Group #17953846 — subgroups

Subgroups — #17953846

Order 4352.

322 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 116 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	#14544708	C₂	1	2	yes	maximal
2	#14544343	C₂	1	2	yes	maximal
3	#14544340	C₂	1	2	yes	maximal
4	C₈²×D₁₇	C₂	1	1	yes	maximal, characteristic
5	#11763813	C₂²	1	4	yes	—
6	#11763685	—	2	4	no	—
7	#11763673	—	2	4	no	—
8	#11763671	—	2	4	no	—
9	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	C₈×#3622	C₄	1	1	yes	characteristic
13	C₈×#3622	C₄	1	1	yes	characteristic
14	C₈×#3622	C₄	1	1	yes	characteristic
15	C₈²×C₁₇	C₄	1	1	yes	characteristic
16	C₂×#63243	—	2	4	no	—
17	C₂×C₈×D₁₇	D₄	1	2	yes	—
18	C₂×C₈×D₁₇	Q₈	1	2	yes	—
19	C₂×C₈×D₁₇	Q₈	1	2	yes	—
20	C₄²×D₁₇	C₂³	1	1	yes	characteristic
21	C₄⋊#3624	—	2	4	no	—
22	C₄⋊#3624	—	2	4	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₁₇	D₄	1	2	yes	—
27	C₈×Dic₁₇	D₄	1	2	yes	—
28	C₈×Dic₁₇	Q₈	1	2	yes	—
29	C₄×#3622	C₂×C₄	1	1	yes	characteristic
30	C₄×#3622	C₂×C₄	1	1	yes	characteristic
31	C₄×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×#3624	—	4	4	no	—
36	#63243	—	4	8	no	—
37	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
40	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
43	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×#3622	C₄⋊C₄	1	2	yes	—
45	C₂×#3622	C₄⋊C₄	1	2	yes	—
46	C₂×#3622	C₄⋊C₄	1	2	yes	—
47	C₈×D₁₇	—	2	4	no	—
48	C₈×D₁₇	—	2	4	no	—
49	C₈×D₁₇	—	2	4	no	—
50	#8824	—	17	1	no	maximal, Sylow
51	C₂²×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	#127	1	1	yes	characteristic
56	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×D₁₇	—	2	2	no	—
59	C₄×D₁₇	—	2	2	no	—
60	C₄×D₁₇	#135	1	4	yes	—
61	#3624	—	8	4	no	—
62	C₈×C₁₇	—	2	2	no	—
63	C₈×C₁₇	—	2	2	no	—
64	C₈×C₁₇	—	2	2	no	—
65	#3622	—	2	2	no	—
66	#3622	#108	1	4	yes	—
67	#3622	#108	1	4	yes	—
68	#1837	—	17	2	no	—
69	#1567	—	17	2	no	—
70	#1566	—	17	2	no	—
71	C₂×C₈²	—	17	1	no	—
72	C₂²×C₁₇	#384	1	1	yes	characteristic
73	C₂×D₁₇	—	2	1	no	—
74	C₂×D₁₇	—	2	1	no	—
75	C₂×D₁₇	#499	1	2	yes	—
76	C₄×C₁₇	#427	1	2	yes	—
77	C₄×C₁₇	C₂×#108	1	2	yes	—
78	C₄×C₁₇	C₂×#108	1	2	yes	—
79	Dic₁₇	—	2	1	no	—
80	Dic₁₇	—	2	1	no	—
81	Dic₁₇	#495	1	2	yes	—
82	#423	—	17	4	no	—
83	C₂×C₄×C₈	—	17	1	no	—
84	C₂×C₄×C₈	—	17	1	no	—
85	C₂×C₄×C₈	—	17	1	no	—
86	#341	—	34	4	no	—
87	#335	—	34	4	no	—
88	#334	—	34	4	no	—
89	C₈²	—	17	1	no	—
90	C₈²	—	17	1	no	—
91	C₈²	—	17	1	no	—
92	C₈²	#595	1	1	yes	characteristic
93	C₂×C₁₇	#1846	1	1	yes	characteristic
94	C₂×C₁₇	#1848	1	1	yes	characteristic
95	C₂×C₁₇	#1848	1	1	yes	characteristic
96	D₁₇	—	2	2	no	—
97	C₂×M₁₆	—	34	4	no	—
98	C₂²×C₈	—	17	2	no	—
99	C₂²×C₈	—	17	2	no	—
100	C₂²×C₈	—	17	2	no	—
101	C₂×C₄²	—	17	1	no	—
102	C₄⋊C₈	—	34	4	no	—
103	C₄⋊C₈	—	34	4	no	—
104	C₄×C₈	—	17	2	no	—
105	C₄×C₈	—	17	2	no	—
106	C₄×C₈	—	17	2	no	—
107	C₄×C₈	—	17	1	no	—
108	C₄×C₈	—	17	1	no	—
109	C₄×C₈	—	17	1	no	—
110	C₄×C₈	C₂×#595	1	1	yes	characteristic
111	C₄×C₈	C₂×#595	1	1	yes	characteristic
112	C₄×C₈	C₂×#595	1	1	yes	characteristic
113	C₁₇	#8824	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₄	—	17	1	no	—
115	C₂²×C₄	—	17	1	no	—
116	C₂²×C₄	—	17	1	no	—
117	M₁₆	—	68	8	no	—
118	C₂×C₈	—	17	2	no	—
119	C₂×C₈	—	17	2	no	—
120	C₂×C₈	—	17	2	no	—
121	C₂×C₈	—	34	4	no	—
122	C₂×C₈	—	34	4	no	—
123	C₂×C₈	—	34	4	no	—
124	C₂×C₈	—	68	4	no	—
125	C₂×C₈	C₄⋊#595	1	2	yes	—
126	C₂×C₈	C₄⋊#595	1	2	yes	—
127	C₂×C₈	C₄⋊#595	1	2	yes	—
128	C₄²	—	17	1	no	—
129	C₄²	—	17	1	no	—
130	C₄²	—	17	1	no	—
131	C₄²	C₂²×#595	1	1	yes	characteristic
132	C₂³	—	17	1	no	—
133	C₂×C₄	—	17	1	no	—
134	C₂×C₄	—	17	4	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	34	2	no	—
138	C₂×C₄	—	34	2	no	—
139	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	17	4	no	—
146	C₈	—	17	4	no	—
147	C₈	—	34	2	no	—
148	C₈	—	136	4	no	—
149	C₂²	—	17	2	no	—
150	C₂²	—	34	1	no	—
151	C₂²	—	34	1	no	—
152	C₂²	#11763755	1	1	yes	characteristic
153	C₄	—	17	2	no	—
154	C₄	—	34	1	no	—
155	C₄	—	34	1	no	—
156	C₄	#11763819	1	2	yes	—
157	C₄	D₁₇⋊#108	1	2	yes	—
158	C₄	D₁₇⋊#108	1	2	yes	—
159	C₂	—	34	2	no	—
160	C₂	#14544721	1	1	yes	characteristic
161	C₂	#14544724	1	1	yes	characteristic
162	C₂	#14544724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544708

