Group #15674242 — subgroups

Subgroups — #15674242

Order 2944.

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

normal subgroups only

This group has 292 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	C₂×#10625018	C₂	1	4	yes	maximal
3	C₄×#67267	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
6	C₂₃×C₄⋊Q₈	—	2	1	no	—
7	C₂₃×C₄⋊Q₈	—	2	1	no	—
8	C₂₃×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#67267	C₂²	1	2	yes	—
12	C₂×#67267	C₂²	1	2	yes	—
13	#10625018	—	2	8	no	—
14	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
15	#10624985	—	2	1	no	—
16	C₄×C₂₃⋊C₈	—	2	1	no	—
17	C₂×Q₈×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	—	2	2	no	—
19	C₂×Q₈×C₂₃	—	2	4	no	—
20	C₂×Q₈×C₂₃	—	2	4	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₄²×C₂₃	D₄	1	1	yes	characteristic
27	C₄²×C₂₃	D₄	1	1	yes	characteristic
28	C₄²×C₂₃	D₄	1	2	yes	—
29	#67267	—	2	2	no	—
30	#67267	—	2	4	no	—
31	#67267	D₄	1	4	yes	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₂₃	—	4	4	no	—
36	Q₈×C₂₃	—	4	2	no	—
37	Q₈×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	8	no	—
41	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂₃⋊C₈	—	2	4	no	—
48	C₂₃⋊C₈	—	4	2	no	—
49	#1966	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₃	C₄×D₄	1	2	yes	—
53	C₄×C₂₃	—	2	2	no	—
54	C₄×C₂₃	—	4	4	no	—
55	C₄×C₂₃	#101	1	4	yes	complemented
56	C₄×C₂₃	#121	1	2	yes	—
57	C₄×C₂₃	#124	1	1	yes	characteristic
58	C₄×C₂₃	#127	1	1	yes	characteristic
59	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
60	#423	—	23	1	no	—
61	C₂×#101	—	23	4	no	—
62	C₄×M₁₆	—	23	1	no	—
63	C₂×C₂₃	—	2	2	no	—
64	C₂×C₂₃	#390	1	1	yes	characteristic
65	C₂×C₂₃	C₂×#101	1	2	yes	—
66	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
67	C₂×M₁₆	—	23	2	no	—
68	C₂×M₁₆	—	23	2	no	—
69	C₄⋊Q₈	—	2	1	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	2	no	—
72	C₂×C₄⋊C₄	Dic₂₃	1	4	yes	—
73	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
74	C₄⋊C₈	—	46	2	no	—
75	#101	—	46	8	no	—
76	#97	—	46	1	no	—
77	C₄×C₈	—	46	1	no	—
78	C₂₃	#1966	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
84	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
86	M₁₆	—	23	4	no	—
87	M₁₆	—	46	2	no	—
88	M₁₆	—	46	4	no	—
89	C₂×C₈	—	46	2	no	—
90	C₂×C₈	—	46	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
95	C₄²	C₂²⋊D₂₃	1	2	yes	—
96	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
97	Q₈	—	4	2	no	—
98	Q₈	—	4	4	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
105	C₂×C₄	D₄×D₂₃	1	2	yes	—
106	C₂×C₄	#67290	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
109	C₈	—	46	4	no	—
110	C₈	—	92	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	D₄×Dic₂₃	1	2	yes	—
113	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#10625018	1	4	yes	complemented
117	C₄	#10625107	1	1	yes	characteristic
118	C₄	#10625109	1	2	yes	—
119	C₄	#10625111	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#13092320	1	1	yes	characteristic
122	C₂	C₂×#10625018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092441

C₂

yes

maximal, characteristic

C₂×#10625018

C₂

yes

maximal

C₄×#67267

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

—

#10625018

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₄×C₂₃⋊C₈

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

#67267

—

#67267

—

#67267

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1966

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#101

yes

complemented

C₄×C₂₃

#121

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₂×C₄⋊Q₈

D₂₃

yes

characteristic

#423

—

C₂×#101

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

C₂×#101

yes

—

C₂²×Q₈

C₂×D₂₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₂₃

#1966

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

characteristic

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂²×Dic₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₂₃

yes

—

104

C₂×C₄

C₂²⋊Dic₂₃

yes

—

105

C₂×C₄

D₄×D₂₃

yes

—

106

C₂×C₄

#67290

yes

—

107

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

D₄×Dic₂₃

yes

—

113

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#10625018

yes

complemented

117

C₄

#10625107

yes

characteristic

118

C₄

#10625109

yes

—

119

C₄

#10625111

yes

characteristic

120

C₂

—

121

C₂

#13092320

yes

characteristic

122

C₂

C₂×#10625018

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator, prime radical	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#1966
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator, prime radical

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×C₄⋊Q₈

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₃

hypocenter

C₂₃

#1966

LCS

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
55. C₄×C₂₃	#101	semidirect
78. C₂₃	#1966	semidirect
116. C₄	#10625018	semidirect

H ≅ G/N

type

55. C₄×C₂₃

#101

semidirect

78. C₂₃

#1966

semidirect

116. C₄

#10625018

semidirect

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

subgroup	index
#13092441	2
C₂×#10625018	2
C₄×#67267	2
C₂×C₂₃×C₄⋊Q₈	2
#1966	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

Subgroups — #15674242

Conjugacy classes of subgroups

Complements

Maximal subgroups