Group #16937437 — subgroups

Subgroups — #16937437

Order 4352.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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	#14537171	C₂	1	1	yes	maximal, characteristic
2	#14527348	C₂	1	2	yes	maximal
3	#14526644	C₂	1	2	yes	maximal
4	#14526617	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1994	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	#11762707	—	2	2	no	—
8	#11762693	—	2	4	no	—
9	#11762675	C₂²	1	2	yes	—
10	#11762546	—	2	4	no	—
11	C₁₇×#533	C₂²	1	1	yes	characteristic
12	C₁₇×#423	C₂²	1	1	yes	characteristic
13	C₄³×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#339	—	2	2	no	—
15	C₂×M₁₆×C₁₇	—	2	1	no	—
16	C₁₇×C₄⋊Q₈	Q₈	1	2	yes	—
17	C₁₇×#125	—	2	1	no	—
18	C₁₇×C₂²⋊Q₈	—	2	2	no	—
19	C₄×Q₈×C₁₇	—	2	2	no	—
20	C₁₇×#117	—	2	1	no	—
21	C₂×C₄²×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	D₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×#63225	—	2	4	no	—
26	C₁₇×C₄⋊C₈	—	2	2	no	—
27	#10587502	—	2	4	no	—
28	#10587502	—	2	2	no	—
29	#10587502	D₄	1	4	yes	—
30	C₂²⋊#3622	—	2	2	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄×#3622	—	2	2	no	—
33	C₂×Q₈×C₁₇	—	2	2	no	—
34	C₂²×C₄×C₁₇	—	2	2	no	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	2	4	no	—
38	M₁₆×C₁₇	—	4	2	no	—
39	C₂×C₈×C₁₇	—	4	1	no	—
40	C₄⋊C₄×C₁₇	—	2	2	no	—
41	C₄⋊C₄×C₁₇	—	2	2	no	—
42	C₄⋊C₄×C₁₇	—	4	2	no	—
43	C₂²⋊C₄×C₁₇	—	4	1	no	—
44	C₄²×C₁₇	—	2	2	no	—
45	C₄²×C₁₇	—	2	2	no	—
46	C₄²×C₁₇	—	4	4	no	—
47	C₄²×C₁₇	—	4	1	no	—
48	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₇	#41	1	1	yes	characteristic
50	C₄²×C₁₇	#41	1	2	yes	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	#63225	—	4	8	no	—
54	C₂×#3622	—	4	2	no	—
55	#10367	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	#120	1	1	yes	characteristic
57	Q₈×C₁₇	—	8	2	no	—
58	C₂×C₄×C₁₇	—	2	4	no	—
59	C₂×C₄×C₁₇	—	4	2	no	—
60	C₂×C₄×C₁₇	—	4	2	no	—
61	C₂×C₄×C₁₇	—	4	2	no	—
62	C₂×C₄×C₁₇	—	4	2	no	—
63	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#124	1	2	yes	—
66	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#135	1	2	yes	—
68	C₂×C₄×C₁₇	#137	1	2	yes	—
69	C₂×Dic₁₇	—	4	8	no	—
70	C₂×Dic₁₇	—	4	4	no	—
71	C₈×C₁₇	—	8	1	no	—
72	#3622	—	8	2	no	—
73	#2864	—	17	1	no	—
74	#1994	D₁₇	1	1	yes	characteristic
75	#1518	—	17	2	no	—
76	#1286	—	17	2	no	—
77	#1276	—	17	1	no	—
78	C₂²×C₁₇	—	2	2	no	—
79	C₂²×C₁₇	#393	1	1	yes	characteristic
80	C₂²×C₁₇	#452	1	2	yes	—
81	C₄×C₁₇	—	2	2	no	—
82	C₄×C₁₇	—	2	2	no	—
83	C₄×C₁₇	—	4	4	no	—
84	C₄×C₁₇	—	8	2	no	—
85	C₄×C₁₇	#479	1	2	yes	—
86	C₄×C₁₇	#488	1	1	yes	characteristic
87	C₄×C₁₇	#491	1	1	yes	characteristic
88	Dic₁₇	—	8	4	no	—
89	#533	C₂×D₁₇	1	1	yes	characteristic
90	#431	—	17	2	no	—
91	#423	C₂×D₁₇	1	1	yes	characteristic
92	#389	—	34	2	no	—
93	#387	—	34	4	no	—
94	#382	—	17	2	no	—
95	C₄³	C₂×D₁₇	1	1	yes	characteristic
96	#339	—	2	2	no	—
97	#328	—	34	4	no	—
98	C₂×C₁₇	—	2	2	no	—
99	C₂×C₁₇	#2026	1	1	yes	characteristic
100	C₂×C₁₇	#2031	1	2	yes	—
101	C₂×M₁₆	—	2	1	no	—
102	C₄⋊Q₈	Dic₃₄	1	2	yes	—
103	#125	—	2	1	no	—
104	C₂²⋊Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	#117	—	2	1	no	—
107	C₂×C₄⋊C₄	—	34	4	no	—
108	C₂×C₄⋊C₄	—	34	2	no	—
109	C₂×C₄²	—	2	2	no	—
110	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
111	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	34	2	no	—
114	C₂²⋊C₈	—	34	2	no	—
115	C₄×C₈	—	34	2	no	—
116	#95	—	17	4	no	—
117	#95	—	34	4	no	—
118	#95	—	34	2	no	—
119	C₁₇	#10367	1	1	yes	characteristic, complemented, Sylow
120	C₂×Q₈	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	34	4	no	—
123	C₂²×C₄	D₄×D₁₇	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	68	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	68	8	no	—
132	C₄⋊C₄	—	68	2	no	—
133	C₄⋊C₄	—	68	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	4	1	no	—
139	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
140	C₄²	#63252	1	1	yes	characteristic
141	C₄²	#63254	1	2	yes	—
142	C₂³	#10587623	1	1	yes	characteristic
143	Q₈	—	8	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	68	8	no	—
150	C₂×C₄	—	68	4	no	—
151	C₂×C₄	#10587564	1	2	yes	—
152	C₂×C₄	#10587571	1	2	yes	—
153	C₂×C₄	#10587610	1	1	yes	characteristic
154	C₂×C₄	#10587622	1	1	yes	characteristic
155	C₂×C₄	#10587628	1	2	yes	—
156	C₂×C₄	#10587636	1	2	yes	—
157	C₈	—	8	1	no	—
158	C₈	—	136	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#11762721	1	1	yes	characteristic
161	C₂²	#11762969	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	2	no	—
166	C₄	—	136	4	no	—
167	C₄	#11763116	1	2	yes	—
168	C₄	#11763163	1	1	yes	characteristic
169	C₄	#11763177	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#14529935	1	1	yes	characteristic
172	C₂	#14529956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14537171

