Group #15686071 — subgroups

Subgroups — #15686071

Order 2944.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	#13092767	C₂	1	3	yes	maximal
3	#13092712	C₂	1	3	yes	maximal
4	#13092624	C₂	1	3	yes	maximal
5	Q₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#67275	C₂	1	3	yes	maximal
7	C₂₃×#557	C₂	1	1	yes	maximal, characteristic
8	C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₂₃×#125	—	2	3	no	—
10	C₄×Q₈×C₂₃	—	2	1	no	—
11	C₄×Q₈×C₂₃	C₂²	1	3	yes	—
12	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	Q₈×Dic₂₃	—	2	1	no	—
14	C₂×#67275	C₂²	1	3	yes	—
15	#10625059	—	2	3	no	—
16	#10625040	C₂²	1	3	yes	—
17	C₄×Dic₄₆	C₂²	1	3	yes	—
18	#10625017	C₂²	1	6	yes	—
19	#10625017	C₂²	1	3	yes	—
20	#10624992	C₂²	1	1	yes	characteristic
21	#10624989	C₂²	1	3	yes	—
22	C₄⋊(C₂₃⋊C₈)	C₂²	1	3	yes	—
23	C₄×C₂₃⋊C₈	C₂²	1	3	yes	—
24	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₂₃	C₂³	1	3	yes	—
26	C₂×Dic₄₆	—	2	3	no	—
27	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₃	—	2	3	no	—
29	C₄⋊C₄×C₂₃	—	2	3	no	—
30	C₄⋊C₄×C₂₃	—	2	3	no	—
31	C₄⋊C₄×C₂₃	D₄	1	3	yes	—
32	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₄²×C₂₃	—	2	3	no	—
34	C₄²×C₂₃	C₂³	1	3	yes	—
35	#67275	—	2	6	no	—
36	C₄⋊Dic₂₃	—	2	3	no	—
37	C₄⋊Dic₂₃	C₂³	1	3	yes	—
38	Dic₂₃⋊C₄	—	2	3	no	—
39	C₄×Dic₂₃	—	2	3	no	—
40	C₂×C₂₃⋊C₈	C₂³	1	3	yes	—
41	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
42	Q₈×C₂₃	—	2	3	no	—
43	Q₈×C₂₃	—	2	3	no	—
44	Q₈×C₂₃	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₂₃	—	2	1	no	—
46	C₂×C₄×C₂₃	—	2	3	no	—
47	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₂₃	—	2	1	no	—
51	C₂×Dic₂₃	—	2	3	no	—
52	Dic₄₆	—	2	1	no	—
53	Dic₄₆	—	8	3	no	—
54	Dic₄₆	#41	1	2	yes	—
55	C₂₃⋊C₈	—	2	3	no	—
56	C₂₃⋊C₈	—	2	1	no	—
57	#3363	—	23	1	no	maximal, Sylow
58	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₂₃	—	2	3	no	—
60	C₄×C₂₃	—	4	3	no	—
61	C₄×C₂₃	—	4	1	no	—
62	C₄×C₂₃	C₂×Q₁₆	1	6	yes	—
63	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
64	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
65	Dic₂₃	—	2	2	no	—
66	Dic₂₃	—	4	3	no	—
67	Q₈²	—	23	1	no	—
68	#557	D₂₃	1	1	yes	characteristic
69	#487	—	23	3	no	—
70	#477	—	23	3	no	—
71	#462	—	23	3	no	—
72	C₈×Q₈	—	23	1	no	—
73	C₄×Q₁₆	—	23	3	no	—
74	C₂×C₂₃	#548	1	1	yes	characteristic
75	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₂₃	#577	1	1	yes	characteristic
77	C₂×Q₁₆	—	23	3	no	—
78	C₄⋊Q₈	—	23	3	no	—
79	C₄⋊Q₈	—	46	3	no	—
80	C₄⋊Q₈	C₂×D₂₃	1	3	yes	—
81	#125	—	2	3	no	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	23	3	no	—
84	C₄×Q₈	—	46	1	no	—
85	C₄×Q₈	C₂×D₂₃	1	3	yes	—
86	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
87	#107	—	23	3	no	—
88	C₄⋊C₈	—	23	3	no	—
89	#103	—	23	3	no	—
90	#103	—	23	6	no	—
91	#103	—	23	1	no	—
92	C₄×C₈	—	23	3	no	—
93	C₂₃	#3363	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	23	1	no	—
95	C₂×Q₈	—	46	3	no	—
96	C₂×Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₂₃	1	3	yes	—
98	Q₁₆	—	46	6	no	—
99	C₂×C₈	—	23	1	no	—
100	C₂×C₈	—	23	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	23	3	no	—
105	C₄⋊C₄	—	46	3	no	—
106	C₄⋊C₄	—	46	3	no	—
107	C₄⋊C₄	C₂²⋊D₂₃	1	3	yes	—
108	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	46	3	no	—
111	C₄²	C₂²×D₂₃	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	23	2	no	—
115	Q₈	—	46	1	no	—
116	Q₈	—	184	3	no	—
117	Q₈	#67275	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	46	1	no	—
121	C₂×C₄	—	46	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
124	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
125	C₈	—	46	3	no	—
126	C₈	—	46	1	no	—
127	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	1	no	—
130	C₄	—	4	3	no	—
131	C₄	—	46	2	no	—
132	C₄	—	92	3	no	—
133	C₄	C₂×#67275	1	6	yes	—
134	C₄	#41×D₂₃	1	1	yes	characteristic
135	C₄	#10625161	1	1	yes	characteristic
136	C₂	#13093074	1	1	yes	characteristic
137	C₂	C₂²×#67275	1	1	yes	characteristic
138	C₂	#13093225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₄₆

