Group #15673061 — subgroups

Subgroups — #15673061

Order 2944.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#13092794	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092793	C₂	1	2	yes	maximal, complemented
3	#13092765	C₂	1	2	yes	maximal, complemented
4	#13092138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₂₃	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	C₂×#67272	—	2	4	no	—
10	C₂×D₁₈₄	—	2	2	no	—
11	#10625045	C₂²	1	2	yes	—
12	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	#10624991	—	2	4	no	—
14	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂×Q₈×C₂₃	—	2	1	no	—
16	C₂×D₄×C₂₃	—	2	2	no	—
17	C₂×D₄×C₂₃	—	2	1	no	—
18	C₂×D₉₂	—	2	2	no	—
19	C₂×D₉₂	—	2	2	no	—
20	C₂×D₉₂	—	2	2	no	—
21	SD₁₆×C₂₃	—	4	4	no	—
22	C₂×C₈×C₂₃	D₄	1	2	yes	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄²×C₂₃	C₂³	1	1	yes	characteristic
25	#67272	—	4	8	no	—
26	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
27	D₁₈₄	—	4	4	no	—
28	C₂³×C₂₃	—	4	1	no	—
29	C₂²×D₂₃	—	4	2	no	—
30	Q₈×C₂₃	—	4	2	no	—
31	D₄×C₂₃	—	4	2	no	—
32	D₄×C₂₃	—	4	4	no	—
33	C₂×C₄×C₂₃	—	4	1	no	—
34	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	D₉₂	—	4	4	no	—
37	D₉₂	—	4	4	no	—
38	D₉₂	—	4	4	no	—
39	C₈×C₂₃	D₈	1	4	yes	complemented
40	C₂₃⋊C₈	—	2	4	no	—
41	#1721	—	23	1	no	maximal, Sylow
42	C₂²×C₂₃	—	8	1	no	—
43	C₂²×C₂₃	—	8	2	no	—
44	C₂²×C₂₃	#127	1	1	yes	characteristic
45	C₂×D₂₃	—	8	2	no	—
46	C₂×D₂₃	—	8	2	no	—
47	C₂×D₂₃	—	8	2	no	—
48	C₄×C₂₃	—	8	1	no	—
49	C₄×C₂₃	C₂×D₈	1	2	yes	—
50	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
51	#493	—	23	2	no	—
52	#493	—	23	1	no	—
53	#492	D₂₃	1	1	yes	characteristic, complemented
54	#486	—	23	2	no	—
55	#322	—	23	1	no	—
56	C₂×C₂₃	—	16	1	no	—
57	C₂×C₂₃	#493	1	1	yes	characteristic
58	C₂×C₂₃	#496	1	2	yes	—
59	D₂₃	—	16	2	no	—
60	C₂×SD₁₆	—	2	2	no	—
61	C₂×D₈	—	46	4	no	—
62	C₂×D₈	—	46	2	no	—
63	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
64	#127	—	23	2	no	—
65	#127	C₂×D₂₃	1	1	yes	characteristic
66	#102	—	46	4	no	—
67	C₄×C₈	—	23	2	no	—
68	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
69	C₂₃	#1721	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	46	2	no	—
74	C₂×D₄	—	46	2	no	—
75	C₂×D₄	—	46	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	92	8	no	—
78	D₈	—	92	4	no	—
79	C₂×C₈	—	23	4	no	—
80	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₂₃	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	92	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	2	no	—
88	D₄	—	92	4	no	—
89	D₄	—	92	4	no	—
90	D₄	—	92	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₂₃	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
94	C₈	—	46	4	no	—
95	C₈	#67272	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	184	2	no	—
99	C₂²	—	184	2	no	—
100	C₂²	—	184	2	no	—
101	C₂²	#10625111	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	#10625085	1	4	yes	—
104	C₄	C₂×#67272	1	2	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	368	2	no	—
107	C₂	#13092793	1	1	yes	characteristic
108	C₂	#13092803	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092794

C₂

yes

maximal, characteristic, complemented

#13092793

C₂

yes

maximal, complemented

#13092765

C₂

yes

maximal, complemented

#13092138

C₂

yes

maximal, characteristic, complemented

C₂₃×#492

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂×#67272

—

C₂×D₁₈₄

—

#10625045

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624991

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

SD₁₆×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67272

—

C₂×C₂₃⋊C₈

D₄

yes

—

D₁₈₄

—

C₂³×C₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

D₉₂

—

D₉₂

—

D₉₂

—

C₈×C₂₃

D₈

yes

complemented

C₂₃⋊C₈

—

#1721

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#127

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₈

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

#493

—

#493

—

#492

D₂₃

yes

characteristic, complemented

#486

—

#322

—

C₂×C₂₃

—

C₂×C₂₃

#493

yes

characteristic

C₂×C₂₃

#496

yes

—

D₂₃

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

#127

—

#127

C₂×D₂₃

yes

characteristic

#102

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1721

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₄×D₂₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

#67272

yes

complemented

C₂²

—

C₂²

—

C₂²

—

184

—

C₂²

—

184

—

100

C₂²

—

184

—

101

C₂²

#10625111

yes

characteristic

102

C₄

—

103

C₄

#10625085

yes

—

104

C₄

C₂×#67272

yes

—

105

C₂

—

106

C₂

—

368

—

107

C₂

#13092793

yes

characteristic

108

C₂

#13092803

yes

—

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#492	C₂
socle	C₂²×C₂₃	#127
hypercenter	#492	D₂₃
hypocenter	C₂₃	#1721
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#10625111

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#492

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #13092794	C₂	semidirect
2. #13092793	C₂	semidirect
3. #13092765	C₂	semidirect
4. #13092138	C₂	semidirect
5. C₂₃×#492	C₂	semidirect
39. C₈×C₂₃	D₈	semidirect
53. #492	D₂₃	semidirect
69. C₂₃	#1721	semidirect
95. C₈	#67272	semidirect

H ≅ G/N

type

1. #13092794

C₂

semidirect

2. #13092793

C₂

semidirect

3. #13092765

C₂

semidirect

4. #13092138

C₂

semidirect

5. C₂₃×#492

C₂

semidirect

39. C₈×C₂₃

D₈

semidirect

53. #492

D₂₃

semidirect

69. C₂₃

#1721

semidirect

95. C₈

#67272

semidirect

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

subgroup	index
#13092794	2
#13092793	2
#13092765	2
#13092138	2
C₂₃×#492	2
#1721	23

subgroup

index

C₂₃×#492

Subgroups — #15673061

Conjugacy classes of subgroups

Complements

Maximal subgroups