Group #15685201 — subgroups

Subgroups — #15685201

Order 2944.

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

normal subgroups only

This group has 444 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	#13093230	C₂	1	4	yes	maximal, complemented
2	#13092689	C₂	1	4	yes	maximal, complemented
3	#13092599	C₂	1	4	yes	maximal, complemented
4	#13092535	C₂	1	2	yes	maximal, complemented
5	C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625158	C₂²	1	4	yes	—
7	C₂×Q₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	#10625113	C₂²	1	8	yes	—
10	#10625090	—	2	4	no	—
11	#10625089	—	2	8	no	—
12	Q₁₆×D₂₃	—	2	4	no	—
13	#10625075	C₂²	1	2	yes	—
14	#10625072	C₂²	1	4	yes	—
15	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	#10625018	C₂²	1	4	yes	—
17	#10625001	C₂²	1	2	yes	—
18	#10624987	C₂²	1	4	yes	—
19	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
20	#67292	—	2	8	no	—
21	Q₈×D₂₃	—	2	8	no	—
22	#67288	—	4	4	no	—
23	#67288	C₂³	1	4	yes	—
24	Q₁₆×C₂₃	—	2	4	no	—
25	Q₁₆×C₂₃	—	2	4	no	—
26	C₂×C₈×C₂₃	C₂³	1	2	yes	—
27	C₄⋊C₄×C₂₃	—	4	2	no	—
28	C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	#67275	—	2	8	no	—
30	#67274	—	2	8	no	—
31	#67267	C₂³	1	4	yes	—
32	Dic₉₂	D₄	1	2	yes	—
33	D₁₈₄	D₄	1	2	yes	—
34	#67263	D₄	1	4	yes	—
35	#67262	—	2	4	no	—
36	C₈×D₂₃	—	2	4	no	—
37	Q₈×C₂₃	—	2	8	no	—
38	C₂×C₄×C₂₃	—	2	1	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₂₃	—	2	4	no	—
42	D₉₂	—	4	4	no	—
43	D₉₂	C₂×D₄	1	4	yes	—
44	C₄×D₂₃	—	2	4	no	—
45	C₄×D₂₃	—	4	8	no	—
46	Dic₄₆	—	4	4	no	—
47	Dic₄₆	C₂×D₄	1	4	yes	—
48	C₈×C₂₃	C₂×D₄	1	2	yes	—
49	C₈×C₂₃	C₂×D₄	1	2	yes	—
50	C₂₃⋊C₈	—	2	4	no	—
51	#3293	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂×D₂₃	—	2	4	no	—
54	C₄×C₂₃	—	2	2	no	—
55	C₄×C₂₃	—	4	4	no	—
56	C₄×C₂₃	C₂²×D₄	1	2	yes	—
57	Dic₂₃	—	2	4	no	—
58	#578	—	23	4	no	—
59	#494	D₂₃	1	1	yes	characteristic, complemented
60	#473	—	23	4	no	—
61	#456	—	23	4	no	—
62	#443	—	23	2	no	—
63	C₂×C₂₃	—	2	1	no	—
64	C₂×C₂₃	D₄²	1	1	yes	characteristic
65	D₂₃	—	4	4	no	—
66	2₋¹⁺⁴	—	23	4	no	—
67	#137	—	23	8	no	—
68	#137	—	46	8	no	—
69	#135	—	23	2	no	—
70	#135	—	46	4	no	—
71	C₂×Q₁₆	—	46	4	no	—
72	C₂×Q₁₆	C₂×D₂₃	1	4	yes	—
73	#131	—	23	4	no	—
74	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
75	#108	—	23	2	no	—
76	#104	—	23	4	no	—
77	#101	—	23	4	no	—
78	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
79	C₂₃	#3293	1	1	yes	characteristic, complemented, Sylow
80	#41	—	23	4	no	—
81	#41	—	46	8	no	—
82	#41	—	92	4	no	—
83	C₂×Q₈	—	46	8	no	—
84	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	23	2	no	—
88	Q₁₆	—	46	8	no	—
89	SD₁₆	—	23	4	no	—
90	SD₁₆	—	46	8	no	—
91	D₈	—	23	2	no	—
92	M₁₆	—	23	4	no	—
93	M₁₆	—	46	4	no	—
94	C₂×C₈	—	46	4	no	—
95	C₂×C₈	C₂²×D₂₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₂₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	23	4	no	—
100	Q₈	—	92	4	no	—
101	D₄	—	23	4	no	—
102	D₄	—	46	4	no	—
103	D₄	—	92	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	46	4	no	—
107	C₂×C₄	—	92	8	no	—
108	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
109	C₈	—	46	4	no	—
110	C₈	D₄×D₂₃	1	2	yes	—
111	C₈	D₄×D₂₃	1	2	yes	—
112	C₂²	—	46	4	no	—
113	C₂²	#10625155	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	46	4	no	—
117	C₄	C₂×D₄×D₂₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	92	4	no	—
120	C₂	#13093052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093230

