Group #16948953 — subgroups

Subgroups — #16948953

Order 4352.

226 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14530931	C₂	1	2	yes	maximal
2	#14528402	C₂	1	1	yes	maximal, characteristic, complemented
3	#14526848	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526843	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1589	C₂	1	1	yes	maximal, characteristic
6	#11763214	—	2	1	no	—
7	#11763214	C₂²	1	2	yes	—
8	#11763189	—	2	1	no	—
9	#11762636	—	2	2	no	—
10	C₄⋊#63214	C₂²	1	2	yes	—
11	#11762610	—	2	2	no	—
12	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#436	—	2	1	no	—
14	C₁₇×#335	C₂²	1	1	yes	characteristic
15	C₁₇×D₄⋊C₈	—	2	1	no	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	—	4	1	no	—
19	C₄×D₄×C₁₇	—	2	1	no	—
20	C₂×Dic₆₈	—	4	2	no	—
21	#10587555	—	2	2	no	—
22	C₁₇×#107	D₄	1	1	yes	characteristic
23	C₁₇×C₄⋊C₈	—	2	1	no	—
24	C₁₇×#102	—	2	1	no	—
25	C₁₇×C₂²⋊C₈	—	4	1	no	—
26	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
29	#10587489	—	2	2	no	—
30	#10587489	—	2	2	no	—
31	#10587489	—	4	1	no	—
32	#10587489	—	4	1	no	—
33	C₂×#63214	—	2	2	no	—
34	C₂×D₄×C₁₇	—	2	1	no	—
35	C₂²×C₄×C₁₇	—	4	1	no	—
36	C₂×Dic₃₄	—	4	2	no	—
37	D₈×C₁₇	—	2	2	no	—
38	M₁₆×C₁₇	—	8	2	no	—
39	C₂×C₈×C₁₇	—	2	1	no	—
40	C₂×C₈×C₁₇	—	4	1	no	—
41	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₁₇	—	2	1	no	—
46	C₂²⋊C₄×C₁₇	—	4	1	no	—
47	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	Dic₆₈	—	8	4	no	—
51	#63214	—	4	2	no	—
52	#12499	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	—	4	1	no	—
54	D₄×C₁₇	—	4	1	no	—
55	D₄×C₁₇	—	4	1	no	—
56	C₂×C₄×C₁₇	—	4	1	no	—
57	C₂×C₄×C₁₇	—	8	1	no	—
58	C₂×C₄×C₁₇	—	8	1	no	—
59	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×Dic₁₇	—	8	2	no	—
63	Dic₃₄	—	8	4	no	—
64	C₈×C₁₇	—	2	2	no	—
65	C₈×C₁₇	—	2	1	no	—
66	C₈×C₁₇	—	2	1	no	—
67	C₈×C₁₇	—	8	1	no	—
68	C₈×C₁₇	SD₃₂	1	2	yes	—
69	C₈×C₁₇	#137	1	2	yes	—
70	#2226	—	17	2	no	—
71	#1713	—	17	1	no	—
72	#1589	D₁₇	1	1	yes	characteristic
73	#1367	—	17	1	no	—
74	#1365	—	17	2	no	—
75	C₂²×C₁₇	—	8	1	no	—
76	C₂²×C₁₇	—	8	1	no	—
77	C₂²×C₁₇	—	8	1	no	—
78	C₂²×C₁₇	#459	1	1	yes	characteristic
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	8	1	no	—
81	C₄×C₁₇	#470	1	1	yes	characteristic
82	C₄×C₁₇	#473	1	1	yes	characteristic
83	C₄×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
84	C₄×C₁₇	#510	1	1	yes	characteristic
85	Dic₁₇	—	16	2	no	—
86	#500	—	17	2	no	—
87	#500	—	34	1	no	—
88	#494	—	34	1	no	—
89	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
90	#436	—	2	1	no	—
91	#367	—	34	2	no	—
92	C₄⋊C₁₆	—	17	2	no	—
93	#357	—	34	2	no	—
94	#335	C₂×D₁₇	1	1	yes	characteristic
95	D₄⋊C₈	—	2	1	no	—
96	C₈²	C₂×D₁₇	1	1	yes	characteristic
97	C₂×C₁₇	—	16	1	no	—
98	C₂×C₁₇	#1689	1	1	yes	characteristic
99	C₂×C₁₇	#2208	1	1	yes	characteristic
100	C₂×C₁₇	#2213	1	1	yes	characteristic
101	C₂×Q₁₆	—	68	2	no	—
102	C₂×D₈	D₆₈	1	1	yes	characteristic
103	C₂×M₁₆	—	4	1	no	—
104	C₄⋊Q₈	—	34	2	no	—
105	C₄×D₄	—	2	1	no	—
106	C₂×C₁₆	—	34	2	no	—
107	#107	—	34	2	no	—
108	#107	—	34	2	no	—
109	#107	—	68	1	no	—
110	#107	—	68	1	no	—
111	#107	D₆₈	1	1	yes	characteristic
112	C₄⋊C₈	—	2	1	no	—
113	#102	—	2	1	no	—
114	C₂²⋊C₈	—	4	1	no	—
115	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
118	C₁₇	#12499	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₈	—	68	2	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	Q₁₆	—	136	4	no	—
123	D₈	—	2	2	no	—
124	M₁₆	—	8	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	#63228	1	1	yes	characteristic
128	C₂×C₈	#63228	1	1	yes	characteristic
129	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
130	C₂×C₈	#63252	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	68	2	no	—
133	C₄⋊C₄	—	68	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
136	C₁₆	—	68	2	no	—
137	C₂³	—	4	1	no	—
138	Q₈	—	136	4	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	136	2	no	—
145	C₂×C₄	#10587616	1	1	yes	characteristic
146	C₂×C₄	C₂×#63228	1	1	yes	characteristic
147	C₂×C₄	#10587635	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	1	no	—
152	C₈	#10587498	1	2	yes	—
153	C₈	#10587595	1	2	yes	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	#11763004	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	8	1	no	—
160	C₄	—	272	2	no	—
161	C₄	#11763067	1	1	yes	characteristic
162	C₄	#11763084	1	1	yes	characteristic
163	C₄	C₂×#10587498	1	1	yes	characteristic
164	C₄	#11763249	1	1	yes	characteristic
165	C₂	—	16	1	no	—
166	C₂	#14528273	1	1	yes	characteristic
167	C₂	#14530829	1	1	yes	characteristic
168	C₂	#14530860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530931

