Group #15670758 — subgroups

Subgroups — C₂₃×#3139 #15670758

Order 2944.

380 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 192 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	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#491	C₂	1	4	yes	maximal, complemented
4	C₂₃×#487	C₂	1	4	yes	maximal, complemented
5	C₂₃×#419	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂₃×C₄⋊Q₈	—	2	2	no	—
10	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂₃×#125	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊Q₈	—	2	2	no	—
15	C₄×Q₈×C₂₃	—	2	2	no	—
16	C₂₃×#117	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#103	C₂²	1	16	yes	—
20	C₂₃×#97	C₂²	1	2	yes	—
21	C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂×Q₈×C₂₃	—	2	2	no	—
23	C₂×Q₈×C₂₃	—	2	2	no	—
24	C₂×Q₈×C₂₃	—	2	8	no	—
25	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₃	—	2	2	no	—
27	C₂²×C₄×C₂₃	D₄	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	M₁₆×C₂₃	—	2	4	no	—
30	C₂×C₈×C₂₃	C₂³	1	4	yes	—
31	C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₂₃	—	2	2	no	—
37	C₄²×C₂₃	—	2	2	no	—
38	C₄²×C₂₃	D₄	1	2	yes	—
39	C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₂₃	—	2	8	no	—
43	Q₈×C₂₃	—	4	4	no	—
44	Q₈×C₂₃	—	4	4	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	4	4	no	—
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	#41	1	4	yes	—
53	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
54	C₈×C₂₃	—	2	4	no	—
55	#3139	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	#124	1	2	yes	—
58	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	—	4	4	no	—
61	C₄×C₂₃	—	4	4	no	—
62	C₄×C₂₃	#124	1	2	yes	—
63	C₄×C₂₃	#137	1	4	yes	—
64	C₄×C₂₃	C₂×#41	1	2	yes	—
65	#533	C₂×C₂₃	1	1	yes	characteristic
66	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
67	#491	C₂×C₂₃	1	4	yes	complemented
68	#487	C₂×C₂₃	1	4	yes	complemented
69	#419	C₂×C₂₃	1	4	yes	—
70	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
71	C₂×C₂₃	—	2	2	no	—
72	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
73	C₂×C₂₃	C₂×#137	1	2	yes	—
74	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
75	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
78	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
79	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
80	#125	C₂²×C₂₃	1	2	yes	—
81	C₂²⋊Q₈	—	2	2	no	—
82	C₄×Q₈	—	2	2	no	—
83	#117	C₂²×C₂₃	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
86	#103	C₂²×C₂₃	1	16	yes	—
87	#97	C₂²×C₂₃	1	2	yes	—
88	C₄×C₈	C₂²×C₂₃	1	2	yes	—
89	C₂₃	#3139	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	8	no	—
93	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	D₄×C₂₃	1	2	yes	—
96	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
97	M₁₆	—	2	4	no	—
98	C₂×C₈	C₂³×C₂₃	1	4	yes	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
104	C₂²⋊C₄	—	2	2	no	—
105	C₄²	—	2	2	no	—
106	C₄²	D₄×C₂₃	1	2	yes	—
107	C₄²	C₂³×C₂₃	1	1	yes	characteristic
108	C₄²	C₂³×C₂₃	1	1	yes	characteristic
109	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	4	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
120	C₂×C₄	#41×C₂₃	1	4	yes	—
121	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂₃×#124	1	2	yes	—
125	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	4	4	no	—
129	C₄	C₂₃×#124	1	2	yes	—
130	C₄	C₂₃×#137	1	4	yes	—
131	C₄	C₂×#41×C₂₃	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
134	C₂	C₂×C₂₃×#137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#533

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂₃×#491

C₂

yes

maximal, complemented

C₂₃×#487

C₂

yes

maximal, complemented

C₂₃×#419

C₂

yes

maximal

C₄×M₁₆×C₂₃

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#125

C₂²

yes

—

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#103

C₂²

yes

—

C₂₃×#97

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₈×C₂₃

—

#3139

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

#137

yes

—

C₄×C₂₃

C₂×#41

yes

—

#533

C₂×C₂₃

yes

characteristic

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic

#491

C₂×C₂₃

yes

complemented

#487

C₂×C₂₃

yes

complemented

#419

C₂×C₂₃

yes

—

C₄×M₁₆

C₂×C₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

C₂×#137

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

—

C₂×M₁₆

C₂²×C₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

#125

C₂²×C₂₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#103

C₂²×C₂₃

yes

—

#97

C₂²×C₂₃

yes

—

C₄×C₈

C₂²×C₂₃

yes

—

C₂₃

#3139

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

M₁₆

—

C₂×C₈

C₂³×C₂₃

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂³×C₂₃

yes

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

D₄×C₂₃

yes

—

107

C₄²

C₂³×C₂₃

yes

characteristic

108

C₄²

C₂³×C₂₃

yes

characteristic

109

C₂³

C₂×D₄×C₂₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₄×C₂₃

yes

—

118

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

119

C₂×C₄

C₂×D₄×C₂₃

yes

—

120

C₂×C₄

#41×C₂₃

yes

—

121

C₂×C₄

C₂⁴×C₂₃

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

C₂₃×#124

yes

—

125

C₂²

C₂²×D₄×C₂₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₂₃×#124

yes

—

130

C₄

C₂₃×#137

yes

—

131

C₄

C₂×#41×C₂₃

yes

—

132

C₂

—

133

C₂

C₂×C₂₃×#124

yes

characteristic

134

C₂

C₂×C₂₃×#137

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₂₃

prime radical

C₂³×C₂₃

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
3. C₂₃×#491	C₂	semidirect
4. C₂₃×#487	C₂	semidirect
55. #3139	C₂₃	direct (N × H)
67. #491	C₂×C₂₃	semidirect
68. #487	C₂×C₂₃	semidirect
89. C₂₃	#3139	direct (N × H)

H ≅ G/N

type

3. C₂₃×#491

C₂

semidirect

4. C₂₃×#487

C₂

semidirect

55. #3139

C₂₃

direct (N × H)

67. #491

C₂×C₂₃

semidirect

68. #487

C₂×C₂₃

semidirect

89. C₂₃

#3139

direct (N × H)

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

subgroup	index
C₂₃×#533	2
C₂×C₂₃×C₄⋊Q₈	2
C₂₃×#491	2
C₂₃×#487	2
C₂₃×#419	2
C₄×M₁₆×C₂₃	2
#3139	23

subgroup

index

C₂₃×#533

C₂×C₂₃×C₄⋊Q₈

C₂₃×#491

C₂₃×#487

C₂₃×#419

C₄×M₁₆×C₂₃

#3139

Subgroups — C₂₃×#3139 #15670758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#3139 #15670758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3139 #15670758