Group #15686799 — subgroups

Subgroups — #15686799

Order 2944.

996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 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	#13093046	C₂	1	12	yes	maximal, complemented
2	#13093043	C₂	1	3	yes	maximal
3	#13092969	C₂	1	12	yes	maximal, complemented
4	C₂×#10625048	C₂	1	3	yes	maximal
5	C₂₃×#543	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	—
8	C₂×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₂₃	C₂²	1	3	yes	—
10	Q₈×Dic₂₃	C₂²	1	4	yes	complemented
11	#10625115	C₂²	1	12	yes	complemented
12	#10625095	C₂²	1	12	yes	—
13	C₂×Dic₂₃⋊C₄	C₂²	1	6	yes	—
14	C₂×C₄×Dic₂₃	C₂²	1	6	yes	—
15	#10625060	C₂²	1	24	yes	complemented
16	#10625059	C₂²	1	12	yes	complemented
17	#10625058	C₂²	1	12	yes	complemented
18	#10625049	C₂²	1	24	yes	—
19	#10625048	C₂²	1	12	yes	—
20	#10625048	C₂²	1	12	yes	—
21	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
24	C₂²×Dic₂₃	Q₈	1	8	yes	complemented
25	C₂²×Dic₂₃	C₂³	1	6	yes	—
26	C₂²×Dic₂₃	C₂³	1	3	yes	—
27	C₂²×Dic₂₃	C₂³	1	3	yes	—
28	C₂×Dic₄₆	C₂³	1	12	yes	—
29	C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
30	C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
31	C₂²⋊Dic₂₃	C₂³	1	12	yes	—
32	C₄⋊Dic₂₃	C₂³	1	12	yes	—
33	Dic₂₃⋊C₄	C₂³	1	24	yes	—
34	Dic₂₃⋊C₄	C₂³	1	24	yes	—
35	C₄×Dic₂₃	C₂³	1	12	yes	—
36	C₄×Dic₂₃	C₂³	1	12	yes	—
37	C₂³×C₂₃	—	4	2	no	—
38	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
39	C₂³×C₂₃	C₂⁴	1	3	yes	—
40	Q₈×C₂₃	—	4	4	no	—
41	C₂×C₄×C₂₃	—	4	6	no	—
42	C₂×C₄×C₂₃	C₂⁴	1	12	yes	—
43	C₂×Dic₂₃	—	4	6	no	—
44	C₂×Dic₂₃	C₂×Q₈	1	12	yes	—
45	C₂×Dic₂₃	C₂×Q₈	1	12	yes	—
46	C₂×Dic₂₃	C₂⁴	1	12	yes	—
47	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₂₃	C₂⁴	1	3	yes	—
49	Dic₄₆	—	4	12	no	—
50	#3458	—	23	1	no	maximal, Sylow
51	C₂²×C₂₃	—	2	6	no	—
52	C₂²×C₂₃	—	4	4	no	—
53	C₂²×C₂₃	C₂²×Q₈	1	6	yes	—
54	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
55	C₄×C₂₃	—	2	12	no	—
56	Dic₂₃	—	2	12	no	—
57	Dic₂₃	C₂²×Q₈	1	2	yes	—
58	Dic₂₃	C₂²×Q₈	1	6	yes	—
59	#544	—	23	12	no	—
60	#543	—	23	3	no	—
61	#543	D₂₃	1	1	yes	characteristic
62	#533	—	23	12	no	—
63	C₂×#117	—	23	3	no	—
64	C₂×C₂₃	—	2	6	no	—
65	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
66	C₂×C₂₃	#585	1	2	yes	—
67	C₂³×C₄	—	23	1	no	—
68	C₄⋊Q₈	—	23	12	no	—
69	C₄⋊Q₈	—	23	12	no	—
70	#125	—	23	24	no	—
71	C₂²⋊Q₈	—	23	12	no	—
72	C₂²⋊Q₈	—	23	24	no	—
73	C₂²⋊Q₈	C₂×D₂₃	1	12	yes	—
74	C₄×Q₈	—	23	4	no	—
75	C₄×Q₈	—	23	12	no	—
76	#117	—	23	12	no	—
77	#117	—	23	12	no	—
78	C₂×C₄⋊C₄	—	23	6	no	—
79	C₂×C₂²⋊C₄	—	23	3	no	—
80	C₂×C₂²⋊C₄	C₂×D₂₃	1	3	yes	—
81	C₂×C₄²	—	23	6	no	—
82	C₂₃	#3458	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
84	C₂×Q₈	—	23	12	no	—
85	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
86	C₂²×C₄	—	23	8	no	—
87	C₂²×C₄	—	23	3	no	—
88	C₂²×C₄	—	23	3	no	—
89	C₂²×C₄	—	23	6	no	—
90	C₂²×C₄	C₂²×D₂₃	1	6	yes	—
91	C₄⋊C₄	—	23	24	no	—
92	C₄⋊C₄	—	23	24	no	—
93	C₄⋊C₄	—	23	12	no	—
94	C₄⋊C₄	C₂²×D₂₃	1	12	yes	—
95	C₂²⋊C₄	—	23	12	no	—
96	C₂²⋊C₄	C₂²×D₂₃	1	12	yes	—
97	C₄²	—	23	12	no	—
98	C₄²	—	23	12	no	—
99	C₂³	—	4	2	no	—
100	C₂³	Q₈×D₂₃	1	4	yes	—
101	C₂³	C₂³×D₂₃	1	3	yes	—
102	Q₈	—	4	4	no	—
103	Q₈	—	92	12	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	23	12	no	—
106	C₂×C₄	—	23	12	no	—
107	C₂×C₄	—	23	1	no	—
108	C₂×C₄	—	23	3	no	—
109	C₂×C₄	—	23	12	no	—
110	C₂×C₄	—	92	6	no	—
111	C₂×C₄	C₂³×D₂₃	1	12	yes	—
112	C₂²	—	2	6	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×Q₈×D₂₃	1	6	yes	—
115	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
116	C₄	—	2	12	no	—
117	C₄	—	23	2	no	—
118	C₄	—	23	6	no	—
119	C₄	—	46	12	no	—
120	C₂	—	2	6	no	—
121	C₂	C₂²×Q₈×D₂₃	1	1	yes	characteristic
122	C₂	#13093252	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13093046

