Group #16949025 — subgroups

Subgroups — #16949025

Order 4352.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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	#14531013	C₂	1	2	yes	maximal, complemented
2	#14531009	C₂	1	2	yes	maximal
3	#14528390	C₂	1	1	yes	maximal, characteristic
4	#14526701	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763189	C₂²	1	1	yes	characteristic
7	#11763186	C₂²	1	1	yes	characteristic
8	#11763156	—	2	1	no	—
9	#11763156	—	2	1	no	—
10	#11762612	—	2	4	no	—
11	#11762609	—	2	4	no	—
12	C₄×#63214	C₂²	1	2	yes	—
13	C₁₇×#500	—	2	1	no	—
14	C₁₇×#500	C₂²	1	2	yes	—
15	C₁₇×#494	—	2	1	no	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	—	4	2	no	—
18	C₁₇×C₄⋊Q₈	—	2	2	no	—
19	C₂×Dic₆₈	—	2	2	no	—
20	C₂×D₁₃₆	—	2	2	no	—
21	#10587560	—	2	1	no	—
22	#10587555	—	2	1	no	—
23	C₁₇×#107	—	2	2	no	—
24	C₁₇×#107	—	2	2	no	—
25	C₁₇×#107	—	4	2	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	#10587492	—	4	2	no	—
30	#10587492	—	4	2	no	—
31	C₂×#63214	D₄	1	4	yes	—
32	C₂×Q₈×C₁₇	—	4	2	no	—
33	C₂×D₆₈	—	4	1	no	—
34	C₂×D₆₈	—	4	1	no	—
35	C₂×D₆₈	—	4	1	no	—
36	C₂×Dic₃₄	—	4	1	no	—
37	Q₁₆×C₁₇	—	8	4	no	—
38	C₂×C₈×C₁₇	D₈	1	2	yes	—
39	C₂×C₈×C₁₇	D₈	1	2	yes	—
40	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₁₇	—	4	2	no	—
42	C₄⋊C₄×C₁₇	—	4	2	no	—
43	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₁₇	—	4	1	no	—
45	C₄⋊Dic₁₇	—	4	1	no	—
46	Dic₆₈	—	4	4	no	—
47	D₁₃₆	—	4	4	no	—
48	#63214	—	2	4	no	—
49	#12512	—	17	1	no	maximal, Sylow
50	C₂²×D₁₇	—	8	1	no	—
51	Q₈×C₁₇	—	8	4	no	—
52	C₂×C₄×C₁₇	—	8	2	no	—
53	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₁₇	—	8	1	no	—
57	D₆₈	—	8	2	no	—
58	D₆₈	—	8	2	no	—
59	D₆₈	—	8	2	no	—
60	Dic₃₄	—	8	2	no	—
61	C₈×C₁₇	—	2	2	no	—
62	C₈×C₁₇	—	2	2	no	—
63	C₈×C₁₇	#135	1	4	yes	—
64	#2241	—	17	2	no	—
65	#2240	—	17	2	no	—
66	#1713	D₁₇	1	1	yes	characteristic, complemented
67	#1709	—	17	1	no	—
68	#1311	—	17	1	no	—
69	C₂²×C₁₇	#493	1	1	yes	characteristic
70	C₂×D₁₇	—	16	1	no	—
71	C₂×D₁₇	—	16	1	no	—
72	C₂×D₁₇	—	16	1	no	—
73	C₄×C₁₇	—	16	2	no	—
74	C₄×C₁₇	#495	1	2	yes	—
75	C₄×C₁₇	#509	1	2	yes	—
76	C₄×C₁₇	#510	1	2	yes	—
77	Dic₁₇	—	16	1	no	—
78	#500	—	2	1	no	—
79	#500	C₂×D₁₇	1	2	yes	—
80	#494	—	2	1	no	—
81	#494	—	17	1	no	—
82	#493	—	17	1	no	—
83	#486	—	34	1	no	—
84	#486	—	34	1	no	—
85	#358	—	34	4	no	—
86	#357	—	34	4	no	—
87	C₄×C₁₆	—	17	2	no	—
88	C₈²	C₂×D₁₇	1	1	yes	characteristic
89	C₂×C₁₇	#1709	1	1	yes	characteristic
90	C₂×C₁₇	#2250	1	1	yes	characteristic
91	C₂×C₁₇	#2251	1	1	yes	characteristic
92	D₁₇	—	32	1	no	—
93	C₂×Q₁₆	—	4	2	no	—
94	C₂×Q₁₆	—	34	2	no	—
95	C₂×D₈	—	34	2	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄⋊Q₈	—	34	1	no	—
98	#127	—	34	1	no	—
99	C₂×C₁₆	—	17	4	no	—
100	#107	—	2	2	no	—
101	#107	—	2	2	no	—
102	#107	—	4	2	no	—
103	#102	—	68	2	no	—
104	#102	—	68	2	no	—
105	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
108	C₁₇	#12512	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	4	2	no	—
110	C₂×Q₈	—	68	1	no	—
111	C₂×D₄	—	68	1	no	—
112	C₂×D₄	—	68	1	no	—
113	C₂×D₄	—	68	1	no	—
114	Q₁₆	—	8	4	no	—
115	Q₁₆	—	68	4	no	—
116	D₈	—	68	4	no	—
117	C₂×C₈	#63228	1	2	yes	—
118	C₂×C₈	#63228	1	2	yes	—
119	C₂×C₈	D₄×D₁₇	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	68	1	no	—
123	C₄⋊C₄	—	68	1	no	—
124	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
125	C₁₆	—	34	4	no	—
126	C₂³	—	136	1	no	—
127	Q₈	—	8	4	no	—
128	Q₈	—	136	2	no	—
129	D₄	—	136	2	no	—
130	D₄	—	136	2	no	—
131	D₄	—	136	2	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	136	1	no	—
134	C₂×C₄	C₂×#63228	1	1	yes	characteristic
135	C₂×C₄	C₂×#63228	1	1	yes	characteristic
136	C₂×C₄	#10587626	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#10587605	1	4	yes	—
140	C₂²	—	272	1	no	—
141	C₂²	—	272	1	no	—
142	C₂²	—	272	1	no	—
143	C₂²	#11763185	1	1	yes	characteristic
144	C₄	—	16	2	no	—
145	C₄	—	272	1	no	—
146	C₄	#11763194	1	2	yes	—
147	C₄	#11763245	1	2	yes	—
148	C₄	#11763250	1	2	yes	—
149	C₂	—	544	1	no	—
150	C₂	#14528391	1	1	yes	characteristic
151	C₂	#14531048	1	1	yes	characteristic
152	C₂	#14531053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531013

