Group #34009718 — subgroups

Subgroups — #34009718

Order 18688.

306 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 152 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	#22626290	C₂	1	2	yes	maximal
2	#22624846	C₂	1	2	yes	maximal, complemented
3	C₈×#11924031	C₂	1	2	yes	maximal, complemented
4	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989508	C₄	1	4	yes	—
6	#17989482	C₄	1	4	yes	—
7	C₄×#11924031	C₂²	1	4	yes	—
8	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₈×#10596831	C₄	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	C₈×#10596831	C₂²	1	1	yes	characteristic
14	C₈²×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	C₂×C₄	1	8	yes	—
16	C₂×C₈×D₇₃	C₈	1	2	yes	—
17	C₂×C₈×D₇₃	C₈	1	2	yes	—
18	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
19	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₂³	1	1	yes	characteristic
23	C₈×Dic₇₃	C₈	1	2	yes	—
24	C₈×Dic₇₃	C₈	1	2	yes	—
25	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924031	—	2	8	no	—
30	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
31	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×#10596831	C₂×C₈	1	2	yes	—
34	C₂×#10596831	C₂×C₈	1	2	yes	—
35	C₂×#10596831	C₂²×C₄	1	2	yes	—
36	C₈×D₇₃	—	2	4	no	—
37	C₈×D₇₃	—	2	4	no	—
38	C₈×D₇₃	C₁₆	1	8	yes	complemented
39	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
40	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
41	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄²	1	2	yes	—
43	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
45	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
52	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	—	2	2	no	—
55	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	C₂×C₁₆	1	4	yes	—
59	#10596831	C₂×C₁₆	1	4	yes	—
60	#10596831	M₃₂	1	4	yes	—
61	#10596831	#131	1	4	yes	—
62	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂×D₇₃	—	2	1	no	—
64	C₂×D₇₃	—	2	1	no	—
65	C₂×D₇₃	C₄×C₁₆	1	2	yes	—
66	C₄×C₇₃	#405	1	2	yes	—
67	C₄×C₇₃	C₂²×C₁₆	1	2	yes	—
68	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
69	Dic₇₃	—	2	1	no	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	C₄×C₁₆	1	2	yes	—
72	#11410	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#1450	1	1	yes	characteristic
74	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
75	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
76	D₇₃	—	2	2	no	—
77	#2162	—	73	2	no	—
78	C₂×C₈²	—	73	1	no	—
79	#1312	—	73	2	no	—
80	C₈×C₁₆	—	73	2	no	—
81	C₇₃	#11410	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄×C₈	—	73	1	no	—
83	C₂×C₄×C₈	—	73	1	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	C₄⋊C₁₆	—	73	4	no	—
86	C₂²⋊C₁₆	—	73	4	no	—
87	C₄×C₁₆	—	73	4	no	—
88	C₈²	—	73	1	no	—
89	C₈²	—	73	1	no	—
90	C₈²	—	73	1	no	—
91	C₈²	#64408	1	1	yes	characteristic
92	C₂²×C₈	—	73	2	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₂×C₄²	—	73	1	no	—
96	C₂×C₁₆	—	73	8	no	—
97	C₄×C₈	—	73	2	no	—
98	C₄×C₈	—	73	2	no	—
99	C₄×C₈	—	73	2	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	C₂×#64408	1	1	yes	characteristic
104	C₄×C₈	C₂×#64408	1	1	yes	characteristic
105	C₄×C₈	C₂×#64408	1	1	yes	characteristic
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂×C₈	—	73	2	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	73	8	no	—
113	C₂×C₈	—	146	4	no	—
114	C₂×C₈	—	146	4	no	—
115	C₂×C₈	C₄×#64408	1	2	yes	—
116	C₂×C₈	#11924036	1	2	yes	—
117	C₂×C₈	#11924036	1	2	yes	—
118	C₄²	—	73	1	no	—
119	C₄²	—	73	1	no	—
120	C₄²	—	73	1	no	—
121	C₄²	C₂²×#64408	1	1	yes	characteristic
122	C₁₆	—	146	8	no	—
123	C₂³	—	73	1	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	4	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
131	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
132	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	73	4	no	—
136	C₈	—	73	4	no	—
137	C₈	—	73	4	no	—
138	C₈	#14556664	1	4	yes	—
139	C₂²	—	73	2	no	—
140	C₂²	—	146	1	no	—
141	C₂²	—	146	1	no	—
142	C₂²	C₄×#11924036	1	1	yes	characteristic
143	C₄	—	73	2	no	—
144	C₄	—	146	1	no	—
145	C₄	—	146	1	no	—
146	C₄	#17989550	1	2	yes	—
147	C₄	C₂×#14556664	1	2	yes	—
148	C₄	#17989643	1	2	yes	—
149	C₂	—	146	2	no	—
150	C₂	#22625135	1	1	yes	characteristic
151	C₂	C₄×#14556664	1	1	yes	characteristic
152	C₂	#22626138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626290

