Group #22609817 — subgroups

Subgroups — #22609817

Order 9344.

224 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14556527	C₂	1	1	yes	maximal, characteristic
2	#17988543	C₂	1	2	yes	maximal
3	C₂×#14556501	C₂	1	1	yes	maximal, characteristic
4	#17988346	C₂	1	2	yes	maximal
5	#427×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	C₂²	1	2	yes	—
7	C₂²×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
9	#14556527	C₄	1	4	yes	complemented
10	#14556502	—	2	2	no	—
11	#14556501	C₄	1	4	yes	complemented
12	C₄×#10596831	—	2	2	no	—
13	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#117×C₇₃	C₂²	1	2	yes	—
15	#107×C₇₃	C₄	1	2	yes	complemented
16	#106×C₇₃	C₄	1	2	yes	complemented
17	C₂²×Dic₇₃	—	2	1	no	—
18	#11924012	—	2	2	no	—
19	#11924012	C₂×C₄	1	4	yes	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	C₂×C₄	1	2	yes	—
22	C₂×#10596831	D₄	1	2	yes	—
23	C₂×#10596831	Q₈	1	2	yes	—
24	C₄×Dic₇₃	—	2	1	no	—
25	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
32	C₂²⋊C₄×C₇₃	—	2	2	no	—
33	C₄²×C₇₃	—	2	2	no	—
34	C₂³×C₇₃	#41	1	1	yes	characteristic
35	C₂×C₄×C₇₃	—	2	4	no	—
36	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
41	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
42	C₂×Dic₇₃	—	2	2	no	—
43	C₂×Dic₇₃	—	4	2	no	—
44	C₈×C₇₃	C₄²	1	2	yes	—
45	C₈×C₇₃	C₄²	1	2	yes	—
46	#10596831	—	2	4	no	—
47	#10596831	C₄⋊C₄	1	4	yes	—
48	C₂²×C₇₃	—	2	2	no	—
49	C₂²×C₇₃	#117	1	1	yes	characteristic
50	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
51	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
52	C₄×C₇₃	—	4	4	no	—
53	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
56	C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
57	Dic₇₃	—	4	2	no	—
58	C₂×C₇₃	—	2	1	no	—
59	C₂×C₇₃	—	2	1	no	—
60	C₂×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₇₃	#444	1	2	yes	—
62	#1773	—	73	1	no	maximal, Sylow
63	C₇₃	#1773	1	1	yes	characteristic, complemented, Sylow
64	C₂×#108	—	73	1	no	—
65	#427	D₇₃	1	1	yes	characteristic
66	#405	—	73	2	no	—
67	C₂×#97	—	73	1	no	—
68	#339	—	73	2	no	—
69	C₂×M₁₆	—	73	2	no	—
70	C₂²×C₈	—	73	1	no	—
71	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
72	#117	C₂×D₇₃	1	2	yes	—
73	C₂×C₄²	—	73	1	no	—
74	#108	—	73	4	no	—
75	#107	Dic₇₃	1	2	yes	complemented
76	#106	Dic₇₃	1	2	yes	complemented
77	#97	—	73	4	no	—
78	#97	—	146	2	no	—
79	C₄×C₈	—	146	2	no	—
80	C₂²×C₄	—	146	1	no	—
81	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
82	M₁₆	—	73	4	no	—
83	M₁₆	—	146	2	no	—
84	C₂×C₈	—	73	2	no	—
85	C₂×C₈	—	73	2	no	—
86	C₂×C₈	—	73	2	no	—
87	C₂×C₈	—	146	2	no	—
88	C₂×C₈	C₄×D₇₃	1	2	yes	—
89	C₂×C₈	C₄×D₇₃	1	2	yes	—
90	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
91	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
92	C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	73	2	no	—
96	C₄²	—	146	1	no	—
97	C₂³	#11924027	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	292	2	no	—
100	C₂×C₄	—	146	2	no	—
101	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
104	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
106	C₂×C₄	#11924029	1	1	yes	characteristic
107	C₈	—	73	4	no	—
108	C₈	—	146	4	no	—
109	C₈	C₄×Dic₇₃	1	2	yes	—
110	C₈	C₄×Dic₇₃	1	2	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	#14556549	1	1	yes	characteristic
113	C₂²	D₄×Dic₇₃	1	1	yes	characteristic
114	C₂²	#14556556	1	1	yes	characteristic
115	C₄	—	4	4	no	—
116	C₄	—	292	2	no	—
117	C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
119	C₄	#14556553	1	1	yes	characteristic
120	C₄	Q₈×Dic₇₃	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
124	C₂	#17988709	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556527