C₂

yes

maximal

#14528402

C₂

yes

maximal, characteristic, complemented

#14526848

C₂

yes

maximal, characteristic, complemented

#14526843

C₂

yes

maximal, complemented

C₁₇×#1589

C₂

yes

maximal, characteristic

#11763214

—

#11763214

C₂²

yes

—

#11763189

—

#11762636

—

C₄⋊#63214

C₂²

yes

—

#11762610

—

C₄×D₈×C₁₇

C₂²

yes

characteristic

C₁₇×#436

—

C₁₇×#335

C₂²

yes

characteristic

C₁₇×D₄⋊C₈

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×D₈×C₁₇

D₄

yes

characteristic

C₂×M₁₆×C₁₇

—

C₄×D₄×C₁₇

—

C₂×Dic₆₈

—

#10587555

—

C₁₇×#107

D₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×#102

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂×Dic₃₄

—

D₈×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

#63214

—

#12499

—

maximal, Sylow

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

SD₃₂

yes

—

C₈×C₁₇

#137

yes

—

#2226

—

#1713

—

#1589

D₁₇

yes

characteristic

#1367

—

#1365

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#459

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂×SD₃₂

yes

characteristic

C₄×C₁₇

#510

yes

characteristic

Dic₁₇

—

#500

—

#500

—

#494

—

C₄×D₈

C₂×D₁₇

yes

characteristic

#436

—

#367

—

C₄⋊C₁₆

—

#357

—

#335

C₂×D₁₇

yes

characteristic

D₄⋊C₈

—

C₈²

C₂×D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

100

C₂×C₁₇

#2213

yes

characteristic

101

C₂×Q₁₆

—

102

C₂×D₈

D₆₈

yes

characteristic

103

C₂×M₁₆

—

104

C₄⋊Q₈

—

105

C₄×D₄

—

106

C₂×C₁₆

—

107

#107

—

108

#107

—

109

#107

—

110

#107

—

111

#107

D₆₈

yes

characteristic

112

C₄⋊C₈

—

113

#102

—

114

C₂²⋊C₈

—

115

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

116

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

117

C₄×C₈

C₂²×D₁₇

yes

characteristic

118

C₁₇

#12499

yes

characteristic, complemented, Sylow

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

Q₁₆

—

136

—

123

D₈

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#63228

yes

characteristic

128

C₂×C₈

#63228

yes

characteristic

129

C₂×C₈

C₂×D₆₈

yes

characteristic

130

C₂×C₈

#63252

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

136

C₁₆

—

137

C₂³

—

138

Q₈

—

136

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

136

—

145

C₂×C₄

#10587616

yes

characteristic

146

C₂×C₄

C₂×#63228

yes

characteristic

147

C₂×C₄

#10587635

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

#10587498

yes

—

153

C₈

#10587595

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#11763004

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

272

—

161

C₄

#11763067

yes

characteristic

162

C₄

#11763084

yes

characteristic

163

C₄

C₂×#10587498

yes

characteristic

164

C₄

#11763249

yes

characteristic

165

C₂

—

166

C₂

#14528273

yes

characteristic

167

C₂

#14530829

yes

characteristic

168

C₂

#14530860

yes

characteristic

	subgroup	G/H
center	C₂²	#11763004
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1589	C₂
socle	C₂²×C₁₇	#459
prime radical	C₂×D₈×C₁₇	D₄
hypercenter	#1589	D₁₇
hypocenter	C₁₇	#12499
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1589

subgroup

G/H

center

C₂²

#11763004

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1589

C₂

socle

C₂²×C₁₇

#459

prime radical

C₂×D₈×C₁₇

D₄

D₁₇

C₁₇

G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1589

N	H ≅ G/N	type
2. #14528402	C₂	semidirect
3. #14526848	C₂	semidirect
4. #14526843	C₂	semidirect
118. C₁₇	#12499	semidirect

H ≅ G/N

type

2. #14528402

C₂

semidirect

3. #14526848

C₂

semidirect

4. #14526843

C₂

semidirect

118. C₁₇

#12499

semidirect

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

subgroup	index
#14530931	2
#14528402	2
#14526848	2
#14526843	2
C₁₇×#1589	2
#12499	17

subgroup

index

C₁₇×#1589

Subgroups — #16948953

Conjugacy classes of subgroups

Complements

Maximal subgroups