Group #17023653 — subgroups

Subgroups — #17023653

Order 4352.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 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	#14539529	C₂	1	3	yes	maximal, complemented
2	#14539159	C₂	1	3	yes	maximal
3	#14528444	C₂	1	6	yes	maximal, complemented
4	#14528396	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763366	C₂²	1	3	yes	—
7	C₂×#10587560	C₂²	1	1	yes	characteristic
8	#11763207	C₂²	1	3	yes	complemented
9	#11763189	C₂²	1	3	yes	complemented
10	#11763174	—	2	6	no	—
11	#11763160	C₂²	1	6	yes	complemented
12	#11763156	C₂²	1	6	yes	—
13	#11763156	C₂²	1	6	yes	—
14	#11763067	—	2	6	no	—
15	#11762917	—	2	6	no	—
16	#11762815	—	2	6	no	—
17	C₄×M₁₆×C₁₇	C₂²	1	3	yes	—
18	C₁₇×#322	C₂²	1	3	yes	complemented
19	C₈²×C₁₇	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₆₈	—	2	3	no	—
21	C₂×M₁₆×C₁₇	D₄	1	6	yes	—
22	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
23	#10587612	—	2	3	no	—
24	#10587610	—	2	6	no	—
25	C₂×Dic₆₈	—	2	6	no	—
26	#10587560	—	2	3	no	—
27	#10587560	C₂³	1	2	yes	—
28	#10587556	—	2	3	no	—
29	#10587555	C₂³	1	6	yes	—
30	C₄×Dic₃₄	—	2	6	no	—
31	C₁₇×#97	D₄	1	6	yes	—
32	C₄×C₈×C₁₇	C₂³	1	3	yes	—
33	C₄×C₈×C₁₇	C₂³	1	3	yes	—
34	#10587492	—	2	12	no	—
35	#10587492	—	2	12	no	—
36	#10587488	—	2	6	no	—
37	#10587487	—	2	12	no	—
38	C₂³×D₁₇	—	4	1	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
40	C₂×D₆₈	—	2	6	no	—
41	C₂×D₆₈	—	4	12	no	—
42	C₂×D₆₈	—	4	3	no	—
43	C₂×Dic₃₄	—	2	6	no	—
44	M₁₆×C₁₇	—	4	6	no	—
45	C₂×C₈×C₁₇	C₂×D₄	1	6	yes	—
46	C₂×C₈×C₁₇	C₂×D₄	1	6	yes	—
47	C₄²×C₁₇	C₂×D₄	1	3	yes	—
48	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
49	C₂²⋊Dic₁₇	—	4	3	no	—
50	C₄⋊Dic₁₇	—	2	6	no	—
51	C₄⋊Dic₁₇	—	2	6	no	—
52	#63225	—	4	6	no	—
53	C₄×Dic₁₇	—	4	3	no	—
54	Dic₆₈	—	4	12	no	—
55	#19753	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	#127	1	1	yes	characteristic
57	C₂²×D₁₇	—	4	2	no	—
58	C₂²×D₁₇	—	8	6	no	—
59	C₂×C₄×C₁₇	—	2	6	no	—
60	C₂×C₄×C₁₇	#127	1	3	yes	—
61	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
62	C₂×Dic₁₇	—	4	6	no	—
63	D₆₈	—	4	12	no	—
64	D₆₈	—	8	6	no	—
65	Dic₃₄	—	4	12	no	—
66	C₈×C₁₇	—	2	6	no	—
67	C₈×C₁₇	—	2	6	no	—
68	#3185	—	17	3	no	—
69	#3138	—	17	3	no	—
70	#1720	—	17	6	no	—
