Group #34034406 — subgroups

Subgroups — #34034406

Order 18688.

330 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17989695	C₂	1	2	yes	maximal, complemented
2	#22627024	C₂	1	4	yes	maximal, complemented
3	C₂²×#14556664	C₂	1	1	yes	maximal, characteristic
4	#17989695	—	2	4	no	—
5	C₂×#14556681	C₂²	1	4	yes	complemented
6	C₂×#14556664	—	2	4	no	—
7	C₂×#14556664	C₄	1	2	yes	—
8	C₂×#14556664	C₂²	1	2	yes	—
9	C₂²×#11924031	C₄	1	1	yes	characteristic
10	C₂²×#11924031	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
12	#14556681	—	2	4	no	—
13	#14556664	—	2	4	no	—
14	#14556664	—	2	4	no	—
15	C₂×#11924031	—	2	2	no	—
16	C₂×#11924031	C₂×C₄	1	2	yes	—
17	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
18	C₂×#11924031	D₄	1	4	yes	—
19	C₂×#11924031	C₂³	1	1	yes	characteristic
20	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	—	2	2	no	—
23	C₂×C₈×D₇₃	—	2	2	no	—
24	C₂×C₈×D₇₃	C₈	1	2	yes	—
25	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
27	#11924031	—	2	2	no	—
28	#11924031	C₂²⋊C₄	1	2	yes	—
29	#11924031	C₂×D₄	1	2	yes	—
30	C₂³×D₇₃	C₁₆	1	1	yes	characteristic
31	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
32	C₂×C₄×D₇₃	—	2	2	no	—
33	C₂×C₄×D₇₃	—	2	2	no	—
34	C₂×C₄×D₇₃	C₁₆	1	2	yes	—
35	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
36	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
37	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
38	C₂×#10596831	M₁₆	1	2	yes	—
39	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
40	C₈×D₇₃	—	2	4	no	—
41	C₈×D₇₃	—	2	2	no	—
42	C₈×D₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	—	2	2	no	—
45	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
46	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	2	no	—
48	C₂²×D₇₃	—	2	4	no	—
49	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
53	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
56	C₄×D₇₃	—	2	4	no	—
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	2	2	no	—
59	C₈×C₇₃	—	2	2	no	—
60	#10596831	C₂²⋊C₈	1	2	yes	—
61	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	#10596831	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₇₃	—	2	2	no	—
64	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
65	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
66	C₂×D₇₃	—	2	8	no	—
67	C₂×D₇₃	—	2	2	no	—
68	C₂×D₇₃	—	2	2	no	—
69	C₂×D₇₃	M₆₄	1	4	yes	complemented
70	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
71	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
72	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
73	Dic₇₃	—	2	2	no	—
74	#13553	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
77	C₂×C₇₃	C₂×M₆₄	1	2	yes	—
78	D₇₃	—	2	4	no	—
79	C₂³×C₁₆	—	73	1	no	—
80	C₂×M₆₄	—	73	2	no	—
81	C₂²⋊C₃₂	—	73	4	no	—
82	C₇₃	#13553	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₈	—	73	1	no	—
84	C₂²×C₁₆	—	73	1	no	—
85	C₂²×C₁₆	—	73	1	no	—
86	C₂²×C₁₆	—	73	2	no	—
87	C₂²×C₁₆	—	73	2	no	—
88	C₂²×C₁₆	—	146	4	no	—
89	M₆₄	—	146	4	no	—
90	C₂×C₃₂	—	73	4	no	—
91	C₂³×C₄	—	73	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₂²×C₈	—	146	2	no	—
96	C₂²×C₈	—	146	2	no	—
97	C₂²×C₈	#10596842	1	1	yes	characteristic
98	C₂×C₁₆	—	73	1	no	—
99	C₂×C₁₆	—	73	4	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	73	2	no	—
102	C₂×C₁₆	—	146	4	no	—
103	C₂×C₁₆	—	146	4	no	—
104	C₂×C₁₆	—	146	2	no	—
105	C₃₂	—	146	4	no	—
106	C₂⁴	—	73	1	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	73	2	no	—
109	C₂²×C₄	—	73	2	no	—
110	C₂²×C₄	—	146	2	no	—
111	C₂²×C₄	—	146	2	no	—
112	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	—	146	4	no	—
120	C₂×C₈	—	146	2	no	—
121	C₂×C₈	C₂×#10596842	1	2	yes	—
122	C₁₆	—	73	2	no	—
123	C₁₆	—	73	2	no	—
124	C₁₆	—	146	2	no	—
125	C₂³	—	73	2	no	—
126	C₂³	—	146	4	no	—
127	C₂³	—	146	2	no	—
128	C₂³	#14556686	1	1	yes	characteristic
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	—	146	4	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	C₂²⋊#10596842	1	2	yes	—
135	C₂×C₄	#14556686	1	1	yes	characteristic
136	C₂×C₄	#14556688	1	2	yes	—
137	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	73	1	no	—
140	C₈	—	73	2	no	—
141	C₈	—	73	1	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	73	4	no	—
144	C₂²	—	146	8	no	—
145	C₂²	—	146	2	no	—
146	C₂²	—	146	2	no	—
147	C₂²	#17989684	1	2	yes	—
148	C₂²	C₂×#14556686	1	1	yes	characteristic
149	C₄	—	146	2	no	—
150	C₄	#17989684	1	2	yes	—
151	C₄	C₂×C₂²⋊#10596842	1	1	yes	characteristic
152	C₄	C₂×#14556688	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	146	4	no	—
155	C₂	C₂×#17989684	1	1	yes	characteristic
156	C₂	#22627162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989695

