Group #15669179 — subgroups

Subgroups — C₂₃×#1560 #15669179

Order 2944.

254 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 96 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#423	C₂	1	2	yes	maximal, complemented
3	C₂₃×#333	C₂	1	2	yes	maximal, complemented
4	C₂₃×#331	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₂₃	C₄	1	2	yes	—
6	C₂×M₁₆×C₂₃	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	C₂₃×#124	C₄	1	2	yes	—
10	C₂₃×#121	—	2	2	no	—
11	C₄×D₄×C₂₃	—	2	2	no	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
15	#41×C₂₃	—	4	4	no	—
16	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₂₃	—	2	2	no	—
18	C₂×D₄×C₂₃	—	2	2	no	—
19	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	D₄	1	2	yes	—
23	M₁₆×C₂₃	—	4	4	no	—
24	C₂×C₈×C₂₃	—	2	4	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂²⋊C₄×C₂₃	—	2	4	no	—
27	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₃	D₄	1	2	yes	—
29	C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	C₂³×C₂₃	—	2	2	no	—
31	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₂₃	—	4	2	no	—
33	D₄×C₂₃	—	4	2	no	—
34	D₄×C₂₃	—	4	4	no	—
35	D₄×C₂₃	—	4	2	no	—
36	D₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	4	4	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₈×C₂₃	—	4	4	no	—
47	#1560	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂²×C₂₃	—	2	1	no	—
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	—	4	2	no	—
51	C₂²×C₂₃	—	4	4	no	—
52	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃	—	2	1	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	4	2	no	—
56	C₄×C₂₃	#100	1	2	yes	—
57	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
58	C₄×C₂₃	#137	1	2	yes	—
59	#532	C₂×C₂₃	1	1	yes	characteristic
60	#423	C₂×C₂₃	1	2	yes	complemented
61	#333	C₂×C₂₃	1	2	yes	complemented
62	#331	C₂×C₂₃	1	2	yes	complemented
63	C₂×C₂₃	—	4	1	no	—
64	C₂×C₂₃	—	8	2	no	—
65	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
66	C₂×C₂₃	#417	1	2	yes	—
67	C₂×#41	C₄×C₂₃	1	2	yes	—
68	C₂×M₁₆	—	2	2	no	—
69	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
70	#127	C₂²×C₂₃	1	1	yes	characteristic
71	#124	C₄×C₂₃	1	2	yes	—
72	#121	—	2	2	no	—
73	C₄×D₄	—	2	2	no	—
74	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
77	C₂₃	#1560	1	1	yes	characteristic, direct product factor, Sylow
78	#41	—	4	4	no	—
79	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
85	C₂²×C₄	D₄×C₂₃	1	2	yes	—
86	M₁₆	—	4	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
91	C₄²	D₄×C₂₃	1	2	yes	—
92	C₄²	C₂³×C₂₃	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
106	C₂×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
107	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
108	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
109	C₈	—	4	4	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	C₂₃×#100	1	2	yes	—
119	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
120	C₄	C₂₃×#137	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×C₂₃×#100	1	1	yes	characteristic
124	C₂	C₂₃×#417	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#532

C₂

yes

maximal, characteristic

C₂₃×#423

C₂

yes

maximal, complemented

C₂₃×#333

C₂

yes

maximal, complemented

C₂₃×#331

C₂

yes

maximal, complemented

C₂×#41×C₂₃

C₄

yes

—

C₂×M₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#124

C₄

yes

—

C₂₃×#121

—

C₄×D₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

Q₈×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×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

—

#1560

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#100

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

#137

yes

—

#532

C₂×C₂₃

yes

characteristic

#423

C₂×C₂₃

yes

complemented

#333

C₂×C₂₃

yes

complemented

#331

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#100

yes

characteristic

C₂×C₂₃

#417

yes

—

C₂×#41

C₄×C₂₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₂₃

yes

characteristic

#127

C₂²×C₂₃

yes

characteristic

#124

C₄×C₂₃

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₃

yes

—

C₂₃

#1560

yes

characteristic, direct product factor, Sylow

#41

—

C₂×Q₈

C₂×C₄×C₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₄×C₂₃

yes

characteristic

C₄²

D₄×C₂₃

yes

—

C₄²

C₂³×C₂₃

yes

characteristic

C₂³

—

C₂³

C₂²⋊C₄×C₂₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

106

C₂×C₄

C₂²⋊C₄×C₂₃

yes

characteristic

107

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

108

C₂×C₄

C₂×D₄×C₂₃

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂₃×#100

yes

—

119

C₄

C₂₃×AGL(1,ℤ/8)

yes

—

120

C₄

C₂₃×#137

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×C₂₃×#100

yes

characteristic

124

C₂

C₂₃×#417

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₄²	C₂³×C₂₃
prime radical	C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₄²

C₂³×C₂₃

prime radical

C₂₃×#532

C₂

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#423	C₂	semidirect
3. C₂₃×#333	C₂	semidirect
4. C₂₃×#331	C₂	semidirect
47. #1560	C₂₃	direct (N × H)
60. #423	C₂×C₂₃	semidirect
61. #333	C₂×C₂₃	semidirect
62. #331	C₂×C₂₃	semidirect
77. C₂₃	#1560	direct (N × H)

H ≅ G/N

type

2. C₂₃×#423

C₂

semidirect

3. C₂₃×#333

C₂

semidirect

4. C₂₃×#331

C₂

semidirect

47. #1560

C₂₃

direct (N × H)

60. #423

C₂×C₂₃

semidirect

61. #333

C₂×C₂₃

semidirect

62. #331

C₂×C₂₃

semidirect

77. C₂₃

#1560

direct (N × H)

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

subgroup	index
C₂₃×#532	2
C₂₃×#423	2
C₂₃×#333	2
C₂₃×#331	2
#1560	23

subgroup

index

C₂₃×#532

C₂₃×#423

C₂₃×#333

C₂₃×#331

#1560

Subgroups — C₂₃×#1560 #15669179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1560 #15669179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1560 #15669179