Group #15669732 — subgroups

Subgroups — C₂₃×#2113 #15669732

Order 2944.

228 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 120 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×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂×#41×C₂₃	C₄	1	1	yes	characteristic
6	C₂₃×#131	—	2	2	no	—
7	C₂₃×#131	—	2	2	no	—
8	C₂×M₁₆×C₂₃	C₄	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₄	1	2	yes	—
10	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×M₃₂	—	2	2	no	—
13	C₂×C₁₆×C₂₃	—	2	2	no	—
14	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
16	#41×C₂₃	—	2	2	no	—
17	#41×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	C₈	1	1	yes	characteristic
19	C₂×D₄×C₂₃	C₈	1	2	yes	—
20	C₂×D₄×C₂₃	C₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	2	no	—
24	M₁₆×C₂₃	—	2	4	no	—
25	C₂×C₈×C₂₃	—	2	2	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	D₄	1	2	yes	—
30	C₂×C₈×C₂₃	D₄	1	2	yes	—
31	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₁₆×C₂₃	—	2	2	no	—
33	C₁₆×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	C₂×C₈	1	2	yes	—
35	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
36	Q₈×C₂₃	—	2	2	no	—
37	D₄×C₂₃	—	2	4	no	—
38	D₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
45	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₂₃	—	2	2	no	—
47	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
50	#2113	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₂₃	—	2	2	no	—
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	—	2	4	no	—
54	C₂²×C₂₃	C₂²⋊C₈	1	2	yes	—
55	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
58	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×#131	C₂×C₂₃	1	1	yes	characteristic
61	C₂×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
62	C₂²×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
63	C₂²⋊C₁₆	C₂×C₂₃	1	4	yes	complemented
64	C₂×C₂₃	—	2	2	no	—
65	C₂×C₂₃	—	4	2	no	—
66	C₂×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₂₃	#504	1	2	yes	—
68	C₂×#41	C₄×C₂₃	1	1	yes	characteristic
69	#131	—	2	2	no	—
70	#131	—	2	2	no	—
71	C₂×M₁₆	C₄×C₂₃	1	1	yes	characteristic
72	C₂×M₁₆	C₄×C₂₃	1	2	yes	—
73	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
74	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
75	M₃₂	—	2	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	C₂²×C₂₃	1	2	yes	—
78	C₂×C₁₆	C₂²×C₂₃	1	2	yes	—
79	C₂₃	#2113	1	1	yes	characteristic, direct product factor, Sylow
80	#41	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	C₈×C₂₃	1	1	yes	characteristic
83	C₂×D₄	C₈×C₂₃	1	2	yes	—
84	C₂×D₄	C₈×C₂₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
86	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
92	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
93	C₂×C₈	D₄×C₂₃	1	2	yes	—
94	C₂×C₈	D₄×C₂₃	1	2	yes	—
95	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	2	2	no	—
98	C₂³	C₂×C₈×C₂₃	1	2	yes	—
99	C₂³	C₂×C₈×C₂₃	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
106	C₂×C₄	C₂×C₈×C₂₃	1	2	yes	—
107	C₂×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
108	C₂×C₄	M₁₆×C₂₃	1	2	yes	—
109	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
112	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
113	C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	2	4	no	—
117	C₂²	C₂₃×C₂²⋊C₈	1	2	yes	—
118	C₂²	C₂²×C₈×C₂₃	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
121	C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
122	C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
126	C₂	C₂₃×#504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#131