C₂

yes

maximal, characteristic

#17988543

C₂

yes

maximal

C₂×#14556501

C₂

yes

maximal, characteristic

#17988346

C₂

yes

maximal

#427×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

—

C₂²×#10596831

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

#14556527

C₄

yes

complemented

#14556502

—

#14556501

C₄

yes

complemented

C₄×#10596831

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

—

#107×C₇₃

C₄

yes

complemented

#106×C₇₃

C₄

yes

complemented

C₂²×Dic₇₃

—

#11924012

—

#11924012

C₂×C₄

yes

—

C₂×#10596831

—

C₂×#10596831

C₂×C₄

yes

—

C₂×#10596831

D₄

yes

—

C₂×#10596831

Q₈

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

C₄²

yes

—

C₈×C₇₃

C₄²

yes

—

#10596831

—

#10596831

C₄⋊C₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#117

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄²

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₄×Q₈

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₃

#444

yes

—

#1773

—

maximal, Sylow

C₇₃

#1773

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

D₇₃

yes

characteristic

#405

—

C₂×#97

—

#339

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

C₂×D₇₃

yes

—

C₂×C₄²

—

#108

—

#107

Dic₇₃

yes

complemented

#106

Dic₇₃

yes

complemented

#97

—

#97

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₄×D₇₃

yes

—

C₂×C₈

C₄×D₇₃

yes

—

C₂×C₈

C₂×Dic₇₃

yes

characteristic

C₂×C₈

C₂×Dic₇₃

yes

characteristic

C₄⋊C₄

C₂×Dic₇₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

146

—

C₂³

#11924027

yes

characteristic

C₂×C₄

—

C₂×C₄

—

292

—

100

C₂×C₄

—

146

—

101

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

102

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

103

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

104

C₂×C₄

D₄×D₇₃

yes

characteristic

105

C₂×C₄

Q₈×D₇₃

yes

characteristic

106

C₂×C₄

#11924029

yes

characteristic

107

C₈

—

108

C₈

—

146

—

109

C₈

C₄×Dic₇₃

yes

—

110

C₈

C₄×Dic₇₃

yes

—

111

C₂²

—

112

C₂²

#14556549

yes

characteristic

113

C₂²

D₄×Dic₇₃

yes

characteristic

114

C₂²

#14556556

yes

characteristic

115

C₄

—

116

C₄

—

292

—

117

C₄

C₂×C₄×Dic₇₃

yes

characteristic

118

C₄

C₄⋊C₄×D₇₃

yes

characteristic

119

C₄

#14556553

yes

characteristic

120

C₄

Q₈×Dic₇₃

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₄⋊C₄×Dic₇₃

yes

characteristic

124

C₂

#17988709

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#427×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₄
hypercenter	#427	D₇₃
hypocenter	C₇₃	#1773
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₇₃

commutator

C₄×C₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#427×C₇₃

C₂

socle

C₂²×C₇₃

C₄×D₄

D₇₃

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #14556527	C₄	semidirect
11. #14556501	C₄	semidirect
15. #107×C₇₃	C₄	semidirect
16. #106×C₇₃	C₄	semidirect
63. C₇₃	#1773	semidirect
75. #107	Dic₇₃	semidirect
76. #106	Dic₇₃	semidirect

H ≅ G/N

type

9. #14556527

C₄

semidirect

11. #14556501

C₄

semidirect

15. #107×C₇₃

C₄

semidirect

16. #106×C₇₃

C₄

semidirect

63. C₇₃

#1773

semidirect

75. #107

Dic₇₃

semidirect

76. #106

Dic₇₃

semidirect

Subgroups — #22609817

Conjugacy classes of subgroups

Complements