Group #22607819 — subgroups

Subgroups — #22607819

Order 9344.

190 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17988551	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924012	C₄	1	2	yes	—
5	C₂²×#10596831	—	2	1	no	—
6	C₂²⋊#10596831	—	2	1	no	—
7	C₂²⋊#10596831	—	2	1	no	—
8	C₂²⋊#10596831	—	2	1	no	—
9	C₂²⋊#10596831	C₄	1	2	yes	—
10	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₇₃	—	2	1	no	—
12	C₂²⋊C₈×C₇₃	—	2	1	no	—
13	C₂²⋊C₈×C₇₃	C₄	1	2	yes	—
14	#11924012	—	4	2	no	—
15	C₂×#10596831	—	2	2	no	—
16	C₂×#10596831	—	2	2	no	—
17	C₂×#10596831	—	4	2	no	—
18	C₂⁴×C₇₃	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₇₃	—	4	2	no	—
20	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₃	—	2	2	no	—
27	C₂×C₈×C₇₃	—	4	2	no	—
28	C₂³×C₇₃	—	2	2	no	—
29	C₂³×C₇₃	—	2	2	no	—
30	C₂³×C₇₃	—	4	1	no	—
31	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂³×C₇₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₃	—	2	2	no	—
35	C₂×C₄×C₇₃	—	2	2	no	—
36	C₂×C₄×C₇₃	—	2	2	no	—
37	C₂×C₄×C₇₃	—	4	1	no	—
38	C₂×C₄×C₇₃	—	4	1	no	—
39	C₂×C₄×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₈×C₇₃	—	4	2	no	—
45	#10596831	—	4	2	no	—
46	#10596831	—	4	2	no	—
47	C₂²×C₇₃	—	2	1	no	—
48	C₂²×C₇₃	—	2	1	no	—
49	C₂²×C₇₃	—	2	1	no	—
50	C₂²×C₇₃	—	2	1	no	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	—	4	4	no	—
55	C₂²×C₇₃	#95	1	1	yes	characteristic
56	C₂²×C₇₃	#108	1	2	yes	—
57	C₂²×C₇₃	#108	1	2	yes	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	#99	1	2	yes	—
62	C₂×C₇₃	—	2	2	no	—
63	C₂×C₇₃	—	2	2	no	—
64	C₂×C₇₃	—	4	1	no	—
65	C₂×C₇₃	#341	1	1	yes	characteristic
66	C₂×C₇₃	#341	1	1	yes	characteristic
67	C₂×C₇₃	#342	1	1	yes	characteristic
68	#1298	—	73	1	no	maximal, Sylow
69	C₇₃	#1298	1	1	yes	characteristic, complemented, Sylow
70	#407	—	73	1	no	—
71	C₂×C₂²⋊C₈	—	73	1	no	—
72	C₂×C₂²⋊C₈	D₇₃	1	1	yes	characteristic
73	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
74	C₂×M₁₆	—	73	2	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	146	1	no	—
77	C₂²⋊C₈	—	2	1	no	—
78	C₂²⋊C₈	—	73	2	no	—
79	C₂²⋊C₈	—	146	1	no	—
80	C₂²⋊C₈	—	146	1	no	—
81	C₂²⋊C₈	—	146	1	no	—
82	C₂²⋊C₈	Dic₇₃	1	2	yes	—
83	C₂⁴	Dic₁₄₆	1	1	yes	characteristic
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
87	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
91	M₁₆	—	292	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	292	2	no	—
95	C₂×C₈	—	146	2	no	—
96	C₂×C₈	—	146	2	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	4	1	no	—
100	C₂³	#11924007	1	1	yes	characteristic
101	C₂³	#11924007	1	1	yes	characteristic
102	C₂³	C₄⋊Dic₇₃	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	C₄×Dic₇₃	1	1	yes	characteristic
110	C₂×C₄	#11924005	1	1	yes	characteristic
111	C₂×C₄	#11924005	1	1	yes	characteristic
112	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	292	2	no	—
115	C₈	—	292	2	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	4	4	no	—
124	C₂²	#14556498	1	1	yes	characteristic
125	C₂²	#14556526	1	2	yes	—
126	C₂²	#14556527	1	2	yes	—
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	#14556505	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	—	2	2	no	—
133	C₂	—	4	1	no	—
134	C₂	#17988351	1	1	yes	characteristic
135	C₂	#17988352	1	1	yes	characteristic
136	C₂	#17988353	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988551

C₂

yes

maximal, characteristic

C₂×C₂²⋊#10596831

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂³×C₄×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

C₄

yes

—

—

—

—

—

C₂⁴×C₇₃

Q₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂³×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#95

yes

characteristic

C₂²×C₇₃

#108

yes

—

C₂²×C₇₃

#108

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#99

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#407

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₇₃

yes

characteristic

C₂³×C₄

C₂×D₇₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

Dic₇₃

yes

—

C₂⁴

Dic₁₄₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₇₃

yes

characteristic

C₂²×C₄

C₄×D₇₃

yes

characteristic

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

#11924007

yes

characteristic

101

C₂³

#11924007

yes

characteristic

102

C₂³

C₄⋊Dic₇₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄×Dic₇₃

yes

characteristic

110

C₂×C₄

#11924005

yes

characteristic

111

C₂×C₄

#11924005

yes

characteristic

112

C₂×C₄

C₂²⋊Dic₇₃

yes

characteristic

113

C₈

—

114

C₈

—

292

—

115

C₈

—

292

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#14556498

yes

characteristic

125

C₂²

#14556526

yes

—

126

C₂²

#14556527

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#14556505

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

134

C₂

#17988351

yes

characteristic

135

C₂

#17988352

yes

characteristic

136

C₂

#17988353

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11924005
commutator	C₂×C₄×C₇₃	C₄²
Frattini	C₂³×C₄	C₂×D₇₃
Fitting	C₂×C₂²⋊C₈×C₇₃	C₂
socle	C₂²×C₇₃	#95
hypercenter	C₂×C₂²⋊C₈	D₇₃
hypocenter	C₇₃	#1298
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂×C₄

#11924005

commutator

C₂×C₄×C₇₃

C₄²

Frattini

C₂³×C₄

C₂×D₇₃

Fitting

C₂×C₂²⋊C₈×C₇₃

C₂

socle

C₂²×C₇₃

#95

hypercenter

C₂×C₂²⋊C₈

D₇₃

hypocenter

C₇₃

#1298

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
69. C₇₃	#1298	semidirect

H ≅ G/N

type

69. C₇₃

#1298

semidirect

Subgroups — #22607819

Conjugacy classes of subgroups

Complements