Group #17189826 — subgroups

Subgroups — #17189826

Order 4352.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#14539298	C₂	1	2	yes	maximal
2	#14531070	C₂	1	8	yes	maximal, complemented
3	#14530568	C₂	1	4	yes	maximal
4	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	#11763366	—	2	2	no	—
6	#11763214	C₂²	1	4	yes	—
7	#11763211	—	2	2	no	—
8	#11763211	C₂²	1	4	yes	—
9	#11763207	—	2	2	no	—
10	#11762847	—	2	2	no	—
11	#11762847	C₂²	1	4	yes	—
12	#11762637	C₂²	1	8	yes	—
13	#11762635	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#346	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
20	#10587612	—	2	4	no	—
21	#10587610	—	4	4	no	—
22	#10587556	—	2	4	no	—
23	#10587555	—	2	4	no	—
24	C₄×Dic₃₄	—	4	4	no	—
25	C₂×C₁₆×C₁₇	Q₈	1	8	yes	complemented
26	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
27	C₄×C₈×C₁₇	D₄	1	2	yes	—
28	C₄×C₈×C₁₇	C₂³	1	2	yes	—
29	#10587489	—	2	4	no	—
30	#10587489	C₂³	1	8	yes	—
31	#10587488	—	2	4	no	—
32	#10587488	—	2	4	no	—
33	C₂²×C₄×C₁₇	D₈	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₃₄	—	4	4	no	—
36	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
37	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
41	C₄²×C₁₇	D₈	1	2	yes	—
42	C₄²×C₁₇	C₂×D₄	1	2	yes	—
43	C₂²⋊Dic₁₇	—	4	4	no	—
44	C₄⋊Dic₁₇	—	2	8	no	—
45	C₄⋊Dic₁₇	—	4	4	no	—
46	#63225	—	4	8	no	—
47	C₄×Dic₁₇	—	4	4	no	—
48	C₁₆×C₁₇	C₂×Q₈	1	8	yes	—
49	#33728	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	—	4	2	no	—
52	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
53	C₂×C₄×C₁₇	C₂×D₈	1	4	yes	—
54	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
56	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×Dic₁₇	—	4	8	no	—
58	Dic₃₄	—	8	8	no	—
59	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
60	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
61	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
62	#3160	—	17	2	no	—
63	#2255	—	17	8	no	—
64	#2158	—	17	4	no	—
65	C₂×#346	D₁₇	1	1	yes	characteristic
66	C₂²×C₁₇	—	2	2	no	—
67	C₂²×C₁₇	#500	1	2	yes	—
68	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
69	C₄×C₁₇	—	2	4	no	—
70	C₄×C₁₇	#500	1	2	yes	—
71	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
72	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
73	Dic₁₇	—	8	8	no	—
74	#533	—	34	2	no	—
75	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
76	#500	—	17	4	no	—
77	#499	—	17	4	no	—
78	#499	—	34	2	no	—
79	#498	—	34	2	no	—
80	#427	—	17	4	no	—
81	#427	—	34	2	no	—
82	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
83	#367	—	17	8	no	—
84	#366	—	17	8	no	—
85	#346	C₂×D₁₇	1	4	yes	—
86	C₂×C₁₇	—	2	2	no	—
87	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
88	C₂×C₁₇	#3414	1	2	yes	—
89	C₂²×C₈	D₆₈	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
91	C₄⋊Q₈	—	34	4	no	—
92	#125	—	34	4	no	—
93	C₂²⋊Q₈	—	68	4	no	—
94	C₄×Q₈	—	68	4	no	—
95	#117	—	34	4	no	—
96	C₂×C₄²	D₆₈	1	1	yes	characteristic
97	C₂×C₁₆	Dic₃₄	1	8	yes	complemented
98	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
99	#107	—	17	8	no	—
100	#107	—	34	4	no	—
101	#106	—	34	4	no	—
102	#106	—	34	4	no	—
103	C₄×C₈	D₆₈	1	2	yes	—
104	C₄×C₈	C₂²×D₁₇	1	2	yes	—
105	C₁₇	#33728	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	68	4	no	—
107	C₂²×C₄	D₁₃₆	1	2	yes	—
108	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
109	C₂×C₈	Dic₆₈	1	4	yes	—
110	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
111	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
112	C₂×C₈	C₂×D₆₈	1	2	yes	—
113	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
114	C₄⋊C₄	—	34	8	no	—
115	C₄⋊C₄	—	68	8	no	—
116	C₄⋊C₄	—	68	4	no	—
117	C₂²⋊C₄	—	68	4	no	—
118	C₄²	—	68	4	no	—
119	C₄²	D₁₃₆	1	2	yes	—
120	C₄²	C₂×D₆₈	1	2	yes	—
121	C₁₆	C₂×Dic₃₄	1	8	yes	—
122	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
123	Q₈	—	136	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	68	8	no	—
126	C₂×C₄	#10587555	1	2	yes	—
127	C₂×C₄	C₂×D₁₃₆	1	4	yes	—
128	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
130	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
131	C₈	#10587555	1	2	yes	—
132	C₈	C₂×Dic₆₈	1	4	yes	—
133	C₈	C₂²×Dic₃₄	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#11763214	1	2	yes	—
136	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	136	8	no	—
139	C₄	#11763214	1	2	yes	—
140	C₄	C₂×#10587555	1	1	yes	characteristic
141	C₄	C₂²×Dic₆₈	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#11763214	1	1	yes	characteristic
144	C₂	#14541920	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539298

