Group #17948580 — subgroups

Subgroups — #17948580

Order 4352.

216 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 84 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	#14545141	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587538	C₂	1	1	yes	maximal, characteristic
3	#14527074	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	—	2	1	no	—
5	C₄⋊#63216	C₄	1	2	yes	—
6	C₂²⋊#63216	—	2	1	no	—
7	#11763697	C₄	1	2	yes	—
8	C₄×#63216	—	2	1	no	—
9	C₄×#63216	—	2	1	no	—
10	C₄×#63223	C₄	1	1	yes	characteristic
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	#11762531	—	2	1	no	—
13	#11762530	—	2	1	no	—
14	C₁₇×#431	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#63223	C₈	1	2	yes	—
17	C₂²×#3622	D₄	1	1	yes	characteristic
18	C₂²×#3622	Q₈	1	1	yes	characteristic
19	#10587538	—	2	4	no	—
20	C₂×#63216	—	2	2	no	—
21	C₂×#63216	—	2	2	no	—
22	C₁₇×C₄⋊C₈	C₈	1	2	yes	—
23	C₁₇×C₂²⋊C₈	—	2	1	no	—
24	C₄×C₈×C₁₇	—	2	1	no	—
25	#10587474	—	2	1	no	—
26	C₄×#3622	—	2	1	no	—
27	C₄×#3622	C₂×C₄	1	1	yes	characteristic
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	D₄	1	1	yes	characteristic
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₁₇	—	2	2	no	—
35	C₄²×C₁₇	C₄²	1	1	yes	characteristic
36	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
39	#63223	—	4	2	no	—
40	C₂×#3622	—	2	2	no	—
41	C₂×#3622	—	2	2	no	—
42	C₂×#3622	—	2	2	no	—
43	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
47	#63216	—	2	4	no	—
48	#63216	—	4	2	no	—
49	#7166	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	—	2	1	no	—
52	C₂×C₄×C₁₇	—	2	1	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₁₇	—	4	2	no	—
62	#3622	—	2	1	no	—
63	#3622	—	2	1	no	—
64	#3622	—	4	2	no	—
65	#3622	#104	1	2	yes	—
66	#3622	#108	1	2	yes	—
67	#2162	—	17	1	no	—
68	C₄×M₃₂	—	17	1	no	—
69	#1451	—	17	1	no	—
70	C₂²×C₁₇	—	2	1	no	—
71	C₂²×C₁₇	—	2	1	no	—
72	C₂²×C₁₇	—	2	1	no	—
73	C₂²×C₁₇	#336	1	1	yes	characteristic
74	C₄×C₁₇	—	2	1	no	—
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	—	2	1	no	—
77	C₄×C₁₇	#339	1	1	yes	characteristic
78	C₄×C₁₇	#341	1	1	yes	characteristic
79	C₄×C₁₇	#349	1	2	yes	—
80	C₄×C₁₇	#350	1	2	yes	—
81	C₂×M₃₂	—	34	1	no	—
82	#431	#595	1	1	yes	characteristic
83	C₄×M₁₆	—	17	1	no	—
84	C₂×C₄×C₈	—	17	1	no	—
85	C₄⋊C₁₆	—	17	2	no	—
86	C₂²⋊C₁₆	—	34	1	no	—
87	#346	—	17	2	no	—
88	C₄×C₁₆	—	34	1	no	—
89	C₄×C₁₆	—	34	1	no	—
90	#322	—	34	1	no	—
91	#322	—	34	1	no	—
92	C₂×C₁₇	—	4	1	no	—
93	C₂×C₁₇	#1278	1	1	yes	characteristic
94	C₂×C₁₇	#1377	1	1	yes	characteristic
95	C₂×C₁₇	#1379	1	1	yes	characteristic
96	C₂×M₁₆	—	17	2	no	—
97	C₂²×C₈	—	17	1	no	—
98	C₂²×C₈	—	17	1	no	—
99	C₂×C₄²	C₂×#595	1	1	yes	characteristic
100	M₃₂	—	34	4	no	—
101	C₂×C₁₆	—	34	2	no	—
102	C₂×C₁₆	—	34	2	no	—
103	C₄⋊C₈	#3624	1	2	yes	—
104	C₂²⋊C₈	—	2	1	no	—
105	#97	—	34	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	17	1	no	—
108	C₄×C₈	—	17	1	no	—
109	C₄×C₈	—	17	1	no	—
110	C₄×C₈	—	17	1	no	—
111	C₄×C₈	—	34	1	no	—
112	C₁₇	#7166	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₄	#63242	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
115	C₂²×C₄	#63247	1	1	yes	characteristic
116	M₁₆	—	68	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	17	1	no	—
119	C₂×C₈	—	17	1	no	—
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	—	34	2	no	—
124	C₂×C₈	—	34	2	no	—
125	C₄²	#63243	1	1	yes	characteristic
126	C₄²	C₄×#595	1	1	yes	characteristic
127	C₄²	C₂×#3624	1	1	yes	characteristic
128	C₄²	C₂²⋊#595	1	1	yes	characteristic
129	C₁₆	—	34	4	no	—
130	C₁₆	—	68	2	no	—
131	C₂³	#10587544	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₄×#3624	1	1	yes	characteristic
137	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
138	C₂×C₄	#10587542	1	1	yes	characteristic
139	C₂×C₄	#10587543	1	1	yes	characteristic
140	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
142	C₈	—	4	2	no	—
143	C₈	—	17	2	no	—
144	C₈	—	17	2	no	—
145	C₈	—	34	1	no	—
146	C₈	—	34	1	no	—
147	C₈	—	68	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#11763675	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#11763681	1	1	yes	characteristic
156	C₄	#11763684	1	1	yes	characteristic
157	C₄	#11763706	1	2	yes	—
158	C₄	#11763707	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#14543617	1	1	yes	characteristic
161	C₂	#14543832	1	1	yes	characteristic
162	C₂	#14543838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545141

