Group #17000072 — subgroups

Subgroups — #17000072

Order 4352.

712 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 323 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	#14537415	C₂	1	3	yes	maximal
2	#14527087	C₂	1	3	yes	maximal, complemented
3	#14527085	C₂	1	6	yes	maximal, complemented
4	#14527083	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×D₁₇	C₂²	1	1	yes	characteristic
7	#11762864	C₄	1	12	yes	—
8	#11762860	C₂²	1	6	yes	—
9	#11762860	C₂²	1	6	yes	—
10	#11762832	C₄	1	6	yes	—
11	#11762772	C₄	1	6	yes	—
12	C₄×#63223	C₂²	1	3	yes	—
13	#11762531	C₂²	1	6	yes	complemented
14	#11762530	C₂²	1	3	yes	complemented
15	C₈×#3622	C₂²	1	3	yes	complemented
16	C₄×M₁₆×C₁₇	C₂²	1	3	yes	—
17	C₁₇×#322	C₂²	1	3	yes	complemented
18	C₈²×C₁₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄×D₁₇	C₂×C₄	1	3	yes	—
20	C₂×M₁₆×C₁₇	C₂×C₄	1	6	yes	—
21	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	C₂×C₄	1	3	yes	—
23	C₂×#63223	C₂×C₄	1	6	yes	—
24	C₄²×D₁₇	C₂×C₄	1	6	yes	—
25	C₄²×D₁₇	C₂³	1	2	yes	—
26	C₁₇×#97	C₂×C₄	1	6	yes	—
27	C₄×C₈×C₁₇	C₂³	1	3	yes	—
28	C₄×C₈×C₁₇	C₂³	1	3	yes	—
29	#10587491	C₂×C₄	1	12	yes	—
30	#10587491	C₂×C₄	1	12	yes	—
31	#10587485	C₂×C₄	1	12	yes	—
32	#10587485	C₂×C₄	1	12	yes	—
33	#10587474	C₂×C₄	1	6	yes	—
34	C₄×#3622	C₂³	1	6	yes	—
35	C₂³×D₁₇	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
37	C₂²×Dic₁₇	C₄²	1	3	yes	—
38	C₂×C₄×D₁₇	—	4	6	no	—
39	C₂×C₄×D₁₇	C₄²	1	6	yes	—
40	C₂×C₄×D₁₇	C₂²×C₄	1	6	yes	—
41	M₁₆×C₁₇	—	4	6	no	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
43	C₂×C₈×C₁₇	C₂²×C₄	1	6	yes	—
44	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
45	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
46	C₄×Dic₁₇	C₄²	1	6	yes	—
47	C₄×Dic₁₇	C₂²×C₄	1	6	yes	—
48	#63223	—	4	6	no	—
49	C₂×#3622	C₂²×C₄	1	12	yes	—
50	#17339	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
52	C₂²×D₁₇	—	4	3	no	—
53	C₂²×D₁₇	C₂×C₄²	1	2	yes	—
54	C₂×C₄×C₁₇	—	2	6	no	—
55	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
56	C₂×C₄×C₁₇	C₂³×C₄	1	3	yes	—
57	C₂×Dic₁₇	—	4	3	no	—
58	C₂×Dic₁₇	C₂×C₄²	1	6	yes	—
59	C₄×D₁₇	—	2	12	no	—
60	C₄×D₁₇	—	4	6	no	—
61	C₈×C₁₇	—	2	6	no	—
62	C₈×C₁₇	—	2	6	no	—
63	#3622	—	2	12	no	—
