Group #34000228 — subgroups

Subgroups — #34000228

Order 18688.

238 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 104 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	#22626288	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
3	#22608212	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	C₄×#11924031	—	2	1	no	—
8	C₄×#11924031	—	2	1	no	—
9	C₄×#11924031	C₄	1	2	yes	—
10	C₄×#11924012	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	#17988309	—	2	1	no	—
13	C₈×#10596831	—	2	1	no	—
14	#431×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	C₂×#11924012	C₈	1	2	yes	—
19	C₂²×#10596831	D₄	1	1	yes	characteristic
20	C₂²×#10596831	Q₈	1	1	yes	characteristic
21	#14556502	—	2	1	no	—
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×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	C₈	1	2	yes	—
29	C₂²⋊C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	—	2	1	no	—
31	#11924031	—	2	4	no	—
32	#11924031	—	4	2	no	—
33	#11924012	—	2	2	no	—
34	#11924012	C₁₆	1	4	yes	complemented
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	2	no	—
46	C₄²×C₇₃	C₄²	1	1	yes	characteristic
47	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
50	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	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	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
62	C₈×C₇₃	—	4	2	no	—
63	#10596831	—	2	1	no	—
64	#10596831	—	2	1	no	—
65	#10596831	—	2	4	no	—
66	#10596831	#104	1	2	yes	—
67	#10596831	#108	1	2	yes	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	#336	1	1	yes	characteristic
71	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	#339	1	1	yes	characteristic
75	C₄×C₇₃	#341	1	1	yes	characteristic
76	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
77	C₄×C₇₃	#346	1	1	yes	characteristic
78	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
79	C₄×C₇₃	#350	1	2	yes	—
80	#7163	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	2	2	no	—
82	C₂×C₇₃	#1278	1	1	yes	characteristic
83	C₂×C₇₃	#1374	1	1	yes	characteristic
84	C₂×C₇₃	#1379	1	1	yes	characteristic
85	#2162	—	73	1	no	—
86	C₂×C₄×C₁₆	—	73	1	no	—
87	C₈×M₁₆	—	73	1	no	—
88	C₇₃	#7163	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	146	1	no	—
90	#431	#64408	1	1	yes	characteristic
91	C₄×M₁₆	—	73	1	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	C₄⋊C₁₆	—	73	2	no	—
94	C₂²⋊C₁₆	—	146	1	no	—
95	C₄×C₁₆	—	73	2	no	—
96	C₄×C₁₆	—	146	1	no	—
97	C₄×C₁₆	—	146	1	no	—
98	#322	—	146	1	no	—
99	C₈²	—	146	1	no	—
100	C₂×M₁₆	—	73	2	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
104	C₂×C₁₆	—	146	2	no	—
105	C₂×C₁₆	—	146	2	no	—
106	C₂×C₁₆	—	146	4	no	—
107	C₄⋊C₈	#10596833	1	2	yes	—
108	C₂²⋊C₈	—	2	1	no	—
109	#97	—	146	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	146	1	no	—
116	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
117	C₂²×C₄	#11924036	1	1	yes	characteristic
118	C₂²×C₄	#11924037	1	1	yes	characteristic
119	M₁₆	—	73	4	no	—
120	M₁₆	—	146	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	—	146	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₄²	C₄×#64408	1	1	yes	characteristic
130	C₄²	C₂²⋊#64408	1	1	yes	characteristic
131	C₄²	C₂×#10596833	1	1	yes	characteristic
132	C₄²	#11924038	1	1	yes	characteristic
133	C₁₆	—	292	2	no	—
134	C₁₆	—	146	4	no	—
135	C₂³	#14556658	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
141	C₂×C₄	#14556646	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
143	C₂×C₄	#14556648	1	1	yes	characteristic
144	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
145	C₂×C₄	#14556664	1	2	yes	—
146	C₂×C₄	#14556666	1	2	yes	—
147	C₈	—	4	2	no	—
148	C₈	—	73	2	no	—
149	C₈	—	73	2	no	—
150	C₈	—	146	1	no	—
151	C₈	—	146	1	no	—
152	C₈	—	146	4	no	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#17989453	1	1	yes	characteristic
156	C₂²	#17989508	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#17989459	1	1	yes	characteristic
160	C₄	#17989462	1	1	yes	characteristic
161	C₄	#17989473	1	1	yes	characteristic
162	C₄	#17989476	1	1	yes	characteristic
163	C₄	#17989482	1	2	yes	—
164	C₄	#17989485	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#22624764	1	1	yes	characteristic
167	C₂	#22624970	1	1	yes	characteristic
168	C₂	#22624985	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626288

