Group #15673254 — subgroups

Subgroups — #15673254

Order 2944.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	2	yes	maximal
5	C₂₃×#376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
11	C₂₃×#95	C₄	1	4	yes	—
12	C₄⋊(C₂₃⋊C₈)	C₄	1	8	yes	complemented
13	#10624985	—	2	2	no	—
14	C₄×C₂₃⋊C₈	—	2	2	no	—
15	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₄²×C₂₃	—	2	2	no	—
20	C₄²×C₂₃	C₂×C₄	1	4	yes	—
21	C₄²×C₂₃	D₄	1	2	yes	—
22	C₄²×C₂₃	Q₈	1	2	yes	—
23	C₂×C₂₃⋊C₈	—	2	2	no	—
24	C₂×C₂₃⋊C₈	—	2	2	no	—
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	—	2	2	no	—
27	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
28	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃	—	2	4	no	—
31	C₂×C₄×C₂₃	—	2	4	no	—
32	C₂×C₄×C₂₃	—	2	2	no	—
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	C₄²	1	4	yes	—
35	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₂₃	#41	1	2	yes	—
40	C₂₃⋊C₈	—	2	4	no	—
41	C₂₃⋊C₈	—	2	4	no	—
42	#1768	—	23	1	no	maximal, Sylow
43	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₂₃	#117	1	1	yes	characteristic
46	C₂²×C₂₃	#131	1	4	yes	—
47	C₄×C₂₃	—	2	4	no	—
48	C₄×C₂₃	—	2	4	no	—
49	C₄×C₂₃	—	4	2	no	—
50	C₄×C₂₃	C₄×D₄	1	2	yes	—
51	C₄×C₂₃	C₄×Q₈	1	2	yes	—
52	C₂×C₄⋊C₈	—	23	2	no	—
53	C₂×#97	—	23	1	no	—
54	C₂×C₄×C₈	—	23	1	no	—
55	#376	D₂₃	1	1	yes	characteristic
56	#336	—	23	2	no	—
57	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₂₃	#405	1	2	yes	—
59	C₂×C₂₃	#424	1	2	yes	—
60	C₂×C₂₃	#433	1	2	yes	—
61	C₂²×C₈	—	23	2	no	—
62	C₂²×C₈	—	23	2	no	—
63	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
66	C₄⋊C₈	—	23	8	no	—
67	#97	—	46	2	no	—
68	C₄×C₈	—	46	2	no	—
69	#95	Dic₂₃	1	4	yes	—
70	C₂₃	#1768	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
72	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
73	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
75	C₂×C₈	—	23	4	no	—
76	C₂×C₈	—	23	4	no	—
77	C₂×C₈	—	46	2	no	—
78	C₂×C₈	—	46	2	no	—
79	C₂×C₈	—	46	2	no	—
80	C₂×C₈	—	46	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₄₆	1	2	yes	—
83	C₄²	C₄×D₂₃	1	4	yes	—
84	C₄²	D₉₂	1	2	yes	—
85	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
92	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
94	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
95	C₂×C₄	#67288	1	2	yes	—
96	C₈	—	46	4	no	—
97	C₈	—	46	4	no	—
98	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
100	C₂²	#10625097	1	1	yes	characteristic
101	C₂²	#10625119	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×Dic₄₆	1	2	yes	—
106	C₄	C₄×D₉₂	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₂₃	1	1	yes	characteristic
108	C₂	#13092373	1	2	yes	—
109	C₂	#13092444	1	2	yes	—
110	C₂	#13092478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal

C₂₃×#376

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×#95

C₄

yes

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

complemented

#10624985

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1768

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

D₂₃

yes

characteristic

#336

—

C₂×C₂₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₃

#405

yes

—

C₂×C₂₃

#424

yes

—

C₂×C₂₃

#433

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

#95

Dic₂₃

yes

—

C₂₃

#1768

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

Dic₄₆

yes

—

C₄²

C₄×D₂₃

yes

—

C₄²

D₉₂

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃

yes

—

C₂×C₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

#67288

yes

—

C₈

—

C₈

—

C₂²

C₂×C₄×Dic₂₃

yes

characteristic

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

100

C₂²

#10625097

yes

characteristic

101

C₂²

#10625119

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×Dic₄₆

yes

—

106

C₄

C₄×D₉₂

yes

—

107

C₂

C₄×C₄⋊Dic₂₃

yes

characteristic

108

C₂

#13092373

yes

—

109

C₂

#13092444

yes

—

110

C₂

#13092478

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂₃×#376	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#376	D₂₃
hypocenter	C₂₃	#1768
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂×C₄²

Fitting

C₂₃×#376

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
12. C₄⋊(C₂₃⋊C₈)	C₄	semidirect
70. C₂₃	#1768	semidirect

H ≅ G/N

type

12. C₄⋊(C₂₃⋊C₈)

C₄

semidirect

70. C₂₃

#1768

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
C₂×#10624985	2
C₂×C₄×C₂₃⋊C₈	2
#13092172	2
C₂₃×#376	2
#1768	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×#10624985

C₂×C₄×C₂₃⋊C₈

#13092172

C₂₃×#376

#1768

Subgroups — #15673254

Conjugacy classes of subgroups

Complements

Maximal subgroups