Group #34033132 — subgroups

Subgroups — #34033132

Order 18688.

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

normal subgroups only

This group has 182 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×#17989696	C₂	1	2	yes	maximal
2	C₂×#17989474	C₂	1	1	yes	maximal, characteristic
3	#17989696	—	2	2	no	—
4	#17989696	—	2	2	no	—
5	C₂×#14556681	C₄	1	4	yes	complemented
6	C₂×#14556664	C₂²	1	1	yes	characteristic
7	C₂×C₈×#64408	C₄	1	1	yes	characteristic
8	#17989474	—	2	2	no	—
9	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
10	#14556681	—	2	4	no	—
11	C₂×#11924036	C₈	1	1	yes	characteristic
12	C₂×C₄×#64408	C₈	1	1	yes	characteristic
13	#14556664	D₄	1	2	yes	—
14	#14556664	D₄	1	1	yes	characteristic
15	#14556664	Q₈	1	1	yes	characteristic
16	C₂×#11924031	C₂×C₄	1	2	yes	—
17	C₈×#64408	—	2	1	no	—
18	C₈×#64408	—	2	1	no	—
19	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924003	C₈	1	1	yes	characteristic
21	C₁₆×D₇₃	—	2	1	no	—
22	C₁₆×D₇₃	—	2	1	no	—
23	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
24	C₂²×#64408	—	2	1	no	—
25	#11924036	—	2	1	no	—
26	#11924036	C₁₆	1	2	yes	—
27	C₂×#10596833	—	2	1	no	—
28	C₄×#64408	—	2	1	no	—
29	C₄×#64408	C₁₆	1	2	yes	—
30	#11924031	C₂²⋊C₄	1	2	yes	—
31	#11924031	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×#10596831	C₄²	1	1	yes	characteristic
34	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₈×D₇₃	M₁₆	1	1	yes	characteristic
37	C₈×D₇₃	M₁₆	1	1	yes	characteristic
38	#11924003	—	2	1	no	—
39	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₇₃	—	2	1	no	—
41	C₂²×D₇₃	M₃₂	1	1	yes	characteristic
42	C₂×#64408	—	2	2	no	—
43	C₂×#64408	—	4	1	no	—
44	C₂×#64408	—	4	1	no	—
45	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	M₃₂	1	1	yes	characteristic
47	C₄×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
50	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
51	#10596833	—	2	2	no	—
52	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
53	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
54	#10596831	#95	1	1	yes	characteristic
55	#10596831	#97	1	1	yes	characteristic
56	C₂²×C₇₃	#346	1	1	yes	characteristic
57	C₂×D₇₃	—	2	1	no	—
58	C₂×D₇₃	—	2	1	no	—
59	C₂×D₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
60	C₂×D₇₃	C₄⋊C₁₆	1	1	yes	characteristic
61	#64408	—	4	2	no	—
62	C₄×C₇₃	#336	1	1	yes	characteristic
63	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
64	Dic₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
65	Dic₇₃	C₄⋊C₁₆	1	1	yes	characteristic
66	#7466	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#1374	1	1	yes	characteristic
68	C₂×C₇₃	#1398	1	1	yes	characteristic
69	C₂×C₇₃	#1398	1	1	yes	characteristic
70	D₇₃	—	2	1	no	—
71	D₇₃	—	2	1	no	—
72	C₂×M₆₄	—	73	2	no	—
73	C₂×#346	—	73	1	no	—
74	C₇₃	#7466	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₁₆	—	73	1	no	—
76	C₂²×C₁₆	—	73	1	no	—
77	C₂×C₄×C₈	—	73	1	no	—
78	M₆₄	—	146	2	no	—
79	M₆₄	—	146	2	no	—
80	C₂×C₃₂	—	73	4	no	—
81	#346	—	146	2	no	—
82	C₂²×C₈	—	73	1	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂×C₄²	—	73	1	no	—
85	C₂×C₁₆	—	73	2	no	—
86	C₂×C₁₆	—	73	2	no	—
87	C₂×C₁₆	—	73	1	no	—
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	146	1	no	—
91	C₂×C₁₆	—	146	1	no	—
92	C₂×C₁₆	#10596842	1	1	yes	characteristic
93	C₄×C₈	—	146	1	no	—
94	C₄×C₈	—	146	1	no	—
95	C₃₂	—	146	4	no	—
96	C₂²×C₄	—	73	1	no	—
97	C₂²×C₄	—	146	1	no	—
98	C₂×C₈	—	73	1	no	—
99	C₂×C₈	—	73	1	no	—
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
107	C₄²	—	73	2	no	—
108	C₄²	—	146	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	73	2	no	—
111	C₁₆	—	73	2	no	—
112	C₁₆	—	146	1	no	—
113	C₂³	—	73	1	no	—
114	C₂×C₄	—	292	1	no	—
115	C₂×C₄	—	292	1	no	—
116	C₂×C₄	—	73	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
123	C₈	—	73	1	no	—
124	C₈	—	73	1	no	—
125	C₈	—	146	2	no	—
126	C₈	C₂²⋊#10596842	1	1	yes	characteristic
127	C₈	C₄⋊#10596842	1	1	yes	characteristic
128	C₂²	—	73	1	no	—
129	C₂²	—	73	1	no	—
130	C₂²	—	146	1	no	—
131	C₂²	—	146	1	no	—
132	C₂²	#17989682	1	1	yes	characteristic
133	C₄	—	292	2	no	—
134	C₄	—	73	1	no	—
135	C₄	—	73	1	no	—
136	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
137	C₄	#64408⋊C₁₆	1	1	yes	characteristic
138	C₂	—	146	1	no	—
139	C₂	—	146	1	no	—
140	C₂	#22627002	1	1	yes	characteristic
141	C₂	#22627021	1	1	yes	characteristic
142	C₂	#22627022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989696

