Group #17016404 — subgroups

Subgroups — #17016404

Order 4352.

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

normal subgroups only

This group has 876 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	#14539160	C₂	1	3	yes	maximal
2	#14528086	C₂	1	8	yes	maximal, complemented
3	#14527379	C₂	1	3	yes	maximal, complemented
4	C₁₇×#3556	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10587560	C₂²	1	1	yes	characteristic
6	#11763175	C₂²	1	12	yes	—
7	#11763157	—	2	6	no	—
8	#11762813	—	2	6	no	—
9	C₄×#63223	C₂²	1	3	yes	—
10	#11762551	C₂²	1	12	yes	complemented
11	C₁₇×#564	C₂²	1	4	yes	complemented
12	C₁₇×#557	—	2	6	no	—
13	C₁₇×#541	—	2	6	no	—
14	C₁₇×#533	C₂²	1	3	yes	—
15	C₂²×D₆₈	—	2	3	no	—
16	C₁₇×C₄⋊Q₈	—	2	6	no	—
17	C₁₇×C₄⋊Q₈	D₄	1	6	yes	—
18	C₁₇×#126	—	2	8	no	—
19	C₁₇×#125	—	2	6	no	—
20	C₁₇×#125	—	2	24	no	—
21	C₁₇×#125	C₂³	1	6	yes	—
22	C₁₇×C₂²⋊Q₈	—	2	6	no	—
23	C₄×Q₈×C₁₇	—	2	12	no	—
24	C₄×Q₈×C₁₇	—	2	6	no	—
25	C₁₇×#117	D₄	1	6	yes	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×#63223	—	2	3	no	—
28	#10587560	—	2	3	no	—
29	#10587560	C₂³	1	2	yes	—
30	#10587480	—	2	24	no	—
31	#10587474	C₂³	1	6	yes	—
32	C₄×#3622	—	2	3	no	—
33	C₂³×D₁₇	—	4	1	no	—
34	C₂×Q₈×C₁₇	—	2	6	no	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	3	yes	—
36	C₂×D₆₈	—	2	6	no	—
37	C₂×D₆₈	—	4	12	no	—
38	C₂×D₆₈	—	4	3	no	—
39	C₄⋊C₄×C₁₇	—	2	24	no	—
40	C₄⋊C₄×C₁₇	—	2	6	no	—
41	C₄⋊C₄×C₁₇	—	2	12	no	—
42	C₄⋊C₄×C₁₇	C₂×D₄	1	12	yes	—
43	C₂²⋊C₄×C₁₇	—	2	6	no	—
44	C₄²×C₁₇	—	2	8	no	—
45	C₄²×C₁₇	—	2	6	no	—
46	C₄²×C₁₇	C₂×D₄	1	3	yes	—
47	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
48	#63223	—	4	6	no	—
49	C₂×#3622	—	2	6	no	—
50	#19083	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	#120	1	1	yes	characteristic
52	C₂²×D₁₇	—	4	2	no	—
53	C₂²×D₁₇	—	8	6	no	—
54	Q₈×C₁₇	—	4	12	no	—
55	C₂×C₄×C₁₇	—	2	6	no	—
56	C₂×C₄×C₁₇	—	2	12	no	—
57	C₂×C₄×C₁₇	#120	1	3	yes	—
58	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
59	D₆₈	—	4	12	no	—
60	D₆₈	—	8	6	no	—
61	#3622	—	4	6	no	—
62	#3556	D₁₇	1	1	yes	characteristic, complemented
63	#3138	—	17	3	no	—
64	#1658	—	17	8	no	—
65	#1526	—	17	3	no	—
66	C₂²×C₁₇	—	2	3	no	—
67	C₂²×C₁₇	C₂×#120	1	1	yes	characteristic
68	C₂×D₁₇	—	8	6	no	—
69	C₂×D₁₇	—	16	4	no	—
70	C₄×C₁₇	—	2	3	no	—
71	C₄×C₁₇	—	4	12	no	—
72	C₄×C₁₇	#454	1	6	yes	—
73	#564	C₂×D₁₇	1	4	yes	complemented
74	#557	—	2	6	no	—
75	#541	—	2	6	no	—
76	#533	C₂×D₁₇	1	3	yes	—
77	C₂×#127	—	17	1	no	—
78	#490	—	17	12	no	—
79	#486	—	34	6	no	—
80	#418	—	34	6	no	—
81	C₄×M₁₆	—	17	3	no	—
82	#330	—	17	12	no	—
83	C₂×C₁₇	—	4	1	no	—
84	C₂×C₁₇	C₂×#454	1	3	yes	—
85	D₁₇	—	16	2	no	—
86	C₂²×D₄	—	34	3	no	—
87	C₂×M₁₆	—	34	3	no	—
88	C₄⋊Q₈	—	2	6	no	—
89	C₄⋊Q₈	C₂²⋊D₁₇	1	6	yes	—
90	#127	—	17	2	no	—
91	#127	—	34	3	no	—
92	#126	—	2	8	no	—
93	#125	—	2	6	no	—
94	#125	—	2	24	no	—
95	#125	C₂²×D₁₇	1	6	yes	—
96	C₂²⋊Q₈	—	2	6	no	—
97	C₄×Q₈	—	2	12	no	—
98	C₄×Q₈	—	2	6	no	—
99	#117	C₂²⋊D₁₇	1	6	yes	—
100	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
101	#102	—	34	24	no	—
102	#97	—	17	6	no	—
103	C₄×C₈	—	34	3	no	—
104	C₁₇	#19083	1	1	yes	characteristic, complemented, Sylow
105	C₂⁴	—	68	1	no	—
106	C₂×Q₈	—	2	6	no	—
107	C₂×D₄	—	34	6	no	—
108	C₂×D₄	—	68	12	no	—
109	C₂×D₄	—	68	3	no	—
110	C₂²×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
111	M₁₆	—	68	6	no	—
112	C₂×C₈	—	34	6	no	—
113	C₄⋊C₄	—	2	24	no	—
114	C₄⋊C₄	—	2	12	no	—
115	C₄⋊C₄	—	2	6	no	—
116	C₄⋊C₄	C₂×C₂²⋊D₁₇	1	12	yes	—
117	C₂²⋊C₄	—	2	6	no	—
118	C₄²	—	2	8	no	—
119	C₄²	—	2	6	no	—
120	C₄²	C₂×C₂²⋊D₁₇	1	3	yes	—
121	C₄²	C₂³×D₁₇	1	1	yes	characteristic
122	C₂³	—	68	2	no	—
123	C₂³	—	136	6	no	—
124	C₂³	#10587639	1	1	yes	characteristic
125	Q₈	—	4	12	no	—
126	D₄	—	68	12	no	—
127	D₄	—	136	6	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	12	no	—
130	C₂×C₄	#10587639	1	3	yes	—
131	C₂×C₄	C₂²×C₂²⋊D₁₇	1	3	yes	—
132	C₈	—	68	6	no	—
133	C₂²	—	2	3	no	—
134	C₂²	—	272	4	no	—
135	C₂²	—	136	6	no	—
136	C₂²	C₂×#10587639	1	1	yes	characteristic
137	C₄	—	2	3	no	—
138	C₄	—	4	12	no	—
139	C₄	#11762982	1	6	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	272	2	no	—
142	C₂	C₂×#11762982	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539160