64	#2920	—	17	3	no	—
65	#1455	—	17	3	no	—
66	#1455	D₁₇	1	1	yes	characteristic, complemented
67	#1454	—	17	6	no	—
68	#1454	—	17	2	no	—
69	C₂²×C₁₇	—	2	3	no	—
70	C₂²×C₁₇	C₂²×C₄²	1	1	yes	characteristic
71	C₂×D₁₇	—	2	6	no	—
72	C₂×D₁₇	—	4	4	no	—
73	C₄×C₁₇	—	2	3	no	—
74	C₄×C₁₇	#568	1	6	yes	—
75	Dic₁₇	—	2	6	no	—
76	C₂²×C₄²	—	17	1	no	—
77	#432	—	17	12	no	—
78	#431	—	17	6	no	—
79	#431	—	17	6	no	—
80	#423	—	17	6	no	—
81	#407	—	17	6	no	—
82	C₄×M₁₆	—	17	3	no	—
83	C₄×M₁₆	C₂×D₁₇	1	3	yes	—
84	#322	—	17	3	no	—
85	#322	—	17	6	no	—
86	#322	C₂×D₁₇	1	3	yes	complemented
87	C₈²	—	17	3	no	—
88	C₈²	C₂×D₁₇	1	1	yes	characteristic, complemented
89	C₂×C₁₇	—	4	1	no	—
90	C₂×C₁₇	#2873	1	3	yes	—
91	D₁₇	—	4	2	no	—
92	C₂³×C₄	—	17	3	no	—
93	C₂×M₁₆	—	17	6	no	—
94	C₂×M₁₆	C₄×D₁₇	1	6	yes	—
95	C₂×C₄²	—	17	3	no	—
96	C₂×C₄²	—	17	6	no	—
97	C₂×C₄²	—	17	2	no	—
98	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
99	C₄⋊C₈	—	17	12	no	—
100	C₄⋊C₈	—	17	12	no	—
101	C₂²⋊C₈	—	17	12	no	—
102	C₂²⋊C₈	—	17	12	no	—
103	#97	—	17	6	no	—
104	#97	C₄×D₁₇	1	6	yes	—
105	C₄×C₈	—	17	6	no	—
106	C₄×C₈	C₂²×D₁₇	1	3	yes	—
107	C₄×C₈	C₂²×D₁₇	1	3	yes	—
108	C₁₇	#17339	1	1	yes	characteristic, complemented, Sylow
109	C₂⁴	—	17	1	no	—
110	C₂²×C₄	—	17	3	no	—
111	C₂²×C₄	—	17	6	no	—
112	C₂²×C₄	—	17	6	no	—
113	C₂²×C₄	—	68	6	no	—
114	C₂²×C₄	C₂×C₄×D₁₇	1	3	yes	—
115	M₁₆	—	4	6	no	—
116	M₁₆	—	68	6	no	—
117	C₂×C₈	—	17	12	no	—
118	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
119	C₂×C₈	C₂×C₄×D₁₇	1	6	yes	—
120	C₄²	—	17	6	no	—
121	C₄²	—	17	6	no	—
122	C₄²	C₂×C₄×D₁₇	1	3	yes	—
123	C₄²	C₂³×D₁₇	1	1	yes	characteristic
124	C₂³	—	17	2	no	—
125	C₂³	—	68	3	no	—
126	C₂³	C₄²×D₁₇	1	1	yes	characteristic
127	C₂×C₄	—	2	6	no	—
128	C₂×C₄	—	17	6	no	—
129	C₂×C₄	—	34	12	no	—
130	C₂×C₄	—	68	6	no	—
131	C₂×C₄	—	68	3	no	—
132	C₂×C₄	C₄²×D₁₇	1	3	yes	—
133	C₂×C₄	C₂²×C₄×D₁₇	1	3	yes	—
134	C₈	—	2	6	no	—
135	C₈	—	2	6	no	—
136	C₈	—	34	12	no	—
137	C₂²	—	2	3	no	—
138	C₂²	—	34	6	no	—
139	C₂²	—	68	4	no	—
140	C₂²	C₂×C₄²×D₁₇	1	1	yes	characteristic
141	C₄	—	2	3	no	—
142	C₄	—	34	6	no	—
143	C₄	#11763575	1	6	yes	—
144	C₂	—	4	1	no	—
145	C₂	—	68	2	no	—
146	C₂	#14537201	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14537415

