Group #13093016 — subgroups

Subgroups — #13093016

Order 1472.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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, complemented
2	#10625117	C₂	1	2	yes	maximal, complemented
3	#10625101	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625069	C₂	1	2	yes	maximal, complemented
6	#10625065	C₂	1	2	yes	maximal, complemented
7	#10625061	C₂	1	2	yes	maximal, complemented
8	#10625057	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₂₃)⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
13	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×D₉₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₃	C₂²	1	2	yes	—
17	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	C₄⋊Dic₂₃	C₂²	1	2	yes	—
25	C₄⋊Dic₂₃	C₂²	1	2	yes	—
26	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₂₃	C₂²	1	2	yes	—
31	C₂³×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×D₂₃	C₂³	1	2	yes	—
33	Q₈×C₂₃	—	4	1	no	—
34	C₂×C₄×C₂₃	—	4	1	no	—
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×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²⋊D₂₃	—	2	4	no	—
40	C₂×Dic₂₃	—	2	2	no	—
41	C₂×Dic₂₃	—	4	1	no	—
42	C₂×Dic₂₃	C₂³	1	2	yes	—
43	C₂×Dic₂₃	C₂³	1	2	yes	—
44	C₂×Dic₂₃	C₂³	1	2	yes	—
45	D₉₂	—	4	1	no	—
46	C₄×D₂₃	—	4	2	no	—
47	C₂²×C₂₃	—	2	1	no	—
48	C₂²×C₂₃	—	2	1	no	—
49	C₂²×C₂₃	#41	1	2	yes	—
50	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₂₃	—	2	2	no	—
52	C₂×D₂₃	—	2	2	no	—
53	C₂×D₂₃	—	2	2	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	2	1	no	—
56	C₄×C₂₃	—	2	1	no	—
57	C₄×C₂₃	—	2	2	no	—
58	Dic₂₃	—	2	2	no	—
59	Dic₂₃	—	2	2	no	—
60	Dic₂₃	—	2	2	no	—
61	#539	—	23	1	no	maximal, Sylow
62	C₂×C₂₃	—	2	2	no	—
63	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
66	D₂₃	—	4	2	no	—
67	#126	—	23	2	no	—
68	#125	—	23	2	no	—
69	#123	—	23	2	no	—
70	#123	—	23	2	no	—
71	C₂²⋊Q₈	—	23	2	no	—
72	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
73	#121	—	23	1	no	—
74	C₄×D₄	—	23	2	no	—
75	C₂×C₄⋊C₄	—	23	1	no	—
76	C₂₃	#539	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
78	C₂×D₄	—	23	2	no	—
79	C₂×D₄	—	23	1	no	—
80	C₂²×C₄	—	23	1	no	—
81	C₂²×C₄	—	23	1	no	—
82	C₂²×C₄	—	23	2	no	—
83	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
84	C₄⋊C₄	—	23	2	no	—
85	C₄⋊C₄	—	23	2	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	1	no	—
88	C₄⋊C₄	—	23	2	no	—
89	C₄⋊C₄	—	23	2	no	—
90	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
91	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
92	C₂²⋊C₄	—	23	2	no	—
93	C₂²⋊C₄	—	23	2	no	—
94	C₂²⋊C₄	—	23	2	no	—
95	C₂²⋊C₄	—	23	2	no	—
96	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
97	C₄²	—	23	2	no	—
98	C₂³	—	23	2	no	—
99	C₂³	C₂²×D₂₃	1	1	yes	characteristic
100	Q₈	—	4	1	no	—
101	D₄	—	46	4	no	—
102	D₄	—	92	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	23	2	no	—
105	C₂×C₄	—	23	2	no	—
106	C₂×C₄	—	23	2	no	—
107	C₂×C₄	—	46	2	no	—
108	C₂×C₄	—	92	2	no	—
109	C₂×C₄	—	92	1	no	—
110	C₂×C₄	C₂²×D₂₃	1	2	yes	—
111	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₂₃	1	2	yes	—
113	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	46	2	no	—
117	C₂²	—	46	2	no	—
118	C₂²	—	46	2	no	—
119	C₂²	#67292	1	2	yes	—
120	C₂²	C₂³×D₂₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	46	2	no	—
126	C₄	—	46	2	no	—
127	C₄	—	46	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	92	2	no	—
130	C₂	#10625155	1	1	yes	characteristic
131	C₂	C₂×#67292	1	1	yes	characteristic
132	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#10625117

C₂

yes

maximal, complemented

#10625101

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₄

C₂

yes

maximal, characteristic, complemented

#10625069

C₂

yes

maximal, complemented

#10625065

C₂

yes

maximal, complemented

#10625061

C₂

yes

maximal, complemented

#10625057

C₂

yes

maximal, complemented

(C₂²⋊D₂₃)⋊C₄

C₂

yes

maximal, complemented

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#41

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#539

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₂₃

yes

characteristic, complemented

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂₃

#539

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₄²

—

C₂³

—

C₂³

C₂²×D₂₃

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×D₂₃

yes

—

111

C₂×C₄

C₂²×D₂₃

yes

characteristic

112

C₂×C₄

C₂²×D₂₃

yes

—

113

C₂×C₄

C₂²×D₂₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#67292

yes

—

120

C₂²

C₂³×D₂₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

—

130

C₂

#10625155

yes

characteristic

131

C₂

C₂×#67292

yes

characteristic

132

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	#10625101	C₂
hypercenter	C₂²⋊Q₈	D₂₃
hypocenter	C₂₃	#539
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

#10625101

C₂

hypercenter

C₂²⋊Q₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₂₃×C₂²⋊Q₈	C₂	semidirect
2. #10625117	C₂	semidirect
4. C₂×Dic₂₃⋊C₄	C₂	semidirect
5. #10625069	C₂	semidirect
6. #10625065	C₂	semidirect
7. #10625061	C₂	semidirect
8. #10625057	C₂	semidirect
9. (C₂²⋊D₂₃)⋊C₄	C₂	semidirect
72. C₂²⋊Q₈	D₂₃	semidirect
76. C₂₃	#539	semidirect

H ≅ G/N

type

1. C₂₃×C₂²⋊Q₈

C₂

semidirect

2. #10625117

C₂

semidirect

4. C₂×Dic₂₃⋊C₄

C₂

semidirect

5. #10625069

C₂

semidirect

6. #10625065

C₂

semidirect

7. #10625061

C₂

semidirect

8. #10625057

C₂

semidirect

9. (C₂²⋊D₂₃)⋊C₄

C₂

semidirect

72. C₂²⋊Q₈

D₂₃

semidirect

76. C₂₃

#539

semidirect

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

subgroup	index
C₂₃×C₂²⋊Q₈	2
#10625117	2
#10625101	2
C₂×Dic₂₃⋊C₄	2
#10625069	2
#10625065	2
#10625061	2
#10625057	2
(C₂²⋊D₂₃)⋊C₄	2
#539	23

subgroup

index

C₂₃×C₂²⋊Q₈

C₂×Dic₂₃⋊C₄

(C₂²⋊D₂₃)⋊C₄

Subgroups — #13093016

Conjugacy classes of subgroups

Complements

Maximal subgroups