Group #15672456 — subgroups

Subgroups — #15672456

Order 2944.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13092793	C₂	1	1	yes	maximal, characteristic
2	#13092617	C₂	1	2	yes	maximal, complemented
3	#13092161	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#127	C₂²	1	1	yes	characteristic
7	C₂₃×#124	—	2	2	no	—
8	C₂₃×#123	—	2	2	no	—
9	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×#67274	—	2	2	no	—
12	C₂×#67272	—	2	2	no	—
13	#10625045	C₂²	1	1	yes	characteristic
14	#10624991	—	2	2	no	—
15	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
16	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₃	D₄	1	2	yes	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂×D₉₂	—	2	1	no	—
22	C₂×D₉₂	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
25	C₂²⋊C₄×C₂₃	—	2	4	no	—
26	C₂²⋊C₄×C₂₃	—	2	2	no	—
27	C₄²×C₂₃	—	2	2	no	—
28	C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	#67274	—	4	4	no	—
30	#67272	—	4	4	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂³×C₂₃	—	2	2	no	—
34	C₂²×D₂₃	—	4	1	no	—
35	Q₈×C₂₃	—	2	2	no	—
36	Q₈×C₂₃	—	2	2	no	—
37	D₄×C₂₃	—	4	2	no	—
38	D₄×C₂₃	—	4	2	no	—
39	D₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	8	2	no	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	D₉₂	—	4	2	no	—
46	D₉₂	—	4	2	no	—
47	D₉₂	—	4	2	no	—
48	C₂₃⋊C₈	—	2	2	no	—
49	C₂₃⋊C₈	—	4	2	no	—
50	#1620	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	4	4	no	—
52	C₂²×C₂₃	—	4	2	no	—
53	C₂²×C₂₃	#120	1	1	yes	characteristic
54	C₂×D₂₃	—	8	2	no	—
55	C₂×D₂₃	—	8	1	no	—
56	C₄×C₂₃	—	2	1	no	—
57	C₄×C₂₃	—	4	2	no	—
58	C₄×C₂₃	—	4	2	no	—
59	C₄×C₂₃	#135	1	2	yes	—
60	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
61	#559	D₂₃	1	1	yes	characteristic, complemented
62	#493	—	23	1	no	—
63	#460	—	23	2	no	—
64	#331	—	23	1	no	—
65	Q₈⋊C₈	—	23	2	no	—
66	C₂×C₂₃	—	8	2	no	—
67	C₂×C₂₃	#449	1	2	yes	—
68	C₂×C₂₃	#453	1	1	yes	characteristic
69	D₂₃	—	16	1	no	—
70	C₂×SD₁₆	—	46	2	no	—
71	C₂×D₈	—	46	2	no	—
72	#127	—	23	1	no	—
73	#127	C₂×D₂₃	1	1	yes	characteristic
74	#124	—	2	2	no	—
75	#123	—	2	2	no	—
76	C₄×Q₈	C₂×D₂₃	1	2	yes	—
77	C₄×D₄	—	2	2	no	—
78	C₄⋊C₈	—	23	2	no	—
79	#102	—	46	2	no	—
80	C₄×C₈	—	23	1	no	—
81	C₂₃	#1620	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	C₂²⋊D₂₃	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	46	2	no	—
85	C₂×D₄	—	46	1	no	—
86	C₂×D₄	C₂²⋊D₂₃	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	92	4	no	—
89	D₈	—	92	4	no	—
90	C₂×C₈	—	46	1	no	—
91	C₂×C₈	—	46	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₂₃	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₂₃	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	92	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	92	2	no	—
106	D₄	—	92	2	no	—
107	D₄	—	92	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
113	C₈	—	46	2	no	—
114	C₈	—	92	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	184	2	no	—
118	C₂²	—	184	1	no	—
119	C₂²	#10625124	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#10625120	1	2	yes	—
124	C₄	#10625121	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	368	1	no	—
127	C₂	#13092562	1	2	yes	—
128	C₂	#13092585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092793

C₂

yes

maximal, characteristic

#13092617

C₂

yes

maximal, complemented

#13092161

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₂₃⋊C₈)

C₂

yes

maximal, complemented

C₂₃×#559

C₂

yes

maximal, characteristic, complemented

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#124

—

C₂₃×#123

—

C₄×Q₈×C₂₃

C₂²

yes

—

C₄×D₄×C₂₃

—

C₂×#67274

—

C₂×#67272

—

#10625045

C₂²

yes

characteristic

#10624991

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67274

—

#67272

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

D₉₂

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1620

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

#559

D₂₃

yes

characteristic, complemented

#493

—

#460

—

#331

—

Q₈⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

#449

yes

—

C₂×C₂₃

#453

yes

characteristic

D₂₃

—

C₂×SD₁₆

—

C₂×D₈

—

#127

—

#127

C₂×D₂₃

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₂₃

yes

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂₃

#1620

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

C₂³

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

184

—

118

C₂²

—

184

—

119

C₂²

#10625124

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#10625120

yes

—

124

C₄

#10625121

yes

—

125

C₂

—

126

C₂

—

368

—

127

C₂

#13092562

yes

—

128

C₂

#13092585

yes

characteristic

	subgroup	G/H
center	C₂²	#10625124
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#559	C₂
socle	C₂²×C₂₃	#120
prime radical	#13092793	C₂
hypercenter	#559	D₂₃
hypocenter	C₂₃	#1620
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#10625124

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#559

C₂

C₂²×C₂₃

C₂

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #13092617	C₂	semidirect
3. #13092161	C₂	semidirect
4. Q₈⋊(C₂₃⋊C₈)	C₂	semidirect
5. C₂₃×#559	C₂	semidirect
61. #559	D₂₃	semidirect
81. C₂₃	#1620	semidirect

H ≅ G/N

type

2. #13092617

C₂

semidirect

3. #13092161

C₂

semidirect

4. Q₈⋊(C₂₃⋊C₈)

C₂

semidirect

5. C₂₃×#559

C₂

semidirect

61. #559

D₂₃

semidirect

81. C₂₃

#1620

semidirect

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

subgroup	index
#13092793	2
#13092617	2
#13092161	2
Q₈⋊(C₂₃⋊C₈)	2
C₂₃×#559	2
#1620	23

subgroup

index

#13092793

#13092617

#13092161

Q₈⋊(C₂₃⋊C₈)

C₂₃×#559

#1620

Subgroups — #15672456

Conjugacy classes of subgroups

Complements

Maximal subgroups