C₂

yes

maximal, characteristic

C₂×C₄×#11924031

C₂

yes

maximal, characteristic

#22608212

C₂

yes

maximal, characteristic

C₂²×#11924031

—

C₄⋊#11924031

C₄

yes

—

—

—

—

C₄

yes

—

C₄×#11924012

C₄

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

characteristic

#17988309

—

C₈×#10596831

—

#431×C₇₃

C₄

yes

—

—

—

C₈

yes

—

C₂²×#10596831

D₄

yes

characteristic

C₂²×#10596831

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

C₈

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

—

—

—

C₁₆

yes

—

—

—

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

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

#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

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₈×C₇₃

—

—

—

—

yes

—

#10596831

#108

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₂²×C₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₄×C₁₆

yes

characteristic

C₄×C₇₃

#346

yes

characteristic

C₄×C₇₃

C₂²⋊C₁₆

yes

—

C₄×C₇₃

#350

yes

—

#7163

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₄×C₁₆

—

C₈×M₁₆

—

C₇₃

#7163

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

#431

#64408

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

146

—

C₄×C₁₆

—

146

—

#322

—

146

—

C₈²

—

146

—

100

C₂×M₁₆

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

C₂×#64408

yes

characteristic

104

C₂×C₁₆

—

146

—

105

C₂×C₁₆

—

146

—

106

C₂×C₁₆

—

146

—

107

C₄⋊C₈

#10596833

yes

—

108

C₂²⋊C₈

—

109

#97

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

146

—

116

C₂²×C₄

C₄⋊#64408

yes

characteristic

117

C₂²×C₄

#11924036

yes

characteristic

118

C₂²×C₄

#11924037

yes

characteristic

119

M₁₆

—

120

M₁₆

—

146

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₄²

C₄×#64408

yes

characteristic

130

C₄²

C₂²⋊#64408

yes

characteristic

131

C₄²

C₂×#10596833

yes

characteristic

132

C₄²

#11924038

yes

characteristic

133

C₁₆

—

292

—

134

C₁₆

—

146

—

135

C₂³

#14556658

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

140

C₂×C₄

C₄×#10596833

yes

characteristic

141

C₂×C₄

#14556646

yes

characteristic

142

C₂×C₄

C₂²⋊#10596833

yes

characteristic

143

C₂×C₄

#14556648

yes

characteristic

144

C₂×C₄

C₄⋊#10596833

yes

characteristic

145

C₂×C₄

#14556664

yes

—

146

C₂×C₄

#14556666

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

146

—

151

C₈

—

146

—

152

C₈

—

146

—

153

C₂²

—

154

C₂²

—

155

C₂²

#17989453

yes

characteristic

156

C₂²

#17989508

yes

—

157

C₄

—

158

C₄

—

159

C₄

#17989459

yes

characteristic

160

C₄

#17989462

yes

characteristic

161

C₄

#17989473

yes

characteristic

162

C₄

#17989476

yes

characteristic

163

C₄

#17989482

yes

—

164

C₄

#17989485

yes

—

165

C₂

—

166

C₂

#22624764

yes

characteristic

167

C₂

#22624970

yes

characteristic

168

C₂

#22624985

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊#64408
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#64408
Fitting	#431×C₇₃	C₄
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₂×C₄×#10596831	C₂²
hypercenter	#431	#64408
hypocenter	C₇₃	#7163
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊#64408

commutator

C₄×C₇₃

C₄×C₁₆

Frattini

C₂×C₄²

C₂×#64408

Fitting

#431×C₇₃

C₄

socle

C₂²×C₇₃

#336

prime radical

C₂³×C₇₃

C₄⋊C₈

C₂²

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
34. #11924012	C₁₆	semidirect
88. C₇₃	#7163	semidirect

H ≅ G/N

type

34. #11924012

C₁₆

semidirect

88. C₇₃

#7163

semidirect

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

subgroup	index
#22626288	2
C₂×C₄×#11924031	2
#22608212	2
#7163	73

subgroup

index

Subgroups — #34000228

Conjugacy classes of subgroups

Complements

Maximal subgroups