Group #34032773 — subgroups

Subgroups — #34032773

Order 18688.

200 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 45 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	#22626964	C₂	1	2	yes	maximal
2	#22625863	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11924040	—	2	2	no	—
4	C₂×C₄⋊#10596833	—	2	2	no	—
5	C₂×C₄⋊#10596833	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#10596833	C₂²	1	1	yes	characteristic
7	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
8	C₂×#11924040	—	4	2	no	—
9	C₂²×#10596833	—	2	1	no	—
10	C₂²×#10596833	D₄	1	2	yes	—
11	C₄⋊#10596833	—	4	1	no	—
12	C₄⋊#10596833	—	4	4	no	—
13	C₄⋊#10596833	—	4	1	no	—
14	C₂×C₄×Dic₇₃	—	2	1	no	—
15	C₂×C₄×Dic₇₃	—	2	1	no	—
16	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
19	#11924040	—	8	2	no	—
20	C₂×#10596833	—	2	2	no	—
21	C₂×#10596833	—	4	2	no	—
22	C₂×#10596833	—	4	2	no	—
23	C₂×#10596833	—	4	1	no	—
24	C₂²×Dic₇₃	—	2	1	no	—
25	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
27	C₄×Dic₇₃	—	2	2	no	—
28	C₄×Dic₇₃	—	2	2	no	—
29	C₄×Dic₇₃	—	2	1	no	—
30	C₄×Dic₇₃	—	2	1	no	—
31	C₄×Dic₇₃	—	4	2	no	—
32	C₄×Dic₇₃	—	4	1	no	—
33	C₂²×C₄×C₇₃	—	2	1	no	—
34	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₇₃	—	2	1	no	—
36	C₄²×C₇₃	C₁₆	1	2	yes	complemented
37	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₇₃	—	2	2	no	—
39	C₂×C₄×C₇₃	—	2	2	no	—
40	C₂×C₄×C₇₃	—	2	1	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
45	C₂×Dic₇₃	—	2	1	no	—
46	C₂×Dic₇₃	—	2	1	no	—
47	C₂×Dic₇₃	—	2	2	no	—
48	C₂×Dic₇₃	—	2	1	no	—
49	C₂×Dic₇₃	—	4	1	no	—
50	C₂×Dic₇₃	—	4	1	no	—
51	C₂×Dic₇₃	—	4	1	no	—
52	C₂×Dic₇₃	#99	1	1	yes	characteristic
53	C₂×Dic₇₃	#101	1	1	yes	characteristic
54	#10596833	—	4	2	no	—
55	#10596833	—	8	1	no	—
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	#324	1	1	yes	characteristic
59	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
60	C₂²×C₇₃	#349	1	1	yes	characteristic
61	C₄×C₇₃	—	2	4	no	—
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	2	1	no	—
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	4	2	no	—
66	Dic₇₃	—	4	1	no	—
67	Dic₇₃	#353	1	1	yes	characteristic
68	Dic₇₃	#356	1	1	yes	characteristic
69	#7063	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	#1322	1	1	yes	characteristic
73	C₂×C₇₃	#1325	1	1	yes	characteristic
74	C₂×C₇₃	#1327	1	1	yes	characteristic
75	#1842	—	73	1	no	—
76	#1322	—	73	2	no	—
77	C₇₃	#7063	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄⋊C₈	—	73	1	no	—
79	C₂×C₄⋊C₈	—	73	1	no	—
80	C₂×C₄⋊C₈	—	146	2	no	—
81	C₄³	—	73	1	no	—
82	C₂²⋊C₁₆	—	146	2	no	—
83	C₂²×C₈	—	73	2	no	—
84	C₂²×C₈	—	146	1	no	—
85	C₂×C₄²	—	73	1	no	—
86	C₂×C₄²	—	73	1	no	—
87	C₂×C₄²	—	146	1	no	—
88	C₂×C₄²	—	146	1	no	—
89	C₂×C₄²	#10596842	1	1	yes	characteristic
90	C₂×C₁₆	—	292	2	no	—
91	C₄⋊C₈	—	292	1	no	—
92	C₄⋊C₈	—	292	4	no	—
93	C₄⋊C₈	—	292	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	73	1	no	—
96	C₂²×C₄	—	73	1	no	—
97	C₂²×C₄	—	146	1	no	—
98	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
99	C₂×C₈	—	292	2	no	—
100	C₂×C₈	—	292	1	no	—
101	C₂×C₈	—	292	2	no	—
102	C₂×C₈	—	146	2	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	292	1	no	—
105	C₄²	—	292	2	no	—
106	C₄²	—	146	1	no	—
107	C₄²	—	146	1	no	—
108	C₄²	—	146	2	no	—
109	C₄²	—	146	2	no	—
110	C₄²	#11924040	1	2	yes	complemented
111	C₁₆	—	584	2	no	—
112	C₂³	C₂²⋊#10596842	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	292	1	no	—
119	C₂×C₄	—	292	1	no	—
120	C₂×C₄	—	292	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	—	146	1	no	—
125	C₂×C₄	—	146	1	no	—
126	C₂×C₄	—	146	1	no	—
127	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
128	C₂×C₄	#14556687	1	1	yes	characteristic
129	C₈	—	292	2	no	—
130	C₈	—	584	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#17989673	1	1	yes	characteristic
134	C₂²	C₂²⋊#11924040	1	1	yes	characteristic
135	C₂²	#17989686	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	292	2	no	—
140	C₄	—	292	1	no	—
141	C₄	—	73	1	no	—
142	C₄	—	73	1	no	—
143	C₄	—	146	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#22626964	1	1	yes	characteristic
147	C₂	#22626968	1	1	yes	characteristic
148	C₂	#22626971	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626964

