Group #22624992 — subgroups

Subgroups — #22624992

Order 9344.

132 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 52 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×#14556664	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556662	C₂	1	1	yes	maximal, characteristic
3	#17988624	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924038	—	2	1	no	—
5	#14556664	—	2	1	no	—
6	#14556664	C₄	1	2	yes	complemented
7	C₂×#11924031	—	2	1	no	—
8	#14556662	—	2	1	no	—
9	#14556662	C₄	1	2	yes	complemented
10	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	#14556549	—	2	1	no	—
12	#14556525	C₄	1	1	yes	characteristic
13	#14556522	—	2	1	no	—
14	#107×C₇₃	C₄	1	1	yes	characteristic
15	#11924038	—	2	2	no	—
16	#11924038	—	4	1	no	—
17	C₂×#10596833	—	4	1	no	—
18	#11924031	—	2	2	no	—
19	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
20	C₂×#10596831	D₄	1	1	yes	characteristic
21	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₈×D₇₃	D₄	1	1	yes	characteristic
24	C₈×D₇₃	Q₈	1	1	yes	characteristic
25	C₄⋊Dic₇₃	—	2	1	no	—
26	#11924005	—	4	1	no	—
27	C₄×Dic₇₃	—	4	1	no	—
28	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	2	1	no	—
30	C₂²×D₇₃	SD₁₆	1	1	yes	characteristic
31	C₂×C₄×C₇₃	—	4	1	no	—
32	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₇₃	—	4	1	no	—
34	C₂×Dic₇₃	D₈	1	1	yes	characteristic
35	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
38	C₄×D₇₃	Q₁₆	1	1	yes	characteristic
39	#10596833	—	4	2	no	—
40	C₈×C₇₃	C₄²	1	1	yes	characteristic
41	C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	#10596831	C₂²⋊C₄	1	1	yes	characteristic
43	#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₇₃	#102	1	1	yes	characteristic
45	C₂×D₇₃	—	2	1	no	—
46	C₂×D₇₃	—	2	1	no	—
47	C₂×D₇₃	#103	1	1	yes	characteristic
48	C₂×D₇₃	#106	1	1	yes	characteristic
49	C₄×C₇₃	—	8	1	no	—
50	C₄×C₇₃	#95	1	1	yes	characteristic
51	C₄×C₇₃	#103	1	1	yes	characteristic
52	Dic₇₃	—	8	1	no	—
53	Dic₇₃	#102	1	1	yes	characteristic
54	Dic₇₃	#107	1	1	yes	characteristic
55	C₂×C₇₃	#340	1	1	yes	characteristic
56	C₂×C₇₃	#359	1	1	yes	characteristic
57	C₂×C₇₃	#359	1	1	yes	characteristic
58	D₇₃	—	2	1	no	—
59	D₇₃	—	2	1	no	—
60	#1382	—	73	1	no	maximal, Sylow
61	C₇₃	#1382	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	—	73	1	no	—
63	C₂×#108	—	73	1	no	—
64	#427	—	73	1	no	—
65	C₂×M₁₆	—	146	1	no	—
66	C₂²×C₈	—	73	1	no	—
67	#117	—	146	1	no	—
68	C₂×C₁₆	—	73	2	no	—
69	C₂×C₁₆	—	146	1	no	—
70	C₂×C₁₆	—	146	1	no	—
71	#108	—	73	2	no	—
72	#108	—	146	1	no	—
73	#107	—	73	1	no	—
74	#107	#64408	1	1	yes	characteristic
75	#106	—	146	1	no	—
76	C₂²×C₄	—	73	1	no	—
77	M₁₆	—	292	1	no	—
78	M₁₆	—	146	2	no	—
79	C₂×C₈	—	292	1	no	—
80	C₂×C₈	—	73	1	no	—
81	C₂×C₈	—	73	1	no	—
82	C₂×C₈	—	73	1	no	—
83	C₂×C₈	—	73	1	no	—
84	C₂×C₈	—	73	1	no	—
85	C₂×C₈	C₂×#64408	1	1	yes	characteristic
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	146	1	no	—
88	C₂²⋊C₄	—	292	1	no	—
89	C₄²	—	292	1	no	—
90	C₁₆	—	146	2	no	—
91	C₂³	—	73	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	292	1	no	—
94	C₂×C₄	—	73	1	no	—
95	C₂×C₄	—	73	1	no	—
96	C₂×C₄	—	73	1	no	—
97	C₂×C₄	—	73	1	no	—
98	C₂×C₄	—	73	1	no	—
99	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
100	C₈	—	292	2	no	—
101	C₈	—	73	1	no	—
102	C₈	—	73	1	no	—
103	C₈	C₄×#64408	1	1	yes	characteristic
104	C₈	C₄⋊#64408	1	1	yes	characteristic
105	C₂²	—	73	1	no	—
106	C₂²	—	73	1	no	—
107	C₂²	—	146	1	no	—
108	C₂²	—	146	1	no	—
109	C₂²	#14556653	1	1	yes	characteristic
110	C₄	—	8	1	no	—
111	C₄	—	584	1	no	—
112	C₄	—	73	1	no	—
113	C₄	—	73	1	no	—
114	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
115	C₄	#14556654	1	1	yes	characteristic
116	C₂	—	146	1	no	—
117	C₂	—	146	1	no	—
118	C₂	#17989460	1	1	yes	characteristic
119	C₂	#17989501	1	1	yes	characteristic
120	C₂	#17989502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556664

