Group #14530197 — subgroups

Subgroups — #14530197

Order 2176.

258 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 81 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×#10587503	C₂	1	2	yes	maximal
2	#11762675	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762571	C₂	1	2	yes	maximal, complemented
4	#11762564	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#390	C₂	1	1	yes	maximal, characteristic
6	C₂²×D₄×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂²×#3622	C₂²	1	2	yes	—
12	#10587503	—	2	4	no	—
13	#10587502	—	2	2	no	—
14	C₂⁴×C₁₇	—	2	1	no	—
15	C₂×D₄×C₁₇	—	2	2	no	—
16	C₂×D₄×C₁₇	—	2	2	no	—
17	C₂×D₄×C₁₇	—	2	1	no	—
18	C₂×D₄×C₁₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₇	—	2	1	no	—
20	C₂²×C₄×C₁₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₂²⋊C₄×C₁₇	—	4	4	no	—
25	C₄²×C₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×#3622	—	2	2	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	—	2	2	no	—
31	C₂³×C₁₇	—	2	2	no	—
32	C₂³×C₁₇	—	4	1	no	—
33	C₂³×C₁₇	—	4	1	no	—
34	C₂³×C₁₇	—	4	2	no	—
35	C₂³×C₁₇	—	4	2	no	—
36	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
37	D₄×C₁₇	—	2	4	no	—
38	D₄×C₁₇	—	4	2	no	—
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	4	1	no	—
43	C₂×C₄×C₁₇	—	4	2	no	—
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₂×C₄×C₁₇	#41	1	2	yes	—
48	C₂×Dic₁₇	—	4	2	no	—
49	C₂×Dic₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	#3622	—	4	2	no	—
52	#2073	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	—	4	2	no	—
54	C₂²×C₁₇	—	4	4	no	—
55	C₂²×C₁₇	—	8	2	no	—
56	C₂²×C₁₇	—	8	1	no	—
57	C₂²×C₁₇	—	8	1	no	—
58	C₂²×C₁₇	#121	1	1	yes	characteristic
59	C₂²×C₁₇	#123	1	1	yes	characteristic
60	C₂²×C₁₇	#124	1	1	yes	characteristic
61	C₂²×C₁₇	#135	1	2	yes	—
62	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₇	—	4	2	no	—
64	C₄×C₁₇	—	8	1	no	—
65	C₄×C₁₇	#123	1	2	yes	—
66	C₄×C₁₇	#126	1	2	yes	—
67	Dic₁₇	—	8	2	no	—
68	C₂×#102	—	17	2	no	—
69	#390	D₁₇	1	1	yes	characteristic
70	#382	—	17	1	no	—
71	#340	—	17	2	no	—
72	#336	—	17	1	no	—
73	C₂×C₁₇	—	8	2	no	—
74	C₂×C₁₇	#397	1	1	yes	characteristic
75	C₂×C₁₇	#463	1	2	yes	—
76	C₂×C₁₇	#482	1	2	yes	—
77	C₂×C₁₇	#488	1	1	yes	characteristic
78	C₂×C₁₇	#490	1	1	yes	characteristic
79	C₂²×D₄	C₂×D₁₇	1	1	yes	characteristic
80	C₂²×C₈	—	17	2	no	—
81	C₂×C₄⋊C₄	—	17	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
83	C₂×C₂²⋊C₄	—	2	2	no	—
84	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
85	#102	—	34	4	no	—
86	#95	—	34	2	no	—
87	C₁₇	#2073	1	1	yes	characteristic, complemented, Sylow
88	C₂⁴	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	D₆₈	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	34	2	no	—
95	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
97	C₂×C₈	—	34	2	no	—
98	C₂×C₈	—	34	2	no	—
99	C₂×C₈	—	34	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	34	2	no	—
102	C₄⋊C₄	—	34	2	no	—
103	C₂²⋊C₄	—	4	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	2	no	—
110	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	68	2	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	—	68	2	no	—
121	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
122	C₂×C₄	#63252	1	1	yes	characteristic
123	C₂×C₄	#63254	1	2	yes	—
124	C₂×C₄	#63254	1	2	yes	—
125	C₈	—	68	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	8	1	no	—
131	C₂²	#10587616	1	1	yes	characteristic
132	C₂²	#10587618	1	2	yes	—
133	C₂²	#10587621	1	1	yes	characteristic
134	C₂²	#10587622	1	1	yes	characteristic
135	C₂²	#10587636	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	136	2	no	—
139	C₄	#10587565	1	2	yes	—
140	C₄	#10587572	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	#11762734	1	1	yes	characteristic
143	C₂	#11763024	1	2	yes	—
144	C₂	#11763130	1	2	yes	—
145	C₂	#11763163	1	1	yes	characteristic
146	C₂	#11763173	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587503

C₂

yes

maximal

#11762675

C₂

yes

maximal, characteristic, complemented

#11762571

C₂

yes

maximal, complemented

#11762564

C₂

yes

maximal, characteristic, complemented

C₁₇×#390

C₂

yes

maximal, characteristic

C₂²×D₄×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

#10587503

—

#10587502

—

C₂⁴×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

D₄×C₁₇

—

D₄×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

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#3622

—

#2073

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#123

yes

—

C₄×C₁₇

#126

yes

—

Dic₁₇

—

C₂×#102

—

#390

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#397

yes

characteristic

C₂×C₁₇

#463

yes

—

C₂×C₁₇

#482

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×D₄

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

#102

—

#95

—

C₁₇

#2073

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₆₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

111

D₄

—

112

D₄

—

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

C₂×D₆₈

yes

characteristic

122

C₂×C₄

#63252

yes

characteristic

123

C₂×C₄

#63254

yes

—

124

C₂×C₄

#63254

yes

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#10587616

yes

characteristic

132

C₂²

#10587618

yes

—

133

C₂²

#10587621

yes

characteristic

134

C₂²

#10587622

yes

characteristic

135

C₂²

#10587636

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

136

—

139

C₄

#10587565

yes

—

140

C₄

#10587572

yes

—

141

C₂

—

142

C₂

#11762734

yes

characteristic

143

C₂

#11763024

yes

—

144

C₂

#11763130

yes

—

145

C₂

#11763163

yes

characteristic

146

C₂

#11763173

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#390	C₂
socle	C₂³×C₁₇	C₂×D₄
prime radical	C₂²×D₄×C₁₇	C₂²
hypercenter	#390	D₁₇
hypocenter	C₁₇	#2073
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#390

C₂

socle

C₂³×C₁₇

C₂×D₄

prime radical

C₂²×D₄×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
2. #11762675	C₂	semidirect
3. #11762571	C₂	semidirect
4. #11762564	C₂	semidirect
87. C₁₇	#2073	semidirect

H ≅ G/N

type

2. #11762675

C₂

semidirect

3. #11762571

C₂

semidirect

4. #11762564

C₂

semidirect

87. C₁₇

#2073

semidirect

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

subgroup	index
C₂×#10587503	2
#11762675	2
#11762571	2
#11762564	2
C₁₇×#390	2
#2073	17

subgroup

index

C₁₇×#390

Subgroups — #14530197

Conjugacy classes of subgroups

Complements

Maximal subgroups