C₂

yes

maximal, characteristic

C₂×C₂₃×M₃₂

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂₃×C₂²⋊C₁₆

C₂

yes

maximal, complemented

C₂×#41×C₂₃

C₄

yes

characteristic

C₂₃×#131

—

C₂₃×#131

—

C₂×M₁₆×C₂₃

C₄

yes

characteristic

C₂×M₁₆×C₂₃

C₄

yes

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂²

yes

—

C₂×C₁₆×C₂₃

C₂²

yes

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₈

yes

characteristic

C₂×D₄×C₂₃

C₈

yes

—

C₂×D₄×C₂₃

C₈

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₁₆×C₂₃

—

C₁₆×C₂₃

—

C₂³×C₂₃

C₂×C₈

yes

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

C₂×D₄

yes

characteristic

C₈×C₂₃

C₂×D₄

yes

characteristic

#2113

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²⋊C₈

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₂×M₁₆

yes

characteristic

C₂×#131

C₂×C₂₃

yes

characteristic

C₂×M₃₂

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₁₆

C₂×C₂₃

yes

characteristic, complemented

C₂²⋊C₁₆

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₃

#504

yes

—

C₂×#41

C₄×C₂₃

yes

characteristic

#131

—

#131

—

C₂×M₁₆

C₄×C₂₃

yes

characteristic

C₂×M₁₆

C₄×C₂₃

yes

—

C₂²×C₈

C₂²×C₂₃

yes

—

C₂²×C₈

C₂²×C₂₃

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₂₃

yes

—

C₂×C₁₆

C₂²×C₂₃

yes

—

C₂₃

#2113

yes

characteristic, direct product factor, Sylow

#41

—

#41

—

C₂×Q₈

C₈×C₂₃

yes

characteristic

C₂×D₄

C₈×C₂₃

yes

—

C₂×D₄

C₈×C₂₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

C₂×C₄×C₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₈

C₂×C₄×C₂₃

yes

characteristic

C₂×C₈

D₄×C₂₃

yes

—

C₂×C₈

D₄×C₂₃

yes

—

C₂×C₈

C₂³×C₂₃

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₂×C₈×C₂₃

yes

—

C₂³

C₂×C₈×C₂₃

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

106

C₂×C₄

C₂×C₈×C₂₃

yes

—

107

C₂×C₄

C₂×C₈×C₂₃

yes

characteristic

108

C₂×C₄

M₁₆×C₂₃

yes

—

109

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

110

C₈

—

111

C₈

C₂²⋊C₄×C₂₃

yes

—

112

C₈

C₂×D₄×C₂₃

yes

characteristic

113

C₈

C₂×D₄×C₂₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂₃×C₂²⋊C₈

yes

—

118

C₂²

C₂²×C₈×C₂₃

yes

characteristic

119

C₄

—

120

C₄

C₂₃×C₂²⋊C₈

yes

—

121

C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

122

C₄

C₂×M₁₆×C₂₃

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×C₂₃×C₂²⋊C₈

yes

characteristic

126

C₂

C₂₃×#504

yes

—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂³
commutator	C₂²	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂²×C₈
prime radical	C₂×#41×C₂₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₈×C₂₃

C₂³

commutator

C₂²

C₂²×C₈×C₂₃

Frattini, cosocle

C₂×C₈

C₂³×C₂₃

socle

C₂²×C₂₃

C₂²×C₈

prime radical

C₂×#41×C₂₃

C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂×C₂₃×M₃₂	C₂	semidirect
3. C₂²×C₁₆×C₂₃	C₂	semidirect
4. C₂₃×C₂²⋊C₁₆	C₂	semidirect
50. #2113	C₂₃	direct (N × H)
61. C₂×M₃₂	C₂×C₂₃	semidirect
62. C₂²×C₁₆	C₂×C₂₃	semidirect
63. C₂²⋊C₁₆	C₂×C₂₃	semidirect
79. C₂₃	#2113	direct (N × H)

H ≅ G/N

type

2. C₂×C₂₃×M₃₂

C₂

semidirect

3. C₂²×C₁₆×C₂₃

C₂

semidirect

4. C₂₃×C₂²⋊C₁₆

C₂

semidirect

50. #2113

C₂₃

direct (N × H)

61. C₂×M₃₂

C₂×C₂₃

semidirect

62. C₂²×C₁₆

C₂×C₂₃

semidirect

63. C₂²⋊C₁₆

C₂×C₂₃

semidirect

79. C₂₃

#2113

direct (N × H)

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

subgroup	index
C₂×C₂₃×#131	2
C₂×C₂₃×M₃₂	2
C₂²×C₁₆×C₂₃	2
C₂₃×C₂²⋊C₁₆	2
#2113	23

subgroup

index

C₂×C₂₃×#131

C₂×C₂₃×M₃₂

C₂²×C₁₆×C₂₃

C₂₃×C₂²⋊C₁₆

#2113

Subgroups — C₂₃×#2113 #15669732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2113 #15669732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2113 #15669732