C₂

yes

maximal, characteristic

C₄×#10587538

C₂

yes

maximal, characteristic

#14527074

C₂

yes

maximal, characteristic

C₂×#10587538

—

C₄⋊#63216

C₄

yes

—

C₂²⋊#63216

—

#11763697

C₄

yes

—

C₄×#63216

—

C₄×#63216

—

C₄×#63223

C₄

yes

characteristic

C₂×C₄×#3622

C₂²

yes

characteristic

#11762531

—

#11762530

—

C₁₇×#431

C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×#63223

C₈

yes

—

C₂²×#3622

D₄

yes

characteristic

C₂²×#3622

Q₈

yes

—

—

—

C₁₇×C₄⋊C₈

C₈

yes

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

—

#10587474

—

C₄×#3622

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

D₄

yes

characteristic

C₄×#3622

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

—

—

C₂³×C₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₈×C₁₇

—

—

—

—

yes

—

#3622

#108

yes

—

#2162

—

C₄×M₃₂

—

#1451

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#336

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₄×C₁₇

#350

yes

—

C₂×M₃₂

—

#431

#595

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

#322

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#595

yes

characteristic

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₄⋊C₈

#3624

yes

—

104

C₂²⋊C₈

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₁₇

#7166

yes

characteristic, complemented, Sylow

113

C₂²×C₄

#63242

yes

characteristic

114

C₂²×C₄

C₄⋊#595

yes

characteristic

115

C₂²×C₄

#63247

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄²

#63243

yes

characteristic

126

C₄²

C₄×#595

yes

characteristic

127

C₄²

C₂×#3624

yes

characteristic

128

C₄²

C₂²⋊#595

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₂³

#10587544

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₄×#3624

yes

characteristic

137

C₂×C₄

C₄⋊#3624

yes

characteristic

138

C₂×C₄

#10587542

yes

characteristic

139

C₂×C₄

#10587543

yes

characteristic

140

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

141

C₂×C₄

C₂²⋊#3624

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#11763675

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#11763681

yes

characteristic

156

C₄

#11763684

yes

characteristic

157

C₄

#11763706

yes

—

158

C₄

#11763707

yes

—

159

C₂

—

160

C₂

#14543617

yes

characteristic

161

C₂

#14543832

yes

characteristic

162

C₂

#14543838

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₁₇×#431	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	#431	#595
hypocenter	C₁₇	#7166
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₁₇×#431

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂³×C₁₇

C₄⋊C₈

cosocle

C₂×C₄×#3622

C₂²

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
112. C₁₇	#7166	semidirect

H ≅ G/N

type

112. C₁₇

#7166

semidirect

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

subgroup	index
#14545141	2
C₄×#10587538	2
#14527074	2
#7166	17

subgroup

index

Subgroups — #17948580

Conjugacy classes of subgroups

Complements

Maximal subgroups