Group #34034222 — subgroups

Subgroups — #34034222

Order 18688.

382 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 133 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×#17989692	C₂	1	4	yes	maximal
2	#22627120	C₂	1	2	yes	maximal
3	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556688	C₂²	1	4	yes	—
5	#17989692	—	2	8	no	—
6	C₄⋊#11924040	—	2	4	no	—
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	C₂×C₈×#64408	C₄	1	2	yes	—
9	C₂×C₈×#64408	C₂²	1	2	yes	—
10	C₄²×#64408	C₄	1	1	yes	characteristic
11	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
12	#14556688	—	2	8	no	—
13	C₂×#11924040	—	2	4	no	—
14	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
15	C₂×#11924036	C₂³	1	1	yes	characteristic
16	C₂×C₄×#64408	—	2	2	no	—
17	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
18	C₄×#10596833	D₄	1	2	yes	—
19	C₄×#10596833	Q₈	1	2	yes	—
20	C₈×#64408	—	2	4	no	—
21	C₈×#64408	D₄	1	4	yes	—
22	C₈×#64408	Q₈	1	4	yes	—
23	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₈	1	2	yes	—
27	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
28	#11924040	—	4	4	no	—
29	C₂²×#64408	—	2	2	no	—
30	#11924036	—	2	2	no	—
31	#11924036	C₂×D₄	1	2	yes	—
32	#11924036	C₂×Q₈	1	2	yes	—
33	C₂×#10596833	C₄⋊C₄	1	2	yes	—
34	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
35	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
36	C₄×#64408	—	2	8	no	—
37	C₄×#64408	—	2	2	no	—
38	C₄×#64408	C₄⋊C₄	1	4	yes	—
39	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
40	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×#10596831	C₂×C₈	1	2	yes	—
42	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
43	C₈×D₇₃	M₁₆	1	4	yes	—
44	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
45	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
47	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
49	C₂×#64408	—	2	8	no	—
50	C₂×#64408	—	2	4	no	—
51	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
52	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
54	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×D₇₃	—	2	4	no	—
56	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
58	#10596833	—	2	2	no	—
59	#10596833	#127	1	1	yes	characteristic
60	#10596833	C₄⋊Q₈	1	1	yes	characteristic
61	#10596833	C₄⋊Q₈	1	2	yes	—
62	C₈×C₇₃	C₄⋊C₈	1	4	yes	—
63	#10596831	C₄⋊C₈	1	4	yes	—
64	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	#423	1	2	yes	—
67	#64408	—	2	8	no	—
68	C₄×C₇₃	#364	1	4	yes	complemented
69	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
70	Dic₇₃	—	2	2	no	—
71	Dic₇₃	#384	1	1	yes	characteristic
72	Dic₇₃	#423	1	1	yes	characteristic
73	#12900	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	#1842	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#364	1	2	yes	—
76	D₇₃	—	2	2	no	—
77	C₂×#364	—	73	4	no	—
78	#2150	—	73	2	no	—
79	C₄²×C₈	—	73	1	no	—
80	C₇₃	#12900	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	73	4	no	—
82	C₂×C₄×C₈	—	73	1	no	—
83	C₂×C₄×C₈	—	73	2	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	C₂×C₄×C₈	—	73	2	no	—
86	C₄³	—	73	1	no	—
87	#364	—	146	8	no	—
88	C₄⋊C₁₆	—	146	4	no	—
89	C₂²×C₈	—	73	1	no	—
90	C₂²×C₈	—	73	2	no	—
91	C₂²×C₈	—	73	1	no	—
92	C₂×C₄²	—	73	1	no	—
93	C₂×C₄²	—	73	2	no	—
94	C₂×C₄²	—	146	2	no	—
95	M₃₂	—	146	8	no	—
96	C₂×C₁₆	—	146	4	no	—
97	C₄×C₈	—	73	2	no	—
98	C₄×C₈	—	73	4	no	—
99	C₄×C₈	—	73	4	no	—
100	C₄×C₈	—	73	2	no	—
101	C₄×C₈	—	73	2	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	146	4	no	—
104	C₄×C₈	#10596842	1	1	yes	characteristic
105	C₂²×C₄	—	73	1	no	—
106	C₂²×C₄	—	73	2	no	—
107	C₂²×C₄	—	146	2	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	146	2	no	—
117	C₂×C₈	C₂×#10596842	1	2	yes	—
118	C₄²	—	73	2	no	—
119	C₄²	—	73	1	no	—
120	C₄²	—	73	4	no	—
121	C₄²	—	146	2	no	—
122	C₄²	—	146	8	no	—
123	C₄²	C₂×#10596842	1	1	yes	characteristic
124	C₁₆	—	292	4	no	—
125	C₂³	—	73	1	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	146	4	no	—
131	C₂×C₄	—	146	8	no	—
132	C₂×C₄	—	146	4	no	—
133	C₂×C₄	C₄⋊#10596842	1	2	yes	—
134	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
135	C₈	—	73	1	no	—
136	C₈	—	73	1	no	—
137	C₈	—	73	2	no	—
138	C₈	—	73	4	no	—
139	C₈	—	146	2	no	—
140	C₈	C₄⋊#10596842	1	4	yes	—
141	C₂²	—	73	2	no	—
142	C₂²	—	146	2	no	—
143	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
144	C₄	—	73	1	no	—
145	C₄	—	73	1	no	—
146	C₄	—	146	8	no	—
147	C₄	—	146	2	no	—
148	C₄	#17989692	1	4	yes	complemented
149	C₄	C₂×C₄⋊#10596842	1	2	yes	—
150	C₂	—	146	2	no	—
151	C₂	#22627075	1	1	yes	characteristic
152	C₂	C₂×#17989692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989692