C₂

yes

maximal, characteristic

#14527348

C₂

yes

maximal

#14526644

C₂

yes

maximal

#14526617

C₂

yes

maximal, characteristic

C₁₇×#1994

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762707

—

#11762693

—

#11762675

C₂²

yes

—

#11762546

—

C₁₇×#533

C₂²

yes

characteristic

C₁₇×#423

C₂²

yes

characteristic

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#339

—

C₂×M₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

Q₈

yes

—

C₁₇×#125

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×#63225

—

C₁₇×C₄⋊C₈

—

#10587502

—

#10587502

—

#10587502

D₄

yes

—

C₂²⋊#3622

—

C₄⋊#3622

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

C₄²×C₁₇

yes

C₄²×C₁₇

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

#124

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#3622

—

#2864

—

#1994

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#393

yes

characteristic

C₂²×C₁₇

#452

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#479

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#533

C₂×D₁₇

yes

characteristic

#431

—

#423

C₂×D₁₇

yes

—

—

—

C₄³

C₂×D₁₇

yes

characteristic

#339

—

#328

—

C₂×C₁₇

—

C₂×C₁₇

#2026

yes

characteristic

100

C₂×C₁₇

#2031

yes

—

101

C₂×M₁₆

—

102

C₄⋊Q₈

Dic₃₄

yes

—

103

#125

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

—

106

#117

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₂²⋊D₁₇

yes

—

111

C₂×C₄²

C₂²×D₁₇

yes

characteristic

112

C₄⋊C₈

—

113

C₄⋊C₈

—

114

C₂²⋊C₈

—

115

C₄×C₈

—

116

#95

—

117

#95

—

118

#95

—

119

C₁₇

#10367

yes

characteristic, complemented, Sylow

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

D₄×D₁₇

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂×Dic₃₄

yes

characteristic

140

C₄²

#63252

yes

characteristic

141

C₄²

#63254

yes

—

142

C₂³

#10587623

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#10587564

yes

—

152

C₂×C₄

#10587571

yes

—

153

C₂×C₄

#10587610

yes

characteristic

154

C₂×C₄

#10587622

yes

characteristic

155

C₂×C₄

#10587628

yes

—

156

C₂×C₄

#10587636

yes

—

157

C₈

—

158

C₈

—

136

—

159

C₂²

—

160

C₂²

#11762721

yes

characteristic

161

C₂²

#11762969

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

136

—

167

C₄

#11763116

yes

—

168

C₄

#11763163

yes

characteristic

169

C₄

#11763177

yes

characteristic

170

C₂

—

171

C₂

#14529935

yes

characteristic

172

C₂

#14529956

yes

—

	subgroup	G/H
center	C₂²	#11762721
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1994	C₂
socle	C₂²×C₁₇	#393
prime radical	C₂³×C₁₇	#120
hypercenter	#1994	D₁₇
hypocenter	C₁₇	#10367
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#11762721

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1994

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

N	H ≅ G/N	type
119. C₁₇	#10367	semidirect

H ≅ G/N

type

119. C₁₇

#10367

semidirect

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

subgroup	index
#14537171	2
#14527348	2
#14526644	2
#14526617	2
C₁₇×#1994	2
#10367	17

subgroup

index

C₁₇×#1994

Subgroups — #16937437

Conjugacy classes of subgroups

Complements

Maximal subgroups