Group #15672131 — subgroups

Subgroups — #15672131

Order 2944.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#13092968	C₂	1	1	yes	maximal, characteristic
2	#13092439	C₂	1	2	yes	maximal, complemented
3	#13092166	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092160	C₂	1	2	yes	maximal, complemented
5	C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67292	C₄	1	2	yes	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	#10625118	C₄	1	2	yes	—
9	C₂×#67262	—	2	2	no	—
10	#10625066	—	2	2	no	—
11	#10625064	—	2	2	no	—
12	#10625045	C₂²	1	1	yes	characteristic
13	C₄²×D₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	Dic₂₃⋊C₈	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
18	#67292	—	4	4	no	—
19	C₂×D₉₂	—	2	2	no	—
20	C₂×D₉₂	—	2	2	no	—
21	C₂×D₉₂	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₂₃	—	2	2	no	—
23	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₂₃	D₄	1	2	yes	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	(C₂×D₂₃)⋊C₄	—	2	4	no	—
29	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₂₃	D₄	1	2	yes	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	#67262	—	4	4	no	—
33	C₂²×D₂₃	—	2	2	no	—
34	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	Q₈×C₂₃	—	4	2	no	—
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
39	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
41	D₉₂	—	4	2	no	—
42	D₉₂	—	4	4	no	—
43	D₉₂	—	4	2	no	—
44	D₉₂	—	4	2	no	—
45	C₄×D₂₃	—	2	2	no	—
46	C₄×D₂₃	—	2	2	no	—
47	C₄×D₂₃	—	2	2	no	—
48	C₄×D₂₃	—	4	4	no	—
49	C₈×C₂₃	—	4	2	no	—
50	C₂₃⋊C₈	—	4	2	no	—
51	#1560	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂×D₂₃	—	2	1	no	—
54	C₂×D₂₃	—	2	2	no	—
55	C₂×D₂₃	—	4	2	no	—
56	C₂×D₂₃	—	4	4	no	—
57	C₄×C₂₃	—	2	1	no	—
58	C₄×C₂₃	—	4	2	no	—
59	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₄×C₂₃	#137	1	2	yes	—
61	Dic₂₃	—	2	2	no	—
62	Dic₂₃	#100	1	2	yes	—
63	#532	—	23	1	no	—
64	#423	—	23	2	no	—
65	#333	—	23	1	no	—
66	#333	D₂₃	1	1	yes	characteristic, complemented
67	#331	—	23	2	no	—
68	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
69	C₂×C₂₃	#417	1	2	yes	—
70	D₂₃	—	4	1	no	—
71	D₂₃	—	8	2	no	—
72	C₂×#41	—	23	2	no	—
73	C₂×M₁₆	—	46	2	no	—
74	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
75	#127	—	23	1	no	—
76	#124	—	23	2	no	—
77	#121	—	46	2	no	—
78	C₄×D₄	—	46	2	no	—
79	C₂×C₄²	—	23	1	no	—
80	C₄⋊C₈	—	23	2	no	—
81	C₄⋊C₈	—	46	2	no	—
82	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
83	C₂₃	#1560	1	1	yes	characteristic, complemented, Sylow
84	#41	—	92	4	no	—
85	C₂×Q₈	C₄×D₂₃	1	2	yes	—
86	C₂×D₄	—	23	2	no	—
87	C₂×D₄	—	46	2	no	—
88	C₂×D₄	—	46	2	no	—
89	C₂²×C₄	—	23	1	no	—
90	C₂²×C₄	—	23	2	no	—
91	C₂²×C₄	—	46	2	no	—
92	M₁₆	—	92	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	46	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	46	4	no	—
97	C₄²	—	23	1	no	—
98	C₄²	—	23	2	no	—
99	C₄²	C₂²×D₂₃	1	1	yes	characteristic
100	C₂³	—	23	1	no	—
101	C₂³	—	46	2	no	—
102	Q₈	—	4	2	no	—
103	D₄	—	92	2	no	—
104	D₄	—	92	4	no	—
105	D₄	—	92	2	no	—
106	D₄	—	92	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	23	1	no	—
109	C₂×C₄	—	23	2	no	—
110	C₂×C₄	—	46	2	no	—
111	C₂×C₄	—	46	2	no	—
112	C₂×C₄	—	46	2	no	—
113	C₂×C₄	—	92	4	no	—
114	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
115	C₂×C₄	D₄×D₂₃	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	92	2	no	—
118	C₂²	—	46	1	no	—
119	C₂²	—	46	2	no	—
120	C₂²	—	92	2	no	—
121	C₂²	—	92	4	no	—
122	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	23	2	no	—
126	C₄	—	46	2	no	—
127	C₄	#10625085	1	2	yes	—
128	C₄	#10625089	1	2	yes	—
129	C₂	—	92	1	no	—
130	C₂	—	184	2	no	—
131	C₂	#100×D₂₃	1	1	yes	characteristic
132	C₂	#13092417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092968

C₂

yes

maximal, characteristic

#13092439

C₂

yes

maximal, complemented

#13092166

C₂

yes

maximal, characteristic, complemented

#13092160

C₂

yes

maximal, complemented

C₂₃×#333

C₂

yes

maximal, characteristic, complemented

C₂×#67292

C₄

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

#10625118

C₄

yes

—

—

—

—

C₂²

yes

characteristic

C₄²×D₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

—

Dic₂₃⋊C₈

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

#67292

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

C₂×C₄

yes

—

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

(C₂×D₂₃)⋊C₄

—

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

#67262

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

C₂²⋊C₄

yes

—

C₂×Dic₂₃

C₂²⋊C₄

yes

characteristic

D₉₂

—

D₉₂

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1560

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

#100

yes

—

—

—

—

D₂₃

yes

characteristic, complemented

#331

—

C₂×C₂₃

C₂×#100

yes

characteristic

C₂×C₂₃

#417

yes

—

D₂₃

—

D₂₃

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₂₃

yes

—

—

—

C₄×D₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

C₂₃

#1560

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₄×D₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

100

C₂³

—

101

C₂³

—

102

Q₈

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

115

C₂×C₄

D₄×D₂₃

yes

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂²⋊C₄×D₂₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#10625085

yes

—

128

C₄

#10625089

yes

—

129

C₂

—

130

C₂

—

184

—

131

C₂

#100×D₂₃

yes

characteristic

132

C₂

#13092417

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#333	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	#13092968	C₂
cosocle	C₄²×C₂₃	C₂³
hypercenter	#333	D₂₃
hypocenter	C₂₃	#1560
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#333

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

#13092968

C₂

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #13092439	C₂	semidirect
3. #13092166	C₂	semidirect
4. #13092160	C₂	semidirect
5. C₂₃×#333	C₂	semidirect
66. #333	D₂₃	semidirect
83. C₂₃	#1560	semidirect

H ≅ G/N

type

2. #13092439

C₂

semidirect

3. #13092166

C₂

semidirect

4. #13092160

C₂

semidirect

5. C₂₃×#333

C₂

semidirect

66. #333

D₂₃

semidirect

83. C₂₃

#1560

semidirect

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

subgroup	index
#13092968	2
#13092439	2
#13092166	2
#13092160	2
C₂₃×#333	2
#1560	23

subgroup

index

C₂₃×#333

Subgroups — #15672131

Conjugacy classes of subgroups

Complements

Maximal subgroups