Group #15674653 — subgroups

Subgroups — #15674653

Order 2944.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×D₁₈₄	C₂	1	1	yes	maximal, characteristic
2	#13092421	C₂	1	2	yes	maximal, complemented
3	C₂×#10625005	C₂	1	2	yes	maximal, complemented
4	#13092182	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₉₂	C₂²	1	2	yes	—
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	2	yes	—
11	C₂×D₁₈₄	—	2	4	no	—
12	C₂×D₁₈₄	—	2	2	no	—
13	C₂₃×C₄⋊C₈	—	2	2	no	—
14	#10625005	—	2	4	no	—
15	#10624991	—	2	4	no	—
16	C₂³×D₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×D₉₂	—	2	2	no	—
19	C₂×D₉₂	—	2	2	no	—
20	C₂×D₉₂	—	2	2	no	—
21	C₂×D₉₂	—	2	2	no	—
22	C₂×D₉₂	—	2	2	no	—
23	C₂×D₉₂	D₄	1	4	yes	—
24	M₁₆×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂×C₈×C₂₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₂₃	—	2	1	no	—
30	C₄²×C₂₃	—	2	1	no	—
31	#67267	—	2	2	no	—
32	#67267	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	D₁₈₄	—	2	8	no	—
35	C₂³×C₂₃	#41	1	1	yes	characteristic
36	C₂²×D₂₃	—	2	4	no	—
37	C₂²×D₂₃	—	4	2	no	—
38	C₂²×D₂₃	—	4	2	no	—
39	C₂²×D₂₃	—	8	4	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	2	yes	—
45	D₉₂	—	2	4	no	—
46	D₉₂	—	2	4	no	—
47	D₉₂	—	4	2	no	—
48	D₉₂	—	4	2	no	—
49	C₈×C₂₃	—	2	2	no	—
50	C₈×C₂₃	—	4	1	no	—
51	C₂₃⋊C₈	—	4	2	no	—
52	#2045	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	#121	1	1	yes	characteristic
55	C₂²×C₂₃	#123	1	2	yes	—
56	C₂×D₂₃	—	4	4	no	—
57	C₂×D₂₃	—	8	4	no	—
58	C₂×D₂₃	—	8	4	no	—
59	C₂×D₂₃	—	8	4	no	—
60	C₄×C₂₃	—	4	2	no	—
61	C₄×C₂₃	#120	1	1	yes	characteristic
62	C₄×C₂₃	#121	1	2	yes	—
63	C₄×C₂₃	#124	1	1	yes	characteristic
64	C₂²×D₈	—	23	1	no	—
65	#424	D₂₃	1	1	yes	characteristic, complemented
66	#418	—	23	2	no	—
67	C₂×#100	—	23	2	no	—
68	#341	—	23	1	no	—
69	C₂×C₂₃	—	2	2	no	—
70	C₂×C₂₃	#394	1	1	yes	characteristic
71	C₂×C₂₃	#472	1	2	yes	—
72	D₂₃	—	8	4	no	—
73	C₂²×D₄	—	23	2	no	—
74	C₂×D₈	—	46	4	no	—
75	C₂×D₈	—	46	2	no	—
76	C₂×M₁₆	—	23	2	no	—
77	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
79	#117	C₂×D₂₃	1	1	yes	characteristic
80	C₄⋊C₈	—	2	2	no	—
81	#102	—	46	4	no	—
82	#100	—	46	4	no	—
83	C₂₃	#2045	1	1	yes	characteristic, complemented, Sylow
84	C₂⁴	—	46	2	no	—
85	C₂×D₄	—	23	4	no	—
86	C₂×D₄	—	46	2	no	—
87	C₂×D₄	—	46	2	no	—
88	C₂×D₄	—	46	2	no	—
89	C₂×D₄	—	46	2	no	—
90	C₂×D₄	—	46	2	no	—
91	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
92	D₈	—	46	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	46	2	no	—
95	M₁₆	—	46	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	46	2	no	—
99	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
100	C₄⋊C₄	D₉₂	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	46	4	no	—
104	C₂³	—	92	2	no	—
105	C₂³	—	92	2	no	—
106	C₂³	—	184	4	no	—
107	C₂³	#67288	1	1	yes	characteristic
108	D₄	—	46	4	no	—
109	D₄	—	46	4	no	—
110	D₄	—	92	2	no	—
111	D₄	—	92	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
114	C₂×C₄	D₄×D₂₃	1	2	yes	—
115	C₂×C₄	#67292	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	92	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	92	4	no	—
122	C₂²	—	184	4	no	—
123	C₂²	—	184	4	no	—
124	C₂²	—	184	4	no	—
125	C₂²	#10625065	1	2	yes	—
126	C₂²	#10625101	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	#10625066	1	2	yes	—
129	C₄	#10625108	1	1	yes	characteristic
130	C₄	#10625118	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	184	4	no	—
133	C₂	#13092334	1	1	yes	characteristic
134	C₂	#13092687	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₈₄

