Group #15673446 — subgroups

Subgroups — #15673446

Order 2944.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#10625078	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625004	C₂	1	2	yes	maximal, complemented
3	#13092399	C₂	1	2	yes	maximal
4	C₂₃⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67288	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	#10625078	—	2	4	no	—
12	C₂×#67262	C₄	1	2	yes	—
13	C₂×C₈×D₂₃	C₄	1	2	yes	—
14	#10625006	—	2	2	no	—
15	#10625005	—	2	2	no	—
16	#10625004	—	2	4	no	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₂²⋊D₂₃	Q₈	1	2	yes	—
19	#67288	—	2	2	no	—
20	#67288	—	2	2	no	—
21	C₂×D₉₂	D₄	1	1	yes	characteristic
22	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
23	C₂×Dic₄₆	D₄	1	1	yes	characteristic
24	M₁₆×C₂₃	—	2	2	no	—
25	M₁₆×C₂₃	—	2	2	no	—
26	M₁₆×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	2	no	—
28	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	#67267	—	2	2	no	—
30	#67267	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
35	#67262	—	2	4	no	—
36	C₈×D₂₃	—	2	4	no	—
37	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₂₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#41	1	2	yes	—
43	C₂²⋊D₂₃	—	2	4	no	—
44	C₂×Dic₂₃	C₄⋊C₄	1	2	yes	—
45	D₉₂	—	2	2	no	—
46	C₄×D₂₃	—	2	2	no	—
47	C₄×D₂₃	—	2	2	no	—
48	Dic₄₆	—	2	2	no	—
49	C₈×C₂₃	—	2	2	no	—
50	C₈×C₂₃	—	4	2	no	—
51	C₂₃⋊C₈	—	4	2	no	—
52	C₂₃⋊C₈	C₂²⋊C₄	1	2	yes	—
53	C₂₃⋊C₈	C₂²⋊C₄	1	2	yes	—
54	#1813	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	—	4	1	no	—
56	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
58	C₂×D₂₃	—	2	2	no	—
59	C₂×D₂₃	—	4	2	no	—
60	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₂₃	#117	1	1	yes	characteristic
62	C₄×C₂₃	#121	1	2	yes	—
63	Dic₂₃	—	2	2	no	—
64	C₂×#131	—	23	1	no	—
65	C₂×#108	—	23	2	no	—
66	#413	—	23	2	no	—
67	#338	—	23	1	no	—
68	#338	D₂₃	1	1	yes	characteristic, complemented
69	C₂×C₂₃	—	2	1	no	—
70	C₂×C₂₃	—	2	2	no	—
71	C₂×C₂₃	#380	1	1	yes	characteristic
72	D₂₃	—	4	2	no	—
73	C₂×#41	—	23	1	no	—
74	#131	—	46	4	no	—
75	C₂×M₁₆	—	23	2	no	—
76	C₂×M₁₆	—	23	2	no	—
77	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
78	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
79	C₂²×C₈	—	23	1	no	—
80	C₂²×C₈	—	23	2	no	—
81	#108	—	46	4	no	—
82	#101	—	46	2	no	—
83	#100	—	46	2	no	—
84	C₂₃	#1813	1	1	yes	characteristic, complemented, Sylow
85	#41	—	46	2	no	—
86	#41	—	46	2	no	—
87	C₂×Q₈	—	23	1	no	—
88	C₂×D₄	—	23	2	no	—
89	C₂×D₄	—	23	1	no	—
90	C₂²×C₄	—	23	2	no	—
91	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	46	4	no	—
96	M₁₆	—	46	2	no	—
97	M₁₆	—	46	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	23	1	no	—
100	C₂×C₈	—	23	1	no	—
101	C₂×C₈	—	23	4	no	—
102	C₂×C₈	—	46	4	no	—
103	C₂×C₈	—	46	2	no	—
104	C₂×C₈	C₄×D₂₃	1	2	yes	—
105	C₂³	—	23	2	no	—
106	C₂³	Q₈×D₂₃	1	1	yes	characteristic
107	Q₈	—	46	2	no	—
108	D₄	—	46	4	no	—
109	D₄	—	46	2	no	—
110	C₂×C₄	—	23	2	no	—
111	C₂×C₄	—	46	2	no	—
112	C₂×C₄	—	46	2	no	—
113	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
114	C₂×C₄	#67288	1	2	yes	—
115	C₂×C₄	D₄×D₂₃	1	2	yes	—
116	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	23	2	no	—
120	C₈	—	23	2	no	—
121	C₈	—	92	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	46	2	no	—
124	C₂²	—	92	2	no	—
125	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
126	C₂²	#10625067	1	2	yes	—
127	C₄	—	46	2	no	—
128	C₄	#10625043	1	1	yes	characteristic
129	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
130	C₄	#10625055	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	92	2	no	—
134	C₂	#13092276	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625078

C₂

yes

maximal, characteristic

C₂×#10625004

C₂

yes

maximal, complemented

#13092399

C₂

yes

maximal

C₂₃⋊#338

C₂

yes

maximal, characteristic, complemented

C₂₃×#338

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625078

—

C₂×#67262

C₄

yes

—

C₂×C₈×D₂₃

C₄

yes

—

#10625006

—

#10625005

—

#10625004

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

Q₈

yes

—

#67288

—

#67288

—

C₂×D₉₂

D₄

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×Dic₄₆

D₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

D₄

yes

—

#67262

—

C₈×D₂₃

—

C₂³×C₂₃

C₂×Q₈

yes

characteristic

C₂²×D₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂²⋊C₄

yes

—

C₂₃⋊C₈

C₂²⋊C₄

yes

—

#1813

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

D₂₃

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₂₃

#1813

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₄×D₂₃

yes

—

105

C₂³

—

106

C₂³

Q₈×D₂₃

yes

characteristic

107

Q₈

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

114

C₂×C₄

#67288

yes

—

115

C₂×C₄

D₄×D₂₃

yes

—

116

C₂×C₄

D₄×D₂₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

126

C₂²

#10625067

yes

—

127

C₄

—

128

C₄

#10625043

yes

characteristic

129

C₄

C₂²⋊C₄×D₂₃

yes

characteristic

130

C₄

#10625055

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

134

C₂

#13092276

yes

characteristic

	subgroup	G/H
center	C₄	#10625043
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#338	C₂
socle	C₂×C₂₃	#380
prime radical	C₂×#67288	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#338	D₂₃
hypocenter	C₂₃	#1813
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#10625043

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#338

C₂

C₂×C₂₃

C₂²

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#10625004	C₂	semidirect
4. C₂₃⋊#338	C₂	semidirect
5. C₂₃×#338	C₂	semidirect
68. #338	D₂₃	semidirect
84. C₂₃	#1813	semidirect

H ≅ G/N

type

2. C₂×#10625004

C₂

semidirect

4. C₂₃⋊#338

C₂

semidirect

5. C₂₃×#338

C₂

semidirect

68. #338

D₂₃

semidirect

84. C₂₃

#1813

semidirect

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

subgroup	index
C₂×#10625078	2
C₂×#10625004	2
#13092399	2
C₂₃⋊#338	2
C₂₃×#338	2
#1813	23

subgroup

index

C₂×#10625078

C₂×#10625004

#13092399

C₂₃⋊#338

C₂₃×#338

#1813

Subgroups — #15673446

Conjugacy classes of subgroups

Complements

Maximal subgroups