C₂

yes

maximal

#22627120

C₂

yes

maximal

C₄×C₈×#64408

C₂

yes

maximal, characteristic

C₂×#14556688

C₂²

yes

—

#17989692

—

C₄⋊#11924040

—

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

D₄

yes

—

C₈×#64408

Q₈

yes

—

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

C₂×D₄

yes

—

#11924036

C₂×Q₈

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×#10596833

C₂×D₄

yes

characteristic

C₂×#10596833

C₂×Q₈

yes

—

—

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

M₁₆

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

C₄⋊C₄

yes

—

C₈×D₇₃

M₁₆

yes

—

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

C₂×C₄⋊C₄

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

—

yes

C₄⋊Q₈

yes

characteristic

#10596833

C₄⋊Q₈

yes

—

C₈×C₇₃

C₄⋊C₈

yes

—

#10596831

C₄⋊C₈

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#423

yes

—

#64408

—

C₄×C₇₃

#364

yes

complemented

C₄×C₇₃

C₂×C₄⋊C₈

yes

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₂×#364

—

#2150

—

C₄²×C₈

—

C₇₃

#12900

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

#364

—

146

—

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

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

146

—

104

C₄×C₈

#10596842

yes

characteristic

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

146

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

146

—

117

C₂×C₈

C₂×#10596842

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

146

—

122

C₄²

—

146

—

123

C₄²

C₂×#10596842

yes

characteristic

124

C₁₆

—

292

—

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

C₄⋊#10596842

yes

—

134

C₂×C₄

C₂²×#10596842

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

146

—

140

C₈

C₄⋊#10596842

yes

—

141

C₂²

—

142

C₂²

—

146

—

143

C₂²

C₂×C₄⋊#10596842

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

146

—

147

C₄

—

146

—

148

C₄

#17989692

yes

complemented

149

C₄

C₂×C₄⋊#10596842

yes

—

150

C₂

—

146

—

151

C₂

#22627075

yes

characteristic

152

C₂

C₂×#17989692

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	#12900
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
68. C₄×C₇₃	#364	semidirect
80. C₇₃	#12900	semidirect
148. C₄	#17989692	semidirect

H ≅ G/N

type

68. C₄×C₇₃

#364

semidirect

80. C₇₃

#12900

semidirect

148. C₄

#17989692

semidirect

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

subgroup	index
C₂×#17989692	2
#22627120	2
C₄×C₈×#64408	2
#12900	73

subgroup

index

C₂×#17989692

#22627120

C₄×C₈×#64408

#12900

Subgroups — #34034222

Conjugacy classes of subgroups

Complements

Maximal subgroups