C₂

yes

maximal

#14528086

C₂

yes

maximal, complemented

#14527379

C₂

yes

maximal, complemented

C₁₇×#3556

C₂

yes

maximal, characteristic, complemented

C₂×#10587560

C₂²

yes

characteristic

#11763175

C₂²

yes

—

#11763157

—

#11762813

—

C₄×#63223

C₂²

yes

—

#11762551

C₂²

yes

complemented

C₁₇×#564

C₂²

yes

complemented

C₁₇×#557

—

C₁₇×#541

—

C₁₇×#533

C₂²

yes

—

C₂²×D₆₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

D₄

yes

—

C₁₇×#126

—

C₁₇×#125

—

C₁₇×#125

—

C₁₇×#125

C₂³

yes

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₁₇×#117

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

—

—

C₂³

yes

—

#10587480

—

#10587474

C₂³

yes

—

C₄×#3622

—

C₂³×D₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂⁴

yes

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

—

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#120

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

—

D₆₈

—

D₆₈

—

#3622

—

#3556

D₁₇

yes

characteristic, complemented

#3138

—

#1658

—

#1526

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#120

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#454

yes

—

#564

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

—

—

—

—

—

C₄×M₁₆

—

#330

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#454

yes

—

D₁₇

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₁₇

yes

—

—

—

—

—

—

C₂²×D₁₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂²⋊D₁₇

yes

—

100

C₂×C₄²

C₂²×D₁₇

yes

characteristic

101

#102

—

102

#97

—

103

C₄×C₈

—

104

C₁₇

#19083

yes

characteristic, complemented, Sylow

105

C₂⁴

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

—

111

M₁₆

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂×C₂²⋊D₁₇

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂×C₂²⋊D₁₇

yes

—

121

C₄²

C₂³×D₁₇

yes

characteristic

122

C₂³

—

123

C₂³

—

136

—

124

C₂³

#10587639

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

136

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10587639

yes

—

131

C₂×C₄

C₂²×C₂²⋊D₁₇

yes

—

132

C₈

—

133

C₂²

—

134

C₂²

—

272

—

135

C₂²

—

136

—

136

C₂²

C₂×#10587639

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#11762982

yes

—

140

C₂

—

141

C₂

—

272

—

142

C₂

C₂×#11762982

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587639
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#3556	C₂
socle	C₂²×C₁₇	C₂×#120
prime radical	C₂×#10587560	C₂²
hypercenter	#3556	D₁₇
hypocenter	C₁₇	#19083
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

subgroup

G/H

center

C₂²

C₂×#10587639

commutator, cosocle

C₄²×C₁₇

C₂⁴

Frattini

C₄²

C₂³×D₁₇

Fitting

C₁₇×#3556

C₂

C₂²×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

N	H ≅ G/N	type
2. #14528086	C₂	semidirect
3. #14527379	C₂	semidirect
4. C₁₇×#3556	C₂	semidirect
10. #11762551	C₂²	semidirect
11. C₁₇×#564	C₂²	semidirect
62. #3556	D₁₇	semidirect
73. #564	C₂×D₁₇	semidirect
104. C₁₇	#19083	semidirect

H ≅ G/N

type

2. #14528086

C₂

semidirect

3. #14527379

C₂

semidirect

4. C₁₇×#3556

C₂

semidirect

10. #11762551

C₂²

semidirect

11. C₁₇×#564

C₂²

semidirect

62. #3556

D₁₇

semidirect

73. #564

C₂×D₁₇

semidirect

104. C₁₇

#19083

semidirect

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

subgroup	index
#14539160	2
#14528086	2
#14527379	2
C₁₇×#3556	2
#19083	17

subgroup

index

C₁₇×#3556

Subgroups — #17016404

Conjugacy classes of subgroups

Complements

Maximal subgroups