C₂

yes

maximal, characteristic

#13092767

C₂

yes

maximal

#13092712

C₂

yes

maximal

#13092624

C₂

yes

maximal

Q₈×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₄×#67275

C₂

yes

maximal

C₂₃×#557

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#125

—

C₄×Q₈×C₂₃

—

C₄×Q₈×C₂₃

C₂²

yes

—

C₄×Q₈×C₂₃

C₂²

yes

characteristic

Q₈×Dic₂₃

—

C₂×#67275

C₂²

yes

—

#10625059

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

C₂²

yes

—

#10625017

C₂²

yes

—

#10625017

C₂²

yes

—

#10624992

C₂²

yes

characteristic

#10624989

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×Q₈×C₂₃

C₂³

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

—

#67275

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

Q₁₆

yes

complemented

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#3363

—

maximal, Sylow

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

C₂×#41

yes

characteristic

C₄×C₂₃

2₋¹⁺⁴

yes

characteristic

Dic₂₃

—

Dic₂₃

—

Q₈²

—

#557

D₂₃

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×C₂₃

#548

yes

characteristic

C₂×C₂₃

C₂²×Q₁₆

yes

characteristic

C₂×C₂₃

#577

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₃

yes

—

C₄×Q₈

C₂×D₂₃

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₂₃

#3363

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

characteristic

C₂×Q₈

C₂²×D₂₃

yes

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²⋊D₂₃

yes

—

108

C₄⋊C₄

C₂²×D₂₃

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₂₃

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

184

—

117

Q₈

#67275

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

123

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

124

C₂×C₄

C₂³×D₂₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

C₂×#67275

yes

—

134

C₄

#41×D₂₃

yes

characteristic

135

C₄

#10625161

yes

characteristic

136

C₂

#13093074

yes

characteristic

137

C₂

C₂²×#67275

yes

characteristic

138

C₂

#13093225

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#557	C₂
socle, prime radical	C₂²×C₂₃	C₂²×D₄
hypercenter	#557	D₂₃
hypocenter	C₂₃	#3363
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#557

C₂

socle, prime radical

C₂²×C₂₃

C₂²×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
44. Q₈×C₂₃	Q₁₆	semidirect
93. C₂₃	#3363	semidirect
117. Q₈	#67275	semidirect

H ≅ G/N

type

44. Q₈×C₂₃

Q₁₆

semidirect

93. C₂₃

#3363

semidirect

117. Q₈

#67275

semidirect

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

subgroup	index
Q₈×Dic₄₆	2
#13092767	2
#13092712	2
#13092624	2
Q₈×C₂₃⋊C₈	2
C₄×#67275	2
C₂₃×#557	2
#3363	23

subgroup

index

Q₈×Dic₄₆

Q₈×C₂₃⋊C₈

C₂₃×#557

Subgroups — #15686071

Conjugacy classes of subgroups

Complements

Maximal subgroups