C₂

yes

maximal

#22624846

C₂

yes

maximal, complemented

C₈×#11924031

C₂

yes

maximal, complemented

C₈²×D₇₃

C₂

yes

maximal, characteristic

#17989508

C₄

yes

—

#17989482

C₄

yes

—

C₄×#11924031

C₂²

yes

—

C₄×C₈×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

C₈²×C₇₃

C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₈

yes

—

C₂×C₈×D₇₃

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

C₈

yes

—

C₈×Dic₇₃

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

C₁₆

yes

complemented

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

—

#10596831

C₂×C₁₆

yes

—

#10596831

M₃₂

yes

—

#10596831

#131

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₄×C₁₆

yes

—

C₄×C₇₃

#405

yes

—

C₄×C₇₃

C₂²×C₁₆

yes

—

C₄×C₇₃

C₂×M₃₂

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₄×C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×C₄×C₁₆

yes

characteristic

C₂×C₇₃

C₄×M₃₂

yes

characteristic

D₇₃

—

#2162

—

C₂×C₈²

—

#1312

—

C₈×C₁₆

—

C₇₃

#11410

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#64408

yes

characteristic

104

C₄×C₈

C₂×#64408

yes

characteristic

105

C₄×C₈

C₂×#64408

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

146

—

114

C₂×C₈

—

146

—

115

C₂×C₈

C₄×#64408

yes

—

116

C₂×C₈

#11924036

yes

—

117

C₂×C₈

#11924036

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂²×#64408

yes

characteristic

122

C₁₆

—

146

—

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

146

—

129

C₂×C₄

—

146

—

130

C₂×C₄

C₂×C₄×#64408

yes

characteristic

131

C₂×C₄

C₂×#11924036

yes

characteristic

132

C₂×C₄

C₂×#11924036

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#14556664

yes

—

139

C₂²

—

140

C₂²

—

146

—

141

C₂²

—

146

—

142

C₂²

C₄×#11924036

yes

characteristic

143

C₄

—

144

C₄

—

146

—

145

C₄

—

146

—

146

C₄

#17989550

yes

—

147

C₄

C₂×#14556664

yes

—

148

C₄

#17989643

yes

—

149

C₂

—

146

—

150

C₂

#22625135

yes

characteristic

151

C₂

C₄×#14556664

yes

characteristic

152

C₂

#22626138

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#11410
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₄×C₈

C₂×#64408

commutator

C₂×C₇₃

C₂×C₄×C₁₆

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄×C₈

prime radical

C₂²×D₇₃

C₄×C₈

C₂³

C₈²

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. #22624846	C₂	semidirect
3. C₈×#11924031	C₂	semidirect
38. C₈×D₇₃	C₁₆	semidirect
81. C₇₃	#11410	semidirect

H ≅ G/N

type

2. #22624846

C₂

semidirect

3. C₈×#11924031

C₂

semidirect

38. C₈×D₇₃

C₁₆

semidirect

81. C₇₃

#11410

semidirect

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

subgroup	index
#22626290	2
#22624846	2
C₈×#11924031	2
C₈²×D₇₃	2
#11410	73

subgroup

index

C₈²×D₇₃

Subgroups — #34009718

Conjugacy classes of subgroups

Complements

Maximal subgroups