71	#1711	—	17	2	no	—
72	#1455	D₁₇	1	1	yes	characteristic, complemented
73	C₂²×C₁₇	—	2	3	no	—
74	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
75	C₂×D₁₇	—	8	6	no	—
76	C₂×D₁₇	—	16	4	no	—
77	C₄×C₁₇	—	2	3	no	—
78	C₄×C₁₇	#575	1	6	yes	—
79	Dic₁₇	—	8	6	no	—
80	#533	—	17	3	no	—
81	C₂×#127	—	17	1	no	—
82	#498	—	17	3	no	—
83	#494	—	17	3	no	—
84	#490	—	34	6	no	—
85	#487	—	17	6	no	—
86	#486	—	17	6	no	—
87	#486	—	17	6	no	—
88	#470	—	34	6	no	—
89	#441	—	34	6	no	—
90	#418	—	34	6	no	—
91	C₄×M₁₆	C₂×D₁₇	1	3	yes	—
92	#322	C₂×D₁₇	1	3	yes	complemented
93	C₈²	C₂×D₁₇	1	1	yes	characteristic, complemented
94	C₂×C₁₇	—	4	1	no	—
95	C₂×C₁₇	#3153	1	3	yes	—
96	D₁₇	—	16	2	no	—
97	C₂²×D₄	—	34	3	no	—
98	C₂×Q₁₆	—	34	6	no	—
99	C₂×M₁₆	D₆₈	1	6	yes	—
100	C₄⋊Q₈	—	17	6	no	—
101	#127	—	17	2	no	—
102	#127	—	34	3	no	—
103	#125	—	34	3	no	—
104	C₂²⋊Q₈	—	34	6	no	—
105	C₄×Q₈	—	34	6	no	—
106	#117	—	34	3	no	—
107	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
108	#106	—	34	6	no	—
109	#103	—	34	12	no	—
110	#102	—	34	12	no	—
111	#102	—	34	12	no	—
112	#97	D₆₈	1	6	yes	—
113	C₄×C₈	C₂²×D₁₇	1	3	yes	—
114	C₄×C₈	C₂²×D₁₇	1	3	yes	—
115	C₁₇	#19753	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	—	68	1	no	—
117	C₂×Q₈	—	34	6	no	—
118	C₂×D₄	—	34	6	no	—
119	C₂×D₄	—	68	12	no	—
120	C₂×D₄	—	68	3	no	—
121	C₂²×C₄	C₂×D₆₈	1	3	yes	—
122	Q₁₆	—	68	12	no	—
123	M₁₆	—	4	6	no	—
124	C₂×C₈	C₂×D₆₈	1	6	yes	—
125	C₂×C₈	C₂×D₆₈	1	6	yes	—
126	C₄⋊C₄	—	34	6	no	—
127	C₄⋊C₄	—	34	6	no	—
128	C₄⋊C₄	—	68	6	no	—
129	C₂²⋊C₄	—	68	3	no	—
130	C₄²	—	68	3	no	—
131	C₄²	C₂×D₆₈	1	3	yes	—
132	C₄²	C₂³×D₁₇	1	1	yes	characteristic
133	C₂³	—	68	2	no	—
134	C₂³	—	136	6	no	—
135	C₂³	#10587560	1	1	yes	characteristic
136	Q₈	—	68	12	no	—
137	D₄	—	68	12	no	—
138	D₄	—	136	6	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	68	6	no	—
141	C₂×C₄	#10587560	1	3	yes	—
142	C₂×C₄	C₂²×D₆₈	1	3	yes	—
143	C₈	—	2	6	no	—
144	C₈	—	2	6	no	—
145	C₂²	—	2	3	no	—
146	C₂²	—	272	4	no	—
147	C₂²	—	136	6	no	—
148	C₂²	C₂×#10587560	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	136	6	no	—
151	C₄	#11763609	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	272	2	no	—
154	C₂	#14539250	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539529

