Group #15683911 — subgroups

Subgroups — #15683911

Order 2944.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#13092974	C₂	1	2	yes	maximal
2	#13092822	C₂	1	2	yes	maximal, complemented
3	#13092819	C₂	1	4	yes	maximal, complemented
4	#13092815	C₂	1	2	yes	maximal, complemented
5	#13092455	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	#10625097	C₂²	1	4	yes	—
10	#10625095	—	2	4	no	—
11	#10625041	C₂²	1	4	yes	—
12	#10625040	C₂²	1	4	yes	—
13	C₄×Dic₄₆	—	2	4	no	—
14	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	#10625000	C₂²	1	8	yes	—
18	#10624999	C₂²	1	8	yes	—
19	C₂²×C₄×C₂₃	D₄	1	2	yes	—
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₆	—	2	4	no	—
22	C₂×C₈×C₂₃	Q₈	1	8	yes	complemented
23	C₂×C₈×C₂₃	C₂³	1	2	yes	—
24	C₂×C₈×C₂₃	C₂³	1	2	yes	—
25	C₄²×C₂₃	D₄	1	2	yes	—
26	C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₂₃	—	2	4	no	—
29	C₄⋊Dic₂₃	—	2	4	no	—
30	C₄⋊Dic₂₃	C₂³	1	8	yes	—
31	Dic₂₃⋊C₄	—	2	8	no	—
32	C₄×Dic₂₃	—	2	4	no	—
33	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₂₃	—	2	8	no	—
41	Dic₄₆	—	4	8	no	—
42	C₈×C₂₃	C₂×Q₈	1	4	yes	—
43	C₈×C₂₃	C₂×Q₈	1	4	yes	—
44	#3160	—	23	1	no	maximal, Sylow
45	C₂²×C₂₃	—	2	2	no	—
46	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
47	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₂₃	—	2	2	no	—
49	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
50	C₄×C₂₃	#135	1	4	yes	—
51	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
52	Dic₂₃	—	4	8	no	—
53	#533	—	23	2	no	—
54	#500	—	23	2	no	—
55	#499	—	23	4	no	—
56	#498	—	23	2	no	—
57	#427	—	23	4	no	—
58	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
59	C₂×C₂₃	—	2	2	no	—
60	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₂₃	C₂×#135	1	2	yes	—
62	C₂²×C₈	C₂×D₂₃	1	2	yes	—
63	C₄⋊Q₈	—	23	4	no	—
64	#125	—	23	4	no	—
65	C₂²⋊Q₈	—	46	4	no	—
66	C₄×Q₈	—	46	4	no	—
67	#117	—	23	4	no	—
68	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
69	#107	—	23	8	no	—
70	#106	—	23	8	no	—
71	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
72	C₄×C₈	C₂×D₂₃	1	2	yes	—
73	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
74	C₂₃	#3160	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	46	4	no	—
76	C₂²×C₄	D₉₂	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	C₂×C₈	Dic₄₆	1	8	yes	complemented
79	C₂×C₈	C₂²×D₂₃	1	2	yes	—
80	C₂×C₈	C₂²×D₂₃	1	2	yes	—
81	C₄⋊C₄	—	23	8	no	—
82	C₄⋊C₄	—	46	8	no	—
83	C₄⋊C₄	—	46	4	no	—
84	C₂²⋊C₄	—	46	4	no	—
85	C₄²	—	46	4	no	—
86	C₄²	D₉₂	1	2	yes	—
87	C₄²	C₂²×D₂₃	1	1	yes	characteristic
88	C₄²	C₂²×D₂₃	1	1	yes	characteristic
89	C₂³	C₂×D₉₂	1	1	yes	characteristic
90	Q₈	—	92	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	46	8	no	—
93	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
94	C₂×C₄	C₂×D₉₂	1	2	yes	—
95	C₂×C₄	C₂×D₉₂	1	2	yes	—
96	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
97	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
98	C₈	C₂×Dic₄₆	1	4	yes	—
99	C₈	C₂×Dic₄₆	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#10625040	1	2	yes	—
102	C₂²	C₂²×D₉₂	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	92	8	no	—
105	C₄	#10625040	1	2	yes	—
106	C₄	#10625075	1	4	yes	—
107	C₄	C₂²×Dic₄₆	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#10625040	1	1	yes	characteristic
110	C₂	C₂×#10625075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal

#13092822

C₂

yes

maximal, complemented

#13092819

C₂

yes

maximal, complemented

#13092815

C₂

yes

maximal, complemented

#13092455

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

#10625041

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₂×C₈×C₂₃

Q₈

yes

complemented

C₂×C₈×C₂₃

C₂³

yes

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

—

C₈×C₂₃

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

C₂²×Q₈

yes

—

Dic₂₃

—

—

—

—

—

—

C₂×C₄×C₈

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

C₂×#135

yes

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₄×C₈

C₂×D₂₃

yes

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

Dic₄₆

yes

complemented

C₂×C₈

C₂²×D₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₉₂

yes

—

C₄²

C₂²×D₂₃

yes

characteristic

C₄²

C₂²×D₂₃

yes

characteristic

C₂³

C₂×D₉₂

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₄₆

yes

—

C₂×C₄

C₂×D₉₂

yes

—

C₂×C₄

C₂×D₉₂

yes

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

C₂³×D₂₃

yes

characteristic

C₈

C₂×Dic₄₆

yes

—

C₈

C₂×Dic₄₆

yes

—

100

C₂²

—

101

C₂²

#10625040

yes

—

102

C₂²

C₂²×D₉₂

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

#10625040

yes

—

106

C₄

#10625075

yes

—

107

C₄

C₂²×Dic₄₆

yes

—

108

C₂

—

109

C₂

C₂×#10625040

yes

characteristic

110

C₂

C₂×#10625075

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#3160
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₉₂

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

#3160

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #13092822	C₂	semidirect
3. #13092819	C₂	semidirect
4. #13092815	C₂	semidirect
22. C₂×C₈×C₂₃	Q₈	semidirect
74. C₂₃	#3160	semidirect
78. C₂×C₈	Dic₄₆	semidirect

H ≅ G/N

type

2. #13092822

C₂

semidirect

3. #13092819

C₂

semidirect

4. #13092815

C₂

semidirect

22. C₂×C₈×C₂₃

Q₈

semidirect

74. C₂₃

#3160

semidirect

78. C₂×C₈

Dic₄₆

semidirect

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

subgroup	index
#13092974	2
#13092822	2
#13092819	2
#13092815	2
#13092455	2
C₂×C₄×C₈×C₂₃	2
#3160	23

subgroup

index

C₂×C₄×C₈×C₂₃

Subgroups — #15683911

Conjugacy classes of subgroups

Complements

Maximal subgroups