Group #15683949 — subgroups

Subgroups — #15683949

Order 2944.

348 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 165 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	#13092974	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625041	C₂	1	1	yes	maximal, characteristic
3	#13092826	C₂	1	4	yes	maximal, complemented
4	#13092819	C₂	1	4	yes	maximal, complemented
5	#13092459	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	#10625097	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	#10625095	—	2	2	no	—
12	C₂×Dic₂₃⋊C₄	—	2	2	no	—
13	#10625041	—	2	2	no	—
14	#10625041	C₂²	1	2	yes	—
15	#10625041	C₂²	1	2	yes	—
16	#10625041	C₂²	1	2	yes	—
17	#10625040	C₂²	1	2	yes	—
18	C₄×Dic₄₆	—	2	2	no	—
19	C₂₃×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	#10625000	C₂²	1	8	yes	—
22	#10624999	C₂²	1	8	yes	—
23	C₂²×C₄×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₂₃	—	2	2	no	—
26	C₂×Dic₄₆	—	2	2	no	—
27	M₁₆×C₂₃	Q₈	1	8	yes	complemented
28	C₂×C₈×C₂₃	C₂³	1	4	yes	—
29	C₄²×C₂₃	D₄	1	2	yes	—
30	C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₂₃	—	2	2	no	—
33	C₄⋊Dic₂₃	—	2	2	no	—
34	C₄⋊Dic₂₃	—	2	2	no	—
35	C₄⋊Dic₂₃	C₂³	1	4	yes	—
36	C₄⋊Dic₂₃	C₂³	1	4	yes	—
37	Dic₂₃⋊C₄	—	2	4	no	—
38	Dic₂₃⋊C₄	—	2	4	no	—
39	Dic₂₃⋊C₄	—	2	4	no	—
40	C₄×Dic₂₃	—	2	2	no	—
41	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	4	2	no	—
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₂₃	—	2	4	no	—
49	C₂×Dic₂₃	—	2	4	no	—
50	C₂×Dic₂₃	—	4	4	no	—
51	Dic₄₆	—	4	4	no	—
52	C₈×C₂₃	C₂×Q₈	1	8	yes	—
53	#3166	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₂₃	—	2	2	no	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
60	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
61	Dic₂₃	—	4	4	no	—
62	Dic₂₃	—	4	4	no	—
63	#533	—	23	1	no	—
64	C₂×#125	—	23	1	no	—
65	#501	—	23	4	no	—
66	#499	—	23	4	no	—
67	#428	—	23	4	no	—
68	C₄×M₁₆	D₂₃	1	1	yes	characteristic
69	C₂×C₂₃	—	2	2	no	—
70	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₂₃	#575	1	2	yes	—
72	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
73	C₄⋊Q₈	—	23	2	no	—
74	#125	—	23	2	no	—
75	#125	—	23	2	no	—
76	#125	—	23	2	no	—
77	#125	—	46	2	no	—
78	C₂²⋊Q₈	—	46	2	no	—
79	C₄×Q₈	—	46	2	no	—
80	#117	—	23	2	no	—
81	C₂×C₄⋊C₄	—	23	2	no	—
82	C₂×C₄⋊C₄	—	46	2	no	—
83	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
84	#107	—	23	8	no	—
85	#106	—	23	8	no	—
86	#97	C₂×D₂₃	1	2	yes	—
87	C₄×C₈	C₂×D₂₃	1	2	yes	—
88	C₂₃	#3166	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	46	2	no	—
90	C₂²×C₄	—	46	2	no	—
91	C₂²×C₄	D₉₂	1	2	yes	—
92	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
93	M₁₆	Dic₄₆	1	8	yes	complemented
94	C₂×C₈	C₂²×D₂₃	1	4	yes	—
95	C₄⋊C₄	—	23	4	no	—
96	C₄⋊C₄	—	23	4	no	—
97	C₄⋊C₄	—	46	2	no	—
98	C₄⋊C₄	—	46	4	no	—
99	C₄⋊C₄	—	46	4	no	—
100	C₄⋊C₄	—	46	4	no	—
101	C₄⋊C₄	—	46	2	no	—
102	C₂²⋊C₄	—	46	2	no	—
103	C₄²	—	46	2	no	—
104	C₄²	D₉₂	1	2	yes	—
105	C₄²	C₂²×D₂₃	1	1	yes	characteristic
106	C₄²	C₂²×D₂₃	1	1	yes	characteristic
107	C₂³	C₂×D₉₂	1	1	yes	characteristic
108	Q₈	—	92	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	46	4	no	—
111	C₂×C₄	—	46	4	no	—
112	C₂×C₄	—	92	4	no	—
113	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
114	C₂×C₄	C₂×D₉₂	1	2	yes	—
115	C₂×C₄	C₂×D₉₂	1	2	yes	—
116	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
118	C₈	C₂×Dic₄₆	1	8	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	#10625040	1	2	yes	—
121	C₂²	C₂²×D₉₂	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	92	4	no	—
125	C₄	—	92	4	no	—
126	C₄	#10625040	1	2	yes	—
127	C₄	C₂²×Dic₄₆	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#10625040	1	1	yes	characteristic
130	C₂	#13093217	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

