Group #22612215 — subgroups

Subgroups — #22612215

Order 9344.

120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17988991	C₂	1	1	yes	maximal, characteristic
2	#17988988	C₂	1	1	yes	maximal, characteristic
3	#17988415	C₂	1	1	yes	maximal, characteristic
4	#17988414	C₂	1	1	yes	maximal, characteristic
5	#17988412	C₂	1	1	yes	maximal, characteristic
6	#17988362	C₂	1	1	yes	maximal, characteristic
7	#366×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556590	—	2	1	no	—
9	#14556582	—	2	1	no	—
10	C₂×#11924003	C₂²	1	1	yes	characteristic
11	#14556525	C₂²	1	1	yes	characteristic
12	#14556525	C₂²	1	1	yes	characteristic
13	#14556524	—	2	1	no	—
14	#14556522	—	2	1	no	—
15	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	#107×C₇₃	C₂²	1	1	yes	characteristic
18	#107×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₆	—	4	1	no	—
20	C₂×#10596831	D₄	1	1	yes	characteristic
21	C₄⋊Dic₇₃	—	2	1	no	—
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	4	1	no	—
24	#11924007	—	4	1	no	—
25	#11924007	—	4	1	no	—
26	C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	#11924003	Q₈	1	2	yes	complemented
28	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	2	1	no	—
30	C₄⋊C₄×C₇₃	—	2	1	no	—
31	C₁₆×C₇₃	Q₈	1	2	yes	complemented
32	C₂×C₄×C₇₃	—	4	1	no	—
33	C₂×C₄×C₇₃	—	4	1	no	—
34	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₇₃	—	4	1	no	—
36	C₂×Dic₇₃	—	4	1	no	—
37	C₂×Dic₇₃	D₈	1	1	yes	characteristic
38	C₂×Dic₇₃	D₈	1	1	yes	characteristic
39	Dic₁₄₆	—	8	2	no	—
40	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
42	#10596831	Q₁₆	1	2	yes	—
43	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
44	C₄×C₇₃	—	8	1	no	—
45	C₄×C₇₃	—	8	1	no	—
46	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₇₃	—	2	1	no	—
49	Dic₇₃	—	2	1	no	—
50	Dic₇₃	—	8	1	no	—
51	Dic₇₃	—	8	1	no	—
52	C₂×C₇₃	#500	1	1	yes	characteristic
53	C₂×C₇₃	#509	1	1	yes	characteristic
54	C₂×C₇₃	#510	1	1	yes	characteristic
55	#2255	—	73	1	no	maximal, Sylow
56	C₇₃	#2255	1	1	yes	characteristic, complemented, Sylow
57	#500	—	73	1	no	—
58	#499	—	73	1	no	—
59	#367	—	73	1	no	—
60	#367	—	73	1	no	—
61	#366	—	73	1	no	—
62	#366	D₇₃	1	1	yes	characteristic
63	#346	—	73	1	no	—
64	C₄⋊Q₈	—	146	1	no	—
65	#125	—	146	1	no	—
66	C₂×C₁₆	—	73	1	no	—
67	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
68	#107	—	73	1	no	—
69	#107	—	73	1	no	—
70	#107	—	146	1	no	—
71	#107	C₂×D₇₃	1	1	yes	characteristic
72	#107	C₂×D₇₃	1	1	yes	characteristic
73	#106	—	146	1	no	—
74	C₄×C₈	—	73	1	no	—
75	C₂×Q₈	—	292	1	no	—
76	C₂×C₈	—	73	1	no	—
77	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	292	1	no	—
81	C₄⋊C₄	—	292	1	no	—
82	C₄⋊C₄	—	292	1	no	—
83	C₄⋊C₄	—	146	1	no	—
84	C₄⋊C₄	—	146	1	no	—
85	C₄²	—	73	1	no	—
86	C₁₆	—	73	2	no	—
87	C₁₆	Dic₁₄₆	1	2	yes	complemented
88	Q₈	—	584	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	292	1	no	—
92	C₂×C₄	—	292	1	no	—
93	C₂×C₄	—	73	1	no	—
94	C₂×C₄	—	73	1	no	—
95	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
96	C₈	—	73	2	no	—
97	C₈	Q₈×D₇₃	1	1	yes	characteristic
98	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
99	C₂²	D₈×D₇₃	1	1	yes	characteristic
100	C₄	—	8	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	584	1	no	—
103	C₄	—	584	1	no	—
104	C₄	—	146	1	no	—
105	C₄	—	146	1	no	—
106	C₄	#14556590	1	1	yes	characteristic
107	C₄	Q₁₆×D₇₃	1	1	yes	characteristic
108	C₂	#17988991	1	1	yes	characteristic
109	C₂	#17989021	1	1	yes	characteristic
110	C₂	#17989029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988991

C₂

yes

maximal, characteristic

#17988988

C₂

yes

maximal, characteristic

#17988415

C₂

yes

maximal, characteristic

#17988414

C₂

yes

maximal, characteristic

#17988412

C₂

yes

maximal, characteristic

#17988362

C₂

yes

maximal, characteristic

#366×C₇₃

C₂

yes

maximal, characteristic

#14556590

—

#14556582

—

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556524

—

#14556522

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

Q₈

yes

complemented

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

Q₈

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₂×Dic₇₃

D₈

yes

characteristic

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

C₂×Q₈

yes

characteristic

#10596831

Q₁₆

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic

#346

—

C₄⋊Q₈

—

146

—

#125

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

—

—

—

146

—

#107

C₂×D₇₃

yes

characteristic

#107

C₂×D₇₃

yes

characteristic

#106

—

146

—

C₄×C₈

—

C₂×Q₈

—

292

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄²

—

C₁₆

—

C₁₆

Dic₁₄₆

yes

complemented

Q₈

—

584

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₇₃

yes

characteristic

C₈

—

C₈

Q₈×D₇₃

yes

characteristic

C₈

C₂×Dic₁₄₆

yes

characteristic

C₂²

D₈×D₇₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

584

—

103

C₄

—

584

—

104

C₄

—

146

—

105

C₄

—

146

—

106

C₄

#14556590

yes

characteristic

107

C₄

Q₁₆×D₇₃

yes

characteristic

108

C₂

#17988991

yes

characteristic

109

C₂

#17989021

yes

characteristic

110

C₂

#17989029

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#366×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#366	D₇₃
hypocenter	C₇₃	#2255
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#366×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
27. #11924003	Q₈	semidirect
31. C₁₆×C₇₃	Q₈	semidirect
56. C₇₃	#2255	semidirect
87. C₁₆	Dic₁₄₆	semidirect

H ≅ G/N

type

27. #11924003

Q₈

semidirect

31. C₁₆×C₇₃

Q₈

semidirect

56. C₇₃

#2255

semidirect

87. C₁₆

Dic₁₄₆

semidirect

Subgroups — #22612215

Conjugacy classes of subgroups

Complements