Group #22625580 — subgroups

Subgroups — #22625580

Order 9344.

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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	#17989621	C₂	1	2	yes	maximal, complemented
2	#17989448	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989435	C₂	1	2	yes	maximal, complemented
4	#17989428	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988669	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924038	—	2	2	no	—
7	C₄⋊#10596833	C₂²	1	2	yes	—
8	#14556648	—	2	2	no	—
9	C₄×#10596833	C₂²	1	1	yes	characteristic
10	C₄×#10596833	C₂²	1	1	yes	characteristic
11	C₂×D₅₈₄	C₄	1	1	yes	characteristic
12	#14556553	C₂²	1	2	yes	—
13	#14556511	C₄	1	2	yes	—
14	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	#107×C₇₃	C₄	1	1	yes	characteristic
16	#11924038	—	4	4	no	—
17	C₂×#10596833	—	2	2	no	—
18	C₂×#10596833	—	2	1	no	—
19	C₂×#10596833	D₄	1	2	yes	—
20	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₇₃	—	2	2	no	—
22	D₅₈₄	—	2	2	no	—
23	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
24	#11924005	—	2	2	no	—
25	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
28	C₂²×D₇₃	—	2	2	no	—
29	C₂×C₄×C₇₃	—	2	2	no	—
30	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₇₃	#41	1	1	yes	characteristic
33	D₂₉₂	—	2	2	no	—
34	D₂₉₂	—	2	2	no	—
35	C₄×D₇₃	—	4	2	no	—
36	C₄×D₇₃	—	4	2	no	—
37	#10596833	—	2	2	no	—
38	#10596833	—	4	1	no	—
39	#10596833	—	4	2	no	—
40	C₈×C₇₃	M₁₆	1	2	yes	complemented
41	#10596831	—	2	1	no	—
42	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
43	C₂×D₇₃	—	4	2	no	—
44	C₂×D₇₃	—	4	2	no	—
45	C₂×D₇₃	—	4	2	no	—
46	C₄×C₇₃	—	4	2	no	—
47	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₇₃	#131	1	1	yes	characteristic
49	Dic₇₃	—	2	1	no	—
50	Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₇₃	#436	1	1	yes	characteristic
52	C₂×C₇₃	#442	1	1	yes	characteristic
53	C₂×C₇₃	#444	1	1	yes	characteristic
54	D₇₃	—	8	2	no	—
55	#1594	—	73	1	no	maximal, Sylow
56	C₇₃	#1594	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	—	73	1	no	—
58	#436	—	73	2	no	—
59	#334	—	73	1	no	—
60	D₄⋊C₈	—	73	2	no	—
61	#322	—	73	1	no	—
62	C₂×D₈	—	73	1	no	—
63	C₂×M₁₆	—	146	2	no	—
64	C₄×D₄	—	73	2	no	—
65	#107	#64408	1	1	yes	characteristic
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	—	73	2	no	—
73	C₂²×C₄	—	146	2	no	—
74	D₈	—	146	2	no	—
75	M₁₆	—	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	C₂×#64408	1	2	yes	—
82	C₂²⋊C₄	—	146	2	no	—
83	C₄²	—	73	1	no	—
84	C₂³	—	146	2	no	—
85	D₄	—	146	2	no	—
86	D₄	—	146	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	292	2	no	—
89	C₂×C₄	—	292	2	no	—
90	C₂×C₄	—	73	1	no	—
91	C₂×C₄	—	73	1	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	#11924038	1	2	yes	complemented
98	C₂²	—	292	2	no	—
99	C₂²	—	292	2	no	—
100	C₂²	—	292	2	no	—
101	C₂²	D₄×#64408	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	73	2	no	—
104	C₄	—	146	1	no	—
105	C₄	C₂×#11924038	1	1	yes	characteristic
106	C₄	#14556679	1	1	yes	characteristic
107	C₂	—	584	2	no	—
108	C₂	#17989621	1	1	yes	characteristic
109	C₂	#17989628	1	1	yes	characteristic
110	C₂	#17989633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989621

C₂

yes

maximal, complemented

#17989448

C₂

yes

maximal, characteristic, complemented

#17989435

C₂

yes

maximal, complemented

#17989428

C₂

yes

maximal, characteristic, complemented

#17988669

C₂

yes

maximal, characteristic

C₂×#11924038

—

C₄⋊#10596833

C₂²

yes

—

#14556648

—

C₄×#10596833

C₂²

yes

characteristic

C₄×#10596833

C₂²

yes

characteristic

C₂×D₅₈₄

C₄

yes

characteristic

#14556553

C₂²

yes

—

#14556511

C₄

yes

—

C₈×Dic₇₃

C₂²

yes

characteristic

#107×C₇₃

C₄

yes

—

—

—

D₄

yes

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

D₅₈₄

—

C₂×#10596831

C₂×C₄

yes

characteristic

#11924005

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

#41

yes

characteristic

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

M₁₆

yes

complemented

#10596831

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

#131

yes

characteristic

Dic₇₃

—

Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1594

—

maximal, Sylow

C₇₃

#1594

yes

characteristic, complemented, Sylow

C₄×D₈

—

#436

—

#334

—

D₄⋊C₈

—

#322

—

C₂×D₈

—

C₂×M₁₆

—

146

—

C₄×D₄

—

#107

#64408

yes

characteristic

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

146

—

D₈

—

146

—

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂×#64408

yes

characteristic

C₄⋊C₄

C₂×#64408

yes

—

C₂²⋊C₄

—

146

—

C₄²

—

C₂³

—

146

—

D₄

—

146

—

D₄

—

146

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#64408

yes

characteristic

C₈

—

292

—

C₈

—

292

—

C₈

—

146

—

C₈

—

146

—

C₈

#11924038

yes

complemented

C₂²

—

292

—

C₂²

—

292

—

100

C₂²

—

292

—

101

C₂²

D₄×#64408

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

146

—

105

C₄

C₂×#11924038

yes

characteristic

106

C₄

#14556679

yes

characteristic

107

C₂

—

584

—

108

C₂

#17989621

yes

characteristic

109

C₂

#17989628

yes

characteristic

110

C₂

#17989633

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	#107×C₇₃	C₄
socle	C₂²×C₇₃	C₄×D₄
hypercenter	#107	#64408
hypocenter	C₇₃	#1594
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

#107×C₇₃

C₄

socle

C₂²×C₇₃

C₄×D₄

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. #17989621	C₂	semidirect
2. #17989448	C₂	semidirect
3. #17989435	C₂	semidirect
4. #17989428	C₂	semidirect
40. C₈×C₇₃	M₁₆	semidirect
56. C₇₃	#1594	semidirect
97. C₈	#11924038	semidirect

H ≅ G/N

type

1. #17989621

C₂

semidirect

2. #17989448

C₂

semidirect

3. #17989435

C₂

semidirect

4. #17989428

C₂

semidirect

40. C₈×C₇₃

M₁₆

semidirect

56. C₇₃

#1594

semidirect

97. C₈

#11924038

semidirect

Subgroups — #22625580

Conjugacy classes of subgroups

Complements