C₂

yes

maximal

#14527087

C₂

yes

maximal, complemented

#14527085

C₂

yes

maximal, complemented

#14527083

C₂

yes

maximal, complemented

C₁₇×#1455

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₁₇

C₂²

yes

characteristic

#11762864

C₄

yes

—

#11762860

C₂²

yes

—

#11762860

C₂²

yes

—

#11762832

C₄

yes

—

#11762772

C₄

yes

—

C₄×#63223

C₂²

yes

—

#11762531

C₂²

yes

complemented

#11762530

C₂²

yes

complemented

C₈×#3622

C₂²

yes

complemented

C₄×M₁₆×C₁₇

C₂²

yes

—

C₁₇×#322

C₂²

yes

complemented

C₈²×C₁₇

C₂²

yes

characteristic, complemented

C₂²×C₄×D₁₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄×Dic₁₇

C₂×C₄

yes

—

C₂×#63223

C₂×C₄

yes

—

C₄²×D₁₇

C₂×C₄

yes

—

C₄²×D₁₇

C₂³

yes

—

C₁₇×#97

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

#10587491

C₂×C₄

yes

—

#10587491

C₂×C₄

yes

—

#10587485

C₂×C₄

yes

—

#10587485

C₂×C₄

yes

—

#10587474

C₂×C₄

yes

—

C₄×#3622

C₂³

yes

—

C₂³×D₁₇

C₄²

yes

characteristic

C₂²×C₄×C₁₇

C₂²×C₄

yes

—

C₂²×Dic₁₇

C₄²

yes

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₄²

yes

—

C₂×C₄×D₁₇

C₂²×C₄

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

—

C₄²×C₁₇

C₂⁴

yes

characteristic

C₄×Dic₁₇

C₄²

yes

—

C₄×Dic₁₇

C₂²×C₄

yes

—

#63223

—

C₂×#3622

C₂²×C₄

yes

—

#17339

—

maximal, Sylow

C₂³×C₁₇

C₂×C₄²

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

C₂×C₄²

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄²

yes

—

C₂×C₄×C₁₇

C₂³×C₄

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×C₄²

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

characteristic, complemented

#1454

—

#1454

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²×C₄²

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#568

yes

—

Dic₁₇

—

C₂²×C₄²

—

—

—

—

—

—

C₄×M₁₆

—

C₄×M₁₆

C₂×D₁₇

yes

—

#322

—

#322

—

#322

C₂×D₁₇

yes

complemented

C₈²

—

C₈²

C₂×D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#2873

yes

—

D₁₇

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₁₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

#97

—

104

#97

C₄×D₁₇

yes

—

105

C₄×C₈

—

106

C₄×C₈

C₂²×D₁₇

yes

—

107

C₄×C₈

C₂²×D₁₇

yes

—

108

C₁₇

#17339

yes

characteristic, complemented, Sylow

109

C₂⁴

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₄×D₁₇

yes

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

C₂×C₄×D₁₇

yes

—

119

C₂×C₈

C₂×C₄×D₁₇

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×C₄×D₁₇

yes

—

123

C₄²

C₂³×D₁₇

yes

characteristic

124

C₂³

—

125

C₂³

—

126

C₂³

C₄²×D₁₇

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₄²×D₁₇

yes

—

133

C₂×C₄

C₂²×C₄×D₁₇

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×C₄²×D₁₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#11763575

yes

—

144

C₂

—

145

C₂

—

146

C₂

#14537201

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂²×C₄²
Fitting	C₁₇×#1455	C₂
prime radical	C₂³×D₁₇	C₄²
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#1455	D₁₇
hypocenter	C₁₇	#17339
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ #1455

subgroup

G/H

center, Frattini

C₄²

C₂³×D₁₇

commutator, socle

C₂²×C₁₇

C₂²×C₄²

Fitting

C₁₇×#1455

C₂

prime radical

C₂³×D₁₇

C₄²

cosocle

C₄²×C₁₇

C₂⁴

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₄² ◁ #1455

N	H ≅ G/N	type
2. #14527087	C₂	semidirect
3. #14527085	C₂	semidirect
4. #14527083	C₂	semidirect
5. C₁₇×#1455	C₂	semidirect
13. #11762531	C₂²	semidirect
14. #11762530	C₂²	semidirect
15. C₈×#3622	C₂²	semidirect
17. C₁₇×#322	C₂²	semidirect
18. C₈²×C₁₇	C₂²	semidirect
66. #1455	D₁₇	semidirect
86. #322	C₂×D₁₇	semidirect
88. C₈²	C₂×D₁₇	semidirect
108. C₁₇	#17339	semidirect

H ≅ G/N

type

2. #14527087

C₂

semidirect

3. #14527085

C₂

semidirect

4. #14527083

C₂

semidirect

5. C₁₇×#1455

C₂

semidirect

13. #11762531

C₂²

semidirect

14. #11762530

C₂²

semidirect

15. C₈×#3622

C₂²

semidirect

17. C₁₇×#322

C₂²

semidirect

18. C₈²×C₁₇

C₂²

semidirect

66. #1455

D₁₇

semidirect

86. #322

C₂×D₁₇

semidirect

88. C₈²

C₂×D₁₇

semidirect

108. C₁₇

#17339

semidirect

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

subgroup	index
#14537415	2
#14527087	2
#14527085	2
#14527083	2
C₁₇×#1455	2
#17339	17

subgroup

index

C₁₇×#1455

Subgroups — #17000072

Conjugacy classes of subgroups

Complements

Maximal subgroups