Group #22609812 — subgroups

Subgroups — #22609812

Order 9344.

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

normal subgroups only

This group has 232 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×C₈×Dic₇₃	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	C₄×#10596831	—	2	2	no	—
12	C₈×Dic₇₃	C₄	1	4	yes	complemented
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	1	yes	characteristic
23	C₂×#10596831	D₄	1	1	yes	characteristic
24	C₂×#10596831	Q₈	1	2	yes	—
25	C₄×Dic₇₃	—	2	1	no	—
26	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₇₃	—	2	2	no	—
34	C₄²×C₇₃	—	2	2	no	—
35	C₂³×C₇₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₇₃	—	2	4	no	—
37	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
43	C₂×Dic₇₃	—	2	4	no	—
44	C₂×Dic₇₃	—	2	2	no	—
45	C₈×C₇₃	C₄²	1	2	yes	—
46	C₈×C₇₃	C₄²	1	2	yes	—
47	#10596831	—	2	4	no	—
48	#10596831	C₄⋊C₄	1	2	yes	—
49	#10596831	C₄⋊C₄	1	2	yes	—
50	C₂²×C₇₃	—	2	2	no	—
51	C₂²×C₇₃	#117	1	1	yes	characteristic
52	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₇₃	—	4	4	no	—
55	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
58	C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
59	Dic₇₃	—	2	4	no	—
60	C₂×C₇₃	—	2	1	no	—
61	C₂×C₇₃	—	2	1	no	—
62	C₂×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₇₃	#443	1	2	yes	—
64	#1772	—	73	1	no	maximal, Sylow
65	C₇₃	#1772	1	1	yes	characteristic, complemented, Sylow
66	C₂×#108	—	73	1	no	—
67	#427	D₇₃	1	1	yes	characteristic
68	#405	—	73	2	no	—
69	C₂×C₄×C₈	—	73	1	no	—
70	#339	—	73	2	no	—
71	C₂×M₁₆	—	73	2	no	—
72	C₂²×C₈	—	73	1	no	—
73	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
74	#117	C₂×D₇₃	1	2	yes	—
75	C₂×C₄²	—	73	1	no	—
76	#108	—	73	4	no	—
77	#107	Dic₇₃	1	2	yes	complemented
78	#106	Dic₇₃	1	2	yes	complemented
79	#97	—	146	2	no	—
80	C₄×C₈	—	73	4	no	—
81	C₄×C₈	—	146	2	no	—
82	C₂²×C₄	—	146	1	no	—
83	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
84	M₁₆	—	73	4	no	—
85	M₁₆	—	146	2	no	—
86	C₂×C₈	—	73	2	no	—
87	C₂×C₈	—	73	1	no	—
88	C₂×C₈	—	73	1	no	—
89	C₂×C₈	—	73	2	no	—
90	C₂×C₈	—	146	2	no	—
91	C₂×C₈	C₄×D₇₃	1	2	yes	—
92	C₂×C₈	C₄×D₇₃	1	2	yes	—
93	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₇₃	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	73	2	no	—
99	C₄²	—	146	1	no	—
100	C₂³	#11924027	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	146	4	no	—
103	C₂×C₄	—	146	2	no	—
104	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
107	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
109	C₂×C₄	#11924029	1	1	yes	characteristic
110	C₈	—	73	2	no	—
111	C₈	—	73	2	no	—
112	C₈	—	146	4	no	—
113	C₈	C₄×Dic₇₃	1	2	yes	—
114	C₈	C₄×Dic₇₃	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#14556549	1	1	yes	characteristic
117	C₂²	D₄×Dic₇₃	1	1	yes	characteristic
118	C₂²	#14556556	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	146	4	no	—
121	C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
123	C₄	#14556553	1	1	yes	characteristic
124	C₄	Q₈×Dic₇₃	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
128	C₂	#17988704	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556527

C₂

yes

maximal, characteristic

#17988543

C₂

yes

maximal

C₂×C₈×Dic₇₃

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

—

C₄×#10596831

—

C₈×Dic₇₃

C₄

yes

complemented

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

characteristic

C₂×#10596831

D₄

yes

characteristic

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

—

#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

#443

yes

—

#1772

—

maximal, Sylow

C₇₃

#1772

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

D₇₃

yes

characteristic

#405

—

C₂×C₄×C₈

—

#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

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

146

—

C₂×C₈

—

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

—

100

C₂³

#11924027

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

146

—

103

C₂×C₄

—

146

—

104

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

105

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

106

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

107

C₂×C₄

D₄×D₇₃

yes

characteristic

108

C₂×C₄

Q₈×D₇₃

yes

characteristic

109

C₂×C₄

#11924029

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

146

—

113

C₈

C₄×Dic₇₃

yes

—

114

C₈

C₄×Dic₇₃

yes

—

115

C₂²

—

116

C₂²

#14556549

yes

characteristic

117

C₂²

D₄×Dic₇₃

yes

characteristic

118

C₂²

#14556556

yes

characteristic

119

C₄

—

120

C₄

—

146

—

121

C₄

C₂×C₄×Dic₇₃

yes

characteristic

122

C₄

C₄⋊C₄×D₇₃

yes

characteristic

123

C₄

#14556553

yes

characteristic

124

C₄

Q₈×Dic₇₃

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₇₃

yes

characteristic

128

C₂

#17988704

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	#1772
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
12. C₈×Dic₇₃	C₄	semidirect
15. #107×C₇₃	C₄	semidirect
16. #106×C₇₃	C₄	semidirect
65. C₇₃	#1772	semidirect
77. #107	Dic₇₃	semidirect
78. #106	Dic₇₃	semidirect

H ≅ G/N

type

9. #14556527

C₄

semidirect

12. C₈×Dic₇₃

C₄

semidirect

15. #107×C₇₃

C₄

semidirect

16. #106×C₇₃

C₄

semidirect

65. C₇₃

#1772

semidirect

77. #107

Dic₇₃

semidirect

78. #106

Dic₇₃

semidirect

Subgroups — #22609812

Conjugacy classes of subgroups

Complements