Group #14528404 — subgroups

Subgroups — #14528404

Order 2176.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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, complemented
2	#11763205	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763181	C₂	1	1	yes	maximal, characteristic
4	#11763160	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763155	C₂	1	1	yes	maximal, characteristic
6	#11762531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#492	C₂	1	1	yes	maximal, characteristic
8	C₂×SD₁₆×C₁₇	—	2	2	no	—
9	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₁₇×#127	C₂²	1	1	yes	characteristic
11	C₂×#63229	—	2	2	no	—
12	#10587555	C₂²	1	1	yes	characteristic
13	#10587555	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	#10587503	—	2	2	no	—
16	#10587487	—	2	2	no	—
17	#10587479	—	2	2	no	—
18	#10587478	—	2	2	no	—
19	C₄×#3622	C₂²	1	1	yes	characteristic
20	C₄×#3622	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	C₂×D₄×C₁₇	—	2	1	no	—
23	C₂×D₄×C₁₇	—	2	1	no	—
24	C₂×D₄×C₁₇	—	2	1	no	—
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×Dic₃₄	—	2	1	no	—
27	SD₁₆×C₁₇	—	4	4	no	—
28	C₂×C₈×C₁₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₄²×C₁₇	C₂³	1	1	yes	characteristic
32	#63229	—	4	4	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₂×#3622	D₄	1	2	yes	—
38	C₂×#3622	D₄	1	2	yes	—
39	C₂³×C₁₇	—	4	1	no	—
40	Q₈×C₁₇	—	4	2	no	—
41	D₄×C₁₇	—	4	2	no	—
42	D₄×C₁₇	—	4	2	no	—
43	D₄×C₁₇	—	4	2	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	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₇	—	4	1	no	—
49	C₂×Dic₁₇	—	4	1	no	—
50	Dic₃₄	—	4	2	no	—
51	Dic₃₄	—	4	2	no	—
52	C₈×C₁₇	—	2	2	no	—
53	#3622	—	2	2	no	—
54	#3622	SD₁₆	1	4	yes	complemented
55	#1714	—	17	1	no	maximal, Sylow
56	C₂²×C₁₇	—	8	1	no	—
57	C₂²×C₁₇	—	8	1	no	—
58	C₂²×C₁₇	—	8	1	no	—
59	C₂²×C₁₇	#127	1	1	yes	characteristic
60	C₄×C₁₇	—	8	1	no	—
61	C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
62	C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₇	#137	1	2	yes	—
64	Dic₁₇	—	8	1	no	—
65	Dic₁₇	—	8	1	no	—
66	#500	—	17	1	no	—
67	#498	—	17	1	no	—
68	#492	—	17	1	no	—
69	#492	D₁₇	1	1	yes	characteristic
70	#487	—	17	1	no	—
71	#486	—	17	1	no	—
72	#322	—	17	1	no	—
73	C₂×C₁₇	—	16	1	no	—
74	C₂×C₁₇	#492	1	1	yes	characteristic
75	C₂×C₁₇	#496	1	1	yes	characteristic
76	C₂×C₁₇	#497	1	1	yes	characteristic
77	C₂×SD₁₆	—	2	2	no	—
78	C₂×SD₁₆	—	34	2	no	—
79	C₄⋊Q₈	—	17	1	no	—
80	C₄⋊Q₈	—	17	1	no	—
81	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
82	#127	C₂×D₁₇	1	1	yes	characteristic
83	#107	—	34	2	no	—
84	#106	—	34	2	no	—
85	#103	—	34	2	no	—
86	#102	—	34	2	no	—
87	C₄×C₈	—	17	1	no	—
88	C₄×C₈	—	17	1	no	—
89	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
90	C₁₇	#1714	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	34	1	no	—
93	C₂×Q₈	—	34	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	68	4	no	—
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	34	1	no	—
105	C₄⋊C₄	—	34	1	no	—
106	C₄⋊C₄	—	34	1	no	—
107	C₄⋊C₄	—	34	1	no	—
108	C₄²	C₂²×D₁₇	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	68	2	no	—
112	Q₈	—	68	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	68	1	no	—
118	C₂×C₄	—	68	1	no	—
119	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	17	4	no	—
124	C₈	—	34	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#10587626	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	136	1	no	—
131	C₄	—	136	1	no	—
132	C₄	SD₁₆×D₁₇	1	2	yes	—
133	C₄	#10587601	1	2	yes	—
134	C₄	#10587618	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#11763180	1	1	yes	characteristic
137	C₂	#11763196	1	1	yes	characteristic
138	C₂	#11763200	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763212

C₂

yes

maximal, characteristic, complemented

#11763205

C₂

yes

maximal, characteristic, complemented

#11763181

C₂

yes

maximal, characteristic

#11763160

C₂

yes

maximal, characteristic, complemented

#11763155

C₂

yes

maximal, characteristic

#11762531

C₂

yes

maximal, characteristic, complemented

C₁₇×#492

C₂

yes

maximal, characteristic

C₂×SD₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₁₇×#127

C₂²

yes

characteristic

C₂×#63229

—

#10587555

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

SD₁₆×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63229

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

D₄

yes

—

C₂×#3622

D₄

yes

—

C₂³×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#3622

SD₁₆

yes

complemented

#1714

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#127

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

#137

yes

—

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

#127

C₂×D₁₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1714

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₁₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²×D₁₇

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₁₇

yes

characteristic

120

C₂×C₄

D₄×D₁₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#10587626

yes

characteristic

129

C₄

—

130

C₄

—

136

—

131

C₄

—

136

—

132

C₄

SD₁₆×D₁₇

yes

—

133

C₄

#10587601

yes

—

134

C₄

#10587618

yes

—

135

C₂

—

136

C₂

#11763180

yes

characteristic

137

C₂

#11763196

yes

characteristic

138

C₂

#11763200

yes

characteristic

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#492	C₂
socle	C₂²×C₁₇	#127
prime radical	C₁₇×#127	C₂²
hypercenter	#492	D₁₇
hypocenter	C₁₇	#1714
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#10587626

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#492

C₂

socle

C₂²×C₁₇

#127

prime radical

C₁₇×#127

C₂²

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #11763212	C₂	semidirect
2. #11763205	C₂	semidirect
4. #11763160	C₂	semidirect
6. #11762531	C₂	semidirect
54. #3622	SD₁₆	semidirect
90. C₁₇	#1714	semidirect

H ≅ G/N

type

1. #11763212

C₂

semidirect

2. #11763205

C₂

semidirect

4. #11763160

C₂

semidirect

6. #11762531

C₂

semidirect

54. #3622

SD₁₆

semidirect

90. C₁₇

#1714

semidirect

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

subgroup	index
#11763212	2
#11763205	2
#11763181	2
#11763160	2
#11763155	2
#11762531	2
C₁₇×#492	2
#1714	17

subgroup

index

C₁₇×#492

Subgroups — #14528404

Conjugacy classes of subgroups

Complements

Maximal subgroups