Group #34000236 — subgroups

Subgroups — #34000236

Order 18688.

222 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 80 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	#22626292	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989475	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	—	2	1	no	—
5	C₄⋊#11924031	C₄	1	2	yes	—
6	C₂²⋊#11924031	—	2	1	no	—
7	#17989475	—	2	1	no	—
8	#17989475	—	2	1	no	—
9	#17989475	C₄	1	2	yes	—
10	#17988640	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	#17988308	—	2	1	no	—
13	C₈×#10596831	—	2	1	no	—
14	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924031	—	2	4	no	—
16	C₂×#11924031	—	2	2	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₂²×#10596831	D₄	1	1	yes	characteristic
19	C₂²×#10596831	Q₈	1	1	yes	characteristic
20	C₄⋊#10596831	—	2	1	no	—
21	C₂²⋊#10596831	C₈	1	2	yes	—
22	C₄×#10596831	—	2	1	no	—
23	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	D₄	1	1	yes	characteristic
26	C₄×#10596831	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₇₃	—	2	1	no	—
28	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#97×C₇₃	C₈	1	2	yes	—
30	C₄×C₈×C₇₃	—	2	1	no	—
31	#11924031	—	2	4	no	—
32	#11924031	—	4	2	no	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	—	2	2	no	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
43	M₁₆×C₇₃	—	2	4	no	—
44	C₂×C₈×C₇₃	—	2	2	no	—
45	C₄²×C₇₃	C₄²	1	1	yes	characteristic
46	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
49	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₇₃	—	2	4	no	—
61	#10596831	—	2	1	no	—
62	#10596831	—	2	1	no	—
63	#10596831	—	4	2	no	—
64	#10596831	#104	1	2	yes	—
65	#10596831	#108	1	2	yes	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	—	2	1	no	—
68	C₂²×C₇₃	#336	1	1	yes	characteristic
69	C₂²×C₇₃	#364	1	2	yes	—
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	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#339	1	1	yes	characteristic
76	C₄×C₇₃	#341	1	1	yes	characteristic
77	#7165	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	#1278	1	1	yes	characteristic
80	C₂×C₇₃	#1376	1	1	yes	characteristic
81	C₂×C₇₃	#1378	1	1	yes	characteristic
82	#2163	—	73	1	no	—
83	C₂×#346	—	73	1	no	—
84	C₈×M₁₆	—	73	1	no	—
85	C₇₃	#7165	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₁₆	—	146	1	no	—
87	#431	—	73	1	no	—
88	C₄×M₁₆	#64408	1	1	yes	characteristic
89	C₂×C₄×C₈	—	73	1	no	—
90	C₄⋊C₁₆	—	73	2	no	—
91	C₂²⋊C₁₆	—	146	1	no	—
92	#346	—	73	2	no	—
93	#346	—	146	1	no	—
94	#346	—	146	1	no	—
95	#322	—	146	1	no	—
96	C₈²	—	146	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
101	C₂×C₁₆	—	146	2	no	—
102	C₂×C₁₆	—	146	2	no	—
103	C₂×C₁₆	—	146	4	no	—
104	C₄⋊C₈	—	146	1	no	—
105	C₂²⋊C₈	—	73	2	no	—
106	#97	#10596833	1	2	yes	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	73	1	no	—
109	C₄×C₈	—	73	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	146	1	no	—
113	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
114	C₂²×C₄	#11924036	1	1	yes	characteristic
115	C₂²×C₄	#11924037	1	1	yes	characteristic
116	M₁₆	—	2	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	—	146	2	no	—
125	C₄²	C₄×#64408	1	1	yes	characteristic
126	C₄²	C₂²⋊#64408	1	1	yes	characteristic
127	C₄²	C₂×#10596833	1	1	yes	characteristic
128	C₄²	#11924038	1	1	yes	characteristic
129	C₁₆	—	292	2	no	—
130	C₁₆	—	146	4	no	—
131	C₂³	#14556658	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
138	C₂×C₄	#14556646	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
140	C₂×C₄	#14556648	1	1	yes	characteristic
141	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	292	2	no	—
144	C₈	—	73	2	no	—
145	C₈	—	73	2	no	—
146	C₈	—	146	1	no	—
147	C₈	—	146	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#17989453	1	1	yes	characteristic
151	C₂²	#17989510	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#17989459	1	1	yes	characteristic
158	C₄	#17989462	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#22624764	1	1	yes	characteristic
161	C₂	#22624975	1	1	yes	characteristic
162	C₂	#22624982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626292

