Group #15669932 — subgroups

Subgroups — C₂₃×#2313 #15669932

Order 2944.

512 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 280 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	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#387	C₂	1	8	yes	maximal, complemented
3	C₂₃×#381	C₂	1	2	yes	maximal
4	C₂₃×#380	C₂	1	2	yes	maximal
5	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₃	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
14	C₂₃×#95	C₂²	1	4	yes	—
15	C₂₃×#95	C₂²	1	4	yes	—
16	C₂₃×#95	C₂²	1	8	yes	—
17	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	—	2	4	no	—
19	C₂²×C₄×C₂₃	—	4	2	no	—
20	C₂²×C₄×C₂₃	—	4	2	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₄⋊C₄×C₂₃	—	4	4	no	—
29	C₄⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
30	C₂²⋊C₄×C₂₃	—	2	8	no	—
31	C₄²×C₂₃	—	4	4	no	—
32	C₂³×C₂₃	—	2	2	no	—
33	C₂³×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
35	C₂³×C₂₃	#41	1	4	yes	—
36	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	8	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	8	no	—
42	C₂×C₄×C₂₃	—	2	8	no	—
43	C₂×C₄×C₂₃	—	2	8	no	—
44	C₂×C₄×C₂₃	—	4	4	no	—
45	C₂×C₄×C₂₃	—	4	4	no	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
49	#2313	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	—	2	8	no	—
53	C₂²×C₂₃	#117	1	4	yes	—
54	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
55	C₂²×C₂₃	C₂×#41	1	2	yes	—
56	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
58	C₄×C₂₃	—	2	4	no	—
59	C₄×C₂₃	—	2	4	no	—
60	C₄×C₂₃	—	2	4	no	—
61	C₄×C₂₃	—	4	8	no	—
62	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
63	#387	C₂×C₂₃	1	8	yes	complemented
64	#381	C₂×C₂₃	1	2	yes	—
65	#380	C₂×C₂₃	1	2	yes	—
66	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
67	C₂×C₂₃	—	2	4	no	—
68	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
69	C₂×C₂₃	#518	1	1	yes	characteristic
70	C₂×C₂₃	#519	1	1	yes	characteristic
71	C₂×C₂₃	#539	1	4	yes	—
72	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
73	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
74	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	C₄×C₂₃	1	8	yes	complemented
76	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
77	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
79	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
80	#95	C₂²×C₂₃	1	4	yes	—
81	#95	C₂²×C₂₃	1	4	yes	—
82	#95	C₂²×C₂₃	1	8	yes	—
83	C₂₃	#2313	1	1	yes	characteristic, direct product factor, Sylow
84	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
89	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
90	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
91	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
92	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
93	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
94	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
95	C₄⋊C₄	—	4	4	no	—
96	C₄⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
97	C₂²⋊C₄	—	2	8	no	—
98	C₄²	—	4	4	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
102	C₂³	#41×C₂₃	1	4	yes	—
103	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
104	C₂×C₄	—	2	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
114	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
115	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	C₂₃×#117	1	4	yes	—
120	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
121	C₂²	C₂×#41×C₂₃	1	2	yes	—
122	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
123	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	8	no	—
128	C₂	—	2	4	no	—
129	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
130	C₂	C₂₃×#518	1	1	yes	characteristic
131	C₂	C₂₃×#519	1	1	yes	characteristic
132	C₂	C₂₃×#539	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂₃×#387

C₂

yes

maximal, complemented

C₂₃×#381

C₂

yes

maximal

C₂₃×#380

C₂

yes

maximal

C₄×C₂²⋊C₄×C₂₃

C₂

yes

maximal

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

#41

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

#2313

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₂²×C₂₃

2₋¹⁺⁴

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₂²×C₄⋊C₄

C₂×C₂₃

yes

characteristic

#387

C₂×C₂₃

yes

complemented

#381

C₂×C₂₃

yes

—

#380

C₂×C₂₃

yes

—

C₄×C₂²⋊C₄

C₂×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₄

C₄×C₂₃

yes

complemented

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

#95

C₂²×C₂₃

yes

—

C₂₃

#2313

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂³×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

100

C₂³

—

101

C₂³

C₂²×C₄×C₂₃

yes

—

102

C₂³

#41×C₂₃

yes

—

103

C₂³

C₂⁴×C₂₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×C₄×C₂₃

yes

—

114

C₂×C₄

C₂²×C₄×C₂₃

yes

—

115

C₂×C₄

C₂²×C₄×C₂₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂₃×#117

yes

—

120

C₂²

C₂³×C₄×C₂₃

yes

characteristic

121

C₂²

C₂×#41×C₂₃

yes

—

122

C₂²

C₂₃×H(5,𝔽₂)

yes

—

123

C₂²

C₂₃×2₋¹⁺⁴

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

C₂×C₂₃×#117

yes

characteristic

130

C₂

C₂₃×#518

yes

characteristic

131

C₂

C₂₃×#519

yes

characteristic

132

C₂

C₂₃×#539

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂⁴

commutator

C₂²

C₂³×C₄×C₂₃

Frattini, cosocle

C₂³

C₂⁴×C₂₃

prime radical

C₂⁴×C₂₃

C₂³

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#387	C₂	semidirect
9. C₂×C₄⋊C₄×C₂₃	C₄	semidirect
49. #2313	C₂₃	direct (N × H)
63. #387	C₂×C₂₃	semidirect
75. C₂×C₄⋊C₄	C₄×C₂₃	semidirect
83. C₂₃	#2313	direct (N × H)

H ≅ G/N

type

2. C₂₃×#387

C₂

semidirect

9. C₂×C₄⋊C₄×C₂₃

C₄

semidirect

49. #2313

C₂₃

direct (N × H)

63. #387

C₂×C₂₃

semidirect

75. C₂×C₄⋊C₄

C₄×C₂₃

semidirect

83. C₂₃

#2313

direct (N × H)

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

subgroup	index
C₂²×C₄⋊C₄×C₂₃	2
C₂₃×#387	2
C₂₃×#381	2
C₂₃×#380	2
C₄×C₂²⋊C₄×C₂₃	2
#2313	23

subgroup

index

C₂²×C₄⋊C₄×C₂₃

C₂₃×#387

C₂₃×#381

C₂₃×#380

C₄×C₂²⋊C₄×C₂₃

#2313

Subgroups — C₂₃×#2313 #15669932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2313 #15669932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2313 #15669932