Group #22625527 — subgroups

Subgroups — #22625527

Order 9344.

192 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₄×#10596833	C₂	1	2	yes	maximal, complemented
2	#17989448	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#10596833	C₂	1	2	yes	maximal, complemented
4	#17989428	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596833	—	2	2	no	—
7	C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂²⋊#10596833	—	2	2	no	—
9	C₄×#10596833	C₂²	1	1	yes	characteristic
10	C₄×#10596833	C₂²	1	1	yes	characteristic
11	D₄×Dic₇₃	C₂²	1	2	yes	—
12	#14556525	C₄	1	1	yes	characteristic
13	#14556512	C₄	1	2	yes	—
14	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×D₈×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#10596833	—	2	2	no	—
17	C₂×#10596833	—	2	1	no	—
18	C₂×#10596833	—	4	4	no	—
19	C₂×#10596833	D₄	1	2	yes	—
20	C₂²×Dic₇₃	—	2	2	no	—
21	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₇₃	—	2	2	no	—
24	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₇₃	C₂×C₄	1	2	yes	—
26	D₈×C₇₃	C₈	1	4	yes	complemented
27	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₇₃	—	2	2	no	—
29	D₄×C₇₃	—	2	2	no	—
30	D₄×C₇₃	C₂×C₈	1	4	yes	complemented
31	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₇₃	—	2	2	no	—
33	C₂×Dic₇₃	—	4	2	no	—
34	C₂×Dic₇₃	—	4	2	no	—
35	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₇₃	#41	1	1	yes	characteristic
37	#10596833	—	2	2	no	—
38	#10596833	—	4	1	no	—
39	#10596833	—	4	2	no	—
40	C₈×C₇₃	C₂×C₈	1	2	yes	complemented
41	#10596831	—	2	1	no	—
42	C₂²×C₇₃	—	2	4	no	—
43	C₂²×C₇₃	—	4	2	no	—
44	C₂²×C₇₃	—	4	2	no	—
45	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
46	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
47	C₄×C₇₃	#131	1	1	yes	characteristic
48	Dic₇₃	—	2	1	no	—
49	Dic₇₃	—	4	2	no	—
50	Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₇₃	—	4	4	no	—
52	C₂×C₇₃	C₈×D₄	1	1	yes	characteristic
53	C₂×C₇₃	#442	1	1	yes	characteristic
54	C₂×C₇₃	#444	1	1	yes	characteristic
55	#1580	—	73	1	no	maximal, Sylow
56	C₇₃	#1580	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	—	73	1	no	—
58	C₈×D₄	—	73	2	no	—
59	#334	—	73	1	no	—
60	D₄⋊C₈	—	73	2	no	—
61	#322	—	73	1	no	—
62	C₂×D₈	#64408	1	1	yes	characteristic
63	C₂²×C₈	—	146	2	no	—
64	C₄×D₄	—	73	2	no	—
65	#107	—	73	1	no	—
66	C₄⋊C₈	—	73	2	no	—
67	#102	—	73	2	no	—
68	C₂²⋊C₈	—	146	2	no	—
69	C₄×C₈	—	73	1	no	—
70	C₄×C₈	—	73	1	no	—
71	C₄×C₈	—	73	1	no	—
72	C₂×D₄	C₂×#64408	1	2	yes	—
73	C₂²×C₄	—	146	2	no	—
74	D₈	#10596833	1	4	yes	complemented
75	C₂×C₈	—	292	4	no	—
76	C₂×C₈	—	73	2	no	—
77	C₂×C₈	—	73	1	no	—
78	C₂×C₈	—	146	1	no	—
79	C₂×C₈	—	146	2	no	—
80	C₂×C₈	C₂×#64408	1	1	yes	characteristic
81	C₄⋊C₄	—	73	2	no	—
82	C₂²⋊C₄	—	146	2	no	—
83	C₄²	—	73	1	no	—
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×#10596833	1	4	yes	complemented
87	C₂×C₄	—	292	2	no	—
88	C₂×C₄	—	292	2	no	—
89	C₂×C₄	—	73	1	no	—
90	C₂×C₄	—	73	1	no	—
91	C₂×C₄	—	146	2	no	—
92	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
93	C₈	—	292	2	no	—
94	C₈	—	292	1	no	—
95	C₈	—	146	2	no	—
96	C₈	—	146	1	no	—
97	C₈	C₂×#10596833	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	D₄×#64408	1	1	yes	characteristic
102	C₄	—	292	2	no	—
103	C₄	—	73	2	no	—
104	C₄	—	146	1	no	—
105	C₄	C₂²×#10596833	1	1	yes	characteristic
106	C₄	#14556678	1	1	yes	characteristic
107	C₂	—	4	4	no	—
108	C₂	D₄×#10596833	1	1	yes	characteristic
109	C₂	#17989628	1	1	yes	characteristic
110	C₂	#17989634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#10596833

