Group #14528186 — subgroups

Subgroups — #14528186

Order 2176.

144 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 57 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	#11763212	C₂	1	1	yes	maximal, characteristic
2	#11763104	C₂	1	1	yes	maximal, characteristic
3	Q₈×#3622	C₂	1	1	yes	maximal, characteristic
4	#11762561	C₂	1	1	yes	maximal, characteristic
5	#11762558	C₂	1	1	yes	maximal, characteristic
6	#11762557	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#477	C₂	1	1	yes	maximal, characteristic
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587555	C₂²	1	1	yes	characteristic
11	C₁₇×#107	—	2	1	no	—
12	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₁₇×#103	—	2	1	no	—
14	#10587506	—	2	1	no	—
15	#10587478	—	2	1	no	—
16	#10587478	—	2	1	no	—
17	#10587478	—	2	1	no	—
18	C₄⋊#3622	—	2	1	no	—
19	C₄⋊#3622	C₂²	1	1	yes	characteristic
20	C₄⋊#3622	C₂²	1	1	yes	characteristic
21	C₄×#3622	—	2	1	no	—
22	C₄×#3622	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	—	2	1	no	—
24	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₂×#3622	—	2	1	no	—
36	C₂×#3622	—	2	1	no	—
37	C₂×#3622	D₄	1	1	yes	characteristic
38	C₂×#3622	D₄	1	1	yes	characteristic
39	Q₈×C₁₇	—	2	1	no	—
40	Q₈×C₁₇	—	8	1	no	—
41	Q₈×C₁₇	Q₁₆	1	2	yes	—
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	—	2	1	no	—
44	C₂×C₄×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
48	C₂×Dic₁₇	—	4	1	no	—
49	Dic₃₄	—	8	1	no	—
50	C₈×C₁₇	—	4	1	no	—
51	#3622	—	2	1	no	—
52	#3622	—	4	1	no	—
53	#3622	—	4	1	no	—
54	#3622	Q₁₆	1	2	yes	—
55	#1676	—	17	1	no	maximal, Sylow
56	C₂²×C₁₇	#121	1	1	yes	characteristic
57	C₄×C₁₇	—	2	2	no	—
58	C₄×C₁₇	—	2	1	no	—
59	C₄×C₁₇	—	4	1	no	—
60	C₄×C₁₇	—	8	1	no	—
61	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₁₇	#135	1	1	yes	characteristic
64	C₄×C₁₇	#137	1	1	yes	characteristic
65	Dic₁₇	—	8	1	no	—
66	#500	—	17	1	no	—
67	#477	—	17	1	no	—
68	#477	D₁₇	1	1	yes	characteristic
69	C₈×Q₈	—	17	1	no	—
70	#335	—	17	1	no	—
71	#333	—	17	1	no	—
72	#333	—	17	1	no	—
73	C₂×C₁₇	#462	1	1	yes	characteristic
74	C₂×C₁₇	#467	1	1	yes	characteristic
75	C₂×C₁₇	#472	1	1	yes	characteristic
76	C₄⋊Q₈	—	17	1	no	—
77	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
78	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
79	#107	—	2	1	no	—
80	#107	—	34	1	no	—
81	#107	—	34	1	no	—
82	#107	—	34	1	no	—
83	C₄⋊C₈	—	17	1	no	—
84	C₄⋊C₈	—	17	1	no	—
85	C₄⋊C₈	—	34	1	no	—
86	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	34	1	no	—
89	C₄×C₈	—	17	1	no	—
90	C₄×C₈	—	34	1	no	—
91	C₁₇	#1676	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	34	1	no	—
94	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	17	1	no	—
97	C₂×C₈	—	17	1	no	—
98	C₂×C₈	—	34	1	no	—
99	C₂×C₈	—	34	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	34	1	no	—
104	C₄⋊C₄	—	34	1	no	—
105	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₂²×D₁₇	1	1	yes	characteristic
108	Q₈	—	2	1	no	—
109	Q₈	—	8	1	no	—
110	Q₈	—	136	1	no	—
111	Q₈	#63231	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	68	1	no	—
116	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
117	C₂×C₄	#63254	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	17	2	no	—
121	C₈	—	34	1	no	—
122	C₈	—	68	1	no	—
123	C₈	—	68	1	no	—
124	C₂²	#10587625	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	4	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	136	1	no	—
130	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
131	C₄	#10587604	1	1	yes	characteristic
132	C₄	C₂×#63231	1	1	yes	characteristic
133	C₄	#10587636	1	1	yes	characteristic
134	C₂	#11763017	1	1	yes	characteristic
135	C₂	#11763048	1	1	yes	characteristic
136	C₂	#11763077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763212

C₂

yes

maximal, characteristic

#11763104

C₂

yes

maximal, characteristic

Q₈×#3622

C₂

yes

maximal, characteristic

#11762561

C₂

yes

maximal, characteristic

#11762558

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal, characteristic

C₁₇×#477

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₁₇

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₁₇×#107

—

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×#103

—

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊#3622

C₂²

yes

characteristic

C₄×#3622

—

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₄

yes

characteristic

C₂×#3622

D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

—

—

—

Q₁₆

yes

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#500

—

#477

—

#477

D₁₇

yes

characteristic

C₈×Q₈

—

#335

—

#333

—

#333

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₄×Q₈

C₂×D₁₇

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1676

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

C₂²⋊D₁₇

yes

characteristic

106

C₄²

—

107

C₄²

C₂²×D₁₇

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

136

—

111

Q₈

#63231

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

D₄×D₁₇

yes

characteristic

117

C₂×C₄

#63254

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₂²

#10587625

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

136

—

130

C₄

Q₁₆×D₁₇

yes

characteristic

131

C₄

#10587604

yes

characteristic

132

C₄

C₂×#63231

yes

characteristic

133

C₄

#10587636

yes

characteristic

134

C₂

#11763017

yes

characteristic

135

C₂

#11763048

yes

characteristic

136

C₂

#11763077

yes

characteristic

	subgroup	G/H
center	C₂²	#10587625
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#477	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	#477	D₁₇
hypocenter	C₁₇	#1676
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#10587625

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#477

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
91. C₁₇	#1676	semidirect

H ≅ G/N

type

91. C₁₇

#1676

semidirect

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

subgroup	index
#11763212	2
#11763104	2
Q₈×#3622	2
#11762561	2
#11762558	2
#11762557	2
C₁₇×#477	2
#1676	17

subgroup

index

C₁₇×#477

Subgroups — #14528186

Conjugacy classes of subgroups

Complements

Maximal subgroups