C₂

yes

maximal, complemented

#22627024

C₂

yes

maximal, complemented

C₂²×#14556664

C₂

yes

maximal, characteristic

#17989695

—

C₂×#14556681

C₂²

yes

complemented

C₂×#14556664

—

C₂×#14556664

C₄

yes

—

C₂×#14556664

C₂²

yes

—

C₂²×#11924031

C₄

yes

characteristic

C₂²×#11924031

C₂²

yes

characteristic

C₂²×C₈×D₇₃

C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

D₄

yes

—

C₂×#11924031

C₂³

yes

characteristic

C₂²×C₄×D₇₃

C₈

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₈

yes

characteristic

#11924031

—

#11924031

C₂²⋊C₄

yes

—

#11924031

C₂×D₄

yes

—

C₂³×D₇₃

C₁₆

yes

characteristic

C₂²×Dic₇₃

C₁₆

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₁₆

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂²⋊C₄

yes

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

#10596831

C₂²⋊C₈

yes

—

#10596831

C₂×C₂²⋊C₄

yes

characteristic

#10596831

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

M₆₄

yes

complemented

C₄×C₇₃

C₂²⋊C₁₆

yes

—

C₄×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

#13553

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

—

D₇₃

—

C₂³×C₁₆

—

C₂×M₆₄

—

C₂²⋊C₃₂

—

C₇₃

#13553

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

146

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂²×C₈

#10596842

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

146

—

103

C₂×C₁₆

—

146

—

104

C₂×C₁₆

—

146

—

105

C₃₂

—

146

—

106

C₂⁴

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

—

146

—

112

C₂²×C₄

C₂×#10596842

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₂×C₈

C₂×#10596842

yes

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

146

—

125

C₂³

—

126

C₂³

—

146

—

127

C₂³

—

146

—

128

C₂³

#14556686

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

C₂²⋊#10596842

yes

—

135

C₂×C₄

#14556686

yes

characteristic

136

C₂×C₄

#14556688

yes

—

137

C₂×C₄

C₂²×#10596842

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

146

—

145

C₂²

—

146

—

146

C₂²

—

146

—

147

C₂²

#17989684

yes

—

148

C₂²

C₂×#14556686

yes

characteristic

149

C₄

—

146

—

150

C₄

#17989684

yes

—

151

C₄

C₂×C₂²⋊#10596842

yes

characteristic

152

C₄

C₂×#14556688

yes

characteristic

153

C₂

—

154

C₂

—

146

—

155

C₂

C₂×#17989684

yes

characteristic

156

C₂

#22627162

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂²×C₈×C₇₃	C₈
prime radical	C₂³×D₇₃	C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	#13553
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂²×C₈×C₇₃

C₈

prime radical

C₂³×D₇₃

C₁₆

C₂³

C₂²×C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
1. C₂×#17989695	C₂	semidirect
2. #22627024	C₂	semidirect
5. C₂×#14556681	C₂²	semidirect
69. C₂×D₇₃	M₆₄	semidirect
82. C₇₃	#13553	semidirect

H ≅ G/N

type

1. C₂×#17989695

C₂

semidirect

2. #22627024

C₂

semidirect

5. C₂×#14556681

C₂²

semidirect

69. C₂×D₇₃

M₆₄

semidirect

82. C₇₃

#13553

semidirect

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

subgroup	index
C₂×#17989695	2
#22627024	2
C₂²×#14556664	2
#13553	73

subgroup

index

Subgroups — #34034406

Conjugacy classes of subgroups

Complements

Maximal subgroups