Group #13093027 — subgroups

Subgroups — #13093027

Order 1472.

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

normal subgroups only

This group has 192 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
2	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625115	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625097	C₂	1	1	yes	maximal, characteristic
5	#10625095	C₂	1	1	yes	maximal, characteristic
6	#10625061	C₂	1	2	yes	maximal, complemented
7	#10625061	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625060	C₂	1	2	yes	maximal, complemented
9	#10625058	C₂	1	1	yes	maximal, characteristic, complemented
10	#10625050	C₂	1	2	yes	maximal
11	#10625050	C₂	1	2	yes	maximal
12	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₂₃	C₂²	1	2	yes	—
22	C₄⋊Dic₂₃	C₂²	1	2	yes	—
23	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	Dic₂₃⋊C₄	C₂²	1	2	yes	—
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	Dic₂₃⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₂₃	C₂²	1	2	yes	—
32	C₄×Dic₂₃	C₂²	1	2	yes	—
33	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂³×C₂₃	C₂³	1	1	yes	characteristic
36	Q₈×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₂₃	C₂³	1	2	yes	—
46	C₂×Dic₂₃	C₂³	1	2	yes	—
47	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
48	Dic₄₆	—	2	2	no	—
49	C₂²×C₂₃	—	2	1	no	—
50	C₂²×C₂₃	—	2	1	no	—
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	#41	1	2	yes	—
57	Dic₂₃	—	2	2	no	—
58	Dic₂₃	—	2	1	no	—
59	Dic₂₃	—	2	2	no	—
60	Dic₂₃	—	2	1	no	—
61	Dic₂₃	—	2	1	no	—
62	Dic₂₃	—	2	1	no	—
63	#541	—	23	1	no	maximal, Sylow
64	C₂×C₂₃	—	4	1	no	—
65	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
66	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
67	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
68	C₄⋊Q₈	—	23	1	no	—
69	#126	—	23	2	no	—
70	#126	—	23	2	no	—
71	#125	—	23	2	no	—
72	#125	—	23	1	no	—
73	#125	—	23	2	no	—
74	C₂²⋊Q₈	—	23	1	no	—
75	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic
76	C₄×Q₈	—	23	1	no	—
77	C₄×Q₈	—	23	1	no	—
78	#117	—	23	1	no	—
79	C₂₃	#541	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₈	—	23	1	no	—
81	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
83	C₄⋊C₄	—	23	1	no	—
84	C₄⋊C₄	—	23	1	no	—
85	C₄⋊C₄	—	23	1	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	2	no	—
88	C₄⋊C₄	—	23	2	no	—
89	C₄⋊C₄	—	23	2	no	—
90	C₄⋊C₄	—	23	2	no	—
91	C₄⋊C₄	—	23	2	no	—
92	C₄⋊C₄	—	23	2	no	—
93	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
95	C₂²⋊C₄	—	23	1	no	—
96	C₂²⋊C₄	—	23	1	no	—
97	C₂²⋊C₄	—	23	2	no	—
98	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
99	C₄²	—	23	1	no	—
100	C₄²	—	23	1	no	—
101	C₄²	—	23	2	no	—
102	C₄²	—	23	2	no	—
103	C₂³	C₂²×D₂₃	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	46	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	23	1	no	—
108	C₂×C₄	—	23	1	no	—
109	C₂×C₄	—	23	1	no	—
110	C₂×C₄	—	23	2	no	—
111	C₂×C₄	—	23	1	no	—
112	C₂×C₄	—	23	2	no	—
113	C₂×C₄	C₂²×D₂₃	1	2	yes	—
114	C₂×C₄	C₂²×D₂₃	1	2	yes	—
115	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₂₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	46	2	no	—
125	C₄	—	46	1	no	—
126	C₄	—	46	1	no	—
127	C₄	—	46	2	no	—
128	C₄	—	46	1	no	—
129	C₄	—	46	1	no	—
130	C₄	#67290	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#67290	1	1	yes	characteristic
133	C₂	#10625158	1	1	yes	characteristic
134	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic

Q₈×Dic₂₃

C₂

yes

maximal, characteristic, complemented

#10625115

C₂

yes

maximal, characteristic, complemented

#10625097

C₂

yes

maximal, characteristic

#10625095

C₂

yes

maximal, characteristic

#10625061

C₂

yes

maximal, complemented

#10625061

C₂

yes

maximal, characteristic, complemented

#10625060

C₂

yes

maximal, complemented

#10625058

C₂

yes

maximal, characteristic, complemented

#10625050

C₂

yes

maximal

#10625050

C₂

yes

maximal

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

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

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

characteristic

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

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#541

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂₃

#541

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₂₃

yes

characteristic

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

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂³

C₂²×D₂₃

yes

characteristic

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₂₃

yes

—

114

C₂×C₄

C₂²×D₂₃

yes

—

115

C₂×C₄

C₂²×D₂₃

yes

characteristic

116

C₂×C₄

C₂²×D₂₃

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₂₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#67290

yes

—

131

C₂

—

132

C₂

C₂×#67290

yes

characteristic

133

C₂

#10625158

yes

characteristic

134

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	#541
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. Q₈×Dic₂₃	C₂	semidirect
3. #10625115	C₂	semidirect
6. #10625061	C₂	semidirect
7. #10625061	C₂	semidirect
8. #10625060	C₂	semidirect
9. #10625058	C₂	semidirect
79. C₂₃	#541	semidirect

H ≅ G/N

type

2. Q₈×Dic₂₃

C₂

semidirect

3. #10625115

C₂

semidirect

6. #10625061

C₂

semidirect

7. #10625061

C₂

semidirect

8. #10625060

C₂

semidirect

9. #10625058

C₂

semidirect

79. C₂₃

#541

semidirect

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

subgroup	index
C₂₃×C₂²⋊Q₈	2
Q₈×Dic₂₃	2
#10625115	2
#10625097	2
#10625095	2
#10625061	2
#10625060	2
#10625058	2
#10625050	2
#541	23

subgroup

index

C₂₃×C₂²⋊Q₈

Q₈×Dic₂₃

Subgroups — #13093027

Conjugacy classes of subgroups

Complements

Maximal subgroups