Group #14530348 — subgroups

Subgroups — #14530348

Order 2176.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#11762707	C₂	1	1	yes	maximal, characteristic
2	#11762679	C₂	1	1	yes	maximal, characteristic
3	#11762572	C₂	1	2	yes	maximal
4	#11762572	C₂	1	2	yes	maximal
5	C₁₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₇	—	2	1	no	—
9	C₂×C₄⋊Dic₁₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
13	C₂×#63225	—	2	1	no	—
14	#10587502	—	2	1	no	—
15	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	1	no	—
19	C₂²×Dic₁₇	—	2	1	no	—
20	C₂²×Dic₁₇	—	2	2	no	—
21	C₂×C₈×C₁₇	—	2	2	no	—
22	C₂×C₈×C₁₇	—	2	2	no	—
23	C₂×C₈×C₁₇	—	2	2	no	—
24	C₄²×C₁₇	—	2	2	no	—
25	C₄⋊Dic₁₇	—	2	1	no	—
26	C₄⋊Dic₁₇	—	2	4	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₄⋊Dic₁₇	—	4	2	no	—
29	C₄⋊Dic₁₇	—	4	2	no	—
30	C₄⋊Dic₁₇	Q₈	1	2	yes	—
31	#63225	—	4	2	no	—
32	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	—	2	1	no	—
35	C₂×C₄×C₁₇	—	2	1	no	—
36	C₂×C₄×C₁₇	—	2	1	no	—
37	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
38	C₂×C₄×C₁₇	Q₁₆	1	2	yes	—
39	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₇	#41	1	2	yes	—
41	C₂×C₄×C₁₇	#41	1	2	yes	—
42	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×Dic₁₇	—	2	2	no	—
44	C₂×Dic₁₇	—	4	1	no	—
45	C₂×Dic₁₇	—	4	1	no	—
46	C₂×Dic₁₇	—	4	1	no	—
47	C₂×Dic₁₇	—	4	2	no	—
48	C₂×Dic₁₇	—	4	1	no	—
49	C₂×Dic₁₇	—	4	1	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₈×C₁₇	—	4	2	no	—
53	#2100	—	17	1	no	maximal, Sylow
54	C₂²×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₇	#123	1	1	yes	characteristic
56	C₂²×C₁₇	#124	1	1	yes	characteristic
57	C₂²×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₁₇	—	2	2	no	—
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	#125	1	2	yes	—
64	C₄×C₁₇	#126	1	2	yes	—
65	Dic₁₇	—	4	2	no	—
66	Dic₁₇	—	8	1	no	—
67	Dic₁₇	—	8	2	no	—
68	#389	—	17	1	no	—
69	#384	—	17	1	no	—
70	#340	—	17	2	no	—
71	#340	—	17	2	no	—
72	#336	D₁₇	1	1	yes	characteristic
73	C₂×C₁₇	#400	1	1	yes	characteristic
74	C₂×C₁₇	#476	1	1	yes	characteristic
75	C₂×C₁₇	#477	1	1	yes	characteristic
76	C₂×C₁₇	#481	1	1	yes	characteristic
77	C₂×C₁₇	#484	1	1	yes	characteristic
78	C₂×C₁₇	#487	1	1	yes	characteristic
79	C₂×C₁₇	#490	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₁₇	1	2	yes	—
81	C₂×C₄⋊C₄	—	17	1	no	—
82	C₂×C₄⋊C₄	—	17	2	no	—
83	C₂×C₄⋊C₄	—	17	1	no	—
84	C₂×C₄⋊C₄	—	34	1	no	—
85	C₂×C₄⋊C₄	—	34	1	no	—
86	C₂×C₄⋊C₄	—	34	1	no	—
87	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
88	#95	—	34	1	no	—
89	C₁₇	#2100	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	—	34	1	no	—
91	C₂²×C₄	—	34	1	no	—
92	C₂²×C₄	—	34	2	no	—
93	C₂²×C₄	D₆₈	1	1	yes	characteristic
94	C₂²×C₄	D₆₈	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	17	2	no	—
100	C₄⋊C₄	—	34	1	no	—
101	C₄⋊C₄	—	34	4	no	—
102	C₄⋊C₄	—	34	2	no	—
103	C₄⋊C₄	—	68	2	no	—
104	C₄⋊C₄	—	68	2	no	—
105	C₄⋊C₄	—	68	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₆₈	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	34	2	no	—
113	C₂×C₄	—	68	1	no	—
114	C₂×C₄	—	68	1	no	—
115	C₂×C₄	—	68	2	no	—
116	C₂×C₄	—	68	1	no	—
117	C₂×C₄	—	68	1	no	—
118	C₂×C₄	—	68	2	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	—	68	1	no	—
121	C₂×C₄	#63219	1	2	yes	—
122	C₂×C₄	Dic₆₈	1	2	yes	—
123	C₂×C₄	#63252	1	2	yes	—
124	C₂×C₄	#63254	1	1	yes	characteristic
125	C₂×C₄	#63254	1	2	yes	—
126	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	#10587561	1	1	yes	characteristic
129	C₂²	#10587572	1	1	yes	characteristic
130	C₂²	#10587583	1	1	yes	characteristic
131	C₂²	C₂×#63219	1	1	yes	characteristic
132	C₂²	C₂×Dic₆₈	1	1	yes	characteristic
133	C₂²	#10587594	1	1	yes	characteristic
134	C₂²	#10587594	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	68	2	no	—
138	C₄	—	136	1	no	—
139	C₄	—	136	2	no	—
140	C₄	#10587565	1	2	yes	—
141	C₄	#10587575	1	2	yes	—
142	C₂	#11762749	1	1	yes	characteristic
143	C₂	#11763100	1	1	yes	characteristic
144	C₂	#11763105	1	1	yes	characteristic
145	C₂	#11763128	1	1	yes	characteristic
146	C₂	#11763145	1	1	yes	characteristic
147	C₂	#11763160	1	1	yes	characteristic
148	C₂	#11763174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762707

