Group #13092969 — subgroups

Subgroups — #13092969

Order 1472.

222 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 109 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	#10625095	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
6	#10625060	C₂	1	2	yes	maximal, complemented
7	#10625059	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625058	C₂	1	1	yes	maximal, characteristic, complemented
9	#10625058	C₂	1	2	yes	maximal, complemented
10	#10625049	C₂	1	2	yes	maximal
11	#10625048	C₂	1	2	yes	maximal
12	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₆	C₂²	1	2	yes	—
17	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₄⋊Dic₂₃	C₂²	1	2	yes	—
23	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
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	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₃	—	2	2	no	—
43	C₂×Dic₂₃	Q₈	1	4	yes	complemented
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	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
50	Dic₄₆	—	2	2	no	—
51	Dic₄₆	—	2	4	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	1	no	—
54	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	1	no	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	#41	1	2	yes	—
60	Dic₂₃	—	2	2	no	—
61	Dic₂₃	—	2	2	no	—
62	Dic₂₃	—	2	1	no	—
63	Dic₂₃	C₂×Q₈	1	2	yes	—
64	Dic₂₃	C₂×Q₈	1	2	yes	—
65	Dic₂₃	#41	1	2	yes	—
66	#533	—	23	1	no	maximal, Sylow
67	C₂×C₂₃	—	2	2	no	—
68	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
71	C₄⋊Q₈	—	23	1	no	—
72	C₄⋊Q₈	—	23	1	no	—
73	#125	—	23	2	no	—
74	C₂²⋊Q₈	—	23	1	no	—
75	C₂²⋊Q₈	—	23	2	no	—
76	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic
77	C₄×Q₈	—	23	1	no	—
78	C₄×Q₈	—	23	1	no	—
79	C₄×Q₈	—	23	2	no	—
80	#117	—	23	2	no	—
81	C₂×C₄²	—	23	1	no	—
82	C₂₃	#533	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	23	2	no	—
84	C₂×Q₈	—	23	1	no	—
85	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
86	C₂²×C₄	—	23	1	no	—
87	C₂²×C₄	—	23	1	no	—
88	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
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	—	23	2	no	—
94	C₄⋊C₄	—	23	2	no	—
95	C₄⋊C₄	—	23	1	no	—
96	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
97	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
98	C₂²⋊C₄	—	23	2	no	—
99	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
100	C₄²	—	23	1	no	—
101	C₄²	—	23	1	no	—
102	C₄²	—	23	1	no	—
103	C₄²	—	23	1	no	—
104	C₄²	—	23	2	no	—
105	C₄²	—	23	2	no	—
106	C₂³	C₂²×D₂₃	1	1	yes	characteristic
107	Q₈	—	2	2	no	—
108	Q₈	—	46	4	no	—
109	Q₈	—	46	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	23	4	no	—
112	C₂×C₄	—	23	1	no	—
113	C₂×C₄	—	23	1	no	—
114	C₂×C₄	—	23	1	no	—
115	C₂×C₄	—	23	1	no	—
116	C₂×C₄	—	23	2	no	—
117	C₂×C₄	—	23	2	no	—
118	C₂×C₄	—	46	2	no	—
119	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₂₃	1	2	yes	—
121	C₂×C₄	C₂²×D₂₃	1	2	yes	—
122	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	Q₈×D₂₃	1	2	yes	—
126	C₂²	C₂³×D₂₃	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	—	23	2	no	—
131	C₄	—	23	2	no	—
132	C₄	—	23	2	no	—
133	C₄	—	46	2	no	—
134	C₄	—	46	1	no	—
135	C₄	—	46	2	no	—
136	C₄	#67290	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#67290	1	1	yes	characteristic
139	C₂	C₂×Q₈×D₂₃	1	1	yes	characteristic
140	C₂	#41×D₂₃	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

#10625095

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₃

C₂

yes

maximal, characteristic

#10625060

C₂

yes

maximal, complemented

#10625059

C₂

yes

maximal, characteristic, complemented

#10625058

C₂

yes

maximal, characteristic, complemented

#10625058

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

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

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

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

characteristic

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

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

Q₈

yes

complemented

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

Dic₄₆

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×Q₈

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×Q₈

yes

—

Dic₂₃

C₂×Q₈

yes

—

Dic₂₃

#41

yes

—

#533

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂²×Q₈

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂₃

#533

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

C₂×D₂₃

yes

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₂³

C₂²×D₂₃

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×D₂₃

yes

characteristic

120

C₂×C₄

C₂²×D₂₃

yes

—

121

C₂×C₄

C₂²×D₂₃

yes

—

122

C₂×C₄

C₂²×D₂₃

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

Q₈×D₂₃

yes

—

126

C₂²

C₂³×D₂₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#67290

yes

—

137

C₂

—

138

C₂

C₂×#67290

yes

characteristic

139

C₂

C₂×Q₈×D₂₃

yes

characteristic

140

C₂

#41×D₂₃

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	#533
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. #10625060	C₂	semidirect
7. #10625059	C₂	semidirect
8. #10625058	C₂	semidirect
9. #10625058	C₂	semidirect
43. C₂×Dic₂₃	Q₈	semidirect
82. C₂₃	#533	semidirect

H ≅ G/N

type

2. Q₈×Dic₂₃

C₂

semidirect

3. #10625115

C₂

semidirect

6. #10625060

C₂

semidirect

7. #10625059

C₂

semidirect

8. #10625058

C₂

semidirect

9. #10625058

C₂

semidirect

43. C₂×Dic₂₃

Q₈

semidirect

82. C₂₃

#533

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
#10625095	2
C₂×C₄×Dic₂₃	2
#10625060	2
#10625059	2
#10625058	2
#10625049	2
#10625048	2
#533	23

subgroup

index

C₂₃×C₂²⋊Q₈

Q₈×Dic₂₃

C₂×C₄×Dic₂₃

Subgroups — #13092969

Conjugacy classes of subgroups

Complements

Maximal subgroups