Group #16933717 — subgroups

Subgroups — #16933717

Order 4352.

370 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14528915	C₂	1	1	yes	maximal, characteristic
2	#14527348	C₂	1	4	yes	maximal
3	#14527076	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	2	yes	—
6	#11762675	C₂²	1	2	yes	—
7	#11762572	—	2	2	no	—
8	#11762546	—	2	8	no	—
9	#11762531	—	2	2	no	—
10	C₁₇×#533	C₂²	1	2	yes	—
11	C₁₇×#500	C₄	1	2	yes	complemented
12	C₁₇×#499	—	2	2	no	—
13	C₁₇×#498	C₄	1	2	yes	complemented
14	C₁₇×#427	—	2	2	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	D₄	1	2	yes	—
17	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
18	C₁₇×#125	—	2	2	no	—
19	C₁₇×C₂²⋊Q₈	—	2	4	no	—
20	C₄×Q₈×C₁₇	—	2	4	no	—
21	C₁₇×#117	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₁₇×#107	—	2	4	no	—
25	C₁₇×#106	—	2	4	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₇	D₄	1	2	yes	—
29	#10587502	—	2	4	no	—
30	C₂²⋊#3622	D₄	1	4	yes	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄×#3622	—	2	2	no	—
33	C₂×Q₈×C₁₇	—	2	4	no	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₁₇	—	4	2	no	—
37	C₂×C₈×C₁₇	—	2	4	no	—
38	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
40	C₄⋊C₄×C₁₇	—	2	4	no	—
41	C₄⋊C₄×C₁₇	—	2	4	no	—
42	C₄⋊C₄×C₁₇	—	4	4	no	—
43	C₂²⋊C₄×C₁₇	—	4	2	no	—
44	C₄²×C₁₇	—	4	2	no	—
45	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₇	#41	1	2	yes	—
48	C₄⋊Dic₁₇	—	4	4	no	—
49	C₂×#3622	—	2	4	no	—
50	#9759	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	#127	1	1	yes	characteristic
52	Q₈×C₁₇	—	8	4	no	—
53	C₂×C₄×C₁₇	—	4	2	no	—
54	C₂×C₄×C₁₇	—	4	4	no	—
55	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₁₇	#121	1	2	yes	—
58	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#137	1	4	yes	—
60	C₂×Dic₁₇	—	8	4	no	—
61	C₂×Dic₁₇	—	8	2	no	—
62	C₈×C₁₇	—	2	2	no	—
63	C₈×C₁₇	—	2	2	no	—
64	#3622	—	4	4	no	—
65	#3160	D₁₇	1	1	yes	characteristic
66	#1835	—	17	1	no	—
67	#1518	—	17	4	no	—
68	#1452	—	17	1	no	—
69	C₂²×C₁₇	—	2	2	no	—
70	C₂²×C₁₇	#390	1	1	yes	characteristic
71	C₂²×C₁₇	#497	1	2	yes	—
72	C₄×C₁₇	—	2	2	no	—
73	C₄×C₁₇	—	2	2	no	—
74	C₄×C₁₇	—	8	4	no	—
75	C₄×C₁₇	#419	1	2	yes	—
76	C₄×C₁₇	#491	1	2	yes	—
77	Dic₁₇	—	16	2	no	—
78	#533	C₂×D₁₇	1	2	yes	—
79	#500	Dic₁₇	1	2	yes	complemented
80	#499	—	2	2	no	—
81	#498	Dic₁₇	1	2	yes	complemented
82	#431	—	17	2	no	—
83	#427	—	2	2	no	—
84	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
85	#382	—	17	2	no	—
86	#340	—	34	2	no	—
87	#328	—	34	8	no	—
88	#322	—	34	2	no	—
89	C₂×C₁₇	—	2	2	no	—
90	C₂×C₁₇	#1960	1	1	yes	characteristic
91	C₂×C₁₇	#1976	1	2	yes	—
92	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
93	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
94	#125	—	2	2	no	—
95	C₂²⋊Q₈	—	2	4	no	—
96	C₄×Q₈	—	2	4	no	—
97	#117	—	2	2	no	—
98	C₂×C₄⋊C₄	—	34	2	no	—
99	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
100	#107	—	2	4	no	—
101	#106	—	2	4	no	—
102	C₄⋊C₈	—	34	2	no	—
103	C₂²⋊C₈	—	17	4	no	—
104	C₄×C₈	—	34	2	no	—
105	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
108	#95	—	34	4	no	—
109	C₁₇	#9759	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	—	2	4	no	—
111	C₂²×C₄	—	68	2	no	—
112	C₂²×C₄	D₄×D₁₇	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	34	4	no	—
116	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
117	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	68	4	no	—
122	C₂²⋊C₄	—	4	2	no	—
123	C₄²	—	4	2	no	—
124	C₄²	#63254	1	2	yes	—
125	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
126	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
127	C₂³	#10587626	1	1	yes	characteristic
128	Q₈	—	8	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	136	4	no	—
132	C₂×C₄	—	136	2	no	—
133	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
134	C₂×C₄	#10587622	1	1	yes	characteristic
135	C₂×C₄	#10587624	1	2	yes	—
136	C₂×C₄	#10587628	1	4	yes	—
137	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	68	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#11762712	1	1	yes	characteristic
143	C₂²	#11763201	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	8	4	no	—
147	C₄	—	272	2	no	—
148	C₄	#11762818	1	2	yes	—
149	C₄	#11763177	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#14529605	1	1	yes	characteristic
152	C₂	#14529683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528915

