Group #34032858 — subgroups

Subgroups — #34032858

Order 18688.

266 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22627087	C₂	1	2	yes	maximal
2	#22625127	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556686	C₄	1	4	yes	—
4	#17989681	—	2	2	no	—
5	#17989681	—	2	2	no	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	#17989429	—	2	2	no	—
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	#14556686	—	2	4	no	—
11	#14556686	C₈	1	8	yes	complemented
12	C₂×#11924040	—	2	4	no	—
13	C₂×#11924036	—	2	1	no	—
14	C₂×#11924036	C₂×C₄	1	2	yes	—
15	C₄×#10596833	—	2	1	no	—
16	C₄×#10596833	C₈	1	2	yes	—
17	C₄×#10596833	D₄	1	2	yes	—
18	C₄×#10596833	D₄	1	1	yes	characteristic
19	C₄×#10596833	Q₈	1	1	yes	characteristic
20	C₂×C₈×D₇₃	—	2	1	no	—
21	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₈	1	1	yes	characteristic
23	C₈×Dic₇₃	—	2	1	no	—
24	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
25	#11924040	—	2	8	no	—
26	#11924036	—	4	2	no	—
27	#11924036	C₂×C₈	1	4	yes	—
28	#11924036	M₁₆	1	4	yes	—
29	C₂×#10596833	—	2	2	no	—
30	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
31	C₂×#10596833	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
34	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
35	C₂×#10596831	—	2	1	no	—
36	C₈×D₇₃	—	4	2	no	—
37	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₇₃	#97	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	#95	1	1	yes	characteristic
47	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₄×D₇₃	—	2	2	no	—
50	C₄×D₇₃	—	2	2	no	—
51	C₄×D₇₃	C₄×C₈	1	2	yes	—
52	C₄×D₇₃	#97	1	2	yes	—
53	#10596833	—	4	2	no	—
54	#10596833	C₂²⋊C₈	1	4	yes	—
55	#10596833	C₄⋊C₈	1	4	yes	—
56	C₈×C₇₃	—	4	1	no	—
57	#10596831	—	4	1	no	—
58	C₂²×C₇₃	#336	1	1	yes	characteristic
59	C₂×D₇₃	#322	1	2	yes	—
60	C₂×D₇₃	#347	1	2	yes	—
61	C₂×D₇₃	#347	1	2	yes	—
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	C₈²	1	1	yes	characteristic
65	C₄×C₇₃	#322	1	1	yes	characteristic
66	Dic₇₃	—	2	1	no	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	#336	1	2	yes	—
69	#7137	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	#1273	1	1	yes	characteristic
71	C₂×C₇₃	#1375	1	1	yes	characteristic
72	C₂×C₇₃	#1375	1	1	yes	characteristic
73	D₇₃	#1313	1	4	yes	—
74	C₂×#346	—	73	2	no	—
75	C₂×#322	—	73	1	no	—
76	C₇₃	#7137	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₁₆	—	73	4	no	—
78	C₂×C₄×C₈	—	73	1	no	—
79	C₂×C₄×C₈	—	73	1	no	—
80	C₂×C₄×C₈	—	73	1	no	—
81	#346	—	146	2	no	—
82	#346	—	146	2	no	—
83	#322	—	146	2	no	—
84	C₂²×C₈	—	73	2	no	—
85	C₂²×C₈	—	146	1	no	—
86	C₂²×C₈	—	146	1	no	—
87	C₂×C₄²	—	73	1	no	—
88	C₂×C₁₆	—	73	8	no	—
89	C₂×C₁₆	—	146	4	no	—
90	C₂×C₁₆	—	146	4	no	—
91	C₄×C₈	—	73	2	no	—
92	C₄×C₈	—	73	1	no	—
93	C₄×C₈	—	73	1	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	146	1	no	—
97	C₄×C₈	—	146	1	no	—
98	C₄×C₈	#10596842	1	1	yes	characteristic
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	C₂²×C₄	—	73	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	292	2	no	—
104	C₂×C₈	—	292	2	no	—
105	C₂×C₈	—	73	2	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	—	73	4	no	—
108	C₂×C₈	—	73	4	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	—	146	1	no	—
111	C₄²	—	73	1	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	C₂×#10596842	1	1	yes	characteristic
115	C₁₆	—	146	8	no	—
116	C₂³	—	73	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	73	2	no	—
119	C₂×C₄	—	73	2	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	146	2	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
126	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	292	1	no	—
129	C₈	—	292	2	no	—
130	C₈	—	73	4	no	—
131	C₈	—	73	4	no	—
132	C₂²	—	73	2	no	—
133	C₂²	—	73	2	no	—
134	C₂²	—	73	2	no	—
135	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	73	2	no	—
139	C₄	—	146	1	no	—
140	C₄	—	146	1	no	—
141	C₄	C₈×#10596842	1	1	yes	characteristic
142	C₄	#17989671	1	1	yes	characteristic
143	C₂	—	73	4	no	—
144	C₂	(C₂×#10596842)⋊C₈	1	1	yes	characteristic
145	C₂	#22627003	1	1	yes	characteristic
146	C₂	#22627004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627087

