Group #22612068 — subgroups

Subgroups — #22612068

Order 9344.

142 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988992	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988412	C₂	1	2	yes	maximal, complemented
3	C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988388	C₂	1	2	yes	maximal
5	C₄×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556591	—	2	1	no	—
7	C₂×#11924003	C₂²	1	2	yes	—
8	#14556525	—	2	1	no	—
9	#14556525	C₂²	1	2	yes	—
10	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₄×D₄×C₇₃	—	2	1	no	—
12	#107×C₇₃	C₂²	1	1	yes	characteristic
13	#102×C₇₃	—	2	1	no	—
14	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₆	—	4	1	no	—
16	C₄⋊Dic₇₃	—	2	2	no	—
17	C₄⋊Dic₇₃	—	4	1	no	—
18	#11924003	—	2	2	no	—
19	C₂×D₄×C₇₃	—	2	1	no	—
20	C₂²×C₄×C₇₃	—	4	1	no	—
21	D₈×C₇₃	—	2	1	no	—
22	D₈×C₇₃	Q₈	1	2	yes	complemented
23	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₇₃	—	2	1	no	—
26	C₂²⋊C₄×C₇₃	—	4	1	no	—
27	C₄²×C₇₃	D₄	1	1	yes	characteristic
28	C₂³×C₇₃	—	4	1	no	—
29	D₄×C₇₃	—	2	2	no	—
30	D₄×C₇₃	—	4	1	no	—
31	C₂×C₄×C₇₃	—	4	1	no	—
32	C₂×C₄×C₇₃	—	8	2	no	—
33	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₇₃	—	4	2	no	—
37	Dic₁₄₆	—	8	2	no	—
38	C₈×C₇₃	—	2	1	no	—
39	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₇₃	#41	1	1	yes	characteristic
41	C₂²×C₇₃	—	4	2	no	—
42	C₂²×C₇₃	—	8	1	no	—
43	C₂²×C₇₃	—	8	1	no	—
44	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
45	C₄×C₇₃	—	2	1	no	—
46	C₄×C₇₃	—	8	1	no	—
47	C₄×C₇₃	D₁₆	1	2	yes	—
48	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
49	C₄×C₇₃	#137	1	1	yes	characteristic
50	Dic₇₃	—	8	2	no	—
51	C₂×C₇₃	—	8	2	no	—
52	C₂×C₇₃	#474	1	1	yes	characteristic
53	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
54	C₂×C₇₃	#510	1	1	yes	characteristic
55	#2224	—	73	1	no	maximal, Sylow
56	C₇₃	#2224	1	1	yes	characteristic, complemented, Sylow
57	#500	—	73	1	no	—
58	C₄×D₈	D₇₃	1	1	yes	characteristic
59	#366	—	73	2	no	—
60	C₄⋊C₁₆	—	73	1	no	—
61	#357	—	73	2	no	—
62	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
63	C₄⋊Q₈	—	146	1	no	—
64	C₄×D₄	—	2	1	no	—
65	C₂×C₁₆	—	73	2	no	—
66	#107	—	73	2	no	—
67	#107	—	146	1	no	—
68	#107	C₂×D₇₃	1	1	yes	characteristic
69	#102	—	2	1	no	—
70	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
71	C₂×Q₈	—	292	1	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂²×C₄	—	4	1	no	—
74	D₈	—	2	1	no	—
75	D₈	Dic₁₄₆	1	2	yes	complemented
76	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	292	1	no	—
80	C₄⋊C₄	—	146	2	no	—
81	C₂²⋊C₄	—	4	1	no	—
82	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
83	C₁₆	—	146	2	no	—
84	C₂³	—	4	1	no	—
85	Q₈	—	584	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	8	2	no	—
90	C₂×C₄	—	292	2	no	—
91	C₂×C₄	#11924013	1	1	yes	characteristic
92	C₂×C₄	#11924013	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
96	C₈	#11924028	1	1	yes	characteristic
97	C₂²	—	4	2	no	—
98	C₂²	—	8	1	no	—
99	C₂²	—	8	1	no	—
100	C₂²	C₂×#11924013	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	8	1	no	—
103	C₄	—	584	2	no	—
104	C₄	#14556532	1	2	yes	—
105	C₄	#14556570	1	1	yes	characteristic
106	C₄	#14556621	1	1	yes	characteristic
107	C₂	—	8	2	no	—
108	C₂	#17988867	1	1	yes	characteristic
109	C₂	C₂×#14556532	1	1	yes	characteristic
110	C₂	#17989027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988992

C₂

yes

maximal, characteristic, complemented

#17988412

C₂

yes

maximal, complemented

C₄⋊#11924003

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal

C₄×D₈×C₇₃

C₂

yes

maximal, characteristic

#14556591

—

C₂×#11924003

C₂²

yes

—

#14556525

—

#14556525

C₂²

yes

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

—

#107×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

Q₈

yes

complemented

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₁₆

yes

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#137

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#474

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#500

—

C₄×D₈

D₇₃

yes

characteristic

#366

—

C₄⋊C₁₆

—

#357

—

C₂×D₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

C₄×D₄

—

C₂×C₁₆

—

#107

—

#107

—

146

—

#107

C₂×D₇₃

yes

characteristic

#102

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

Dic₁₄₆

yes

complemented

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

C₂³

—

Q₈

—

584

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

—

C₈

C₂×Dic₁₄₆

yes

characteristic

C₈

#11924028

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×#11924013

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

584

—

104

C₄

#14556532

yes

—

105

C₄

#14556570

yes

characteristic

106

C₄

#14556621

yes

characteristic

107

C₂

—

108

C₂

#17988867

yes

characteristic

109

C₂

C₂×#14556532

yes

characteristic

110

C₂

#17989027

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₄×D₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₄×D₈	D₇₃
hypocenter	C₇₃	#2224
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₄×D₈×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₄×D₈

D₇₃

hypocenter

C₇₃

#2224

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

N	H ≅ G/N	type
1. #17988992	C₂	semidirect
2. #17988412	C₂	semidirect
3. C₄⋊#11924003	C₂	semidirect
22. D₈×C₇₃	Q₈	semidirect
56. C₇₃	#2224	semidirect
75. D₈	Dic₁₄₆	semidirect

H ≅ G/N

type

1. #17988992

C₂

semidirect

2. #17988412

C₂

semidirect

3. C₄⋊#11924003

C₂

semidirect

22. D₈×C₇₃

Q₈

semidirect

56. C₇₃

#2224

semidirect

75. D₈

Dic₁₄₆

semidirect

Subgroups — #22612068

Conjugacy classes of subgroups

Complements