Group #15674375 — subgroups

Subgroups — #15674375

Order 2944.

284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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	#13092441	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092176	C₂	1	4	yes	maximal
4	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
5	C₂₃×C₄⋊Q₈	C₄	1	2	yes	complemented
6	C₂₃×#125	C₄	1	2	yes	complemented
7	C₂₃×C₂²⋊Q₈	—	2	2	no	—
8	C₄×Q₈×C₂₃	—	2	2	no	—
9	C₂₃×#117	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	—	2	2	no	—
12	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
13	C₂×#67267	C₂²	1	2	yes	—
14	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	D₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	2	no	—
19	C₄⋊C₄×C₂₃	—	2	4	no	—
20	C₄⋊C₄×C₂₃	—	2	2	no	—
21	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₂₃	—	2	2	no	—
23	C₄²×C₂₃	—	2	2	no	—
24	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₃	D₄	1	2	yes	—
27	C₄×Dic₂₃	—	2	8	no	—
28	C₄×Dic₂₃	—	2	2	no	—
29	C₄×Dic₂₃	Q₈	1	4	yes	complemented
30	#67267	—	2	4	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₂₃	—	4	4	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₃	#41	1	2	yes	—
42	C₂×Dic₂₃	—	2	8	no	—
43	C₂×Dic₂₃	—	2	4	no	—
44	C₂₃⋊C₈	—	4	2	no	—
45	#1994	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₂₃	—	2	2	no	—
50	C₄×C₂₃	—	4	4	no	—
51	C₄×C₂₃	#104	1	4	yes	—
52	C₄×C₂₃	C₄×Q₈	1	2	yes	—
53	C₄×C₂₃	#124	1	1	yes	characteristic
54	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₂₃	—	2	8	no	—
56	#533	D₂₃	1	1	yes	characteristic
57	#423	—	23	1	no	—
58	C₄³	—	23	1	no	—
59	#339	—	23	4	no	—
60	C₂×C₂₃	—	2	2	no	—
61	C₂×C₂₃	#391	1	1	yes	characteristic
62	C₂×C₂₃	C₂×#104	1	2	yes	—
63	C₂×M₁₆	—	23	2	no	—
64	C₄⋊Q₈	Dic₂₃	1	2	yes	complemented
65	#125	Dic₂₃	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₂₃	1	2	yes	—
69	C₂×C₄²	—	23	2	no	—
70	C₂×C₄²	—	46	2	no	—
71	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
72	C₄⋊C₈	—	46	2	no	—
73	C₂₃	#1994	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	46	2	no	—
76	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	M₁₆	—	46	4	no	—
79	C₂×C₈	—	46	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	23	4	no	—
86	C₄²	—	46	8	no	—
87	C₄²	—	46	2	no	—
88	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
89	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₂₃	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	46	8	no	—
96	C₂×C₄	—	46	4	no	—
97	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
99	C₂×C₄	Q₈×D₂₃	1	2	yes	—
100	C₂×C₄	#67292	1	2	yes	—
101	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
103	C₈	—	92	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#10625117	1	2	yes	—
106	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	46	8	no	—
110	C₄	#10625019	1	4	yes	—
111	C₄	#10625115	1	1	yes	characteristic
112	C₄	Q₈×Dic₂₃	1	2	yes	—
113	C₄	#10625118	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#13092324	1	1	yes	characteristic
116	C₂	C₂×#10625019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, characteristic

C₄²×Dic₂₃

C₂

yes

maximal, characteristic

#13092176

C₂

yes

maximal

C₂₃×#533

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₄

yes

complemented

C₂₃×#125

C₄

yes

complemented

C₂₃×C₂²⋊Q₈

—

C₄×Q₈×C₂₃

—

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

Q₈

yes

complemented

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×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

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1994

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#104

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

#533

D₂₃

yes

characteristic

#423

—

C₄³

—

#339

—

C₂×C₂₃

—

C₂×C₂₃

#391

yes

characteristic

C₂×C₂₃

C₂×#104

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₂₃

yes

complemented

#125

Dic₂₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂₃

#1994

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

Q₈×D₂₃

yes

—

100

C₂×C₄

#67292

yes

—

101

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

#10625117

yes

—

106

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#10625019

yes

—

111

C₄

#10625115

yes

characteristic

112

C₄

Q₈×Dic₂₃

yes

—

113

C₄

#10625118

yes

characteristic

114

C₂

—

115

C₂

#13092324

yes

characteristic

116

C₂

C₂×#10625019

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#533	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#533	D₂₃
hypocenter	C₂₃	#1994
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#533

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
5. C₂₃×C₄⋊Q₈	C₄	semidirect
6. C₂₃×#125	C₄	semidirect
29. C₄×Dic₂₃	Q₈	semidirect
64. C₄⋊Q₈	Dic₂₃	semidirect
65. #125	Dic₂₃	semidirect
73. C₂₃	#1994	semidirect

H ≅ G/N

type

5. C₂₃×C₄⋊Q₈

C₄

semidirect

6. C₂₃×#125

C₄

semidirect

29. C₄×Dic₂₃

Q₈

semidirect

64. C₄⋊Q₈

Dic₂₃

semidirect

65. #125

Dic₂₃

semidirect

73. C₂₃

#1994

semidirect

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

subgroup	index
#13092441	2
C₄²×Dic₂₃	2
#13092176	2
C₂₃×#533	2
#1994	23

subgroup

index

#13092441

C₄²×Dic₂₃

#13092176

C₂₃×#533

#1994

Subgroups — #15674375

Conjugacy classes of subgroups

Complements

Maximal subgroups