C₂

yes

maximal, characteristic

C₂×#17989475

C₂

yes

maximal, characteristic

M₁₆×#10596831

C₂

yes

maximal, characteristic

C₂²×#11924031

—

C₄⋊#11924031

C₄

yes

—

—

—

—

C₄

yes

—

#17988640

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

characteristic

#17988308

—

C₈×#10596831

—

C₄×M₁₆×C₇₃

C₄

yes

—

—

—

D₄

yes

characteristic

C₂²×#10596831

Q₈

yes

characteristic

C₄⋊#10596831

—

C₂²⋊#10596831

C₈

yes

—

C₄×#10596831

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#97×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

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

C₈×C₇₃

—

—

—

—

yes

—

#10596831

#108

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₂²×C₇₃

#364

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₈×M₁₆

—

C₇₃

#7165

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

#431

—

C₄×M₁₆

#64408

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

#346

—

#346

—

146

—

#346

—

146

—

#322

—

146

—

C₈²

—

146

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

C₂×#64408

yes

characteristic

101

C₂×C₁₆

—

146

—

102

C₂×C₁₆

—

146

—

103

C₂×C₁₆

—

146

—

104

C₄⋊C₈

—

146

—

105

C₂²⋊C₈

—

106

#97

#10596833

yes

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

146

—

113

C₂²×C₄

C₄⋊#64408

yes

characteristic

114

C₂²×C₄

#11924036

yes

characteristic

115

C₂²×C₄

#11924037

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₄²

C₄×#64408

yes

characteristic

126

C₄²

C₂²⋊#64408

yes

characteristic

127

C₄²

C₂×#10596833

yes

characteristic

128

C₄²

#11924038

yes

characteristic

129

C₁₆

—

292

—

130

C₁₆

—

146

—

131

C₂³

#14556658

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

137

C₂×C₄

C₄×#10596833

yes

characteristic

138

C₂×C₄

#14556646

yes

characteristic

139

C₂×C₄

C₂²⋊#10596833

yes

characteristic

140

C₂×C₄

#14556648

yes

characteristic

141

C₂×C₄

C₄⋊#10596833

yes

characteristic

142

C₈

—

143

C₈

—

292

—

144

C₈

—

145

C₈

—

146

C₈

—

146

—

147

C₈

—

146

—

148

C₂²

—

149

C₂²

—

150

C₂²

#17989453

yes

characteristic

151

C₂²

#17989510

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#17989459

yes

characteristic

158

C₄

#17989462

yes

characteristic

159

C₂

—

160

C₂

#22624764

yes

characteristic

161

C₂

#22624975

yes

characteristic

162

C₂

#22624982

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#64408)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₄×M₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₂×C₄×#10596831	C₂²
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#7165
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×#64408)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₄×M₁₆×C₇₃

C₄

socle

C₂²×C₇₃

#336

prime radical

C₂³×C₇₃

C₄⋊C₈

C₂²

C₄×M₁₆

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
85. C₇₃	#7165	semidirect

H ≅ G/N

type

85. C₇₃

#7165

semidirect

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

subgroup	index
#22626292	2
C₂×#17989475	2
M₁₆×#10596831	2
#7165	73

subgroup

index

#22626292

C₂×#17989475

M₁₆×#10596831

#7165

Subgroups — #34000236

Conjugacy classes of subgroups

Complements

Maximal subgroups