Group #34000771 — subgroups

Subgroups — #34000771

Order 18688.

170 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22625046	C₂	1	2	yes	maximal, complemented
2	#22611542	C₂	1	1	yes	maximal, characteristic
3	#17989519	—	2	2	no	—
4	C₂×#14556531	C₄	1	1	yes	characteristic
5	C₂×#14556531	C₂²	1	1	yes	characteristic
6	C₂²⋊#11924003	—	2	2	no	—
7	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
8	#14556641	—	4	2	no	—
9	#14556531	—	4	1	no	—
10	#14556531	—	4	1	no	—
11	C₂×#11924003	—	2	1	no	—
12	C₂×#11924003	D₄	1	2	yes	—
13	C₂³×C₄×C₇₃	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	—	2	1	no	—
15	C₂²×C₈×C₇₃	—	2	1	no	—
16	C₂²×C₈×C₇₃	—	4	1	no	—
17	C₂²×C₈×C₇₃	—	4	1	no	—
18	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	#11924003	—	2	2	no	—
21	#11924003	—	4	1	no	—
22	C₂⁴×C₇₃	C₁₆	1	1	yes	characteristic
23	C₂²×C₄×C₇₃	—	2	1	no	—
24	C₂²×C₄×C₇₃	—	2	1	no	—
25	C₂²×C₄×C₇₃	—	4	1	no	—
26	C₂²×C₄×C₇₃	—	4	1	no	—
27	C₂²×C₄×C₇₃	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₇₃	—	2	1	no	—
30	C₂×C₈×C₇₃	—	2	1	no	—
31	C₂×C₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	—	4	2	no	—
33	C₂×C₈×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	4	1	no	—
36	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
38	C₂³×C₇₃	—	2	1	no	—
39	C₂³×C₇₃	—	4	1	no	—
40	C₂³×C₇₃	—	4	2	no	—
41	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	—	4	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
51	C₈×C₇₃	—	2	1	no	—
52	C₈×C₇₃	—	4	1	no	—
53	C₈×C₇₃	#99	1	1	yes	characteristic
54	C₈×C₇₃	#100	1	1	yes	characteristic
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	—	4	1	no	—
58	C₂²×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	—	4	2	no	—
60	C₂²×C₇₃	—	4	2	no	—
61	C₂²×C₇₃	—	4	1	no	—
62	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	#323	1	1	yes	characteristic
66	C₄×C₇₃	#349	1	1	yes	characteristic
67	#7336	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	—	2	1	no	—
69	C₂×C₇₃	—	4	2	no	—
70	C₂×C₇₃	—	4	1	no	—
71	C₂×C₇₃	#1314	1	1	yes	characteristic
72	#2112	—	73	1	no	—
73	#1400	—	73	2	no	—
74	C₇₃	#7336	1	1	yes	characteristic, complemented, Sylow
75	C₂³×C₈	#64408	1	1	yes	characteristic
76	C₂×M₃₂	—	73	1	no	—
77	C₂×M₃₂	—	73	1	no	—
78	M₆₄	—	146	2	no	—
79	C₂²⋊C₁₆	—	146	2	no	—
80	C₂³×C₄	#10596833	1	1	yes	characteristic
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	4	1	no	—
84	C₂²×C₈	—	4	1	no	—
85	C₂²×C₈	#10596833	1	1	yes	characteristic
86	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
87	M₃₂	—	292	1	no	—
88	M₃₂	—	292	1	no	—
89	C₂×C₁₆	—	73	2	no	—
90	C₂×C₁₆	—	146	1	no	—
91	C₃₂	—	292	2	no	—
92	C₂⁴	#11924031	1	1	yes	characteristic
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	—	4	1	no	—
97	C₂²×C₄	#11924031	1	1	yes	characteristic
98	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
107	C₂×C₈	#11924037	1	1	yes	characteristic
108	C₁₆	—	292	1	no	—
109	C₁₆	—	146	2	no	—
110	C₂³	—	2	1	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×#11924031	1	1	yes	characteristic
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
122	C₂×C₄	#14556665	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	#14556649	1	1	yes	characteristic
126	C₈	#14556651	1	1	yes	characteristic
127	C₂²	—	2	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	1	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	1	no	—
134	C₂²	C₂²⋊#11924031	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	4	1	no	—
137	C₄	#17989430	1	1	yes	characteristic
138	C₄	#17989483	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	2	no	—
141	C₂	—	4	1	no	—
142	C₂	#22624851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625046

C₂

yes

maximal, complemented

#22611542

C₂

yes

maximal, characteristic

#17989519

—

C₂×#14556531

C₄

yes

characteristic

C₂×#14556531

C₂²

yes

characteristic

C₂²⋊#11924003

—

C₂³×C₈×C₇₃

C₄

yes

—

—

—

—

D₄

yes

—

C₂³×C₄×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924003

—

#11924003

—

C₂⁴×C₇₃

C₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

C₇₃

#7336

yes

characteristic, complemented, Sylow

C₂³×C₈

#64408

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

M₆₄

—

146

—

C₂²⋊C₁₆

—

146

—

C₂³×C₄

#10596833

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

yes

C₂²×C₈

yes

M₃₂

—

292

—

M₃₂

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₃₂

—

292

—

C₂⁴

#11924031

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²⋊#64408

yes

characteristic

107

C₂×C₈

#11924037

yes

characteristic

108

C₁₆

—

292

—

109

C₁₆

—

146

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×#11924031

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊#10596833

yes

characteristic

122

C₂×C₄

#14556665

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

#14556649

yes

characteristic

126

C₈

#14556651

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂²⋊#11924031

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

#17989430

yes

characteristic

138

C₄

#17989483

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

142

C₂

#22624851

yes

characteristic

	subgroup	G/H
center	C₈	#14556649
commutator	C₂³×C₇₃	C₂×C₁₆
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂×C₇₃	#1314
prime radical	C₂⁴×C₇₃	C₁₆
cosocle	C₂×#14556531	C₂²
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	#7336
LCS	G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂³×C₈

subgroup

G/H

center

C₈

#14556649

commutator

C₂³×C₇₃

C₂×C₁₆

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂³×C₈×C₇₃

C₄

socle

C₂×C₇₃

#1314

prime radical

C₂⁴×C₇₃

C₁₆

C₂²

C₂³×C₈

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
1. #22625046	C₂	semidirect
74. C₇₃	#7336	semidirect

H ≅ G/N

type

1. #22625046

C₂

semidirect

74. C₇₃

#7336

semidirect

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

subgroup	index
#22625046	2
#22611542	2
#7336	73

subgroup

index

#22625046

#22611542

#7336

Subgroups — #34000771

Conjugacy classes of subgroups

Complements

Maximal subgroups