C₂

yes

maximal

#22625863

C₂

yes

maximal, characteristic

C₂²⋊#11924040

—

C₂×C₄⋊#10596833

—

C₂×C₄⋊#10596833

C₄

yes

characteristic

C₂×C₄⋊#10596833

C₂²

yes

characteristic

C₄²×Dic₇₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#10596833

—

C₄⋊#10596833

—

C₄⋊#10596833

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₈

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₈

yes

—

—

—

—

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₇₃

M₁₆

yes

characteristic

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₁₆

yes

complemented

C₂³×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#324

yes

characteristic

C₂²×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₂²×C₇₃

#349

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₇₃

#7063

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

C₄³

—

C₂²⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

C₂×C₄²

#10596842

yes

characteristic

C₂×C₁₆

—

292

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

292

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂×#10596842

yes

characteristic

C₂×C₈

—

292

—

100

C₂×C₈

—

292

—

101

C₂×C₈

—

292

—

102

C₂×C₈

—

146

—

103

C₄²

—

104

C₄²

—

292

—

105

C₄²

—

292

—

106

C₄²

—

146

—

107

C₄²

—

146

—

108

C₄²

—

146

—

109

C₄²

—

146

—

110

C₄²

#11924040

yes

complemented

111

C₁₆

—

584

—

112

C₂³

C₂²⋊#10596842

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

292

—

119

C₂×C₄

—

292

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

—

146

—

127

C₂×C₄

C₂×#11924040

yes

characteristic

128

C₂×C₄

#14556687

yes

characteristic

129

C₈

—

292

—

130

C₈

—

584

—

131

C₂²

—

132

C₂²

—

133

C₂²

#17989673

yes

characteristic

134

C₂²

C₂²⋊#11924040

yes

characteristic

135

C₂²

#17989686

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

292

—

140

C₄

—

292

—

141

C₄

—

142

C₄

—

143

C₄

—

146

—

144

C₂

—

145

C₂

—

146

C₂

#22626964

yes

characteristic

147

C₂

#22626968

yes

characteristic

148

C₂

#22626971

yes

characteristic

	subgroup	G/H
center	C₂²	#17989673
commutator	C₂×C₄×C₇₃	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#10596842
Fitting	C₂×C₄²×C₇₃	C₈
socle	C₂²×C₇₃	#324
prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄⋊#10596833	C₂²
hypercenter	C₂×C₄²	#10596842
hypocenter	C₇₃	#7063
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²

#17989673

commutator

C₂×C₄×C₇₃

C₂×C₁₆

Frattini

C₂²×C₄

C₂×#10596842

Fitting

C₂×C₄²×C₇₃

C₈

socle

C₂²×C₇₃

#324

prime radical

C₂³×C₇₃

C₂²⋊C₈

C₂²

C₂×C₄²

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
36. C₄²×C₇₃	C₁₆	semidirect
77. C₇₃	#7063	semidirect
110. C₄²	#11924040	semidirect

H ≅ G/N

type

36. C₄²×C₇₃

C₁₆

semidirect

77. C₇₃

#7063

semidirect

110. C₄²

#11924040

semidirect

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

subgroup	index
#22626964	2
#22625863	2
#7063	73

subgroup

index

#22626964

#22625863

#7063

Subgroups — #34032773

Conjugacy classes of subgroups

Complements

Maximal subgroups