C₂

yes

maximal, characteristic

#13092421

C₂

yes

maximal, complemented

C₂×#10625005

C₂

yes

maximal, complemented

#13092182

C₂

yes

maximal, characteristic, complemented

C₂₃×#424

C₂

yes

maximal, characteristic, complemented

C₂²×D₉₂

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×D₁₈₄

—

C₂×D₁₈₄

—

C₂₃×C₄⋊C₈

—

#10625005

—

#10624991

—

C₂³×D₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

D₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

D₁₈₄

—

C₂³×C₂₃

#41

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

D₉₂

—

D₉₂

—

D₉₂

—

D₉₂

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#2045

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#121

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

#121

yes

—

C₄×C₂₃

#124

yes

characteristic

C₂²×D₈

—

#424

D₂₃

yes

characteristic, complemented

#418

—

C₂×#100

—

#341

—

C₂×C₂₃

—

C₂×C₂₃

#394

yes

characteristic

C₂×C₂₃

#472

yes

—

D₂₃

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#102

—

#100

—

C₂₃

#2045

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

—

100

C₄⋊C₄

D₉₂

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

184

—

107

C₂³

#67288

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₉₂

yes

characteristic

114

C₂×C₄

D₄×D₂₃

yes

—

115

C₂×C₄

#67292

yes

—

116

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

184

—

123

C₂²

—

184

—

124

C₂²

—

184

—

125

C₂²

#10625065

yes

—

126

C₂²

#10625101

yes

characteristic

127

C₄

—

128

C₄

#10625066

yes

—

129

C₄

#10625108

yes

characteristic

130

C₄

#10625118

yes

characteristic

131

C₂

—

132

C₂

—

184

—

133

C₂

#13092334

yes

characteristic

134

C₂

#13092687

yes

—

	subgroup	G/H
center	C₂²	#10625101
commutator, cosocle	C₂²×C₄×C₂₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#424	C₂
socle	C₂²×C₂₃	#121
prime radical	C₂²×D₁₈₄	C₂
hypercenter	#424	D₂₃
hypocenter	C₂₃	#2045
LCS	G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#10625101

commutator, cosocle

C₂²×C₄×C₂₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#424

C₂

socle

C₂²×C₂₃

#121

prime radical

C₂²×D₁₈₄

C₂

D₂₃

C₂₃

G ▷ C₂²×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #13092421	C₂	semidirect
3. C₂×#10625005	C₂	semidirect
4. #13092182	C₂	semidirect
5. C₂₃×#424	C₂	semidirect
65. #424	D₂₃	semidirect
83. C₂₃	#2045	semidirect

H ≅ G/N

type

2. #13092421

C₂

semidirect

3. C₂×#10625005

C₂

semidirect

4. #13092182

C₂

semidirect

5. C₂₃×#424

C₂

semidirect

65. #424

D₂₃

semidirect

83. C₂₃

#2045

semidirect

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

subgroup	index
C₂²×D₁₈₄	2
#13092421	2
C₂×#10625005	2
#13092182	2
C₂₃×#424	2
#2045	23

subgroup

index

C₂²×D₁₈₄

C₂₃×#424

Subgroups — #15674653

Conjugacy classes of subgroups

Complements

Maximal subgroups