C₂

yes

maximal, complemented

#14539159

C₂

yes

maximal

#14528444

C₂

yes

maximal, complemented

#14528396

C₂

yes

maximal, complemented

C₁₇×#1455

C₂

yes

maximal, characteristic, complemented

#11763366

C₂²

yes

—

C₂×#10587560

C₂²

yes

characteristic

#11763207

C₂²

yes

complemented

#11763189

C₂²

yes

complemented

#11763174

—

#11763160

C₂²

yes

complemented

#11763156

C₂²

yes

—

#11763156

C₂²

yes

—

#11763067

—

#11762917

—

#11762815

—

C₄×M₁₆×C₁₇

C₂²

yes

—

C₁₇×#322

C₂²

yes

complemented

C₈²×C₁₇

C₂²

yes

characteristic, complemented

C₂²×D₆₈

—

C₂×M₁₆×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

#10587612

—

#10587610

—

C₂×Dic₆₈

—

#10587560

—

#10587560

C₂³

yes

—

#10587556

—

#10587555

C₂³

yes

—

C₄×Dic₃₄

—

C₁₇×#97

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

—

C₂³×D₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂⁴

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

Dic₆₈

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#127

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#575

yes

—

Dic₁₇

—

—

—

—

—

—

—

—

—

—

—

—

C₄×M₁₆

C₂×D₁₇

yes

—

#322

C₂×D₁₇

yes

complemented

C₈²

C₂×D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#3153

yes

—

D₁₇

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×M₁₆

D₆₈

yes

—

100

C₄⋊Q₈

—

101

#127

—

102

#127

—

103

#125

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

—

106

#117

—

107

C₂×C₄²

C₂²×D₁₇

yes

characteristic

108

#106

—

109

#103

—

110

#102

—

111

#102

—

112

#97

D₆₈

yes

—

113

C₄×C₈

C₂²×D₁₇

yes

—

114

C₄×C₈

C₂²×D₁₇

yes

—

115

C₁₇

#19753

yes

characteristic, complemented, Sylow

116

C₂⁴

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂²×C₄

C₂×D₆₈

yes

—

122

Q₁₆

—

123

M₁₆

—

124

C₂×C₈

C₂×D₆₈

yes

—

125

C₂×C₈

C₂×D₆₈

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₄²

C₂×D₆₈

yes

—

132

C₄²

C₂³×D₁₇

yes

characteristic

133

C₂³

—

134

C₂³

—

136

—

135

C₂³

#10587560

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

136

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#10587560

yes

—

142

C₂×C₄

C₂²×D₆₈

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

272

—

147

C₂²

—

136

—

148

C₂²

C₂×#10587560

yes

characteristic

149

C₄

—

150

C₄

—

136

—

151

C₄

#11763609

yes

—

152

C₂

—

153

C₂

—

272

—

154

C₂

#14539250

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587560
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#1455	C₂
socle	C₂²×C₁₇	C₂×#127
prime radical	C₂×#10587560	C₂²
hypercenter	#1455	D₁₇
hypocenter	C₁₇	#19753
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

subgroup

G/H

center

C₂²

C₂×#10587560

commutator, cosocle

C₄²×C₁₇

C₂⁴

Frattini

C₄²

C₂³×D₁₇

Fitting

C₁₇×#1455

C₂

C₂²×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

N	H ≅ G/N	type
1. #14539529	C₂	semidirect
3. #14528444	C₂	semidirect
4. #14528396	C₂	semidirect
5. C₁₇×#1455	C₂	semidirect
8. #11763207	C₂²	semidirect
9. #11763189	C₂²	semidirect
11. #11763160	C₂²	semidirect
18. C₁₇×#322	C₂²	semidirect
19. C₈²×C₁₇	C₂²	semidirect
72. #1455	D₁₇	semidirect
92. #322	C₂×D₁₇	semidirect
93. C₈²	C₂×D₁₇	semidirect
115. C₁₇	#19753	semidirect

H ≅ G/N

type

1. #14539529

C₂

semidirect

3. #14528444

C₂

semidirect

4. #14528396

C₂

semidirect

5. C₁₇×#1455

C₂

semidirect

8. #11763207

C₂²

semidirect

9. #11763189

C₂²

semidirect

11. #11763160

C₂²

semidirect

18. C₁₇×#322

C₂²

semidirect

19. C₈²×C₁₇

C₂²

semidirect

72. #1455

D₁₇

semidirect

92. #322

C₂×D₁₇

semidirect

93. C₈²

C₂×D₁₇

semidirect

115. C₁₇

#19753

semidirect

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

subgroup	index
#14539529	2
#14539159	2
#14528444	2
#14528396	2
C₁₇×#1455	2
#19753	17

subgroup

index

C₁₇×#1455

Subgroups — #17023653

Conjugacy classes of subgroups

Complements

Maximal subgroups