C₂

yes

maximal, complemented

#13093043

C₂

yes

maximal

#13092969

C₂

yes

maximal, complemented

C₂×#10625048

C₂

yes

maximal

C₂₃×#543

C₂

yes

maximal, characteristic

C₂³×Dic₂₃

C₂²

yes

characteristic

C₂₃×C₂²⋊Q₈

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₃

C₂²

yes

—

Q₈×Dic₂₃

C₂²

yes

complemented

#10625115

C₂²

yes

complemented

#10625095

C₂²

yes

—

C₂×Dic₂₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

#10625060

C₂²

yes

complemented

#10625059

C₂²

yes

complemented

#10625058

C₂²

yes

complemented

#10625049

C₂²

yes

—

#10625048

C₂²

yes

—

#10625048

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×Dic₂₃

Q₈

yes

complemented

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂²×Dic₂₃

C₂³

yes

—

C₂×Dic₄₆

C₂³

yes

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊Dic₂₃

C₂³

yes

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

C₂³

yes

—

Dic₂₃⋊C₄

C₂³

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×Q₈

yes

—

C₂³×C₂₃

C₂⁴

yes

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂⁴

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

C₂⁴

yes

—

C₂×Dic₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

C₂⁴

yes

—

Dic₄₆

—

#3458

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×Q₈

yes

—

C₂²×C₂₃

C₂⁵

yes

characteristic

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

C₂²×Q₈

yes

—

Dic₂₃

C₂²×Q₈

yes

—

#544

—

#543

—

#543

D₂₃

yes

characteristic

#533

—

C₂×#117

—

C₂×C₂₃

—

C₂×C₂₃

C₂³×Q₈

yes

characteristic

C₂×C₂₃

#585

yes

—

C₂³×C₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×D₂₃

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂₃

#3458

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₂₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₂₃

yes

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

Q₈×D₂₃

yes

—

101

C₂³

C₂³×D₂₃

yes

—

102

Q₈

—

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂³×D₂₃

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×Q₈×D₂₃

yes

—

115

C₂²

C₂⁴×D₂₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₂

—

121

C₂

C₂²×Q₈×D₂₃

yes

characteristic

122

C₂

#13093252

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#543	C₂
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	#543	D₂₃
hypocenter	C₂₃	#3458
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #543

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁵

Fitting

C₂₃×#543

C₂

prime radical

C₂⁴×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #543

N	H ≅ G/N	type
1. #13093046	C₂	semidirect
3. #13092969	C₂	semidirect
10. Q₈×Dic₂₃	C₂²	semidirect
11. #10625115	C₂²	semidirect
15. #10625060	C₂²	semidirect
16. #10625059	C₂²	semidirect
17. #10625058	C₂²	semidirect
24. C₂²×Dic₂₃	Q₈	semidirect
82. C₂₃	#3458	semidirect

H ≅ G/N

type

1. #13093046

C₂

semidirect

3. #13092969

C₂

semidirect

10. Q₈×Dic₂₃

C₂²

semidirect

11. #10625115

C₂²

semidirect

15. #10625060

C₂²

semidirect

16. #10625059

C₂²

semidirect

17. #10625058

C₂²

semidirect

24. C₂²×Dic₂₃

Q₈

semidirect

82. C₂₃

#3458

semidirect

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

subgroup	index
#13093046	2
#13093043	2
#13092969	2
C₂×#10625048	2
C₂₃×#543	2
#3458	23

subgroup

index

C₂₃×#543

Subgroups — #15686799

Conjugacy classes of subgroups

Complements

Maximal subgroups