C₂

yes

maximal, characteristic

#14527348

C₂

yes

maximal

#14527076

C₂

yes

maximal, characteristic

C₁₇×#3160

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762675

C₂²

yes

—

#11762572

—

#11762546

—

#11762531

—

C₁₇×#533

C₂²

yes

—

C₁₇×#500

C₄

yes

complemented

C₁₇×#499

—

C₁₇×#498

C₄

yes

complemented

C₁₇×#427

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₁₇×#125

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₁₇×#107

—

C₁₇×#106

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

#10587502

—

C₂²⋊#3622

D₄

yes

—

C₄⋊#3622

—

C₄×#3622

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3160

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#497

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#419

yes

—

C₄×C₁₇

#491

yes

—

Dic₁₇

—

#533

C₂×D₁₇

yes

—

#500

Dic₁₇

yes

complemented

#499

—

#498

Dic₁₇

yes

complemented

#431

—

#427

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#1960

yes

characteristic

C₂×C₁₇

#1976

yes

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₄⋊Q₈

C₂×Dic₁₇

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

100

#107

—

101

#106

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×Dic₁₇

yes

characteristic

106

C₄×C₈

C₂×Dic₁₇

yes

characteristic

107

C₄×C₈

C₂²⋊D₁₇

yes

—

108

#95

—

109

C₁₇

#9759

yes

characteristic, complemented, Sylow

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

D₄×D₁₇

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊Dic₁₇

yes

—

117

C₂×C₈

C₂²⋊Dic₁₇

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

#63254

yes

—

125

C₄²

C₂²×Dic₁₇

yes

characteristic

126

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

127

C₂³

#10587626

yes

characteristic

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

136

—

132

C₂×C₄

—

136

—

133

C₂×C₄

D₄×Dic₁₇

yes

—

134

C₂×C₄

#10587622

yes

characteristic

135

C₂×C₄

#10587624

yes

—

136

C₂×C₄

#10587628

yes

—

137

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

#11762712

yes

characteristic

143

C₂²

#11763201

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

272

—

148

C₄

#11762818

yes

—

149

C₄

#11763177

yes

—

150

C₂

—

151

C₂

#14529605

yes

characteristic

152

C₂

#14529683

yes

—

	subgroup	G/H
center	C₂²	#11762712
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#9759
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂²

#11762712

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#3160

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
11. C₁₇×#500	C₄	semidirect
13. C₁₇×#498	C₄	semidirect
79. #500	Dic₁₇	semidirect
81. #498	Dic₁₇	semidirect
109. C₁₇	#9759	semidirect

H ≅ G/N

type

11. C₁₇×#500

C₄

semidirect

13. C₁₇×#498

C₄

semidirect

79. #500

Dic₁₇

semidirect

81. #498

Dic₁₇

semidirect

109. C₁₇

#9759

semidirect

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

subgroup	index
#14528915	2
#14527348	2
#14527076	2
C₁₇×#3160	2
#9759	17

subgroup

index

C₁₇×#3160

Subgroups — #16933717

Conjugacy classes of subgroups

Complements

Maximal subgroups