C₂

yes

maximal, complemented

#14531009

C₂

yes

maximal

#14528390

C₂

yes

maximal, characteristic

#14526701

C₂

yes

maximal, characteristic, complemented

C₁₇×#1713

C₂

yes

maximal, characteristic, complemented

#11763189

C₂²

yes

characteristic

#11763186

C₂²

yes

—

—

—

—

C₂²

yes

—

C₁₇×#500

—

C₁₇×#500

C₂²

yes

—

C₁₇×#494

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₂×Dic₆₈

—

C₂×D₁₃₆

—

#10587560

—

#10587555

—

C₁₇×#107

—

C₁₇×#107

—

C₁₇×#107

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

D₁₃₆

—

#63214

—

#12512

—

maximal, Sylow

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#135

yes

—

#2241

—

#2240

—

#1713

D₁₇

yes

characteristic, complemented

#1709

—

#1311

—

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#495

yes

—

C₄×C₁₇

#509

yes

—

C₄×C₁₇

#510

yes

—

Dic₁₇

—

#500

—

#500

C₂×D₁₇

yes

—

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

100

#107

—

101

#107

—

102

#107

—

103

#102

—

104

#102

—

105

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

106

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

107

C₄×C₈

C₂²×D₁₇

yes

characteristic

108

C₁₇

#12512

yes

characteristic, complemented, Sylow

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

Q₁₆

—

115

Q₁₆

—

116

D₈

—

117

C₂×C₈

#63228

yes

—

118

C₂×C₈

#63228

yes

—

119

C₂×C₈

D₄×D₁₇

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

125

C₁₆

—

126

C₂³

—

136

—

127

Q₈

—

128

Q₈

—

136

—

129

D₄

—

136

—

130

D₄

—

136

—

131

D₄

—

136

—

132

C₂×C₄

—

133

C₂×C₄

—

136

—

134

C₂×C₄

C₂×#63228

yes

characteristic

135

C₂×C₄

C₂×#63228

yes

characteristic

136

C₂×C₄

#10587626

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#10587605

yes

—

140

C₂²

—

272

—

141

C₂²

—

272

—

142

C₂²

—

272

—

143

C₂²

#11763185

yes

characteristic

144

C₄

—

145

C₄

—

272

—

146

C₄

#11763194

yes

—

147

C₄

#11763245

yes

—

148

C₄

#11763250

yes

—

149

C₂

—

544

—

150

C₂

#14528391

yes

characteristic

151

C₂

#14531048

yes

characteristic

152

C₂

#14531053

yes

characteristic

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1713	C₂
socle	C₂²×C₁₇	#493
prime radical	#11763186	C₂²
hypercenter	#1713	D₁₇
hypocenter	C₁₇	#12512
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1713

C₂

C₂²×C₁₇

C₂²

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #14531013	C₂	semidirect
4. #14526701	C₂	semidirect
5. C₁₇×#1713	C₂	semidirect
66. #1713	D₁₇	semidirect
108. C₁₇	#12512	semidirect

H ≅ G/N

type

1. #14531013

C₂

semidirect

4. #14526701

C₂

semidirect

5. C₁₇×#1713

C₂

semidirect

66. #1713

D₁₇

semidirect

108. C₁₇

#12512

semidirect

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

subgroup	index
#14531013	2
#14531009	2
#14528390	2
#14526701	2
C₁₇×#1713	2
#12512	17

subgroup

index

C₁₇×#1713

Subgroups — #16949025

Conjugacy classes of subgroups

Complements

Maximal subgroups