Group #13093136 — subgroups

Subgroups — #13093136

Order 1472.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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×#124	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	#10625107	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
6	#10625050	C₂	1	2	yes	maximal, complemented
7	#10625050	C₂	1	2	yes	maximal, complemented
8	#10625049	C₂	1	2	yes	maximal, complemented
9	#10625048	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₃	C₂²	1	2	yes	—
15	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂²⋊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	2	yes	—
26	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
33	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₃	C₂²	1	2	yes	—
35	C₂³×C₂₃	C₂³	1	2	yes	—
36	Q₈×C₂₃	—	2	2	no	—
37	D₄×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	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₃	—	2	4	no	—
43	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₃	C₂³	1	2	yes	—
45	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₂₃	C₂³	1	2	yes	—
47	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
49	Dic₄₆	—	2	2	no	—
50	Dic₄₆	—	2	2	no	—
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
55	C₄×C₂₃	—	2	1	no	—
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	1	no	—
58	C₄×C₂₃	#41	1	2	yes	—
59	Dic₂₃	—	2	2	no	—
60	Dic₂₃	—	2	1	no	—
61	Dic₂₃	—	2	2	no	—
62	Dic₂₃	—	2	1	no	—
63	Dic₂₃	—	2	1	no	—
64	Dic₂₃	#41	1	2	yes	—
65	#556	—	23	1	no	maximal, Sylow
66	C₂×C₂₃	—	4	2	no	—
67	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
70	C₄⋊Q₈	—	23	1	no	—
71	#126	—	23	2	no	—
72	#126	—	23	2	no	—
73	#124	—	23	1	no	—
74	#124	D₂₃	1	1	yes	characteristic
75	C₂²⋊Q₈	—	23	2	no	—
76	C₄×Q₈	—	23	1	no	—
77	C₄×Q₈	—	23	1	no	—
78	C₄×Q₈	—	23	1	no	—
79	C₄×D₄	—	23	1	no	—
80	#117	—	23	2	no	—
81	C₂₃	#556	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	23	1	no	—
83	C₂×Q₈	—	23	1	no	—
84	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
85	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
86	C₂²×C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	1	no	—
88	C₄⋊C₄	—	23	2	no	—
89	C₄⋊C₄	—	23	1	no	—
90	C₄⋊C₄	—	23	2	no	—
91	C₄⋊C₄	—	23	2	no	—
92	C₄⋊C₄	—	23	1	no	—
93	C₄⋊C₄	—	23	1	no	—
94	C₄⋊C₄	—	23	2	no	—
95	C₂²⋊C₄	—	23	2	no	—
96	C₂²⋊C₄	—	23	2	no	—
97	C₂²⋊C₄	—	23	2	no	—
98	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
99	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
100	C₄²	—	23	2	no	—
101	C₄²	—	23	2	no	—
102	C₄²	—	23	1	no	—
103	C₄²	—	23	1	no	—
104	C₄²	C₂×D₂₃	1	1	yes	characteristic
105	C₂³	C₂²×D₂₃	1	2	yes	—
106	Q₈	—	2	2	no	—
107	Q₈	—	46	2	no	—
108	Q₈	—	46	2	no	—
109	D₄	—	2	2	no	—
110	C₂×C₄	—	23	1	no	—
111	C₂×C₄	—	23	1	no	—
112	C₂×C₄	—	23	2	no	—
113	C₂×C₄	—	23	1	no	—
114	C₂×C₄	—	23	2	no	—
115	C₂×C₄	—	23	1	no	—
116	C₂×C₄	—	46	4	no	—
117	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₂₃	1	2	yes	—
119	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂³×D₂₃	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	23	2	no	—
129	C₄	—	46	2	no	—
130	C₄	—	46	2	no	—
131	C₄	—	46	1	no	—
132	C₄	—	46	1	no	—
133	C₄	—	46	1	no	—
134	C₄	#67290	1	2	yes	—
135	C₂	—	4	2	no	—
136	C₂	C₂×#67290	1	1	yes	characteristic
137	C₂	#41×D₂₃	1	1	yes	characteristic
138	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#124

C₂

yes

maximal, characteristic

Q₈×Dic₂₃

C₂

yes

maximal, characteristic, complemented

#10625115

C₂

yes

maximal, characteristic, complemented

#10625107

C₂

yes

maximal, characteristic

D₄×Dic₂₃

C₂

yes

maximal, characteristic

#10625050

C₂

yes

maximal, complemented

#10625050

C₂

yes

maximal, complemented

#10625049

C₂

yes

maximal, complemented

#10625048

C₂

yes

maximal, complemented

C₄×Dic₄₆

C₂

yes

maximal, characteristic, complemented

C₄×Dic₄₆

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×Dic₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

—

Q₈×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

Dic₄₆

—

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

#41

yes

—

#556

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

C₄⋊Q₈

—

—

—

—

D₂₃

yes

characteristic

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂₃

#556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

C₂×D₂₃

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×D₂₃

yes

characteristic

105

C₂³

C₂²×D₂₃

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×D₂₃

yes

characteristic

118

C₂×C₄

C₂²×D₂₃

yes

—

119

C₂×C₄

C₂²×D₂₃

yes

characteristic

120

C₂×C₄

C₂²×D₂₃

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂³×D₂₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#67290

yes

—

135

C₂

—

136

C₂

C₂×#67290

yes

characteristic

137

C₂

#41×D₂₃

yes

characteristic

138

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#124	C₂
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	#124	D₂₃
hypocenter	C₂₃	#556
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#124

C₂

prime radical

C₂×D₄×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. Q₈×Dic₂₃	C₂	semidirect
3. #10625115	C₂	semidirect
6. #10625050	C₂	semidirect
7. #10625050	C₂	semidirect
8. #10625049	C₂	semidirect
9. #10625048	C₂	semidirect
10. C₄×Dic₄₆	C₂	semidirect
11. C₄×Dic₄₆	C₂	semidirect
81. C₂₃	#556	semidirect

H ≅ G/N

type

2. Q₈×Dic₂₃

C₂

semidirect

3. #10625115

C₂

semidirect

6. #10625050

C₂

semidirect

7. #10625050

C₂

semidirect

8. #10625049

C₂

semidirect

9. #10625048

C₂

semidirect

10. C₄×Dic₄₆

C₂

semidirect

11. C₄×Dic₄₆

C₂

semidirect

81. C₂₃

#556

semidirect

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

subgroup	index
C₂₃×#124	2
Q₈×Dic₂₃	2
#10625115	2
#10625107	2
D₄×Dic₂₃	2
#10625050	2
#10625049	2
#10625048	2
C₄×Dic₄₆	2
#556	23

subgroup

index

C₂₃×#124

Q₈×Dic₂₃

D₄×Dic₂₃

C₄×Dic₄₆

Subgroups — #13093136

Conjugacy classes of subgroups

Complements

Maximal subgroups