C₂

yes

maximal

#22625127

C₂

yes

maximal, characteristic

C₂×#14556686

C₄

yes

—

#17989681

—

#17989681

—

C₄×#11924036

C₄

yes

characteristic

C₄×#11924036

C₂²

yes

characteristic

#17989429

—

C₄×C₈×D₇₃

C₄

yes

characteristic

#14556686

—

#14556686

C₈

yes

—

—

C₂×C₄

yes

—

C₄×#10596833

—

C₄×#10596833

C₈

yes

—

C₄×#10596833

D₄

yes

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₈

yes

—

—

C₂×C₈

yes

—

#11924036

M₁₆

yes

—

C₂×#10596833

—

C₂×#10596833

C₂²⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

#95

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₄×C₈

yes

—

C₄×D₇₃

#97

yes

—

#10596833

—

#10596833

C₂²⋊C₈

yes

—

#10596833

C₄⋊C₈

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

#322

yes

—

C₂×D₇₃

#347

yes

—

C₂×D₇₃

#347

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈²

yes

characteristic

C₄×C₇₃

#322

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#336

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

C₂×#346

—

C₂×#322

—

C₇₃

#7137

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

146

—

#346

—

146

—

#322

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₄×C₈

#10596842

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂×C₈

—

103

C₂×C₈

—

292

—

104

C₂×C₈

—

292

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×#10596842

yes

characteristic

115

C₁₆

—

146

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

C₄×#10596842

yes

characteristic

125

C₂×C₄

C₂²⋊#10596842

yes

characteristic

126

C₂×C₄

C₄⋊#10596842

yes

characteristic

127

C₈

—

128

C₈

—

292

—

129

C₈

—

292

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

146

—

140

C₄

—

146

—

141

C₄

C₈×#10596842

yes

characteristic

142

C₄

#17989671

yes

characteristic

143

C₂

—

144

C₂

(C₂×#10596842)⋊C₈

yes

characteristic

145

C₂

#22627003

yes

characteristic

146

C₂

#22627004

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#10596842
commutator	C₄×C₇₃	C₈²
Frattini	C₄²	C₂×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	#97
cosocle	C₄×#11924036	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7137
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄×#10596842

commutator

C₄×C₇₃

C₈²

Frattini

C₄²

C₂×#10596842

Fitting

C₄×C₈×C₇₃

C₈

C₂²×C₇₃

C₂²×D₇₃

C₂²

C₄×C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
11. #14556686	C₈	semidirect
76. C₇₃	#7137	semidirect

H ≅ G/N

type

11. #14556686

C₈

semidirect

76. C₇₃

#7137

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22627087	2
#22625127	2
#7137	73

subgroup

index

#22627087

#22625127

#7137

Subgroups — #34032858

Conjugacy classes of subgroups

Complements

Maximal subgroups