Group #22610909 — subgroups

Subgroups — #22610909

Order 9344.

284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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	#17988611	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988346	C₂	1	4	yes	maximal
4	#533×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924012	C₂²	1	2	yes	—
6	C₂×C₄×Dic₇₃	—	2	2	no	—
7	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
8	C₄⋊#10596831	—	2	2	no	—
9	C₄⋊Q₈×C₇₃	C₄	1	2	yes	complemented
10	#125×C₇₃	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₇₃	—	2	2	no	—
12	C₄×Q₈×C₇₃	—	2	2	no	—
13	#117×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇₃	—	2	2	no	—
16	#11924012	—	2	4	no	—
17	C₂×#10596831	—	2	2	no	—
18	C₄×Dic₇₃	—	2	8	no	—
19	C₄×Dic₇₃	—	2	2	no	—
20	C₄×Dic₇₃	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₇₃	—	2	2	no	—
22	C₂²×C₄×C₇₃	D₄	1	2	yes	—
23	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₇₃	—	2	4	no	—
25	C₄⋊C₄×C₇₃	—	2	2	no	—
26	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₇₃	—	2	2	no	—
28	C₄²×C₇₃	—	2	2	no	—
29	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₇₃	D₄	1	2	yes	—
32	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₇₃	—	4	4	no	—
34	C₂×C₄×C₇₃	—	2	4	no	—
35	C₂×C₄×C₇₃	—	2	4	no	—
36	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₇₃	#41	1	2	yes	—
42	C₂×Dic₇₃	—	2	8	no	—
43	C₂×Dic₇₃	—	2	4	no	—
44	#10596831	—	4	2	no	—
45	C₂²×C₇₃	—	2	2	no	—
46	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₇₃	—	2	2	no	—
49	C₄×C₇₃	—	4	4	no	—
50	C₄×C₇₃	#104	1	4	yes	—
51	C₄×C₇₃	C₄×Q₈	1	2	yes	—
52	C₄×C₇₃	#124	1	1	yes	characteristic
53	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₇₃	—	2	8	no	—
55	C₂×C₇₃	—	2	2	no	—
56	C₂×C₇₃	#391	1	1	yes	characteristic
57	C₂×C₇₃	C₂×#104	1	2	yes	—
58	#1994	—	73	1	no	maximal, Sylow
59	C₇₃	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	D₇₃	1	1	yes	characteristic
61	#423	—	73	1	no	—
62	C₄³	—	73	1	no	—
63	#339	—	73	4	no	—
64	C₂×M₁₆	—	73	2	no	—
65	C₄⋊Q₈	Dic₇₃	1	2	yes	complemented
66	#125	Dic₇₃	1	2	yes	complemented
67	C₂²⋊Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	#117	C₂×D₇₃	1	2	yes	—
70	C₂×C₄²	—	73	2	no	—
71	C₂×C₄²	—	146	2	no	—
72	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
73	C₄⋊C₈	—	146	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	146	2	no	—
76	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
77	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
78	M₁₆	—	146	4	no	—
79	C₂×C₈	—	146	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	73	4	no	—
86	C₄²	—	146	8	no	—
87	C₄²	—	146	2	no	—
88	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
89	C₄²	C₂×Dic₇₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₇₃	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	146	8	no	—
96	C₂×C₄	—	146	4	no	—
97	C₂×C₄	C₂²⋊Dic₇₃	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₇₃	1	2	yes	—
99	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₇₃	1	2	yes	—
102	C₂×C₄	#11924029	1	2	yes	—
103	C₈	—	292	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
106	C₂²	#14556567	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	146	8	no	—
110	C₄	#14556518	1	4	yes	—
111	C₄	Q₈×Dic₇₃	1	2	yes	—
112	C₄	#14556580	1	1	yes	characteristic
113	C₄	#14556589	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#17988494	1	1	yes	characteristic
116	C₂	C₂×#14556518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17988611

C₂

yes

maximal, characteristic

C₄²×Dic₇₃

C₂

yes

maximal, characteristic

#17988346

C₂

yes

maximal

#533×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₄⋊#10596831

—

C₄⋊Q₈×C₇₃

C₄

yes

complemented

#125×C₇₃

C₄

yes

complemented

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

Q₈

yes

complemented

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

—

C₄×C₇₃

C₄×Q₈

yes

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#391

yes

characteristic

C₂×C₇₃

C₂×#104

yes

—

#1994

—

maximal, Sylow

C₇₃

#1994

yes

characteristic, complemented, Sylow

#533

D₇₃

yes

characteristic

#423

—

C₄³

—

#339

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₇₃

yes

complemented

#125

Dic₇₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

146

—

C₂×Q₈

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²⋊D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₇₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

146

—

C₄²

—

146

—

C₄²

C₂×Dic₇₃

yes

characteristic

C₄²

C₂×Dic₇₃

yes

characteristic

C₄²

C₂²⋊D₇₃

yes

—

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂²⋊Dic₇₃

yes

—

C₂×C₄

C₂²⋊Dic₇₃

yes

—

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

101

C₂×C₄

Q₈×D₇₃

yes

—

102

C₂×C₄

#11924029

yes

—

103

C₈

—

292

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

106

C₂²

#14556567

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

146

—

110

C₄

#14556518

yes

—

111

C₄

Q₈×Dic₇₃

yes

—

112

C₄

#14556580

yes

characteristic

113

C₄

#14556589

yes

characteristic

114

C₂

—

115

C₂

#17988494

yes

characteristic

116

C₂

C₂×#14556518

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#533×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#533	D₇₃
hypocenter	C₇₃	#1994
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#533×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₇₃	C₄	semidirect
10. #125×C₇₃	C₄	semidirect
20. C₄×Dic₇₃	Q₈	semidirect
59. C₇₃	#1994	semidirect
65. C₄⋊Q₈	Dic₇₃	semidirect
66. #125	Dic₇₃	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₇₃

C₄

semidirect

10. #125×C₇₃

C₄

semidirect

20. C₄×Dic₇₃

Q₈

semidirect

59. C₇₃

#1994

semidirect

65. C₄⋊Q₈

Dic₇₃

semidirect

66. #125

Dic₇₃

semidirect

Subgroups — #22610909

Conjugacy classes of subgroups

Complements