C₂×#10625041

C₂

yes

maximal, characteristic

#13092826

C₂

yes

maximal, complemented

#13092819

C₂

yes

maximal, complemented

#13092459

C₂

yes

maximal

C₄×M₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

#10625095

—

C₂×Dic₂₃⋊C₄

—

#10625041

—

#10625041

C₂²

yes

—

#10625041

C₂²

yes

—

#10625041

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

—

C₂₃×#97

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

M₁₆×C₂₃

Q₈

yes

complemented

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

—

#3166

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

—

C₄×M₁₆

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

#575

yes

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

—

—

C₂×D₂₃

yes

—

C₄×C₈

C₂×D₂₃

yes

—

C₂₃

#3166

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

Dic₄₆

yes

complemented

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

D₉₂

yes

—

105

C₄²

C₂²×D₂₃

yes

characteristic

106

C₄²

C₂²×D₂₃

yes

characteristic

107

C₂³

C₂×D₉₂

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₄₆

yes

—

114

C₂×C₄

C₂×D₉₂

yes

—

115

C₂×C₄

C₂×D₉₂

yes

—

116

C₂×C₄

C₂×D₉₂

yes

characteristic

117

C₂×C₄

C₂³×D₂₃

yes

characteristic

118

C₈

C₂×Dic₄₆

yes

—

119

C₂²

—

120

C₂²

#10625040

yes

—

121

C₂²

C₂²×D₉₂

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10625040

yes

—

127

C₄

C₂²×Dic₄₆

yes

—

128

C₂

—

129

C₂

C₂×#10625040

yes

characteristic

130

C₂

#13093217

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#3166
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₉₂

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₄×M₁₆×C₂₃

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

hypercenter

C₄×M₁₆

D₂₃

hypocenter

C₂₃

#3166

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #13092826	C₂	semidirect
4. #13092819	C₂	semidirect
27. M₁₆×C₂₃	Q₈	semidirect
88. C₂₃	#3166	semidirect
93. M₁₆	Dic₄₆	semidirect

H ≅ G/N

type

3. #13092826

C₂

semidirect

4. #13092819

C₂

semidirect

27. M₁₆×C₂₃

Q₈

semidirect

88. C₂₃

#3166

semidirect

93. M₁₆

Dic₄₆

semidirect

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

subgroup	index
#13092974	2
C₂×#10625041	2
#13092826	2
#13092819	2
#13092459	2
C₄×M₁₆×C₂₃	2
#3166	23

subgroup

index

C₄×M₁₆×C₂₃

Subgroups — #15683949

Conjugacy classes of subgroups

Complements

Maximal subgroups