C₂

yes

maximal, complemented

#17989448

C₂

yes

maximal, characteristic, complemented

D₄⋊#10596833

C₂

yes

maximal, complemented

#17989428

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₃

C₂

yes

maximal, characteristic

C₂²×#10596833

—

C₄⋊#10596833

C₂²

yes

—

C₂²⋊#10596833

—

C₄×#10596833

C₂²

yes

characteristic

C₄×#10596833

C₂²

yes

characteristic

D₄×Dic₇₃

C₂²

yes

—

#14556525

C₄

yes

characteristic

#14556512

C₄

yes

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₄

yes

—

—

—

D₄

yes

—

C₂²×Dic₇₃

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

—

D₈×C₇₃

C₈

yes

complemented

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

C₂×C₈

yes

complemented

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

C₂×Dic₇₃

yes

—

—

—

C₈×C₇₃

C₂×C₈

yes

complemented

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

#131

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₈×D₄

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₄×D₈

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

—

C₂×D₈

#64408

yes

characteristic

C₂²×C₈

—

146

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

C₂×#64408

yes

—

C₂²×C₄

—

146

—

D₈

#10596833

yes

complemented

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂×#64408

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

146

—

C₄²

—

C₂³

—

D₄

—

D₄

C₂×#10596833

yes

complemented

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

C₂²×#64408

yes

characteristic

C₈

—

292

—

C₈

—

292

—

C₈

—

146

—

C₈

—

146

—

C₈

C₂×#10596833

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

D₄×#64408

yes

characteristic

102

C₄

—

292

—

103

C₄

—

104

C₄

—

146

—

105

C₄

C₂²×#10596833

yes

characteristic

106

C₄

#14556678

yes

characteristic

107

C₂

—

108

C₂

D₄×#10596833

yes

characteristic

109

C₂

#17989628

yes

characteristic

110

C₂

#17989634

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#64408
commutator	C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×D₈×C₇₃	C₄
socle	C₂²×C₇₃	C₄×D₄
hypercenter	C₂×D₈	#64408
hypocenter	C₇₃	#1580
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#64408

commutator

C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×D₈×C₇₃

C₄

socle

C₂²×C₇₃

C₄×D₄

C₂×D₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. D₄×#10596833	C₂	semidirect
2. #17989448	C₂	semidirect
3. D₄⋊#10596833	C₂	semidirect
4. #17989428	C₂	semidirect
26. D₈×C₇₃	C₈	semidirect
30. D₄×C₇₃	C₂×C₈	semidirect
40. C₈×C₇₃	C₂×C₈	semidirect
56. C₇₃	#1580	semidirect
74. D₈	#10596833	semidirect
86. D₄	C₂×#10596833	semidirect
97. C₈	C₂×#10596833	semidirect

H ≅ G/N

type

1. D₄×#10596833

C₂

semidirect

2. #17989448

C₂

semidirect

3. D₄⋊#10596833

C₂

semidirect

4. #17989428

C₂

semidirect

26. D₈×C₇₃

C₈

semidirect

30. D₄×C₇₃

C₂×C₈

semidirect

40. C₈×C₇₃

C₂×C₈

semidirect

56. C₇₃

#1580

semidirect

74. D₈

#10596833

semidirect

86. D₄

C₂×#10596833

semidirect

97. C₈

C₂×#10596833

semidirect

Subgroups — #22625527

Conjugacy classes of subgroups

Complements