Group #22627109 — subgroups

Subgroups — #22627109

Order 9344.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#17989706	C₂	1	2	yes	maximal, complemented
2	#17989686	C₂	1	4	yes	maximal, complemented
3	C₂×#14556679	C₂	1	1	yes	maximal, characteristic
4	#14556688	—	2	2	no	—
5	#14556687	C₂²	1	4	yes	—
6	#14556686	—	2	2	no	—
7	#14556679	—	2	2	no	—
8	#14556679	—	2	2	no	—
9	C₂×#11924038	C₄	1	1	yes	characteristic
10	C₂×#11924038	C₂²	1	2	yes	—
11	C₂×#11924036	C₄	1	2	yes	—
12	C₂×#11924036	C₂²	1	1	yes	characteristic
13	C₂×#11924029	C₄	1	1	yes	characteristic
14	#11924040	—	2	4	no	—
15	#11924038	—	2	2	no	—
16	#11924038	—	2	4	no	—
17	#11924036	—	2	2	no	—
18	#11924036	—	2	2	no	—
19	#11924036	D₄	1	2	yes	—
20	#11924036	D₄	1	2	yes	—
21	C₂×#10596833	C₂×C₄	1	2	yes	—
22	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
23	C₂×#10596833	C₂³	1	1	yes	characteristic
24	#11924029	—	2	2	no	—
25	#11924029	—	2	2	no	—
26	C₂×D₂₉₂	C₈	1	2	yes	—
27	C₂×D₂₉₂	C₈	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₇₃	C₈	1	1	yes	characteristic
31	C₂²×D₇₃	C₂×C₈	1	2	yes	—
32	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₇₃	—	2	2	no	—
34	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
37	D₂₉₂	—	2	4	no	—
38	D₂₉₂	—	2	2	no	—
39	C₄×D₇₃	—	2	2	no	—
40	C₄×D₇₃	—	2	2	no	—
41	C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
42	C₄×D₇₃	M₁₆	1	2	yes	—
43	#10596833	—	2	2	no	—
44	#10596833	C₂²⋊C₄	1	1	yes	characteristic
45	#10596833	C₂²⋊C₄	1	1	yes	characteristic
46	#10596833	C₂×D₄	1	2	yes	—
47	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×D₇₃	—	2	2	no	—
49	C₂×D₇₃	—	4	1	no	—
50	C₂×D₇₃	—	4	2	no	—
51	C₂×D₇₃	C₂²⋊C₈	1	2	yes	—
52	C₄×C₇₃	—	2	2	no	—
53	C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
54	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₇₃	—	2	1	no	—
57	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
58	D₇₃	—	2	2	no	—
59	D₇₃	—	4	2	no	—
60	#2115	—	73	1	no	maximal, Sylow
61	C₇₃	#2115	1	1	yes	characteristic, complemented, Sylow
62	C₂×#131	—	73	1	no	—
63	C₂×M₃₂	—	73	2	no	—
64	#349	—	73	4	no	—
65	C₂×#41	—	73	1	no	—
66	#131	—	146	2	no	—
67	#131	—	146	2	no	—
68	C₂×M₁₆	—	73	1	no	—
69	C₂×M₁₆	—	73	2	no	—
70	C₂²×C₈	—	73	1	no	—
71	C₂²×C₈	—	73	2	no	—
72	M₃₂	—	73	4	no	—
73	M₃₂	—	146	2	no	—
74	C₂×C₁₆	—	146	2	no	—
75	#41	—	146	2	no	—
76	#41	—	146	2	no	—
77	C₂×Q₈	#10596842	1	1	yes	characteristic
78	C₂×D₄	—	73	2	no	—
79	C₂×D₄	—	73	1	no	—
80	C₂²×C₄	—	73	1	no	—
81	C₂²×C₄	—	73	2	no	—
82	M₁₆	—	146	4	no	—
83	M₁₆	—	146	2	no	—
84	C₂×C₈	—	73	1	no	—
85	C₂×C₈	—	73	1	no	—
86	C₂×C₈	—	73	2	no	—
87	C₂×C₈	—	73	2	no	—
88	C₂×C₈	—	73	2	no	—
89	C₂×C₈	—	146	2	no	—
90	C₂×C₈	—	146	2	no	—
91	C₁₆	—	146	4	no	—
92	C₂³	—	73	1	no	—
93	C₂³	—	73	2	no	—
94	Q₈	—	2	2	no	—
95	D₄	—	146	4	no	—
96	D₄	—	146	2	no	—
97	C₂×C₄	—	73	1	no	—
98	C₂×C₄	—	73	2	no	—
99	C₂×C₄	—	73	2	no	—
100	C₂×C₄	—	146	2	no	—
101	C₂×C₄	—	146	2	no	—
102	C₂×C₄	C₂×#10596842	1	2	yes	—
103	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
104	C₈	—	73	2	no	—
105	C₈	—	73	1	no	—
106	C₈	—	73	1	no	—
107	C₈	—	146	2	no	—
108	C₂²	—	292	2	no	—
109	C₂²	—	292	1	no	—
110	C₂²	—	73	2	no	—
111	C₂²	—	146	2	no	—
112	C₂²	C₂²×#10596842	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	73	1	no	—
115	C₄	—	73	1	no	—
116	C₄	C₂²⋊#10596842	1	2	yes	—
117	C₂	—	2	1	no	—
118	C₂	—	292	2	no	—
119	C₂	—	146	2	no	—
120	C₂	C₂×C₂²⋊#10596842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989706

