Group #17022129 — subgroups

Subgroups — #17022129

Order 4352.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#14539370	C₂	1	4	yes	maximal
2	#14539298	C₂	1	2	yes	maximal
3	#14528461	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₇×#322	C₂	1	1	yes	maximal, characteristic
5	#11763366	C₂²	1	4	yes	—
6	#11763214	C₂²	1	4	yes	—
7	#11763214	C₂²	1	4	yes	—
8	#11763211	—	2	4	no	—
9	#11763207	C₂²	1	4	yes	—
10	#11763189	C₂²	1	4	yes	—
11	#11763160	C₂²	1	8	yes	—
12	#11763050	—	2	8	no	—
13	C₄×Dic₆₈	—	2	8	no	—
14	#11762847	—	2	4	no	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
17	C₁₇×#322	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₇	D₄	1	4	yes	—
19	C₂²×C₈×C₁₇	D₄	1	2	yes	—
20	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
21	#10587612	—	2	4	no	—
22	#10587610	—	2	8	no	—
23	C₂×Dic₆₈	—	2	8	no	—
24	#10587556	—	2	4	no	—
25	#10587555	C₂³	1	8	yes	—
26	C₄×Dic₃₄	—	2	8	no	—
27	C₄×C₈×C₁₇	D₄	1	2	yes	—
28	C₄×C₈×C₁₇	D₄	1	4	yes	—
29	C₄×C₈×C₁₇	C₂³	1	2	yes	—
30	C₄×C₈×C₁₇	C₂³	1	4	yes	—
31	#10587489	—	2	8	no	—
32	#10587489	—	2	8	no	—
33	#10587488	—	2	8	no	—
34	#10587487	—	2	16	no	—
35	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₃₄	—	2	8	no	—
38	C₂×C₈×C₁₇	—	4	4	no	—
39	C₂×C₈×C₁₇	Q₁₆	1	8	yes	complemented
40	C₂×C₈×C₁₇	C₂×D₄	1	8	yes	—
41	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
42	C₄²×C₁₇	C₂×D₄	1	2	yes	—
43	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
45	C₂²⋊Dic₁₇	—	4	4	no	—
46	C₄⋊Dic₁₇	—	2	8	no	—
47	C₄⋊Dic₁₇	—	2	8	no	—
48	#63225	—	4	8	no	—
49	C₄×Dic₁₇	—	4	4	no	—
50	Dic₆₈	—	4	16	no	—
51	#19618	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	#127	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	2	4	no	—
54	C₂×C₄×C₁₇	#127	1	2	yes	—
55	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
57	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂²×D₄	1	2	yes	—
59	C₂×Dic₁₇	—	4	8	no	—
60	Dic₃₄	—	4	16	no	—
61	C₈×C₁₇	—	2	8	no	—
62	C₈×C₁₇	C₂×Q₁₆	1	8	yes	—
63	#3170	—	17	4	no	—
64	#3160	—	17	2	no	—
65	#1723	—	17	8	no	—
66	C₂×#322	D₁₇	1	1	yes	characteristic
67	C₂²×C₁₇	—	2	2	no	—
68	C₂²×C₁₇	#494	1	2	yes	—
69	C₂²×C₁₇	C₂×#127	1	1	yes	characteristic
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	#494	1	2	yes	—
72	C₄×C₁₇	C₂²×Q₁₆	1	2	yes	—
73	C₄×C₁₇	#575	1	4	yes	—
74	Dic₁₇	—	8	8	no	—
75	#533	—	17	4	no	—
76	#500	—	17	4	no	—
77	#500	—	17	4	no	—
78	#499	—	34	4	no	—
79	#498	—	17	4	no	—
80	#494	—	17	4	no	—
81	#487	—	17	8	no	—
82	#467	—	34	8	no	—
83	C₄×Q₁₆	—	34	8	no	—
84	#427	—	34	4	no	—
85	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
87	#322	C₂×D₁₇	1	4	yes	—
88	C₂×C₁₇	—	2	2	no	—
89	C₂×C₁₇	C₂×#494	1	1	yes	characteristic
90	C₂×C₁₇	#3150	1	2	yes	—
91	C₂×Q₁₆	—	34	8	no	—
92	C₂²×C₈	D₆₈	1	4	yes	—
93	C₂²×C₈	D₆₈	1	2	yes	—
94	C₄⋊Q₈	—	17	8	no	—
95	#125	—	34	4	no	—
96	C₂²⋊Q₈	—	34	8	no	—
97	C₄×Q₈	—	34	8	no	—
98	#117	—	34	4	no	—
99	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
100	#107	—	34	8	no	—
101	#107	—	34	8	no	—
102	#106	—	34	8	no	—
103	#103	—	34	16	no	—
104	C₄×C₈	D₆₈	1	4	yes	—
105	C₄×C₈	D₆₈	1	2	yes	—
106	C₄×C₈	C₂²×D₁₇	1	4	yes	—
107	C₄×C₈	C₂²×D₁₇	1	2	yes	—
108	C₁₇	#19618	1	1	yes	characteristic, complemented, Sylow
109	C₂×Q₈	—	34	8	no	—
110	C₂²×C₄	C₂×D₆₈	1	2	yes	—
111	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
112	Q₁₆	—	68	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₆₈	1	8	yes	complemented
115	C₂×C₈	C₂×D₆₈	1	8	yes	—
116	C₂×C₈	C₂×D₆₈	1	4	yes	—
117	C₄⋊C₄	—	34	8	no	—
118	C₄⋊C₄	—	34	8	no	—
119	C₄⋊C₄	—	68	8	no	—
120	C₂²⋊C₄	—	68	4	no	—
121	C₄²	—	68	4	no	—
122	C₄²	C₂×D₆₈	1	1	yes	characteristic
123	C₄²	C₂×D₆₈	1	2	yes	—
124	C₄²	C₂³×D₁₇	1	1	yes	characteristic
125	C₂³	#10587560	1	1	yes	characteristic
126	Q₈	—	68	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	68	8	no	—
129	C₂×C₄	#10587560	1	2	yes	—
130	C₂×C₄	#10587560	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₆₈	1	4	yes	—
132	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₆₈	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₆₈	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#11763189	1	2	yes	—
138	C₂²	C₂×#10587560	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	136	8	no	—
141	C₄	#11763189	1	2	yes	—
142	C₄	C₂²×Dic₆₈	1	2	yes	—
143	C₄	#11763609	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#11763189	1	1	yes	characteristic
146	C₂	#14539230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539370

