Group #15673559 — subgroups

Subgroups — #15673559

Order 2944.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#10624989	C₂	1	2	yes	maximal
2	C₂×#10624990	C₂	1	2	yes	maximal
3	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#13092283	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
9	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂₃×#95	—	2	2	no	—
12	C₂₃⋊#95	—	2	2	no	—
13	#10624990	—	2	4	no	—
14	#10624989	—	2	4	no	—
15	C₄×C₂₃⋊C₈	C₄	1	4	yes	complemented
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	—	2	2	no	—
20	C₄⋊C₄×C₂₃	—	2	2	no	—
21	C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₄²×C₂₃	Q₈	1	2	yes	—
24	C₄⋊Dic₂₃	—	2	2	no	—
25	C₄⋊Dic₂₃	—	2	2	no	—
26	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
29	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
30	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
31	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
32	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	4	no	—
34	C₂×C₄×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	—	4	4	no	—
36	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×Dic₂₃	—	4	2	no	—
44	C₂×Dic₂₃	—	4	4	no	—
45	C₂₃⋊C₈	C₄⋊C₄	1	4	yes	—
46	C₂₃⋊C₈	C₄⋊C₄	1	4	yes	—
47	#1846	—	23	1	no	maximal, Sylow
48	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₃	#127	1	1	yes	characteristic
50	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₃	#135	1	4	yes	—
52	C₄×C₂₃	—	2	4	no	—
53	C₄×C₂₃	—	8	2	no	—
54	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₃	—	8	2	no	—
59	C₂×#107	—	23	2	no	—
60	C₂×#106	—	23	2	no	—
61	C₂×C₄×C₈	—	23	1	no	—
62	#382	—	23	1	no	—
63	#382	D₂₃	1	1	yes	characteristic
64	C₂×C₂₃	#384	1	1	yes	characteristic
65	C₂×C₂₃	#427	1	2	yes	—
66	C₂×C₂₃	#495	1	2	yes	—
67	C₂×C₂₃	#499	1	2	yes	—
68	C₂²×C₈	—	23	1	no	—
69	C₂²×C₈	—	23	1	no	—
70	C₂×C₄⋊C₄	—	23	2	no	—
71	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
72	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
73	#107	—	46	4	no	—
74	#106	—	46	4	no	—
75	C₄×C₈	—	23	4	no	—
76	#95	—	2	2	no	—
77	#95	—	46	2	no	—
78	C₂₃	#1846	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	46	2	no	—
81	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
82	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
83	C₂×C₈	—	23	2	no	—
84	C₂×C₈	—	23	2	no	—
85	C₂×C₈	—	23	2	no	—
86	C₂×C₈	—	23	2	no	—
87	C₂×C₈	—	23	2	no	—
88	C₂×C₈	—	23	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	46	2	no	—
92	C₄⋊C₄	—	46	2	no	—
93	C₄²	Dic₄₆	1	2	yes	—
94	C₄²	C₄×D₂₃	1	2	yes	—
95	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	92	2	no	—
100	C₂×C₄	—	92	4	no	—
101	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
102	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
104	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
108	C₈	—	23	4	no	—
109	C₈	—	23	4	no	—
110	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
111	C₂²	#10625111	1	1	yes	characteristic
112	C₂²	#10625115	1	1	yes	characteristic
113	C₂²	#10625121	1	4	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	184	2	no	—
117	C₄	#10625059	1	1	yes	characteristic
118	C₄	#10625059	1	1	yes	characteristic
119	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
120	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
121	C₂	#13092286	1	1	yes	characteristic
122	C₂	#13092453	1	2	yes	—
123	C₂	#13092799	1	2	yes	—
124	C₂	#13092816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624989

C₂

yes

maximal

C₂×#10624990

C₂

yes

maximal

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092283

C₂

yes

maximal, characteristic

C₂₃×#382

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#95

—

C₂₃⋊#95

—

#10624990

—

#10624989

—

C₄×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

C₂₃⋊C₈

C₄⋊C₄

yes

—

#1846

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#127

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

#135

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

#382

—

#382

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#495

yes

—

C₂×C₂₃

#499

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#107

—

#106

—

C₄×C₈

—

#95

—

#95

—

C₂₃

#1846

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

—

C₄²

C₄×D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

Dic₂₃⋊C₄

yes

—

102

C₂×C₄

C₂×Dic₄₆

yes

characteristic

103

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

104

C₂×C₄

D₄×D₂₃

yes

characteristic

105

C₂×C₄

D₄×D₂₃

yes

characteristic

106

C₂×C₄

Q₈×D₂₃

yes

characteristic

107

C₂×C₄

Q₈×D₂₃

yes

characteristic

108

C₈

—

109

C₈

—

110

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

111

C₂²

#10625111

yes

characteristic

112

C₂²

#10625115

yes

characteristic

113

C₂²

#10625121

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

184

—

117

C₄

#10625059

yes

characteristic

118

C₄

#10625059

yes

characteristic

119

C₄

C₄⋊C₄×D₂₃

yes

characteristic

120

C₄

C₄⋊C₄×D₂₃

yes

characteristic

121

C₂

#13092286

yes

characteristic

122

C₂

#13092453

yes

—

123

C₂

#13092799

yes

—

124

C₂

#13092816

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#382	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#382	D₂₃
hypocenter	C₂₃	#1846
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#382

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
15. C₄×C₂₃⋊C₈	C₄	semidirect
78. C₂₃	#1846	semidirect

H ≅ G/N

type

15. C₄×C₂₃⋊C₈

C₄

semidirect

78. C₂₃

#1846

semidirect

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

subgroup	index
C₂×#10624989	2
C₂×#10624990	2
C₂×C₄×C₂₃⋊C₈	2
#13092283	2
C₂₃×#382	2
#1846	23

subgroup

index

C₂×C₄×C₂₃⋊C₈

C₂₃×#382

Subgroups — #15673559

Conjugacy classes of subgroups

Complements

Maximal subgroups