C₂

yes

maximal, complemented

#17989686

C₂

yes

maximal, complemented

C₂×#14556679

C₂

yes

maximal, characteristic

#14556688

—

#14556687

C₂²

yes

—

—

—

—

C₄

yes

characteristic

C₂×#11924038

C₂²

yes

—

C₂×#11924036

C₄

yes

—

C₂×#11924036

C₂²

yes

characteristic

C₂×#11924029

C₄

yes

—

—

—

—

—

D₄

yes

—

#11924036

D₄

yes

—

C₂×#10596833

C₂×C₄

yes

—

C₂×#10596833

C₂×C₄

yes

characteristic

C₂×#10596833

C₂³

yes

characteristic

#11924029

—

#11924029

—

C₂×D₂₉₂

C₈

yes

—

C₂×D₂₉₂

C₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₃

C₈

yes

characteristic

C₂²×D₇₃

C₂×C₈

yes

—

C₂²×D₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²⋊C₄

yes

—

C₄×D₇₃

M₁₆

yes

—

#10596833

—

#10596833

C₂²⋊C₄

yes

characteristic

#10596833

C₂²⋊C₄

yes

characteristic

#10596833

C₂×D₄

yes

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₈

yes

—

Dic₇₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₇₃

C₂×M₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

D₇₃

—

D₇₃

—

#2115

—

maximal, Sylow

C₇₃

#2115

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

#349

—

C₂×#41

—

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

146

—

#41

—

146

—

#41

—

146

—

C₂×Q₈

#10596842

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

146

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₁₆

—

146

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

146

—

D₄

—

146

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

146

—

101

C₂×C₄

—

146

—

102

C₂×C₄

C₂×#10596842

yes

—

103

C₂×C₄

C₂×#10596842

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

146

—

108

C₂²

—

292

—

109

C₂²

—

292

—

110

C₂²

—

111

C₂²

—

146

—

112

C₂²

C₂²×#10596842

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

C₂²⋊#10596842

yes

—

117

C₂

—

118

C₂

—

292

—

119

C₂

—

146

—

120

C₂

C₂×C₂²⋊#10596842

yes

characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#10596842
commutator	C₂²×C₇₃	C₂²×C₈
Frattini	C₂²	C₂²×#10596842
Fitting	C₂×Q₈×C₇₃	C₈
socle	C₂×C₇₃	C₂×C₂²⋊C₈
hypercenter	C₂×Q₈	#10596842
hypocenter	C₇₃	#2115
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

subgroup

G/H

center

C₂

C₂×C₂²⋊#10596842

commutator

C₂²×C₇₃

C₂²×C₈

Frattini

C₂²

C₂²×#10596842

Fitting

C₂×Q₈×C₇₃

C₈

socle

C₂×C₇₃

C₂×C₂²⋊C₈

C₂×Q₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
1. #17989706	C₂	semidirect
2. #17989686	C₂	semidirect
61. C₇₃	#2115	semidirect

H ≅ G/N

type

1. #17989706

C₂

semidirect

2. #17989686

C₂

semidirect

61. C₇₃

#2115

semidirect

Subgroups — #22627109

Conjugacy classes of subgroups

Complements