Group #15673514 — subgroups

Subgroups — #15673514

Order 2944.

260 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
2	#13092172	C₂	1	4	yes	maximal
3	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₂₃	C₄	1	2	yes	—
5	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
10	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
11	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₄²×C₂₃	—	2	4	no	—
17	C₄²×C₂₃	—	2	2	no	—
18	C₄²×C₂₃	—	2	4	no	—
19	C₄²×C₂₃	C₈	1	4	yes	complemented
20	C₄²×C₂₃	D₄	1	2	yes	—
21	C₄²×C₂₃	Q₈	1	2	yes	—
22	C₂×C₂₃⋊C₈	—	2	4	no	—
23	C₂×C₂₃⋊C₈	—	2	8	no	—
24	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	—	2	2	no	—
26	C₂×C₄×C₂₃	—	2	2	no	—
27	C₂×C₄×C₂₃	—	2	2	no	—
28	C₂×C₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	—	2	2	no	—
30	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
33	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
35	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃	#41	1	4	yes	—
39	C₂₃⋊C₈	—	4	4	no	—
40	#1831	—	23	1	no	maximal, Sylow
41	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₂₃	#117	1	2	yes	—
43	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
44	C₂²×C₂₃	C₂×M₁₆	1	1	yes	characteristic
45	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
46	C₄×C₂₃	—	2	2	no	—
47	C₄×C₂₃	—	2	4	no	—
48	C₄×C₂₃	—	2	4	no	—
49	C₄×C₂₃	C₄⋊C₈	1	4	yes	—
50	C₄×C₂₃	#124	1	2	yes	—
51	C₄×C₂₃	#125	1	2	yes	—
52	C₂×C₄⋊C₈	—	23	2	no	—
53	C₄³	D₂₃	1	1	yes	characteristic
54	#336	—	23	4	no	—
55	C₂×C₂₃	#382	1	1	yes	characteristic
56	C₂×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₂₃	#423	1	1	yes	characteristic
58	C₂×C₂₃	#431	1	2	yes	—
59	C₂×C₂₃	#432	1	2	yes	—
60	C₂²×C₈	—	23	4	no	—
61	C₂×C₄²	Dic₂₃	1	2	yes	—
62	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
63	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₂₃	1	2	yes	—
65	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
66	C₄⋊C₈	—	46	4	no	—
67	C₂₃	#1831	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
69	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
70	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
71	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
73	C₂×C₈	—	46	4	no	—
74	C₂×C₈	—	46	8	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	2	4	no	—
78	C₄²	C₂₃⋊C₈	1	4	yes	complemented
79	C₄²	Dic₄₆	1	2	yes	—
80	C₄²	D₉₂	1	2	yes	—
81	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₂×C₂₃⋊C₈	1	4	yes	—
88	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
89	C₂×C₄	#67267	1	4	yes	—
90	C₂×C₄	#67267	1	2	yes	—
91	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
92	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
93	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
94	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
95	C₂×C₄	#67288	1	4	yes	—
96	C₈	—	92	4	no	—
97	C₂²	C₂²×C₂₃⋊C₈	1	1	yes	characteristic
98	C₂²	C₂×#67267	1	1	yes	characteristic
99	C₂²	C₂×#67267	1	2	yes	—
100	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
101	C₂²	#10625097	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	4	no	—
104	C₄	—	2	4	no	—
105	C₄	C₄⋊(C₂₃⋊C₈)	1	4	yes	—
106	C₄	#10625041	1	2	yes	—
107	C₄	#10625046	1	2	yes	—
108	C₂	#13092283	1	1	yes	characteristic
109	C₂	C₂×C₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic
110	C₂	#13092441	1	1	yes	characteristic
111	C₂	#13092470	1	2	yes	—
112	C₂	#13092474	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal

#13092172

C₂

yes

maximal

C₄³×C₂₃

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₄

yes

characteristic

C₂×C₄²×C₂₃

C₄

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₈

yes

complemented

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

#1831

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄⋊C₈

yes

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

#125

yes

—

C₂×C₄⋊C₈

—

C₄³

D₂₃

yes

characteristic

#336

—

C₂×C₂₃

#382

yes

characteristic

C₂×C₂₃

C₂×C₄⋊C₈

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#432

yes

—

C₂²×C₈

—

C₂×C₄²

Dic₂₃

yes

—

C₂×C₄²

Dic₂₃

yes

characteristic

C₂×C₄²

Dic₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂₃

#1831

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂₃⋊C₈

yes

complemented

C₄²

Dic₄₆

yes

—

C₄²

D₉₂

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

C₂×C₄

#67267

yes

—

C₂×C₄

#67267

yes

—

C₂×C₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

#67288

yes

—

C₈

—

C₂²

C₂²×C₂₃⋊C₈

yes

C₂²

yes

C₂²

yes

—

100

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

101

C₂²

#10625097

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄⋊(C₂₃⋊C₈)

yes

—

106

C₄

#10625041

yes

—

107

C₄

#10625046

yes

—

108

C₂

#13092283

yes

characteristic

109

C₂

C₂×C₄⋊(C₂₃⋊C₈)

yes

characteristic

110

C₂

#13092441

yes

characteristic

111

C₂

#13092470

yes

—

112

C₂

#13092474

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂²×C₈
Fitting	C₄³×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄³	D₂₃
hypocenter	C₂₃	#1831
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂²×C₈

Fitting

C₄³×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₄³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
19. C₄²×C₂₃	C₈	semidirect
67. C₂₃	#1831	semidirect
78. C₄²	C₂₃⋊C₈	semidirect

H ≅ G/N

type

19. C₄²×C₂₃

C₈

semidirect

67. C₂₃

#1831

semidirect

78. C₄²

C₂₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092172	2
C₄³×C₂₃	2
#1831	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

#13092172

C₄³×C₂₃

#1831

Subgroups — #15673514

Conjugacy classes of subgroups

Complements

Maximal subgroups