Group #15673466 — subgroups

Subgroups — #15673466

Order 2944.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#10625097	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#13092171	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₂₃	C₄	1	2	yes	—
8	#10625097	—	2	2	no	—
9	#10625097	—	2	2	no	—
10	C₂×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
16	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	—	2	4	no	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
23	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	4	no	—
26	C₂²⋊Dic₂₃	—	2	4	no	—
27	C₄⋊Dic₂₃	—	2	2	no	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄×Dic₂₃	—	2	2	no	—
30	C₄×Dic₂₃	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	4	no	—
33	C₂³×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	—	2	2	no	—
35	C₂³×C₂₃	—	2	2	no	—
36	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	4	yes	—
45	C₂×Dic₂₃	—	2	2	no	—
46	C₂×Dic₂₃	—	2	2	no	—
47	C₂×Dic₂₃	—	2	4	no	—
48	C₂×Dic₂₃	—	2	4	no	—
49	C₈×C₂₃	—	4	2	no	—
50	C₂₃⋊C₈	—	4	2	no	—
51	#1818	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	—	2	2	no	—
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	—	4	4	no	—
56	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₃	#117	1	2	yes	—
58	C₂²×C₂₃	#131	1	2	yes	—
59	C₂²×C₂₃	#131	1	2	yes	—
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	#123	1	2	yes	—
62	C₄×C₂₃	#123	1	2	yes	—
63	Dic₂₃	—	4	2	no	—
64	Dic₂₃	—	4	2	no	—
65	C₂×#117	—	23	1	no	—
66	C₂×C₂²⋊C₈	—	23	1	no	—
67	C₂×C₂²⋊C₈	D₂₃	1	1	yes	characteristic
68	#336	—	23	4	no	—
69	C₂×C₂₃	—	4	2	no	—
70	C₂×C₂₃	#381	1	1	yes	characteristic
71	C₂×C₂₃	#408	1	1	yes	characteristic
72	C₂×C₂₃	#408	1	1	yes	characteristic
73	C₂×C₂₃	#433	1	4	yes	—
74	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
75	C₂²×C₈	—	23	2	no	—
76	C₂²×C₈	C₂×D₂₃	1	2	yes	—
77	#117	—	46	2	no	—
78	#117	—	46	2	no	—
79	C₂×C₄⋊C₄	—	23	1	no	—
80	C₂×C₄⋊C₄	—	23	1	no	—
81	C₂×C₂²⋊C₄	—	23	2	no	—
82	C₂×C₄²	—	23	2	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₂²⋊C₈	—	46	2	no	—
85	C₂₃	#1818	1	1	yes	characteristic, complemented, Sylow
86	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	23	2	no	—
89	C₂²×C₄	—	23	2	no	—
90	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
91	C₂²×C₄	D₉₂	1	2	yes	—
92	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	46	2	no	—
97	C₂×C₈	—	46	4	no	—
98	C₄⋊C₄	—	46	2	no	—
99	C₄⋊C₄	—	46	2	no	—
100	C₂²⋊C₄	—	46	4	no	—
101	C₄²	—	46	2	no	—
102	C₄²	—	46	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
107	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	46	4	no	—
112	C₂×C₄	—	46	4	no	—
113	C₂×C₄	—	46	2	no	—
114	C₂×C₄	—	46	2	no	—
115	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
116	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
117	C₂×C₄	#67290	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	92	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	4	no	—
125	C₂²	#10625048	1	2	yes	—
126	C₂²	#10625072	1	2	yes	—
127	C₂²	#10625078	1	2	yes	—
128	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	92	2	no	—
131	C₄	—	92	2	no	—
132	C₄	#10625057	1	2	yes	—
133	C₄	#10625106	1	2	yes	—
134	C₂	—	4	2	no	—
135	C₂	#13092278	1	1	yes	characteristic
136	C₂	#13092383	1	1	yes	characteristic
137	C₂	#13092385	1	1	yes	characteristic
138	C₂	#13092475	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625097

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal, complemented

C₂×C₂₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₄

yes

—

#10625097

—

#10625097

—

C₂×C₄⋊Dic₂₃

C₄

yes

characteristic

C₂×C₄⋊Dic₂₃

C₄

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1818

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

#131

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₄×C₂₃

#123

yes

—

Dic₂₃

—

Dic₂₃

—

C₂×#117

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₂₃

yes

characteristic

#336

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1818

yes

characteristic, complemented, Sylow

C₂⁴

C₄×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

(C₂×D₂₃)⋊C₄

yes

—

107

C₂³

C₂×C₄×D₂₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

116

C₂×C₄

C₂×D₉₂

yes

characteristic

117

C₂×C₄

#67290

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

#10625048

yes

—

126

C₂²

#10625072

yes

—

127

C₂²

#10625078

yes

—

128

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#10625057

yes

—

133

C₄

#10625106

yes

—

134

C₂

—

135

C₂

#13092278

yes

characteristic

136

C₂

#13092383

yes

characteristic

137

C₂

#13092385

yes

characteristic

138

C₂

#13092475

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×C₂²⋊C₈	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#1818
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₂₃×C₂²⋊C₈

C₂

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #13092171	C₂	semidirect
85. C₂₃	#1818	semidirect

H ≅ G/N

type

3. #13092171

C₂

semidirect

85. C₂₃

#1818

semidirect

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

subgroup	index
C₂×#10625097	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092171	2
C₂×C₂₃×C₂²⋊C₈	2
#1818	23

subgroup

index

C₂×#10625097

C₂×C₂²⋊(C₂₃⋊C₈)

#13092171

C₂×C₂₃×C₂²⋊C₈

#1818

Subgroups — #15673466

Conjugacy classes of subgroups

Complements

Maximal subgroups