C₂

yes

maximal, complemented

#13092689

C₂

yes

maximal, complemented

#13092599

C₂

yes

maximal, complemented

#13092535

C₂

yes

maximal, complemented

C₂₃×#494

C₂

yes

maximal, characteristic, complemented

#10625158

C₂²

yes

—

C₂×Q₁₆×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

#10625113

C₂²

yes

—

#10625090

—

#10625089

—

Q₁₆×D₂₃

—

#10625075

C₂²

yes

—

#10625072

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625018

C₂²

yes

—

#10625001

C₂²

yes

—

#10624987

C₂²

yes

—

C₂×Q₈×C₂₃

C₂³

yes

—

#67292

—

Q₈×D₂₃

—

#67288

—

#67288

C₂³

yes

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

—

—

C₂³

yes

—

Dic₉₂

D₄

yes

—

D₁₈₄

D₄

yes

—

#67263

D₄

yes

—

#67262

—

C₈×D₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂²⋊D₂₃

—

D₉₂

—

D₉₂

C₂×D₄

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

Dic₄₆

C₂×D₄

yes

—

C₈×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

C₂×D₄

yes

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²×D₄

yes

—

Dic₂₃

—

#578

—

#494

D₂₃

yes

characteristic, complemented

#473

—

#456

—

#443

—

C₂×C₂₃

—

C₂×C₂₃

D₄²

yes

characteristic

D₂₃

—

2₋¹⁺⁴

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₂₃

yes

—

#131

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#3293

yes

characteristic, complemented, Sylow

#41

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₂₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

D₄

—

102

D₄

—

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂³×D₂₃

yes

characteristic

109

C₈

—

110

C₈

D₄×D₂₃

yes

—

111

C₈

D₄×D₂₃

yes

—

112

C₂²

—

113

C₂²

#10625155

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×D₄×D₂₃

yes

—

118

C₂

—

119

C₂

—

120

C₂

#13093052

yes

characteristic

	subgroup	G/H
center	C₂	#13093052
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#494	C₂
socle	C₂×C₂₃	D₄²
hypercenter	#494	D₂₃
hypocenter	C₂₃	#3293
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#13093052

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#494

C₂

socle

C₂×C₂₃

D₄²

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #13093230	C₂	semidirect
2. #13092689	C₂	semidirect
3. #13092599	C₂	semidirect
4. #13092535	C₂	semidirect
5. C₂₃×#494	C₂	semidirect
59. #494	D₂₃	semidirect
79. C₂₃	#3293	semidirect

H ≅ G/N

type

1. #13093230

C₂

semidirect

2. #13092689

C₂

semidirect

3. #13092599

C₂

semidirect

4. #13092535

C₂

semidirect

5. C₂₃×#494

C₂

semidirect

59. #494

D₂₃

semidirect

79. C₂₃

#3293

semidirect

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

subgroup	index
#13093230	2
#13092689	2
#13092599	2
#13092535	2
C₂₃×#494	2
#3293	23

subgroup

index

C₂₃×#494

Subgroups — #15685201

Conjugacy classes of subgroups

Complements

Maximal subgroups