Group #16949082 — subgroups

Subgroups — #16949082

Order 4352.

296 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14531041	C₂	1	2	yes	maximal
2	#14531011	C₂	1	2	yes	maximal, complemented
3	#14528376	C₂	1	1	yes	maximal, characteristic
4	#14526702	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10587499	—	2	4	no	—
7	#11763214	C₂²	1	1	yes	characteristic
8	#11763186	C₂²	1	1	yes	characteristic
9	#11763182	—	2	1	no	—
10	#11763156	—	2	1	no	—
11	#11762613	—	2	4	no	—
12	C₄×#63214	C₂²	1	2	yes	—
13	C₁₇×#500	—	2	1	no	—
14	C₁₇×#500	—	2	1	no	—
15	C₁₇×#494	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	—	2	2	no	—
18	C₂×Q₁₆×C₁₇	—	2	2	no	—
19	C₁₇×C₄⋊Q₈	—	2	2	no	—
20	C₂×D₁₃₆	—	2	2	no	—
21	C₂×#63219	—	4	2	no	—
22	#10587560	—	2	1	no	—
23	#10587555	—	2	1	no	—
24	C₁₇×#107	—	4	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	#10587499	—	4	8	no	—
30	#10587492	—	4	2	no	—
31	#10587489	—	2	2	no	—
32	C₂×#63214	D₄	1	4	yes	—
33	C₂×Q₈×C₁₇	—	4	2	no	—
34	C₂×D₆₈	—	4	1	no	—
35	C₂×D₆₈	—	4	1	no	—
36	C₂×D₆₈	—	4	1	no	—
37	C₂×Dic₃₄	—	4	1	no	—
38	Q₁₆×C₁₇	—	4	4	no	—
39	Q₁₆×C₁₇	—	4	4	no	—
40	C₂×C₈×C₁₇	D₈	1	2	yes	—
41	C₂×C₈×C₁₇	D₈	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₄⋊C₄×C₁₇	—	4	2	no	—
45	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₁₇	—	4	1	no	—
47	C₄⋊Dic₁₇	—	4	1	no	—
48	D₁₃₆	—	4	4	no	—
49	#63219	—	8	4	no	—
50	#63214	—	2	4	no	—
51	#12526	—	17	1	no	maximal, Sylow
52	C₂²×D₁₇	—	8	1	no	—
53	Q₈×C₁₇	—	8	2	no	—
54	Q₈×C₁₇	—	8	2	no	—
55	C₂×C₄×C₁₇	—	8	2	no	—
56	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₁₇	—	8	1	no	—
60	D₆₈	—	8	2	no	—
61	D₆₈	—	8	2	no	—
62	D₆₈	—	8	2	no	—
63	Dic₃₄	—	8	2	no	—
64	C₈×C₁₇	—	2	2	no	—
65	C₈×C₁₇	SD₃₂	1	4	yes	complemented
66	C₈×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
67	#2248	—	17	2	no	—
68	#2241	—	17	2	no	—
69	#1713	D₁₇	1	1	yes	characteristic, complemented
70	#1706	—	17	1	no	—
71	#1312	—	17	1	no	—
72	C₂²×C₁₇	#493	1	1	yes	characteristic
73	C₂×D₁₇	—	16	1	no	—
74	C₂×D₁₇	—	16	1	no	—
75	C₂×D₁₇	—	16	1	no	—
76	C₄×C₁₇	—	16	2	no	—
77	C₄×C₁₇	#496	1	2	yes	—
78	C₄×C₁₇	C₂×SD₃₂	1	2	yes	—
79	C₄×C₁₇	#510	1	2	yes	—
80	Dic₁₇	—	16	1	no	—
81	C₂×SD₃₂	—	34	4	no	—
82	#500	—	2	1	no	—
83	#500	—	2	1	no	—
84	#500	—	17	1	no	—
85	#494	C₂×D₁₇	1	2	yes	—
86	#493	—	17	1	no	—
87	#492	—	34	1	no	—
88	#486	—	34	1	no	—
89	#358	—	34	4	no	—
90	C₄×C₁₆	—	17	2	no	—
91	C₈²	C₂×D₁₇	1	1	yes	characteristic
92	C₂×C₁₇	#1721	1	1	yes	characteristic
93	C₂×C₁₇	#2248	1	1	yes	characteristic
94	C₂×C₁₇	#2251	1	1	yes	characteristic
95	D₁₇	—	32	1	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₂×SD₁₆	—	68	2	no	—
99	C₂×D₈	—	34	2	no	—
100	C₄⋊Q₈	—	2	2	no	—
101	C₄⋊Q₈	—	34	1	no	—
102	#127	—	34	1	no	—
103	SD₃₂	—	68	8	no	—
104	C₂×C₁₆	—	17	4	no	—
105	#107	—	4	2	no	—
106	#107	—	4	2	no	—
107	#107	—	34	2	no	—
108	#102	—	68	2	no	—
109	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
111	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
112	C₁₇	#12526	1	1	yes	characteristic, complemented, Sylow
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	68	1	no	—
115	C₂×D₄	—	68	1	no	—
116	C₂×D₄	—	68	1	no	—
117	C₂×D₄	—	68	1	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	4	no	—
120	SD₁₆	—	136	4	no	—
121	D₈	—	68	4	no	—
122	C₂×C₈	#63228	1	2	yes	—
123	C₂×C₈	#63228	1	2	yes	—
124	C₂×C₈	D₄×D₁₇	1	2	yes	—
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	68	1	no	—
128	C₄⋊C₄	—	68	1	no	—
129	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
130	C₁₆	—	34	4	no	—
131	C₂³	—	136	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	2	no	—
134	Q₈	—	136	2	no	—
135	D₄	—	136	2	no	—
136	D₄	—	136	2	no	—
137	D₄	—	136	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	136	1	no	—
140	C₂×C₄	C₂×#63228	1	1	yes	characteristic
141	C₂×C₄	C₂×#63228	1	1	yes	characteristic
142	C₂×C₄	#10587626	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#10587499	1	4	yes	complemented
145	C₈	#10587597	1	4	yes	—
146	C₂²	—	272	1	no	—
147	C₂²	—	272	1	no	—
148	C₂²	—	272	1	no	—
149	C₂²	#11763185	1	1	yes	characteristic
150	C₄	—	16	2	no	—
151	C₄	—	272	1	no	—
152	C₄	#11763197	1	2	yes	—
153	C₄	C₂×#10587499	1	2	yes	—
154	C₄	#11763250	1	2	yes	—
155	C₂	—	544	1	no	—
156	C₂	#14528451	1	1	yes	characteristic
157	C₂	#14531041	1	1	yes	characteristic
158	C₂	#14531053	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531041

