Group #15672478 — subgroups

Subgroups — #15672478

Order 2944.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#13093133	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092609	C₂	1	2	yes	maximal, complemented
3	#13092166	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092146	C₂	1	2	yes	maximal, complemented
5	C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₂₃	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	#10625118	—	2	2	no	—
10	C₂×#67272	—	2	2	no	—
11	#10625066	—	2	2	no	—
12	#10625063	—	2	2	no	—
13	C₄×D₉₂	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	#10625014	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₃	D₄	1	2	yes	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	1	no	—
20	C₂×D₉₂	—	2	2	no	—
21	C₂×D₉₂	D₄	1	2	yes	—
22	C₂×C₄×D₂₃	—	2	2	no	—
23	SD₁₆×C₂₃	—	4	4	no	—
24	C₂×C₈×C₂₃	—	2	1	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	#67272	—	4	4	no	—
28	(C₂×D₂₃)⋊C₄	—	2	4	no	—
29	(C₂×D₂₃)⋊C₄	—	2	2	no	—
30	C₄⋊Dic₂₃	D₄	1	2	yes	—
31	C₄×Dic₂₃	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂³×C₂₃	—	4	1	no	—
34	C₂²×D₂₃	—	2	2	no	—
35	Q₈×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	4	2	no	—
37	D₄×C₂₃	—	4	2	no	—
38	D₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₂₃	—	2	2	no	—
43	D₉₂	—	2	2	no	—
44	D₉₂	—	2	2	no	—
45	D₉₂	—	8	2	no	—
46	C₄×D₂₃	—	4	2	no	—
47	C₄×D₂₃	—	4	2	no	—
48	C₈×C₂₃	—	2	2	no	—
49	C₂₃⋊C₈	—	4	2	no	—
50	#1624	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	8	2	no	—
52	C₂²×C₂₃	—	8	1	no	—
53	C₂²×C₂₃	#120	1	1	yes	characteristic
54	C₂×D₂₃	—	4	2	no	—
55	C₂×D₂₃	—	4	2	no	—
56	C₂×D₂₃	—	4	2	no	—
57	C₄×C₂₃	—	2	1	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	Dic₂₃	—	4	2	no	—
62	#555	—	23	1	no	—
63	#492	D₂₃	1	1	yes	characteristic, complemented
64	#459	—	23	2	no	—
65	#333	—	23	1	no	—
66	D₄⋊C₈	—	23	2	no	—
67	C₂×C₂₃	—	16	1	no	—
68	C₂×C₂₃	#449	1	2	yes	—
69	C₂×C₂₃	#453	1	1	yes	characteristic
70	D₂₃	—	8	2	no	—
71	C₂×SD₁₆	—	2	2	no	—
72	C₂×D₈	—	46	2	no	—
73	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
74	#127	C₂×D₂₃	1	1	yes	characteristic
75	#124	—	46	2	no	—
76	#121	—	46	2	no	—
77	C₄×D₄	—	23	2	no	—
78	#117	—	46	2	no	—
79	C₄⋊C₈	—	23	2	no	—
80	#102	—	46	2	no	—
81	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
82	C₂₃	#1624	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	C₂²⋊D₂₃	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	23	2	no	—
87	C₂×D₄	—	46	2	no	—
88	C₂²×C₄	—	46	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	92	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	46	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	23	2	no	—
95	C₂²⋊C₄	—	46	4	no	—
96	C₂²⋊C₄	—	46	2	no	—
97	C₄²	—	46	2	no	—
98	C₄²	C₂²×D₂₃	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	46	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	46	2	no	—
106	D₄	—	46	2	no	—
107	D₄	—	184	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	46	2	no	—
110	C₂×C₄	—	92	2	no	—
111	C₂×C₄	—	92	2	no	—
112	C₂×C₄	D₄×D₂₃	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	92	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	92	2	no	—
119	C₂²	—	92	2	no	—
120	C₂²	—	92	2	no	—
121	C₂²	#10625108	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	92	2	no	—
125	C₄	#10625085	1	2	yes	—
126	C₄	#10625087	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	184	2	no	—
129	C₂	#13092561	1	2	yes	—
130	C₂	#13092584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093133

C₂

yes

maximal, characteristic, complemented

#13092609

C₂

yes

maximal, complemented

#13092166

C₂

yes

maximal, characteristic, complemented

#13092146

C₂

yes

maximal, complemented

C₂₃×#492

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

—

—

—

—

C₄×D₉₂

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625014

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

D₄

yes

—

C₂×C₄×D₂₃

—

SD₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67272

—

(C₂×D₂₃)⋊C₄

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1624

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

Dic₂₃

—

#555

—

#492

D₂₃

yes

characteristic, complemented

#459

—

#333

—

D₄⋊C₈

—

C₂×C₂₃

—

C₂×C₂₃

#449

yes

—

C₂×C₂₃

#453

yes

characteristic

D₂₃

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

C₂×D₂₃

yes

characteristic

#124

—

#121

—

C₄×D₄

—

#117

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1624

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

100

C₂³

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

184

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

D₄×D₂₃

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#10625108

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#10625085

yes

—

126

C₄

#10625087

yes

—

127

C₂

—

128

C₂

—

184

—

129

C₂

#13092561

yes

—

130

C₂

#13092584

yes

characteristic

	subgroup	G/H
center	C₂²	#10625108
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#492	C₂
socle	C₂²×C₂₃	#120
hypercenter	#492	D₂₃
hypocenter	C₂₃	#1624
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#10625108

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#492

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #13093133	C₂	semidirect
2. #13092609	C₂	semidirect
3. #13092166	C₂	semidirect
4. #13092146	C₂	semidirect
5. C₂₃×#492	C₂	semidirect
63. #492	D₂₃	semidirect
82. C₂₃	#1624	semidirect

H ≅ G/N

type

1. #13093133

C₂

semidirect

2. #13092609

C₂

semidirect

3. #13092166

C₂

semidirect

4. #13092146

C₂

semidirect

5. C₂₃×#492

C₂

semidirect

63. #492

D₂₃

semidirect

82. C₂₃

#1624

semidirect

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

subgroup	index
#13093133	2
#13092609	2
#13092166	2
#13092146	2
C₂₃×#492	2
#1624	23

subgroup

index

C₂₃×#492

Subgroups — #15672478

Conjugacy classes of subgroups

Complements

Maximal subgroups