C₂

yes

maximal, characteristic

C₂×#14556662

C₂

yes

maximal, characteristic

#17988624

C₂

yes

maximal, characteristic

C₂×#11924038

—

#14556664

—

#14556664

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556549

—

#14556525

C₄

yes

characteristic

#14556522

—

#107×C₇₃

C₄

yes

—

—

—

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

C₄⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

Q₁₆

yes

characteristic

#10596833

—

C₈×C₇₃

C₄²

yes

characteristic

C₈×C₇₃

C₄⋊C₄

yes

characteristic

#10596831

C₂²⋊C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

C₂²×C₇₃

#102

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1382

—

maximal, Sylow

C₇₃

#1382

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

#427

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

#117

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#108

—

#108

—

146

—

—

yes

—

146

—

C₂²×C₄

—

M₁₆

—

292

—

M₁₆

—

146

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#64408

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

—

292

—

C₁₆

—

146

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#64408

yes

characteristic

100

C₈

—

292

—

101

C₈

—

102

C₈

—

103

C₈

C₄×#64408

yes

characteristic

104

C₈

C₄⋊#64408

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

146

—

108

C₂²

—

146

—

109

C₂²

#14556653

yes

characteristic

110

C₄

—

111

C₄

—

584

—

112

C₄

—

113

C₄

—

114

C₄

(C₂×#64408)⋊C₄

yes

characteristic

115

C₄

#14556654

yes

characteristic

116

C₂

—

146

—

117

C₂

—

146

—

118

C₂

#17989460

yes

characteristic

119

C₂

#17989501

yes

characteristic

120

C₂

#17989502

yes

characteristic

	subgroup	G/H
center	C₂²	#14556653
commutator	C₈×C₇₃	C₄²
Frattini	C₂×C₈	C₂×#64408
Fitting	#107×C₇₃	C₄
socle	C₂²×C₇₃	#102
hypercenter	#107	#64408
hypocenter	C₇₃	#1382
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

subgroup

G/H

center

C₂²

#14556653

commutator

C₈×C₇₃

C₄²

Frattini

C₂×C₈

C₂×#64408

Fitting

#107×C₇₃

C₄

C₂²×C₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

N	H ≅ G/N	type
6. #14556664	C₄	semidirect
9. #14556662	C₄	semidirect
61. C₇₃	#1382	semidirect

H ≅ G/N

type

6. #14556664

C₄

semidirect

9. #14556662

C₄

semidirect

61. C₇₃

#1382

semidirect

Subgroups — #22624992

Conjugacy classes of subgroups

Complements