Group #14529945 — subgroups

Subgroups — #14529945

Order 2176.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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	#11763440	C₂	1	1	yes	maximal, characteristic
2	#11762818	C₂	1	2	yes	maximal
3	#11762578	C₂	1	1	yes	maximal, characteristic
4	#11762569	C₂	1	2	yes	maximal
5	C₁₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	#10587612	C₂²	1	2	yes	—
10	#10587610	—	2	2	no	—
11	C₂×#63223	C₂²	1	2	yes	—
12	#10587576	—	2	2	no	—
13	#10587574	—	2	2	no	—
14	C₁₇×#95	—	2	2	no	—
15	#10587506	—	2	2	no	—
16	#10587506	—	2	2	no	—
17	C₂×Q₈×C₁₇	—	2	1	no	—
18	C₂×Q₈×C₁₇	—	2	2	no	—
19	C₂×Q₈×C₁₇	—	2	2	no	—
20	C₂×Q₈×C₁₇	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₁₇	—	2	1	no	—
22	C₂²×C₄×C₁₇	—	2	1	no	—
23	C₂²×C₄×C₁₇	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₄	—	2	2	no	—
26	C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₄²×C₁₇	—	2	2	no	—
28	C₂²⋊Dic₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	—	2	2	no	—
30	C₄⋊Dic₁₇	D₄	1	2	yes	—
31	C₄⋊Dic₁₇	D₄	1	2	yes	—
32	#63225	—	2	4	no	—
33	C₄×Dic₁₇	—	2	2	no	—
34	#63223	—	2	2	no	—
35	#63223	—	2	2	no	—
36	C₂×#3622	—	2	2	no	—
37	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₇	—	2	4	no	—
39	Q₈×C₁₇	—	4	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	—	4	1	no	—
44	C₂×C₄×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	—	4	1	no	—
46	C₂×C₄×C₁₇	—	4	1	no	—
47	C₂×C₄×C₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#41	1	2	yes	—
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	—	2	2	no	—
54	Dic₃₄	—	8	2	no	—
55	#3622	—	4	2	no	—
56	#2028	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	—	2	1	no	—
58	C₂²×C₁₇	—	2	1	no	—
59	C₂²×C₁₇	#120	1	1	yes	characteristic
60	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₇	#124	1	1	yes	characteristic
62	C₄×C₁₇	—	4	2	no	—
63	C₄×C₁₇	—	4	2	no	—
64	C₄×C₁₇	—	4	2	no	—
65	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₇	#124	1	2	yes	—
67	Dic₁₇	—	4	2	no	—
68	Dic₁₇	—	4	2	no	—
69	#544	—	17	1	no	—
70	#419	—	17	2	no	—
71	#391	D₁₇	1	1	yes	characteristic
72	#343	—	17	1	no	—
73	#339	—	17	2	no	—
74	C₂×C₁₇	—	2	1	no	—
75	C₂×C₁₇	—	2	1	no	—
76	C₂×C₁₇	#393	1	1	yes	characteristic
77	C₂×C₁₇	#455	1	1	yes	characteristic
78	C₂×C₁₇	#456	1	1	yes	characteristic
79	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
80	C₂×M₁₆	—	17	2	no	—
81	C₄⋊Q₈	—	34	2	no	—
82	#125	—	34	2	no	—
83	C₂²⋊Q₈	—	34	2	no	—
84	#117	—	17	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
86	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
87	#103	—	34	2	no	—
88	#103	—	34	2	no	—
89	#95	—	2	2	no	—
90	C₁₇	#2028	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	34	2	no	—
95	C₂×Q₈	Dic₃₄	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
100	M₁₆	—	34	2	no	—
101	M₁₆	—	34	2	no	—
102	C₂×C₈	—	34	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	17	2	no	—
105	C₄⋊C₄	—	17	2	no	—
106	C₄⋊C₄	—	34	2	no	—
107	C₄⋊C₄	—	34	4	no	—
108	C₂²⋊C₄	—	34	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	34	2	no	—
111	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	136	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	34	2	no	—
125	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
126	C₂×C₄	#63252	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₇	1	2	yes	—
128	C₂×C₄	#63254	1	2	yes	—
129	C₈	—	68	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#10587610	1	1	yes	characteristic
133	C₂²	#10587622	1	1	yes	characteristic
134	C₂²	#10587623	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	68	2	no	—
139	C₄	—	68	2	no	—
140	C₄	#10587564	1	2	yes	—
141	C₄	#10587571	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#11762721	1	1	yes	characteristic
145	C₂	#11762986	1	1	yes	characteristic
146	C₂	#11762991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763440

C₂

yes

maximal, characteristic

#11762818

C₂

yes

maximal

#11762578

C₂

yes

maximal, characteristic

#11762569

C₂

yes

maximal

C₁₇×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

#10587612

C₂²

yes

—

#10587610

—

C₂×#63223

C₂²

yes

—

#10587576

—

#10587574

—

C₁₇×#95

—

#10587506

—

#10587506

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

Q₈

yes

complemented

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

—

C₄⋊Dic₁₇

D₄

yes

—

#63225

—

C₄×Dic₁₇

—

#63223

—

#63223

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

#3622

—

#2028

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#120

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇

#124

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

#124

yes

—

Dic₁₇

—

Dic₁₇

—

#544

—

#419

—

#391

D₁₇

yes

characteristic

#343

—

#339

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

C₂×D₁₇

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄²

C₂×D₁₇

yes

—

—

—

C₁₇

#2028

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₃₄

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

136

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₃₄

yes

characteristic

126

C₂×C₄

#63252

yes

characteristic

127

C₂×C₄

D₄×D₁₇

yes

—

128

C₂×C₄

#63254

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

#10587610

yes

characteristic

133

C₂²

#10587622

yes

characteristic

134

C₂²

#10587623

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#10587564

yes

—

141

C₄

#10587571

yes

—

142

C₂

—

143

C₂

—

144

C₂

#11762721

yes

characteristic

145

C₂

#11762986

yes

characteristic

146

C₂

#11762991

yes

characteristic

	subgroup	G/H
center	C₂²	#10587623
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#391	C₂
socle	C₂²×C₁₇	#120
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#391	D₁₇
hypocenter	C₁₇	#2028
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#10587623

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#391

C₂

socle

C₂²×C₁₇

#120

prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₁₇	Q₈	semidirect
90. C₁₇	#2028	semidirect
95. C₂×Q₈	Dic₃₄	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₁₇

Q₈

semidirect

90. C₁₇

#2028

semidirect

95. C₂×Q₈

Dic₃₄

semidirect

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

subgroup	index
#11763440	2
#11762818	2
#11762578	2
#11762569	2
C₁₇×#391	2
#2028	17

subgroup

index

C₁₇×#391

Subgroups — #14529945

Conjugacy classes of subgroups

Complements

Maximal subgroups