C₂

yes

maximal

#14531070

C₂

yes

maximal, complemented

#14530568

C₂

yes

maximal

C₂×C₁₇×#346

C₂

yes

maximal, characteristic

#11763366

—

#11763214

C₂²

yes

—

#11763211

—

#11763211

C₂²

yes

—

#11763207

—

#11762847

—

#11762847

C₂²

yes

—

#11762637

C₂²

yes

—

#11762635

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#346

C₂²

yes

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

—

—

—

C₄×Dic₃₄

—

C₂×C₁₆×C₁₇

Q₈

yes

complemented

C₂×C₁₆×C₁₇

C₂³

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

#10587489

—

#10587489

C₂³

yes

—

#10587488

—

#10587488

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

C₂⁴

yes

characteristic

C₄²×C₁₇

D₈

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₁₆×C₁₇

C₂×Q₈

yes

—

#33728

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

C₄⋊Q₈

yes

—

C₈×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

C₂²×Q₈

yes

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

#500

yes

—

C₂²×C₁₇

C₂²×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#500

yes

—

C₄×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₁₇

C₂²×Q₁₆

yes

characteristic

Dic₁₇

—

#533

—

C₂²×C₁₆

C₂×D₁₇

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₇

yes

—

—

C₂×D₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#500

yes

characteristic

C₂×C₁₇

#3414

yes

—

C₂²×C₈

D₆₈

yes

characteristic

C₂²×C₈

C₂²×D₁₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₆₈

yes

characteristic

C₂×C₁₆

Dic₃₄

yes

complemented

C₂×C₁₆

C₂²×D₁₇

yes

—

#107

—

100

#107

—

101

#106

—

102

#106

—

103

C₄×C₈

D₆₈

yes

—

104

C₄×C₈

C₂²×D₁₇

yes

—

105

C₁₇

#33728

yes

characteristic, complemented, Sylow

106

C₂×Q₈

—

107

C₂²×C₄

D₁₃₆

yes

—

108

C₂²×C₄

C₂×D₆₈

yes

characteristic

109

C₂×C₈

Dic₆₈

yes

—

110

C₂×C₈

C₂×Dic₃₄

yes

—

111

C₂×C₈

C₂×D₆₈

yes

characteristic

112

C₂×C₈

C₂×D₆₈

yes

—

113

C₂×C₈

C₂³×D₁₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

D₁₃₆

yes

—

120

C₄²

C₂×D₆₈

yes

—

121

C₁₆

C₂×Dic₃₄

yes

—

122

C₂³

C₂×D₁₃₆

yes

characteristic

123

Q₈

—

136

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#10587555

yes

—

127

C₂×C₄

C₂×D₁₃₆

yes

—

128

C₂×C₄

C₂×D₁₃₆

yes

characteristic

129

C₂×C₄

C₂×Dic₆₈

yes

—

130

C₂×C₄

C₂²×D₆₈

yes

characteristic

131

C₈

#10587555

yes

—

132

C₈

C₂×Dic₆₈

yes

—

133

C₈

C₂²×Dic₃₄

yes

—

134

C₂²

—

135

C₂²

#11763214

yes

—

136

C₂²

C₂²×D₁₃₆

yes

characteristic

137

C₄

—

138

C₄

—

136

—

139

C₄

#11763214

yes

—

140

C₄

C₂×#10587555

yes

characteristic

141

C₄

C₂²×Dic₆₈

yes

characteristic

142

C₂

—

143

C₂

C₂×#11763214

yes

characteristic

144

C₂

#14541920

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#33728
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₁₃₆

commutator, cosocle

C₂×C₈×C₁₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₇

Fitting

C₂×C₁₇×#346

C₂

socle

C₂²×C₁₇

C₂²×D₈

prime radical

C₂³×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #14531070	C₂	semidirect
25. C₂×C₁₆×C₁₇	Q₈	semidirect
97. C₂×C₁₆	Dic₃₄	semidirect
105. C₁₇	#33728	semidirect

H ≅ G/N

type

2. #14531070

C₂

semidirect

25. C₂×C₁₆×C₁₇

Q₈

semidirect

97. C₂×C₁₆

Dic₃₄

semidirect

105. C₁₇

#33728

semidirect

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

subgroup	index
#14539298	2
#14531070	2
#14530568	2
C₂×C₁₇×#346	2
#33728	17

subgroup

index

C₂×C₁₇×#346

Subgroups — #17189826

Conjugacy classes of subgroups

Complements

Maximal subgroups