Group #33999900 — subgroups

Subgroups — #33999900

Order 18688.

306 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 92 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	#22625858	C₂	1	2	yes	maximal
2	C₄×#14556501	C₂	1	1	yes	maximal, characteristic
3	#17989453	—	2	2	no	—
4	#17989453	—	2	2	no	—
5	#17989453	C₄	1	4	yes	—
6	C₂×#14556501	—	2	2	no	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂²×#10596833	—	2	4	no	—
11	C₂²×#10596831	—	2	1	no	—
12	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
13	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
14	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
16	#14556501	—	2	8	no	—
17	C₄×#10596831	—	2	1	no	—
18	C₄×#10596831	C₈	1	2	yes	complemented
19	C₂²×C₈×C₇₃	—	2	1	no	—
20	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₈	1	2	yes	complemented
23	C₂×#10596833	—	4	4	no	—
24	C₂×#10596833	—	4	4	no	—
25	C₂×#10596833	—	4	4	no	—
26	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
27	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
28	C₂×#10596831	—	2	4	no	—
29	C₂×#10596831	—	2	2	no	—
30	C₄×Dic₇₃	—	2	4	no	—
31	C₄×Dic₇₃	—	4	2	no	—
32	C₄×Dic₇₃	—	4	1	no	—
33	C₄×Dic₇₃	—	4	1	no	—
34	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	—	2	4	no	—
38	C₂×C₈×C₇₃	—	2	2	no	—
39	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
42	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
43	C₂³×C₇₃	#95	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	2	2	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	2	2	no	—
58	C₂×Dic₇₃	—	2	2	no	—
59	C₂×Dic₇₃	#104	1	4	yes	—
60	#10596833	—	8	4	no	—
61	C₈×C₇₃	—	2	4	no	—
62	#10596831	—	2	4	no	—
63	C₂²×C₇₃	#336	1	1	yes	characteristic
64	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
65	C₂²×C₇₃	#338	1	1	yes	characteristic
66	C₂²×C₇₃	#339	1	2	yes	—
67	C₂²×C₇₃	#344	1	2	yes	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	Dic₇₃	—	2	4	no	—
75	Dic₇₃	—	4	2	no	—
76	#7019	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	#1275	1	1	yes	characteristic
78	C₂×C₇₃	#1276	1	1	yes	characteristic
79	C₂×C₇₃	#1281	1	1	yes	characteristic
80	C₂×C₇₃	#1288	1	2	yes	—
81	C₂×C₇₃	#1291	1	2	yes	—
82	#1839	—	73	2	no	—
83	C₄×#97	—	73	1	no	—
84	C₇₃	#7019	1	1	yes	characteristic, complemented, Sylow
85	C₂×#97	—	146	2	no	—
86	C₂×C₄×C₈	—	73	1	no	—
87	C₂×C₄×C₈	#64408	1	1	yes	characteristic
88	C₄³	—	73	1	no	—
89	#336	—	73	4	no	—
90	#336	—	146	2	no	—
91	#336	—	146	2	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	146	4	no	—
94	C₂²×C₈	—	146	1	no	—
95	C₂×C₄²	—	73	2	no	—
96	C₂×C₄²	—	73	1	no	—
97	C₂×C₄²	—	73	1	no	—
98	C₂×C₄²	—	73	2	no	—
99	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
100	#97	—	146	8	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	73	2	no	—
103	C₄×C₈	—	146	1	no	—
104	C₄×C₈	#10596833	1	2	yes	complemented
105	C₂²×C₄	—	73	2	no	—
106	C₂²×C₄	—	73	2	no	—
107	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
108	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	292	4	no	—
113	C₂×C₈	—	292	4	no	—
114	C₂×C₈	—	292	4	no	—
115	C₂×C₈	—	146	4	no	—
116	C₂×C₈	—	146	2	no	—
117	C₄²	—	292	1	no	—
118	C₄²	—	292	1	no	—
119	C₄²	—	292	2	no	—
120	C₄²	—	146	4	no	—
121	C₄²	C₂×#10596833	1	1	yes	characteristic
122	C₄²	#11924036	1	1	yes	characteristic
123	C₄²	#11924037	1	1	yes	characteristic
124	C₄²	#11924038	1	1	yes	characteristic
125	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	73	4	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
137	C₂×C₄	#14556646	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
139	C₂×C₄	#14556648	1	1	yes	characteristic
140	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
141	C₂×C₄	#14556658	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	584	4	no	—
144	C₈	—	146	4	no	—
145	C₂²	#17989453	1	1	yes	characteristic
146	C₂²	#17989455	1	1	yes	characteristic
147	C₂²	#17989456	1	1	yes	characteristic
148	C₂²	#17989458	1	2	yes	—
149	C₂²	#17989469	1	2	yes	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	292	2	no	—
157	C₄	—	146	4	no	—
158	C₂	#22624757	1	1	yes	characteristic
159	C₂	#22624759	1	1	yes	characteristic
160	C₂	#22624771	1	1	yes	characteristic
161	C₂	#22624791	1	2	yes	—
162	C₂	#22624798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625858

