	8	no	—
103	C₂³	—	4	16	no	—
104	C₂³	—	4	32	no	—
105	C₂³	C₂²×Dic₂₃	1	8	yes	—
106	C₂³	C₂²×Dic₂₃	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
108	C₂³	C₂³×D₂₃	1	1	yes	characteristic
109	D₄	—	2	32	no	—
110	C₂×C₄	—	46	16	no	—
111	C₂×C₄	—	46	8	no	—
112	C₂×C₄	C₂²⋊Dic₂₃	1	16	yes	complemented
113	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
115	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
116	C₂²	—	2	4	no	—
117	C₂²	—	2	8	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	16	no	—
120	C₂²	—	4	32	no	—
121	C₂²	—	4	16	no	—
122	C₂²	C₂×C₂²⋊Dic₂₃	1	4	yes	—
123	C₂²	#10625155	1	4	yes	—
124	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
125	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
126	C₄	—	92	8	no	—
127	C₄	C₂×C₂²⋊Dic₂₃	1	8	yes	—
128	C₂	—	2	4	no	—
129	C₂	—	4	8	no	—
130	C₂	C₂²×C₂²⋊Dic₂₃	1	1	yes	characteristic
131	C₂	#13092892	1	2	yes	—
132	C₂	#13092984	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625097

C₂

yes

maximal, complemented

#13092320

C₂

yes

maximal, complemented

#13092273

C₂

yes

maximal, complemented

C₂³×D₄×C₂₃

C₂

yes

maximal, characteristic

C₂⁵×C₂₃

C₂²

yes

—

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₄

yes

complemented

C₂²×D₄×C₂₃

C₂²

yes

—

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₂²

yes

complemented

#10625097

—

C₂×C₄⋊Dic₂₃

C₂²

yes

complemented

C₂×C₄×Dic₂₃

C₂²

yes

complemented

C₂⁴×C₂₃

—

C₂⁴×C₂₃

—

C₂⁴×C₂₃

C₂×C₄

yes

—

C₂⁴×C₂₃

C₂³

yes

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

complemented

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

#2309

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂²×D₄

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

—

Dic₂₃

—

C₂³×D₄

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₂₃

#518

yes

—

C₂×C₂₃

#535

yes

—

C₂⁵

C₂×D₂₃

yes

—

C₂²×D₄

—

C₂²×D₄

Dic₂₃

yes

complemented

C₂²×D₄

C₂×D₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂₃

#2309

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₂₃

yes

—

C₂⁴

C₂²×D₂₃

yes

—

C₂⁴

C₂²×D₂₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂²×Dic₂₃

yes

—

106

C₂³

C₂²×Dic₂₃

yes

—

107

C₂³

C₂×C₂²⋊D₂₃

yes

—

108

C₂³

C₂³×D₂₃

yes

characteristic

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊Dic₂₃

yes

complemented

113

C₂×C₄

C₂²×Dic₂₃

yes

—

114

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

115

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₂²⋊Dic₂₃

yes

—

123

C₂²

#10625155

yes

—

124

C₂²

C₂³×Dic₂₃

yes

characteristic

125

C₂²

C₂²×C₂²⋊D₂₃

yes

—

126

C₄

—

127

C₄

C₂×C₂²⋊Dic₂₃

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×C₂²⋊Dic₂₃

yes

characteristic

131

C₂

#13092892

yes

—

132

C₂

#13092984

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting, prime radical	C₂³×D₄×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂³×D₄	D₂₃
hypocenter	C₂₃	#2309
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₃

commutator

C₂²×C₂₃

C₂³×C₄

Fitting, prime radical

C₂³×D₄×C₂₃

C₂

socle, cosocle

C₂³×C₂₃

C₂⁴

hypercenter

C₂³×D₄

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂³ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂×#10625097	C₂	semidirect
2. #13092320	C₂	semidirect
3. #13092273	C₂	semidirect
7. C₂²×D₄×C₂₃	C₄	semidirect
11. C₂×C₂²⋊Dic₂₃	C₂²	semidirect
13. C₂×C₄⋊Dic₂₃	C₂²	semidirect
14. C₂×C₄×Dic₂₃	C₂²	semidirect
43. C₂×C₄×C₂₃	C₂²⋊C₄	semidirect
73. C₂²×D₄	Dic₂₃	semidirect
81. C₂₃	#2309	semidirect
112. C₂×C₄	C₂²⋊Dic₂₃	semidirect

H ≅ G/N

type

1. C₂×#10625097

C₂

semidirect

2. #13092320

C₂

semidirect

3. #13092273

C₂

semidirect

7. C₂²×D₄×C₂₃

C₄

semidirect

11. C₂×C₂²⋊Dic₂₃

C₂²

semidirect

13. C₂×C₄⋊Dic₂₃

C₂²

semidirect

14. C₂×C₄×Dic₂₃

C₂²

semidirect

43. C₂×C₄×C₂₃

C₂²⋊C₄

semidirect

73. C₂²×D₄

Dic₂₃

semidirect

81. C₂₃

#2309

semidirect

112. C₂×C₄

C₂²⋊Dic₂₃

semidirect

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

subgroup	index
C₂×#10625097	2
#13092320	2
#13092273	2
C₂³×D₄×C₂₃	2
#2309	23

subgroup

index

C₂³×D₄×C₂₃

Subgroups — #15675979

Conjugacy classes of subgroups

Complements

Maximal subgroups