Group #22609434 — subgroups

Subgroups — #22609434

Order 9344.

146 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17988879	C₂	1	1	yes	maximal, characteristic
2	#17988794	C₂	1	1	yes	maximal, characteristic
3	#17988717	C₂	1	1	yes	maximal, characteristic
4	#17988340	C₂	1	1	yes	maximal, characteristic
5	#17988336	C₂	1	1	yes	maximal, characteristic
6	#17988333	C₂	1	1	yes	maximal, characteristic
7	#498×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924018	—	2	1	no	—
9	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
10	#14556524	—	2	1	no	—
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556520	—	2	1	no	—
15	#14556515	—	2	1	no	—
16	#14556514	—	2	1	no	—
17	#14556501	—	2	1	no	—
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	#106×C₇₃	—	2	1	no	—
21	#106×C₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
24	#11924018	—	4	2	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
29	#11924007	—	2	1	no	—
30	C₄×Dic₇₃	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×Q₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	Q₈×C₇₃	—	4	1	no	—
41	Q₈×C₇₃	—	4	1	no	—
42	Q₈×C₇₃	—	8	1	no	—
43	C₂×C₄×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	—	4	1	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
48	C₂×Dic₇₃	—	2	1	no	—
49	C₂×Dic₇₃	—	2	1	no	—
50	Dic₁₄₆	—	2	1	no	—
51	Dic₁₄₆	—	2	1	no	—
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	Q₁₆	1	2	yes	complemented
54	#10596831	—	4	1	no	—
55	#10596831	—	4	1	no	—
56	#10596831	—	4	1	no	—
57	C₂²×C₇₃	#121	1	1	yes	characteristic
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	8	1	no	—
60	C₄×C₇₃	—	8	1	no	—
61	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₄×C₇₃	#137	1	1	yes	characteristic
64	C₄×C₇₃	#137	1	1	yes	characteristic
65	Dic₇₃	—	4	1	no	—
66	Dic₇₃	—	4	1	no	—
67	C₂×C₇₃	#462	1	1	yes	characteristic
68	C₂×C₇₃	#468	1	1	yes	characteristic
69	C₂×C₇₃	#471	1	1	yes	characteristic
70	#1692	—	73	1	no	maximal, Sylow
71	C₇₃	#1692	1	1	yes	characteristic, complemented, Sylow
72	#498	D₇₃	1	1	yes	characteristic
73	#477	—	73	1	no	—
74	#462	—	73	1	no	—
75	#446	—	73	1	no	—
76	#335	—	73	1	no	—
77	#333	—	73	1	no	—
78	#332	—	73	1	no	—
79	C₂×Q₁₆	—	146	1	no	—
80	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
82	C₄×Q₈	—	73	1	no	—
83	#107	—	146	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	C₄⋊C₈	—	73	1	no	—
87	C₄⋊C₈	—	73	1	no	—
88	C₄⋊C₈	—	73	1	no	—
89	C₄⋊C₈	—	146	1	no	—
90	#103	—	146	1	no	—
91	#103	—	146	1	no	—
92	#97	—	146	1	no	—
93	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	73	1	no	—
97	Q₁₆	—	292	2	no	—
98	C₂×C₈	—	146	1	no	—
99	C₂×C₈	—	146	1	no	—
100	C₂×C₈	—	146	1	no	—
101	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	73	1	no	—
108	C₄⋊C₄	—	146	1	no	—
109	C₄²	—	146	1	no	—
110	C₄²	C₂²×D₇₃	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	146	1	no	—
115	Q₈	—	146	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	146	1	no	—
119	C₂×C₄	—	146	1	no	—
120	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂×C₄	#11924027	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	292	1	no	—
125	C₈	—	292	1	no	—
126	C₈	—	292	1	no	—
127	C₈	#11924018	1	2	yes	complemented
128	C₂²	#14556562	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	292	1	no	—
133	C₄	—	292	1	no	—
134	C₄	C₂×#11924018	1	1	yes	characteristic
135	C₄	#14556618	1	1	yes	characteristic
136	C₄	#14556620	1	1	yes	characteristic
137	C₄	#14556622	1	1	yes	characteristic
138	C₂	#17988794	1	1	yes	characteristic
139	C₂	#17988830	1	1	yes	characteristic
140	C₂	#17988847	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988879

C₂

yes

maximal, characteristic

#17988794

C₂

yes

maximal, characteristic

#17988717

C₂

yes

maximal, characteristic

#17988340

C₂

yes

maximal, characteristic

#17988336

C₂

yes

maximal, characteristic

#17988333

C₂

yes

maximal, characteristic

#498×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924018

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

#14556524

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

—

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#106×C₇₃

—

#106×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

—

—

—

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

Q₁₆

yes

—

—

—

C₂²×C₇₃

#121

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#498

D₇₃

yes

—

—

—

—

—

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×Q₈

—

#107

—

146

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

—

146

—

#103

—

146

—

#97

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

292

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

100

C₂×C₈

—

146

—

101

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

102

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

146

—

109

C₄²

—

146

—

110

C₄²

C₂²×D₇₃

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

146

—

115

Q₈

—

146

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

146

—

119

C₂×C₄

—

146

—

120

C₂×C₄

D₄×D₇₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

122

C₂×C₄

#11924027

yes

characteristic

123

C₈

—

124

C₈

—

292

—

125

C₈

—

292

—

126

C₈

—

292

—

127

C₈

#11924018

yes

complemented

128

C₂²

#14556562

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

292

—

133

C₄

—

292

—

134

C₄

C₂×#11924018

yes

characteristic

135

C₄

#14556618

yes

characteristic

136

C₄

#14556620

yes

characteristic

137

C₄

#14556622

yes

characteristic

138

C₂

#17988794

yes

characteristic

139

C₂

#17988830

yes

characteristic

140

C₂

#17988847

yes

characteristic

	subgroup	G/H
center	C₂²	#14556562
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#498×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#498	D₇₃
hypocenter	C₇₃	#1692
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14556562

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#498×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
53. C₈×C₇₃	Q₁₆	semidirect
71. C₇₃	#1692	semidirect
127. C₈	#11924018	semidirect

H ≅ G/N

type

53. C₈×C₇₃

Q₁₆

semidirect

71. C₇₃

#1692

semidirect

127. C₈

#11924018

semidirect

Subgroups — #22609434

Conjugacy classes of subgroups

Complements