Group #13093046 — subgroups

Subgroups — #13093046

Order 1472.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10625115	C₂	1	2	yes	maximal, complemented
3	#10625095	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#10625060	C₂	1	2	yes	maximal, complemented
6	#10625060	C₂	1	2	yes	maximal, complemented
7	#10625059	C₂	1	2	yes	maximal, complemented
8	#10625049	C₂	1	2	yes	maximal
9	#10625048	C₂	1	2	yes	maximal
10	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₆	C₂²	1	2	yes	—
15	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
20	C₄⋊Dic₂₃	C₂²	1	2	yes	—
21	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	Dic₂₃⋊C₄	C₂²	1	2	yes	—
23	Dic₂₃⋊C₄	C₂²	1	2	yes	—
24	Dic₂₃⋊C₄	C₂²	1	2	yes	—
25	Dic₂₃⋊C₄	C₂²	1	2	yes	—
26	Dic₂₃⋊C₄	C₂²	1	2	yes	—
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₂₃	C₂²	1	2	yes	—
30	C₄×Dic₂₃	C₂²	1	2	yes	—
31	C₂³×C₂₃	C₂³	1	1	yes	characteristic
32	Q₈×C₂₃	—	4	1	no	—
33	C₂×C₄×C₂₃	—	4	1	no	—
34	C₂×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×Dic₂₃	—	4	1	no	—
39	C₂×Dic₂₃	Q₈	1	4	yes	complemented
40	C₂×Dic₂₃	C₂³	1	2	yes	—
41	C₂×Dic₂₃	C₂³	1	2	yes	—
42	C₂×Dic₂₃	C₂³	1	2	yes	—
43	C₂×Dic₂₃	C₂³	1	2	yes	—
44	Dic₄₆	—	4	1	no	—
45	Dic₄₆	—	4	2	no	—
46	C₂²×C₂₃	—	2	1	no	—
47	C₂²×C₂₃	—	2	1	no	—
48	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	—	2	2	no	—
52	C₄×C₂₃	—	2	1	no	—
53	C₄×C₂₃	—	2	2	no	—
54	Dic₂₃	—	2	2	no	—
55	Dic₂₃	—	2	2	no	—
56	Dic₂₃	—	2	2	no	—
57	Dic₂₃	C₂×Q₈	1	4	yes	—
58	#544	—	23	1	no	maximal, Sylow
59	C₂×C₂₃	—	2	2	no	—
60	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
61	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
62	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
63	C₄⋊Q₈	—	23	2	no	—
64	C₄⋊Q₈	—	23	2	no	—
65	#125	—	23	2	no	—
66	#125	—	23	2	no	—
67	C₂²⋊Q₈	—	23	1	no	—
68	C₂²⋊Q₈	—	23	2	no	—
69	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic
70	#117	—	23	2	no	—
71	C₂×C₄⋊C₄	—	23	1	no	—
72	C₂₃	#544	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	23	1	no	—
74	C₂×Q₈	—	23	2	no	—
75	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
76	C₂²×C₄	—	23	1	no	—
77	C₂²×C₄	—	23	1	no	—
78	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
79	C₄⋊C₄	—	23	2	no	—
80	C₄⋊C₄	—	23	2	no	—
81	C₄⋊C₄	—	23	2	no	—
82	C₄⋊C₄	—	23	2	no	—
83	C₄⋊C₄	—	23	2	no	—
84	C₄⋊C₄	—	23	1	no	—
85	C₄⋊C₄	—	23	2	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	2	no	—
88	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
90	C₂²⋊C₄	—	23	2	no	—
91	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
92	C₄²	—	23	2	no	—
93	C₄²	—	23	2	no	—
94	C₂³	C₂²×D₂₃	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	92	1	no	—
97	Q₈	—	92	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	23	4	no	—
100	C₂×C₄	—	23	2	no	—
101	C₂×C₄	—	23	2	no	—
102	C₂×C₄	—	23	2	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	92	1	no	—
105	C₂×C₄	C₂²×D₂₃	1	2	yes	—
106	C₂×C₄	C₂²×D₂₃	1	2	yes	—
107	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₂₃	1	2	yes	—
112	C₂²	C₂³×D₂₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	23	4	no	—
118	C₄	—	46	2	no	—
119	C₄	—	46	2	no	—
120	C₄	—	46	2	no	—
121	C₂	—	2	2	no	—
122	C₂	#10625155	1	1	yes	characteristic
123	C₂	C₂×Q₈×D₂₃	1	1	yes	characteristic
124	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic

#10625115

C₂

yes

maximal, complemented

#10625095

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₄

C₂

yes

maximal, characteristic

#10625060

C₂

yes

maximal, complemented

#10625060

C₂

yes

maximal, complemented

#10625059

C₂

yes

maximal, complemented

#10625049

C₂

yes

maximal

#10625048

C₂

yes

maximal

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

Q₈

yes

complemented

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

Dic₄₆

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×Q₈

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×Q₈

yes

—

#544

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂²×Q₈

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₂₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂₃

#544

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₄²

—

C₄²

—

C₂³

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×D₂₃

yes

—

106

C₂×C₄

C₂²×D₂₃

yes

—

107

C₂×C₄

C₂²×D₂₃

yes

characteristic

108

C₂×C₄

C₂²×D₂₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₂₃

yes

—

112

C₂²

C₂³×D₂₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#10625155

yes

characteristic

123

C₂

C₂×Q₈×D₂₃

yes

characteristic

124

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₂²⋊Q₈	C₂
prime radical	C₂³×C₂₃	C₂³
hypercenter	C₂²⋊Q₈	D₂₃
hypocenter	C₂₃	#544
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×C₂²⋊Q₈

C₂

prime radical

C₂³×C₂₃

C₂³

hypercenter

C₂²⋊Q₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
2. #10625115	C₂	semidirect
5. #10625060	C₂	semidirect
6. #10625060	C₂	semidirect
7. #10625059	C₂	semidirect
39. C₂×Dic₂₃	Q₈	semidirect
72. C₂₃	#544	semidirect

H ≅ G/N

type

2. #10625115

C₂

semidirect

5. #10625060

C₂

semidirect

6. #10625060

C₂

semidirect

7. #10625059

C₂

semidirect

39. C₂×Dic₂₃

Q₈

semidirect

72. C₂₃

#544

semidirect

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

subgroup	index
C₂₃×C₂²⋊Q₈	2
#10625115	2
#10625095	2
C₂×Dic₂₃⋊C₄	2
#10625060	2
#10625059	2
#10625049	2
#10625048	2
#544	23

subgroup

index

C₂₃×C₂²⋊Q₈

C₂×Dic₂₃⋊C₄

Subgroups — #13093046

Conjugacy classes of subgroups

Complements

Maximal subgroups