Group #15673067 — subgroups

Subgroups — #15673067

Order 2944.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#13092820	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092796	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092791	C₂	1	1	yes	maximal, characteristic
4	#13092765	C₂	1	1	yes	maximal, characteristic
5	#13092764	C₂	1	1	yes	maximal, characteristic
6	#13092139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×#67275	—	2	2	no	—
11	C₂×#67274	—	2	2	no	—
12	#10625045	C₂²	1	1	yes	characteristic
13	#10625040	C₂²	1	1	yes	characteristic
14	C₂₃×#107	—	2	2	no	—
15	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	#10625003	—	2	2	no	—
17	#10624991	—	2	2	no	—
18	#10624989	—	2	2	no	—
19	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₂₃	—	2	1	no	—
22	C₂×Q₈×C₂₃	—	2	1	no	—
23	C₂×D₉₂	—	2	1	no	—
24	C₂×D₉₂	—	2	1	no	—
25	C₂×D₉₂	—	2	1	no	—
26	C₂×Dic₄₆	—	2	1	no	—
27	C₂×C₈×C₂₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	#67275	—	4	4	no	—
34	#67274	—	4	4	no	—
35	C₄⋊Dic₂₃	—	2	1	no	—
36	C₄⋊Dic₂₃	—	2	1	no	—
37	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
38	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
39	C₂²×D₂₃	—	4	1	no	—
40	Q₈×C₂₃	—	4	2	no	—
41	Q₈×C₂₃	—	4	2	no	—
42	C₂×C₄×C₂₃	—	4	1	no	—
43	C₂×C₄×C₂₃	—	4	1	no	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₂₃	—	4	1	no	—
48	D₉₂	—	4	2	no	—
49	D₉₂	—	4	2	no	—
50	D₉₂	—	4	2	no	—
51	Dic₄₆	—	4	2	no	—
52	C₈×C₂₃	—	2	2	no	—
53	C₂₃⋊C₈	—	2	2	no	—
54	C₂₃⋊C₈	D₈	1	4	yes	—
55	#1722	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	#127	1	1	yes	characteristic
57	C₂×D₂₃	—	8	1	no	—
58	C₂×D₂₃	—	8	1	no	—
59	C₂×D₂₃	—	8	1	no	—
60	C₄×C₂₃	—	8	1	no	—
61	C₄×C₂₃	—	8	1	no	—
62	C₄×C₂₃	C₂×D₈	1	2	yes	—
63	C₄×C₂₃	#137	1	2	yes	—
64	C₄×C₂₃	#137	1	2	yes	—
65	Dic₂₃	—	8	1	no	—
66	#500	—	23	1	no	—
67	#500	D₂₃	1	1	yes	characteristic, complemented
68	#494	—	23	1	no	—
69	#492	—	23	1	no	—
70	#486	—	23	1	no	—
71	#486	—	23	1	no	—
72	#322	—	23	1	no	—
73	C₂×C₂₃	#493	1	1	yes	characteristic
74	C₂×C₂₃	#497	1	1	yes	characteristic
75	C₂×C₂₃	#497	1	1	yes	characteristic
76	D₂₃	—	16	1	no	—
77	C₂×Q₁₆	—	46	2	no	—
78	C₂×SD₁₆	—	46	2	no	—
79	C₄⋊Q₈	—	23	1	no	—
80	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
82	#127	—	23	1	no	—
83	#107	—	2	2	no	—
84	#107	—	46	2	no	—
85	#102	—	46	2	no	—
86	#102	—	46	2	no	—
87	C₄×C₈	—	23	1	no	—
88	C₄×C₈	—	23	1	no	—
89	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
90	C₂₃	#1722	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	46	1	no	—
94	C₂×D₄	—	46	1	no	—
95	C₂×D₄	—	46	1	no	—
96	C₂×D₄	—	46	1	no	—
97	Q₁₆	—	92	4	no	—
98	SD₁₆	—	92	4	no	—
99	C₂×C₈	—	23	2	no	—
100	C₂×C₈	—	23	2	no	—
101	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	46	1	no	—
107	C₄⋊C₄	—	46	1	no	—
108	C₄²	C₂²×D₂₃	1	1	yes	characteristic
109	C₂³	—	92	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	92	2	no	—
113	D₄	—	92	2	no	—
114	D₄	—	92	2	no	—
115	D₄	—	92	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	92	1	no	—
119	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
120	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	23	4	no	—
124	C₈	—	46	2	no	—
125	C₂²	—	184	1	no	—
126	C₂²	—	184	1	no	—
127	C₂²	—	184	1	no	—
128	C₂²	#10625111	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	184	1	no	—
132	C₄	D₈×D₂₃	1	2	yes	—
133	C₄	#10625089	1	2	yes	—
134	C₄	#10625113	1	2	yes	—
135	C₂	—	368	1	no	—
136	C₂	#13092792	1	1	yes	characteristic
137	C₂	#13092809	1	1	yes	characteristic
138	C₂	#13092810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092820

C₂

yes

maximal, characteristic, complemented

#13092796

C₂

yes

maximal, characteristic, complemented

#13092791

C₂

yes

maximal, characteristic

#13092765

C₂

yes

maximal, characteristic

#13092764

C₂

yes

maximal, characteristic

#13092139

C₂

yes

maximal, characteristic, complemented

C₂₃×#500

C₂

yes

maximal, characteristic, complemented

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

—

C₂²

yes

characteristic

#10625040

C₂²

yes

characteristic

C₂₃×#107

—

C₄×C₈×C₂₃

C₂²

yes

—

—

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67275

—

#67274

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂²×D₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

D₉₂

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

D₈

yes

—

—

C₂²×C₂₃

yes

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₈

yes

—

C₄×C₂₃

#137

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

#500

—

#500

D₂₃

yes

characteristic, complemented

—

—

—

—

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

D₂₃

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄⋊Q₈

C₂×D₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1722

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₂₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²×D₂₃

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₂₃

yes

characteristic

120

C₂×C₄

D₄×D₂₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

184

—

126

C₂²

—

184

—

127

C₂²

—

184

—

128

C₂²

#10625111

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

184

—

132

C₄

D₈×D₂₃

yes

—

133

C₄

#10625089

yes

—

134

C₄

#10625113

yes

—

135

C₂

—

368

—

136

C₂

#13092792

yes

characteristic

137

C₂

#13092809

yes

characteristic

138

C₂

#13092810

yes

characteristic

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#500	C₂
socle	C₂²×C₂₃	#127
prime radical	#10625045	C₂²
hypercenter	#500	D₂₃
hypocenter	C₂₃	#1722
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#10625111

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#500

C₂

C₂²×C₂₃

C₂²

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #13092820	C₂	semidirect
2. #13092796	C₂	semidirect
6. #13092139	C₂	semidirect
7. C₂₃×#500	C₂	semidirect
67. #500	D₂₃	semidirect
90. C₂₃	#1722	semidirect

H ≅ G/N

type

1. #13092820

C₂

semidirect

2. #13092796

C₂

semidirect

6. #13092139

C₂

semidirect

7. C₂₃×#500

C₂

semidirect

67. #500

D₂₃

semidirect

90. C₂₃

#1722

semidirect

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

subgroup	index
#13092820	2
#13092796	2
#13092791	2
#13092765	2
#13092764	2
#13092139	2
C₂₃×#500	2
#1722	23

subgroup

index

C₂₃×#500

Subgroups — #15673067

Conjugacy classes of subgroups

Complements

Maximal subgroups