Group #14539160 — subgroups

Subgroups — #14539160

Order 2176.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#10587560	C₂	1	1	yes	maximal, characteristic
2	#11763175	C₂	1	4	yes	maximal, complemented
3	#11763157	C₂	1	4	yes	maximal, complemented
4	#11762813	C₂	1	4	yes	maximal, complemented
5	C₄×#63223	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₆₈	—	2	2	no	—
8	C₂²×D₆₈	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	complemented
10	C₁₇×#125	C₂²	1	2	yes	complemented
11	C₁₇×C₂²⋊Q₈	—	2	2	no	—
12	C₄×Q₈×C₁₇	—	2	2	no	—
13	C₁₇×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×#63223	C₂²	1	2	yes	—
16	#10587560	—	2	2	no	—
17	#10587560	C₂²	1	2	yes	—
18	#10587560	C₂²	1	2	yes	—
19	#10587480	C₂²	1	16	yes	complemented
20	#10587474	C₂²	1	2	yes	complemented
21	C₄×#3622	C₂²	1	2	yes	complemented
22	C₂³×D₁₇	—	2	2	no	—
23	C₂×Q₈×C₁₇	—	2	2	no	—
24	C₂²×C₄×C₁₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×D₆₈	—	2	8	no	—
27	C₂×D₆₈	—	2	4	no	—
28	C₂×D₆₈	—	2	4	no	—
29	C₂×D₆₈	—	2	2	no	—
30	C₂×D₆₈	—	4	8	no	—
31	C₂×D₆₈	C₂³	1	4	yes	—
32	C₄⋊C₄×C₁₇	—	2	4	no	—
33	C₄⋊C₄×C₁₇	—	2	2	no	—
34	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
35	C₂²⋊C₄×C₁₇	—	2	2	no	—
36	C₄²×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	D₄	1	2	yes	—
38	C₄²×C₁₇	C₂³	1	1	yes	characteristic
39	C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	#63223	—	2	4	no	—
41	C₂×#3622	C₂³	1	4	yes	—
42	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₁₇	—	2	4	no	—
44	C₂²×D₁₇	—	4	8	no	—
45	C₂²×D₁₇	—	4	4	no	—
46	Q₈×C₁₇	—	4	4	no	—
47	C₂×C₄×C₁₇	—	2	4	no	—
48	C₂×C₄×C₁₇	—	2	4	no	—
49	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₇	#41	1	4	yes	—
53	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
54	D₆₈	—	2	8	no	—
55	D₆₈	—	4	8	no	—
56	D₆₈	—	4	4	no	—
57	D₆₈	—	4	8	no	—
58	#3622	—	2	4	no	—
59	#3138	—	17	1	no	maximal, Sylow
60	C₂²×C₁₇	—	2	2	no	—
61	C₂²×C₁₇	#124	1	2	yes	—
62	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×D₁₇	—	4	4	no	—
64	C₂×D₁₇	—	4	8	no	—
65	C₂×D₁₇	—	8	8	no	—
66	C₄×C₁₇	—	2	2	no	—
67	C₄×C₁₇	—	4	4	no	—
68	C₄×C₁₇	#124	1	2	yes	—
69	C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
70	C₄×C₁₇	C₂×#41	1	2	yes	—
71	#533	D₁₇	1	1	yes	characteristic, complemented
72	C₂×#127	—	17	1	no	—
73	#490	—	17	4	no	—
74	#486	—	17	4	no	—
75	#418	—	17	4	no	—
76	C₄×M₁₆	—	17	1	no	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	C₂×#124	1	1	yes	characteristic
79	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
80	D₁₇	—	8	4	no	—
81	C₂²×D₄	—	17	2	no	—
82	C₂²×D₄	—	34	2	no	—
83	C₂×M₁₆	—	17	2	no	—
84	C₄⋊Q₈	C₂×D₁₇	1	2	yes	complemented
85	#127	—	17	2	no	—
86	#127	—	17	2	no	—
87	#127	—	34	2	no	—
88	#125	C₂×D₁₇	1	2	yes	complemented
89	C₂²⋊Q₈	—	2	2	no	—
90	C₄×Q₈	—	2	2	no	—
91	#117	C₂×D₁₇	1	2	yes	—
92	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
93	#102	—	17	16	no	—
94	#97	—	17	2	no	—
95	C₄×C₈	—	17	2	no	—
96	C₁₇	#3138	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	34	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	17	4	no	—
100	C₂×D₄	—	34	8	no	—
101	C₂×D₄	—	34	4	no	—
102	C₂×D₄	—	34	4	no	—
103	C₂×D₄	—	34	2	no	—
104	C₂×D₄	—	68	8	no	—
105	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
107	M₁₆	—	34	4	no	—
108	C₂×C₈	—	17	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₁₇	1	2	yes	—
115	C₄²	C₂²×D₁₇	1	1	yes	characteristic
116	C₄²	C₂²×D₁₇	1	1	yes	characteristic
117	C₂³	—	34	4	no	—
118	C₂³	—	68	8	no	—
119	C₂³	—	68	4	no	—
120	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	34	8	no	—
123	D₄	—	68	8	no	—
124	D₄	—	68	4	no	—
125	D₄	—	68	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	#63256	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
132	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
133	C₈	—	34	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	68	4	no	—
136	C₂²	—	68	8	no	—
137	C₂²	—	136	8	no	—
138	C₂²	#10587633	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#10587633	1	2	yes	—
143	C₄	#10587635	1	4	yes	—
144	C₄	C₂×#63256	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	136	4	no	—
147	C₂	C₂×#10587633	1	1	yes	characteristic
148	C₂	C₂×#10587635	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587560