C₂

yes

maximal

#14531011

C₂

yes

maximal, complemented

#14528376

C₂

yes

maximal, characteristic

#14526702

C₂

yes

maximal, characteristic, complemented

C₁₇×#1713

C₂

yes

maximal, characteristic, complemented

C₂×#10587499

—

#11763214

C₂²

yes

characteristic

#11763186

C₂²

yes

—

—

—

C₂²

yes

—

C₁₇×#500

—

C₁₇×#500

—

C₁₇×#494

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₂×D₁₃₆

—

C₂×#63219

—

#10587560

—

#10587555

—

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

—

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

—

D₁₃₆

—

—

—

—

C₂²×D₁₇

—

Q₈×C₁₇

—

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

SD₃₂

yes

complemented

C₈×C₁₇

AGL(1,ℤ/8)

yes

—

#2248

—

#2241

—

#1713

D₁₇

yes

characteristic, complemented

#1706

—

#1312

—

C₂²×C₁₇

#493

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#496

yes

—

C₄×C₁₇

C₂×SD₃₂

yes

—

C₄×C₁₇

#510

yes

—

Dic₁₇

—

C₂×SD₃₂

—

—

—

—

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

—

102

#127

—

103

SD₃₂

—

104

C₂×C₁₆

—

105

#107

—

106

#107

—

107

#107

—

108

#102

—

109

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

110

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

111

C₄×C₈

C₂²×D₁₇

yes

characteristic

112

C₁₇

#12526

yes

characteristic, complemented, Sylow

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

Q₁₆

—

119

Q₁₆

—

120

SD₁₆

—

136

—

121

D₈

—

122

C₂×C₈

#63228

yes

—

123

C₂×C₈

#63228

yes

—

124

C₂×C₈

D₄×D₁₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

130

C₁₆

—

131

C₂³

—

136

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

136

—

135

D₄

—

136

—

136

D₄

—

136

—

137

D₄

—

136

—

138

C₂×C₄

—

139

C₂×C₄

—

136

—

140

C₂×C₄

C₂×#63228

yes

characteristic

141

C₂×C₄

C₂×#63228

yes

characteristic

142

C₂×C₄

#10587626

yes

characteristic

143

C₈

—

144

C₈

#10587499

yes

complemented

145

C₈

#10587597

yes

—

146

C₂²

—

272

—

147

C₂²

—

272

—

148

C₂²

—

272

—

149

C₂²

#11763185

yes

characteristic

150

C₄

—

151

C₄

—

272

—

152

C₄

#11763197

yes

—

153

C₄

C₂×#10587499

yes

—

154

C₄

#11763250

yes

—

155

C₂

—

544

—

156

C₂

#14528451

yes

characteristic

157

C₂

#14531041

yes

characteristic

158

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	#12526
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
2. #14531011	C₂	semidirect
4. #14526702	C₂	semidirect
5. C₁₇×#1713	C₂	semidirect
65. C₈×C₁₇	SD₃₂	semidirect
69. #1713	D₁₇	semidirect
112. C₁₇	#12526	semidirect
144. C₈	#10587499	semidirect

H ≅ G/N

type

2. #14531011

C₂

semidirect

4. #14526702

C₂

semidirect

5. C₁₇×#1713

C₂

semidirect

65. C₈×C₁₇

SD₃₂

semidirect

69. #1713

D₁₇

semidirect

112. C₁₇

#12526

semidirect

144. C₈

#10587499

semidirect

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

subgroup	index
#14531041	2
#14531011	2
#14528376	2
#14526702	2
C₁₇×#1713	2
#12526	17

subgroup

index

C₁₇×#1713

Subgroups — #16949082

Conjugacy classes of subgroups

Complements

Maximal subgroups