Group #15669243 — subgroups

Subgroups — C₂₃×#1624 #15669243

Order 2944.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#459	C₂	1	2	yes	maximal, complemented
4	C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₂₃	—	2	2	no	—
7	C₂×D₈×C₂₃	—	2	2	no	—
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂₃×#124	—	2	2	no	—
11	C₂₃×#121	—	2	2	no	—
12	C₄×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#117	—	2	2	no	—
14	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂₃×#102	—	2	2	no	—
16	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₃	D₄	1	2	yes	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×D₄×C₂₃	—	2	1	no	—
21	C₂×D₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	—	2	2	no	—
23	SD₁₆×C₂₃	—	4	4	no	—
24	D₈×C₂₃	—	4	4	no	—
25	C₂×C₈×C₂₃	—	2	1	no	—
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
29	C₂²⋊C₄×C₂₃	—	2	4	no	—
30	C₂²⋊C₄×C₂₃	—	2	2	no	—
31	C₄²×C₂₃	—	2	2	no	—
32	C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂³×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	—	4	1	no	—
35	Q₈×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	2	2	no	—
37	D₄×C₂₃	—	2	2	no	—
38	D₄×C₂₃	—	4	2	no	—
39	D₄×C₂₃	—	4	2	no	—
40	D₄×C₂₃	—	4	2	no	—
41	D₄×C₂₃	—	8	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	4	2	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₈×C₂₃	—	2	2	no	—
49	C₈×C₂₃	—	4	2	no	—
50	#1624	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₂₃	—	4	2	no	—
52	C₂²×C₂₃	—	4	2	no	—
53	C₂²×C₂₃	—	4	2	no	—
54	C₂²×C₂₃	—	8	2	no	—
55	C₂²×C₂₃	—	8	1	no	—
56	C₂²×C₂₃	#120	1	1	yes	characteristic
57	C₄×C₂₃	—	2	1	no	—
58	C₄×C₂₃	—	4	2	no	—
59	C₄×C₂₃	—	4	2	no	—
60	C₄×C₂₃	#135	1	2	yes	—
61	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
62	#555	C₂×C₂₃	1	1	yes	characteristic, complemented
63	#492	C₂×C₂₃	1	1	yes	characteristic, complemented
64	#459	C₂×C₂₃	1	2	yes	complemented
65	#333	C₂×C₂₃	1	1	yes	characteristic, complemented
66	D₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
67	C₂×C₂₃	—	8	2	no	—
68	C₂×C₂₃	—	16	1	no	—
69	C₂×C₂₃	#449	1	2	yes	—
70	C₂×C₂₃	#453	1	1	yes	characteristic
71	C₂×SD₁₆	—	2	2	no	—
72	C₂×D₈	—	2	2	no	—
73	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
74	#127	C₂²×C₂₃	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#121	—	2	2	no	—
77	C₄×D₄	C₂²×C₂₃	1	2	yes	—
78	#117	—	2	2	no	—
79	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
80	#102	—	2	2	no	—
81	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
82	C₂₃	#1624	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×Q₈	D₄×C₂₃	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	1	no	—
87	C₂×D₄	D₄×C₂₃	1	2	yes	—
88	C₂²×C₄	—	2	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	4	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂³×C₂₃	1	1	yes	characteristic
99	C₂³	—	2	2	no	—
100	C₂³	—	4	1	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	4	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	8	2	no	—
120	C₂²	—	8	1	no	—
121	C₂²	C₂₃×#120	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	C₂₃×#135	1	2	yes	—
126	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	16	1	no	—
129	C₂	C₂₃×#449	1	2	yes	—
130	C₂	C₂₃×#453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#555

C₂

yes

maximal, characteristic, complemented

C₂₃×#492

C₂

yes

maximal, characteristic, complemented

C₂₃×#459

C₂

yes

maximal, complemented

C₂₃×#333

C₂

yes

maximal, characteristic, complemented

C₂₃×D₄⋊C₈

C₂

yes

maximal, complemented

C₂×SD₁₆×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#124

—

C₂₃×#121

—

C₄×D₄×C₂₃

C₂²

yes

—

C₂₃×#117

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₂₃×#102

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

—

C₂³×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#1624

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#120

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

#555

C₂×C₂₃

yes

characteristic, complemented

#492

C₂×C₂₃

yes

characteristic, complemented

#459

C₂×C₂₃

yes

complemented

#333

C₂×C₂₃

yes

characteristic, complemented

D₄⋊C₈

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#449

yes

—

C₂×C₂₃

#453

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₂₃

yes

characteristic

#127

C₂²×C₂₃

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂²×C₂₃

yes

—

#117

—

C₄⋊C₈

C₂²×C₂₃

yes

—

#102

—

C₄×C₈

C₂²×C₂₃

yes

characteristic

C₂₃

#1624

yes

characteristic, direct product factor, Sylow

C₂×Q₈

D₄×C₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₂₃

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₂₃

yes

characteristic

C₂³

—

100

C₂³

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

113

C₂×C₄

C₂×D₄×C₂₃

yes

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂₃×#120

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂₃×#135

yes

—

126

C₄

C₂₃×AGL(1,ℤ/8)

yes

—

127

C₂

—

128

C₂

—

129

C₂

C₂₃×#449

yes

—

130

C₂

C₂₃×#453

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₂₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂₃×#555	C₂	semidirect
2. C₂₃×#492	C₂	semidirect
3. C₂₃×#459	C₂	semidirect
4. C₂₃×#333	C₂	semidirect
5. C₂₃×D₄⋊C₈	C₂	semidirect
50. #1624	C₂₃	direct (N × H)
62. #555	C₂×C₂₃	semidirect
63. #492	C₂×C₂₃	semidirect
64. #459	C₂×C₂₃	semidirect
65. #333	C₂×C₂₃	semidirect
66. D₄⋊C₈	C₂×C₂₃	semidirect
82. C₂₃	#1624	direct (N × H)

H ≅ G/N

type

1. C₂₃×#555

C₂

semidirect

2. C₂₃×#492

C₂

semidirect

3. C₂₃×#459

C₂

semidirect

4. C₂₃×#333

C₂

semidirect

5. C₂₃×D₄⋊C₈

C₂

semidirect

50. #1624

C₂₃

direct (N × H)

62. #555

C₂×C₂₃

semidirect

63. #492

C₂×C₂₃

semidirect

64. #459

C₂×C₂₃

semidirect

65. #333

C₂×C₂₃

semidirect

66. D₄⋊C₈

C₂×C₂₃

semidirect

82. C₂₃

#1624

direct (N × H)

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

subgroup	index
C₂₃×#555	2
C₂₃×#492	2
C₂₃×#459	2
C₂₃×#333	2
C₂₃×D₄⋊C₈	2
#1624	23

subgroup

index

C₂₃×#555

C₂₃×#492

C₂₃×#459

C₂₃×#333

C₂₃×D₄⋊C₈

#1624

Subgroups — C₂₃×#1624 #15669243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1624 #15669243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1624 #15669243