C₂

yes

maximal, characteristic

#11762679

C₂

yes

maximal, characteristic

#11762572

C₂

yes

maximal

#11762572

C₂

yes

maximal

C₁₇×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×#63225

—

#10587502

—

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

Q₈

yes

—

#63225

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#2100

—

maximal, Sylow

C₂²×C₁₇

C₂²⋊Q₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇

C₂×Q₁₆

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#125

yes

—

C₄×C₁₇

#126

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#95

—

C₁₇

#2100

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

—

107

C₂³

C₂×D₆₈

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#63219

yes

—

122

C₂×C₄

Dic₆₈

yes

—

123

C₂×C₄

#63252

yes

—

124

C₂×C₄

#63254

yes

characteristic

125

C₂×C₄

#63254

yes

—

126

C₂×C₄

Q₈×D₁₇

yes

characteristic

127

C₈

—

128

C₂²

#10587561

yes

characteristic

129

C₂²

#10587572

yes

characteristic

130

C₂²

#10587583

yes

characteristic

131

C₂²

C₂×#63219

yes

characteristic

132

C₂²

C₂×Dic₆₈

yes

characteristic

133

C₂²

#10587594

yes

characteristic

134

C₂²

#10587594

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

136

—

139

C₄

—

136

—

140

C₄

#10587565

yes

—

141

C₄

#10587575

yes

—

142

C₂

#11762749

yes

characteristic

143

C₂

#11763100

yes

characteristic

144

C₂

#11763105

yes

characteristic

145

C₂

#11763128

yes

characteristic

146

C₂

#11763145

yes

characteristic

147

C₂

#11763160

yes

characteristic

148

C₂

#11763174

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#336	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#336	D₁₇
hypocenter	C₁₇	#2100
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#336

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
89. C₁₇	#2100	semidirect

H ≅ G/N

type

89. C₁₇

#2100

semidirect

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

subgroup	index
#11762707	2
#11762679	2
#11762572	2
C₁₇×#336	2
#2100	17

subgroup

index

C₁₇×#336

Subgroups — #14530348

Conjugacy classes of subgroups

Complements

Maximal subgroups