C₂

yes

maximal

C₂×#17989474

C₂

yes

maximal, characteristic

#17989696

—

#17989696

—

C₂×#14556681

C₄

yes

complemented

C₂×#14556664

C₂²

yes

characteristic

C₂×C₈×#64408

C₄

yes

characteristic

#17989474

—

C₂×C₁₆×D₇₃

C₄

yes

characteristic

#14556681

—

C₂×#11924036

C₈

yes

characteristic

C₂×C₄×#64408

C₈

yes

characteristic

#14556664

D₄

yes

—

#14556664

D₄

yes

characteristic

#14556664

Q₈

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₈×#64408

—

C₈×#64408

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×#11924003

C₈

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₂×C₁₆×C₇₃

C₈

yes

characteristic

C₂²×#64408

—

#11924036

—

#11924036

C₁₆

yes

—

C₂×#10596833

—

C₄×#64408

—

C₄×#64408

C₁₆

yes

—

#11924031

C₂²⋊C₄

yes

—

#11924031

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₈×D₇₃

C₂²⋊C₄

yes

characteristic

C₈×D₇₃

C₄⋊C₄

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

M₃₂

yes

—

—

—

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

M₃₂

yes

characteristic

C₄×D₇₃

C₂²⋊C₈

yes

characteristic

C₄×D₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

C₂×C₁₆

yes

characteristic

C₄×D₇₃

C₂×C₁₆

yes

characteristic

#10596833

—

C₈×C₇₃

C₂²⋊C₈

yes

characteristic

C₈×C₇₃

C₄⋊C₈

yes

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₁₆

yes

characteristic

C₂×D₇₃

C₄⋊C₁₆

yes

characteristic

#64408

—

C₄×C₇₃

#336

yes

characteristic

C₄×C₇₃

C₄×C₁₆

yes

characteristic

Dic₇₃

C₂²⋊C₁₆

yes

characteristic

Dic₇₃

C₄⋊C₁₆

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

C₂×M₆₄

—

C₂×#346

—

C₇₃

#7466

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

M₆₄

—

146

—

M₆₄

—

146

—

C₂×C₃₂

—

#346

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

#10596842

yes

characteristic

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₃₂

—

146

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂×#10596842

yes

characteristic

107

C₄²

—

108

C₄²

—

146

—

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

146

—

113

C₂³

—

114

C₂×C₄

—

292

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

146

—

122

C₂×C₄

C₄×#10596842

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

146

—

126

C₈

C₂²⋊#10596842

yes

characteristic

127

C₈

C₄⋊#10596842

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

146

—

131

C₂²

—

146

—

132

C₂²

#17989682

yes

characteristic

133

C₄

—

292

—

134

C₄

—

135

C₄

—

136

C₄

(C₂×#10596842)⋊C₄

yes

characteristic

137

C₄

#64408⋊C₁₆

yes

characteristic

138

C₂

—

146

—

139

C₂

—

146

—

140

C₂

#22627002

yes

characteristic

141

C₂

#22627021

yes

characteristic

142

C₂

#22627022

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#10596842
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₈	C₂×#10596842
Fitting	C₂×C₁₆×C₇₃	C₈
socle	C₂²×C₇₃	#346
prime radical	C₂²×D₇₃	M₃₂
cosocle	C₂×#14556664	C₂²
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	#7466
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

subgroup

G/H

center

C₂×C₄

C₄×#10596842

commutator

C₄×C₇₃

C₄×C₁₆

Frattini

C₂×C₈

C₂×#10596842

Fitting

C₂×C₁₆×C₇₃

C₈

socle

C₂²×C₇₃

#346

prime radical

C₂²×D₇₃

M₃₂

C₂²

C₂×C₁₆

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

N	H ≅ G/N	type
5. C₂×#14556681	C₄	semidirect
74. C₇₃	#7466	semidirect

H ≅ G/N

type

5. C₂×#14556681

C₄

semidirect

74. C₇₃

#7466

semidirect

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

subgroup	index
C₂×#17989696	2
C₂×#17989474	2
#7466	73

subgroup

index

C₂×#17989696

C₂×#17989474

#7466

Subgroups — #34033132

Conjugacy classes of subgroups

Complements

Maximal subgroups