C₂

yes

maximal

C₄×#14556501

C₂

yes

maximal, characteristic

#17989453

—

#17989453

—

#17989453

C₄

yes

—

C₂×#14556501

—

C₂×C₄×#10596831

C₄

yes

characteristic

C₄²×Dic₇₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

C₂²×#10596833

—

C₂²×#10596831

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×C₄×Dic₇₃

Q₈

yes

—

—

C₈

yes

complemented

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₈

yes

—

—

—

C₂²×Dic₇₃

C₂²⋊C₄

yes

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#104

yes

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#344

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#1291

yes

—

#1839

—

C₄×#97

—

C₇₃

#7019

yes

characteristic, complemented, Sylow

C₂×#97

—

146

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄³

—

#336

—

#336

—

146

—

#336

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

100

#97

—

146

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

146

—

104

C₄×C₈

#10596833

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×#64408

yes

characteristic

108

C₂²×C₄

C₂²⋊#64408

yes

characteristic

109

C₂²×C₄

C₄⋊#64408

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

292

—

113

C₂×C₈

—

292

—

114

C₂×C₈

—

292

—

115

C₂×C₈

—

146

—

116

C₂×C₈

—

146

—

117

C₄²

—

292

—

118

C₄²

—

292

—

119

C₄²

—

292

—

120

C₄²

—

146

—

121

C₄²

C₂×#10596833

yes

characteristic

122

C₄²

#11924036

yes

characteristic

123

C₄²

#11924037

yes

characteristic

124

C₄²

#11924038

yes

characteristic

125

C₂³

(C₂×#64408)⋊C₄

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

C₄×#10596833

yes

characteristic

137

C₂×C₄

#14556646

yes

characteristic

138

C₂×C₄

C₂²⋊#10596833

yes

characteristic

139

C₂×C₄

#14556648

yes

characteristic

140

C₂×C₄

C₄⋊#10596833

yes

characteristic

141

C₂×C₄

#14556658

yes

characteristic

142

C₈

—

143

C₈

—

584

—

144

C₈

—

146

—

145

C₂²

#17989453

yes

characteristic

146

C₂²

#17989455

yes

characteristic

147

C₂²

#17989456

yes

characteristic

148

C₂²

#17989458

yes

—

149

C₂²

#17989469

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

292

—

157

C₄

—

146

—

158

C₂

#22624757

yes

characteristic

159

C₂

#22624759

yes

characteristic

160

C₂

#22624771

yes

characteristic

161

C₂

#22624791

yes

—

162

C₂

#22624798

yes

—

	subgroup	G/H
center	C₂³	(C₂×#64408)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	#95
cosocle	C₄²×Dic₇₃	C₂²
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#7019
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

(C₂×#64408)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

#95

cosocle

C₄²×Dic₇₃

C₂²

C₂×C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
18. C₄×#10596831	C₈	semidirect
22. C₄×C₈×C₇₃	C₈	semidirect
84. C₇₃	#7019	semidirect
104. C₄×C₈	#10596833	semidirect

H ≅ G/N

type

18. C₄×#10596831

C₈

semidirect

22. C₄×C₈×C₇₃

C₈

semidirect

84. C₇₃

#7019

semidirect

104. C₄×C₈

#10596833

semidirect

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

subgroup	index
#22625858	2
C₄×#14556501	2
#7019	73

subgroup

index

#22625858

C₄×#14556501

#7019

Subgroups — #33999900

Conjugacy classes of subgroups

Complements

Maximal subgroups