C₂

yes

maximal, characteristic

#11763175

C₂

yes

maximal, complemented

#11763157

C₂

yes

maximal, complemented

#11762813

C₂

yes

maximal, complemented

C₄×#63223

C₂

yes

maximal, characteristic, complemented

C₁₇×#533

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₈

—

C₂²×D₆₈

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₂²

yes

complemented

C₁₇×#125

C₂²

yes

complemented

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

#10587560

—

#10587560

C₂²

yes

—

#10587560

C₂²

yes

—

#10587480

C₂²

yes

complemented

#10587474

C₂²

yes

complemented

C₄×#3622

C₂²

yes

complemented

C₂³×D₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

#63223

—

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

#3622

—

#3138

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#124

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#124

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

C₂×#41

yes

—

#533

D₁₇

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#124

yes

characteristic

C₂×C₁₇

C₂×AGL(1,ℤ/8)

yes

—

D₁₇

—

C₂²×D₄

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₁₇

yes

—

—

—

C₂×D₁₇

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₁₇

#3138

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

C₂²⋊D₁₇

yes

—

106

C₂²×C₄

C₂²×D₁₇

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₁₇

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₁₇

yes

—

115

C₄²

C₂²×D₁₇

yes

characteristic

116

C₄²

C₂²×D₁₇

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#63256

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

132

C₂×C₄

C₂³×D₁₇

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

136

—

138

C₂²

#10587633

yes

—

139

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#10587633

yes

—

143

C₄

#10587635

yes

—

144

C₄

C₂×#63256

yes

—

145

C₂

—

146

C₂

—

136

—

147

C₂

C₂×#10587633

yes

characteristic

148

C₂

C₂×#10587635

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#533	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#10587560	C₂
hypercenter	#533	D₁₇
hypocenter	C₁₇	#3138
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#533

C₂

socle

C₂²×C₁₇

C₂²×D₄

C₂

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #11763175	C₂	semidirect
3. #11763157	C₂	semidirect
4. #11762813	C₂	semidirect
5. C₄×#63223	C₂	semidirect
6. C₁₇×#533	C₂	semidirect
9. C₁₇×C₄⋊Q₈	C₂²	semidirect
10. C₁₇×#125	C₂²	semidirect
19. #10587480	C₂²	semidirect
20. #10587474	C₂²	semidirect
21. C₄×#3622	C₂²	semidirect
71. #533	D₁₇	semidirect
84. C₄⋊Q₈	C₂×D₁₇	semidirect
88. #125	C₂×D₁₇	semidirect
96. C₁₇	#3138	semidirect

H ≅ G/N

type

2. #11763175

C₂

semidirect

3. #11763157

C₂

semidirect

4. #11762813

C₂

semidirect

5. C₄×#63223

C₂

semidirect

6. C₁₇×#533

C₂

semidirect

9. C₁₇×C₄⋊Q₈

C₂²

semidirect

10. C₁₇×#125

C₂²

semidirect

19. #10587480

C₂²

semidirect

20. #10587474

C₂²

semidirect

21. C₄×#3622

C₂²

semidirect

71. #533

D₁₇

semidirect

84. C₄⋊Q₈

C₂×D₁₇

semidirect

88. #125

C₂×D₁₇

semidirect

96. C₁₇

#3138

semidirect

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

subgroup	index
C₂×#10587560	2
#11763175	2
#11763157	2
#11762813	2
C₄×#63223	2
C₁₇×#533	2
#3138	17

subgroup

index

C₁₇×#533

Subgroups — #14539160

Conjugacy classes of subgroups

Complements

Maximal subgroups