Group #22627102 — subgroups

Subgroups — #22627102

Order 9344.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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×#14556688	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556686	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#11924040	C₂	1	4	yes	maximal, complemented
4	C₂×#14556678	C₂	1	1	yes	maximal, characteristic
5	#14556688	—	2	2	no	—
6	#14556686	—	2	2	no	—
7	C₂×#11924040	C₂²	1	2	yes	—
8	C₂×#11924040	C₂²	1	2	yes	—
9	#14556678	—	2	2	no	—
10	#14556678	—	2	2	no	—
11	C₂×#11924038	C₄	1	1	yes	characteristic
12	C₂×#11924037	C₄	1	2	yes	—
13	C₂×#11924036	C₂²	1	1	yes	characteristic
14	C₂²×#10596833	C₂²	1	2	yes	—
15	C₂×#11924027	C₄	1	1	yes	characteristic
16	#11924040	—	2	2	no	—
17	#11924040	—	2	2	no	—
18	#11924038	—	2	2	no	—
19	#11924037	—	2	4	no	—
20	#11924036	D₄	1	2	yes	—
21	#11924036	D₄	1	2	yes	—
22	C₂×#10596833	—	2	2	no	—
23	C₂×#10596833	—	2	2	no	—
24	C₂×#10596833	C₂×C₄	1	2	yes	—
25	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
26	C₂×#10596833	C₂³	1	1	yes	characteristic
27	#11924027	—	2	2	no	—
28	#11924027	—	2	2	no	—
29	C₂×Dic₁₄₆	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊D₇₃	C₈	1	2	yes	—
31	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₇₃	C₈	1	1	yes	characteristic
34	C₂³×C₇₃	C₂×C₈	1	2	yes	—
35	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
36	D₄×C₇₃	—	2	2	no	—
37	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂²⋊D₇₃	—	2	4	no	—
39	C₂×Dic₇₃	—	2	2	no	—
40	C₂×Dic₇₃	—	2	2	no	—
41	C₂×Dic₇₃	C₂×C₈	1	2	yes	—
42	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
44	C₄×D₇₃	M₁₆	1	2	yes	—
45	Dic₁₄₆	—	2	2	no	—
46	#10596833	—	2	2	no	—
47	#10596833	C₂²⋊C₄	1	2	yes	—
48	#10596833	C₂×D₄	1	1	yes	characteristic
49	#10596833	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₇₃	—	2	2	no	—
51	C₂²×C₇₃	—	2	4	no	—
52	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×D₇₃	—	2	2	no	—
54	C₂×D₇₃	C₂²⋊C₈	1	2	yes	—
55	C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
56	Dic₇₃	—	2	2	no	—
57	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₇₃	—	4	2	no	—
60	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
61	C₂×C₇₃	#504	1	2	yes	—
62	D₇₃	—	2	2	no	—
63	#2113	—	73	1	no	maximal, Sylow
64	C₇₃	#2113	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	—	73	1	no	—
66	C₂×M₃₂	—	73	1	no	—
67	C₂²×C₁₆	—	73	1	no	—
68	C₂²⋊C₁₆	—	73	4	no	—
69	C₂×#41	—	73	1	no	—
70	#131	—	146	2	no	—
71	#131	—	146	2	no	—
72	C₂×M₁₆	—	73	2	no	—
73	C₂×M₁₆	—	73	1	no	—
74	C₂²×C₈	—	73	1	no	—
75	C₂²×C₈	—	73	2	no	—
76	M₃₂	—	146	2	no	—
77	C₂×C₁₆	—	73	2	no	—
78	C₂×C₁₆	—	73	2	no	—
79	C₂×C₁₆	—	146	2	no	—
80	#41	—	146	2	no	—
81	#41	—	146	2	no	—
82	C₂×Q₈	—	73	1	no	—
83	C₂×D₄	—	73	2	no	—
84	C₂×D₄	#10596842	1	1	yes	characteristic
85	C₂²×C₄	—	73	1	no	—
86	C₂²×C₄	—	73	2	no	—
87	M₁₆	—	146	4	no	—
88	M₁₆	—	146	2	no	—
89	C₂×C₈	—	73	1	no	—
90	C₂×C₈	—	73	1	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	—	73	2	no	—
93	C₂×C₈	—	73	2	no	—
94	C₂×C₈	—	146	2	no	—
95	C₂×C₈	—	146	2	no	—
96	C₁₆	—	146	2	no	—
97	C₁₆	—	146	2	no	—
98	C₂³	—	73	1	no	—
99	C₂³	C₂×#10596842	1	2	yes	—
100	Q₈	—	146	2	no	—
101	D₄	—	2	2	no	—
102	D₄	—	146	4	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	2	no	—
105	C₂×C₄	—	73	2	no	—
106	C₂×C₄	—	73	2	no	—
107	C₂×C₄	—	146	2	no	—
108	C₂×C₄	—	146	2	no	—
109	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
110	C₈	—	73	1	no	—
111	C₈	—	73	1	no	—
112	C₈	—	73	2	no	—
113	C₈	—	146	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	73	2	no	—
117	C₂²	—	146	2	no	—
118	C₂²	C₂²×#10596842	1	1	yes	characteristic
119	C₄	—	73	1	no	—
120	C₄	—	73	1	no	—
121	C₄	—	146	2	no	—
122	C₄	C₂²⋊#10596842	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	146	2	no	—
125	C₂	C₂×C₂²⋊#10596842	1	1	yes	characteristic
126	C₂	#17989708	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556688