C₂

yes

maximal

#14539298

C₂

yes

maximal

#14528461

C₂

yes

maximal, complemented

C₂×C₁₇×#322

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

—

#11763214

C₂²

yes

—

#11763214

C₂²

yes

—

#11763211

—

#11763207

C₂²

yes

—

#11763189

C₂²

yes

—

#11763160

C₂²

yes

—

#11763050

—

C₄×Dic₆₈

—

#11762847

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₁₇×#322

C₂²

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

#10587612

—

#10587610

—

C₂×Dic₆₈

—

#10587556

—

#10587555

C₂³

yes

—

C₄×Dic₃₄

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

Q₁₆

yes

complemented

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂⁴

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

Dic₆₈

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×D₄

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

#494

yes

—

C₂²×C₁₇

C₂×#127

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#494

yes

—

C₄×C₁₇

C₂²×Q₁₆

yes

—

C₄×C₁₇

#575

yes

—

Dic₁₇

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

—

#322

C₂×D₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#494

yes

characteristic

C₂×C₁₇

#3150

yes

—

C₂×Q₁₆

—

C₂²×C₈

D₆₈

yes

—

C₂²×C₈

D₆₈

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

100

#107

—

101

#107

—

102

#106

—

103

#103

—

104

C₄×C₈

D₆₈

yes

—

105

C₄×C₈

D₆₈

yes

—

106

C₄×C₈

C₂²×D₁₇

yes

—

107

C₄×C₈

C₂²×D₁₇

yes

—

108

C₁₇

#19618

yes

characteristic, complemented, Sylow

109

C₂×Q₈

—

110

C₂²×C₄

C₂×D₆₈

yes

—

111

C₂²×C₄

C₂×D₆₈

yes

characteristic

112

Q₁₆

—

113

C₂×C₈

—

114

C₂×C₈

Dic₆₈

yes

complemented

115

C₂×C₈

C₂×D₆₈

yes

—

116

C₂×C₈

C₂×D₆₈

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

C₂×D₆₈

yes

characteristic

123

C₄²

C₂×D₆₈

yes

—

124

C₄²

C₂³×D₁₇

yes

characteristic

125

C₂³

#10587560

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#10587560

yes

—

130

C₂×C₄

#10587560

yes

characteristic

131

C₂×C₄

C₂×Dic₆₈

yes

—

132

C₂×C₄

C₂²×D₆₈

yes

characteristic

133

C₂×C₄

C₂²×D₆₈

yes

—

134

C₈

—

135

C₈

C₂×Dic₆₈

yes

—

136

C₂²

—

137

C₂²

#11763189

yes

—

138

C₂²

C₂×#10587560

yes

characteristic

139

C₄

—

140

C₄

—

136

—

141

C₄

#11763189

yes

—

142

C₄

C₂²×Dic₆₈

yes

—

143

C₄

#11763609

yes

—

144

C₂

—

145

C₂

C₂×#11763189

yes

characteristic

146

C₂

#14539230

yes

—

	subgroup	G/H
center	C₂×C₄	#10587560
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₁₇×#322	C₂
socle	C₂²×C₁₇	C₂×#127
prime radical	C₂³×C₁₇	#127
hypercenter	C₂×#322	D₁₇
hypocenter	C₁₇	#19618
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#10587560

commutator, cosocle

C₄²×C₁₇

C₂⁴

Frattini

C₄²

C₂³×D₁₇

Fitting

C₂×C₁₇×#322

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #14528461	C₂	semidirect
39. C₂×C₈×C₁₇	Q₁₆	semidirect
108. C₁₇	#19618	semidirect
114. C₂×C₈	Dic₆₈	semidirect

H ≅ G/N

type

3. #14528461

C₂

semidirect

39. C₂×C₈×C₁₇

Q₁₆

semidirect

108. C₁₇

#19618

semidirect

114. C₂×C₈

Dic₆₈

semidirect

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

subgroup	index
#14539370	2
#14539298	2
#14528461	2
C₂×C₁₇×#322	2
#19618	17

subgroup

index

C₂×C₁₇×#322

Subgroups — #17022129

Conjugacy classes of subgroups

Complements

Maximal subgroups