Group #34034210 — subgroups

Subgroups — #34034210

Order 18688.

342 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 101 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	#22627121	C₂	1	2	yes	maximal
2	#22627006	C₂	1	4	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556688	C₂²	1	4	yes	—
5	#17989690	—	2	4	no	—
6	C₄×#11924036	C₂²	1	1	yes	characteristic
7	C₂×C₈×#64408	C₄	1	2	yes	—
8	C₂×C₈×#64408	C₂²	1	2	yes	—
9	C₄²×#64408	C₄	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
11	#14556688	—	2	8	no	—
12	C₂×#11924040	—	2	4	no	—
13	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
14	C₂×#11924036	C₂³	1	1	yes	characteristic
15	C₂×C₄×#64408	—	2	2	no	—
16	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
17	C₄×#10596833	D₄	1	2	yes	—
18	C₄×#10596833	Q₈	1	2	yes	—
19	C₈×#64408	—	2	4	no	—
20	C₈×#64408	—	2	4	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
22	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₈	1	1	yes	characteristic
24	C₈×Dic₇₃	C₈	1	2	yes	—
25	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
26	#11924040	—	4	4	no	—
27	C₂²×#64408	—	2	2	no	—
28	#11924036	—	2	2	no	—
29	#11924036	#41	1	4	yes	—
30	C₂×#10596833	C₄⋊C₄	1	2	yes	—
31	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
32	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
33	C₄×#64408	—	2	8	no	—
34	C₄×#64408	—	2	2	no	—
35	C₄×#64408	M₁₆	1	4	yes	—
36	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
37	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×#10596831	C₂×C₈	1	2	yes	—
39	C₈×D₇₃	—	2	2	no	—
40	C₈×D₇₃	—	2	2	no	—
41	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
46	C₂×#64408	—	2	8	no	—
47	C₂×#64408	—	2	4	no	—
48	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
51	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×D₇₃	—	2	4	no	—
53	C₄×D₇₃	#117	1	2	yes	—
54	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
55	#10596833	—	2	2	no	—
56	#10596833	#124	1	2	yes	—
57	#10596833	#125	1	2	yes	—
58	C₈×C₇₃	—	2	2	no	—
59	#10596831	—	2	2	no	—
60	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₂×D₇₃	—	2	2	no	—
62	C₂×D₇₃	#432	1	2	yes	—
63	#64408	—	2	8	no	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	#431	1	2	yes	—
66	Dic₇₃	#364	1	4	yes	—
67	Dic₇₃	#382	1	1	yes	characteristic
68	Dic₇₃	#423	1	1	yes	characteristic
69	#12883	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	#1831	1	1	yes	characteristic
71	C₂×C₇₃	C₂×#364	1	2	yes	—
72	D₇₃	—	2	2	no	—
73	#2150	—	73	2	no	—
74	C₄²×C₈	—	73	1	no	—
75	#1376	—	73	4	no	—
76	C₇₃	#12883	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	73	4	no	—
78	C₂×C₄×C₈	—	73	1	no	—
79	C₂×C₄×C₈	—	73	1	no	—
80	C₂×C₄×C₈	—	73	2	no	—
81	C₂×C₄×C₈	—	73	2	no	—
82	C₄³	—	73	1	no	—
83	C₄⋊C₁₆	—	146	4	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₂²×C₈	—	73	1	no	—
86	C₂²×C₈	—	73	2	no	—
87	C₂×C₄²	—	73	1	no	—
88	C₂×C₄²	—	73	2	no	—
89	C₂×C₄²	—	146	2	no	—
90	M₃₂	—	146	8	no	—
91	C₂×C₁₆	—	146	4	no	—
92	C₄×C₈	—	73	2	no	—
93	C₄×C₈	—	73	2	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	146	4	no	—
97	C₄×C₈	—	146	4	no	—
98	C₄×C₈	#10596842	1	1	yes	characteristic
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	73	2	no	—
101	C₂²×C₄	—	146	2	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	4	no	—
105	C₂×C₈	—	73	2	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	—	146	2	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	C₂×#10596842	1	2	yes	—
111	C₄²	—	73	2	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	4	no	—
114	C₄²	—	146	2	no	—
115	C₄²	—	146	8	no	—
116	C₄²	C₂×#10596842	1	1	yes	characteristic
117	C₁₆	—	292	4	no	—
118	C₂³	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	2	no	—
121	C₂×C₄	—	73	2	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	146	4	no	—
124	C₂×C₄	—	146	4	no	—
125	C₂×C₄	—	146	8	no	—
126	C₂×C₄	C₄⋊#10596842	1	2	yes	—
127	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	73	2	no	—
130	C₈	—	73	2	no	—
131	C₈	—	146	2	no	—
132	C₈	—	146	2	no	—
133	C₂²	—	73	2	no	—
134	C₂²	—	146	2	no	—
135	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	73	1	no	—
138	C₄	—	73	1	no	—
139	C₄	—	73	4	no	—
140	C₄	—	146	8	no	—
141	C₄	#17989700	1	2	yes	—
142	C₂	—	146	2	no	—
143	C₂	#22627073	1	1	yes	characteristic
144	C₂	#22627125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22627121

C₂

yes

maximal

#22627006

C₂

yes

maximal

C₄×C₈×#64408

C₂

yes

maximal, characteristic

C₂×#14556688

C₂²

yes

—

#17989690

—

C₄×#11924036

C₂²

yes

characteristic

C₂×C₈×#64408

C₄

yes

—

C₂×C₈×#64408

C₂²

yes

—

C₄²×#64408

C₄

yes

characteristic

C₄×C₈×D₇₃

C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#11924036

C₂³

yes

characteristic

C₂×C₄×#64408

—

C₂×C₄×#64408

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

—

C₄×#10596833

Q₈

yes

—

C₈×#64408

—

C₈×#64408

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

—

#11924036

—

#11924036

#41

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×#10596833

C₂×D₄

yes

characteristic

C₂×#10596833

C₂×Q₈

yes

—

—

M₁₆

yes

—

C₂×C₄×D₇₃

M₁₆

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×M₁₆

yes

characteristic

C₂×#64408

—

C₂×#64408

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#117

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

#10596833

—

#10596833

#124

yes

—

#10596833

#125

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#432

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

—

Dic₇₃

#364

yes

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2150

—

C₄²×C₈

—

#1376

—

C₇₃

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₄×C₈

#10596842

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

146

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₂×C₈

C₂×#10596842

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

146

—

115

C₄²

—

146

—

116

C₄²

C₂×#10596842

yes

characteristic

117

C₁₆

—

292

—

118

C₂³

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

C₄⋊#10596842

yes

—

127

C₂×C₄

C₂²×#10596842

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

146

—

132

C₈

—

146

—

133

C₂²

—

134

C₂²

—

146

—

135

C₂²

C₂×C₄⋊#10596842

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

146

—

141

C₄

#17989700

yes

—

142

C₂

—

146

—

143

C₂

#22627073

yes

characteristic

144

C₂

#22627125

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10596842
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₂×M₁₆
cosocle	C₂×#11924036	C₂³
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#12883
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#10596842

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂²×D₇₃

C₂×M₁₆

C₂³

C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
76. C₇₃	#12883	semidirect

H ≅ G/N

type

76. C₇₃

#12883

semidirect

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

subgroup	index
#22627121	2
#22627006	2
C₄×C₈×#64408	2
#12883	73

subgroup

index

#22627121

#22627006

C₄×C₈×#64408

#12883

Subgroups — #34034210

Conjugacy classes of subgroups

Complements

Maximal subgroups