C₂

yes

maximal, characteristic, complemented

C₂×#14556686

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11924040

C₂

yes

maximal, complemented

C₂×#14556678

C₂

yes

maximal, characteristic

#14556688

—

#14556686

—

C₂×#11924040

C₂²

yes

—

C₂×#11924040

C₂²

yes

—

#14556678

—

#14556678

—

C₂×#11924038

C₄

yes

characteristic

C₂×#11924037

C₄

yes

—

C₂×#11924036

C₂²

yes

characteristic

C₂²×#10596833

C₂²

yes

—

C₂×#11924027

C₄

yes

—

—

—

—

D₄

yes

—

#11924036

D₄

yes

—

C₂×#10596833

—

C₂×#10596833

—

C₂×#10596833

C₂×C₄

yes

—

C₂×#10596833

C₂×C₄

yes

characteristic

C₂×#10596833

C₂³

yes

characteristic

#11924027

—

#11924027

—

C₂×Dic₁₄₆

C₈

yes

characteristic

C₂×C₂²⋊D₇₃

C₈

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₈

yes

characteristic

C₂³×C₇₃

C₂×C₈

yes

—

C₂²×D₇₃

C₂×C₈

yes

characteristic

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×D₇₃

C₂²⋊C₄

yes

—

C₄×D₇₃

M₁₆

yes

—

Dic₁₄₆

—

#10596833

—

#10596833

C₂²⋊C₄

yes

—

#10596833

C₂×D₄

yes

characteristic

#10596833

C₂×D₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₈

yes

—

C₄×C₇₃

C₂²⋊C₈

yes

—

Dic₇₃

—

Dic₇₃

C₂×C₂²⋊C₄

yes

characteristic

Dic₇₃

C₂×M₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₃

#504

yes

—

D₇₃

—

#2113

—

maximal, Sylow

C₇₃

#2113

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂×#41

—

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#41

—

146

—

#41

—

146

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

#10596842

yes

characteristic

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

—

C₁₆

—

146

—

C₂³

—

C₂³

C₂×#10596842

yes

—

100

Q₈

—

146

—

101

D₄

—

102

D₄

—

146

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

146

—

108

C₂×C₄

—

146

—

109

C₂×C₄

C₂×#10596842

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

146

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

146

—

118

C₂²

C₂²×#10596842

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

146

—

122

C₄

C₂²⋊#10596842

yes

—

123

C₂

—

124

C₂

—

146

—

125

C₂

C₂×C₂²⋊#10596842

yes

characteristic

126

C₂

#17989708

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₂×D₄×C₇₃	C₈
hypercenter	C₂×D₄	#10596842
hypocenter	C₇₃	#2113
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₈

Fitting

C₂×D₄×C₇₃

C₈

C₂×D₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂×#14556688	C₂	semidirect
2. C₂×#14556686	C₂	semidirect
3. C₂²⋊#11924040	C₂	semidirect
64. C₇₃	#2113	semidirect

H ≅ G/N

type

1. C₂×#14556688

C₂

semidirect

2. C₂×#14556686

C₂

semidirect

3. C₂²⋊#11924040

C₂

semidirect

64. C₇₃

#2113

semidirect

Subgroups — #22627102

Conjugacy classes of subgroups

Complements