C₂

yes

maximal

#14544343

C₂

yes

maximal

#14544340

C₂

yes

maximal

C₈²×D₁₇

C₂

yes

maximal, characteristic

#11763813

C₂²

yes

—

#11763685

—

#11763673

—

#11763671

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈²×C₁₇

C₄

yes

characteristic

C₂×#63243

—

C₂×C₈×D₁₇

D₄

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₄²×D₁₇

C₂³

yes

characteristic

C₄⋊#3624

—

C₄⋊#3624

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

Q₈

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

characteristic

C₂×#3624

—

#63243

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

#8824

—

maximal, Sylow

C₂²×D₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

#127

yes

characteristic

C₂×Dic₁₇

C₄⋊Q₈

yes

characteristic

C₂×Dic₁₇

C₄⋊Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

#135

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

#108

yes

—

#3622

#108

yes

—

#1837

—

#1567

—

#1566

—

C₂×C₈²

—

C₂²×C₁₇

#384

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#499

yes

—

C₄×C₁₇

#427

yes

—

C₄×C₁₇

C₂×#108

yes

—

C₄×C₁₇

C₂×#108

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#495

yes

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#341

—

#335

—

#334

—

C₈²

—

C₈²

—

C₈²

—

C₈²

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×#595

yes

characteristic

111

C₄×C₈

C₂×#595

yes

characteristic

112

C₄×C₈

C₂×#595

yes

characteristic

113

C₁₇

#8824

yes

characteristic, complemented, Sylow

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

C₄⋊#595

yes

—

126

C₂×C₈

C₄⋊#595

yes

—

127

C₂×C₈

C₄⋊#595

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

C₂²×#595

yes

characteristic

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

140

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

141

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

136

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

#11763755

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#11763819

yes

—

157

C₄

D₁₇⋊#108

yes

—

158

C₄

D₁₇⋊#108

yes

—

159

C₂

—

160

C₂

#14544721

yes

characteristic

161

C₂

#14544724

yes

characteristic

162

C₂

#14544724

yes

characteristic

	subgroup	G/H
center	C₂²	#11763755
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	#384
prime radical	C₂²×D₁₇	C₄⋊Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	C₈²	#595
hypocenter	C₁₇	#8824
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11763755

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₈²×C₁₇

C₄

socle

C₂²×C₁₇

#384

prime radical

C₂²×D₁₇

C₄⋊Q₈

cosocle

C₄²×D₁₇

C₂³

C₈²

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
113. C₁₇	#8824	semidirect

H ≅ G/N

type

113. C₁₇

#8824

semidirect

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

subgroup	index
#14544708	2
#14544343	2
#14544340	2
C₈²×D₁₇	2
#8824	17

subgroup

index

C₈²×D₁₇

Subgroups — #17953846

Conjugacy classes of subgroups

Complements

Maximal subgroups