Group #22612022 — subgroups

Subgroups — #22612022

Order 9344.

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

normal subgroups only

This group has 154 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	C₂×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988828	C₂	1	2	yes	maximal
3	#17988413	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988392	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₉₂	C₂²	1	2	yes	—
7	#14556570	—	2	2	no	—
8	Dic₅₈₄	—	2	2	no	—
9	#14556525	C₂²	1	2	yes	—
10	#14556516	—	2	2	no	—
11	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	—	2	2	no	—
13	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₆	—	2	2	no	—
16	Dic₂₉₂	—	2	2	no	—
17	Dic₂₉₂	—	2	2	no	—
18	C₄⋊Dic₇₃	—	2	2	no	—
19	#11924007	—	4	2	no	—
20	C₂²⋊Dic₇₃	—	4	2	no	—
21	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₇₃	—	2	1	no	—
23	C₂×C₈×C₇₃	—	2	1	no	—
24	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₁₆×C₇₃	—	2	1	no	—
27	C₁₆×C₇₃	D₄	1	2	yes	—
28	C₂³×C₇₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₇₃	—	2	1	no	—
30	C₂×C₄×C₇₃	—	2	1	no	—
31	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₇₃	—	4	2	no	—
34	C₂×Dic₇₃	—	4	2	no	—
35	Dic₁₄₆	—	4	2	no	—
36	Dic₁₄₆	—	4	2	no	—
37	C₈×C₇₃	—	2	1	no	—
38	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₇₃	#41	1	1	yes	characteristic
40	C₂²×C₇₃	—	2	1	no	—
41	C₂²×C₇₃	—	2	1	no	—
42	C₂²×C₇₃	Q₃₂	1	2	yes	complemented
43	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
44	C₄×C₇₃	—	2	1	no	—
45	C₄×C₇₃	#121	1	1	yes	characteristic
46	C₄×C₇₃	#135	1	1	yes	characteristic
47	Dic₇₃	—	8	2	no	—
48	Dic₇₃	—	8	2	no	—
49	C₂×C₇₃	—	2	2	no	—
50	C₂×C₇₃	#466	1	1	yes	characteristic
51	C₂×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
52	C₂×C₇₃	#508	1	1	yes	characteristic
53	#2216	—	73	1	no	maximal, Sylow
54	C₇₃	#2216	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₃₂	—	73	1	no	—
56	C₂²×C₁₆	D₇₃	1	1	yes	characteristic
57	#467	—	73	2	no	—
58	#366	—	73	1	no	—
59	#358	—	73	2	no	—
60	C₂×Q₁₆	—	73	2	no	—
61	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
62	C₂²⋊Q₈	—	146	2	no	—
63	Q₃₂	—	146	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
66	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
67	#107	—	73	2	no	—
68	#103	—	146	2	no	—
69	C₂×Q₈	—	146	2	no	—
70	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
71	Q₁₆	—	146	2	no	—
72	Q₁₆	—	146	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	D₂₉₂	1	1	yes	characteristic
76	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
77	C₄⋊C₄	—	292	2	no	—
78	C₄⋊C₄	—	146	2	no	—
79	C₂²⋊C₄	—	292	2	no	—
80	C₁₆	—	2	1	no	—
81	C₁₆	C₂²⋊D₇₃	1	2	yes	—
82	C₂³	D₅₈₄	1	1	yes	characteristic
83	Q₈	—	292	2	no	—
84	Q₈	—	292	2	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	292	2	no	—
88	C₂×C₄	—	292	2	no	—
89	C₂×C₄	D₅₈₄	1	1	yes	characteristic
90	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
93	C₈	#11924028	1	1	yes	characteristic
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	Dic₅₈₄	1	2	yes	complemented
97	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	584	2	no	—
100	C₄	—	584	2	no	—
101	C₄	#14556565	1	1	yes	characteristic
102	C₄	#14556613	1	1	yes	characteristic
103	C₂	—	2	2	no	—
104	C₂	#17988823	1	1	yes	characteristic
105	C₂	C₂×Dic₅₈₄	1	1	yes	characteristic
106	C₂	#17989019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₅₈₄

C₂

yes

maximal, characteristic, complemented

#17988828

C₂

yes

maximal

#17988413

C₂

yes

maximal, characteristic, complemented

#17988392

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

C₂²

yes

—

#14556570

—

Dic₅₈₄

—

#14556525

C₂²

yes

—

#14556516

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

D₄

yes

—

C₂³×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

Q₃₂

yes

complemented

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×Q₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂²×C₁₆

D₇₃

yes

—

—

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂²⋊Q₈

—

146

—

Q₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#103

—

146

—

C₂×Q₈

—

146

—

C₂²×C₄

D₂₉₂

yes

characteristic

Q₁₆

—

146

—

Q₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₉₂

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₁₆

—

C₁₆

C₂²⋊D₇₃

yes

—

C₂³

D₅₈₄

yes

characteristic

Q₈

—

292

—

Q₈

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

D₅₈₄

yes

characteristic

C₂×C₄

C₂×D₂₉₂

yes

characteristic

C₈

—

C₈

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

#11924028

yes

characteristic

C₂²

—

C₂²

—

C₂²

Dic₅₈₄

yes

complemented

C₂²

C₂×D₅₈₄

yes

characteristic

C₄

—

C₄

—

584

—

100

C₄

—

584

—

101

C₄

#14556565

yes

characteristic

102

C₄

#14556613

yes

characteristic

103

C₂

—

104

C₂

#17988823

yes

characteristic

105

C₂

C₂×Dic₅₈₄

yes

characteristic

106

C₂

#17989019

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂²×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂²×C₁₆	D₇₃
hypocenter	C₇₃	#2216
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂²×C₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₂²×C₁₆

D₇₃

hypocenter

C₇₃

#2216

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×Dic₅₈₄	C₂	semidirect
3. #17988413	C₂	semidirect
4. #17988392	C₂	semidirect
42. C₂²×C₇₃	Q₃₂	semidirect
54. C₇₃	#2216	semidirect
96. C₂²	Dic₅₈₄	semidirect

H ≅ G/N

type

1. C₂×Dic₅₈₄

C₂

semidirect

3. #17988413

C₂

semidirect

4. #17988392

C₂

semidirect

42. C₂²×C₇₃

Q₃₂

semidirect

54. C₇₃

#2216

semidirect

96. C₂²

Dic₅₈₄

semidirect

Subgroups — #22612022

Conjugacy classes of subgroups

Complements