Group #16945993 — subgroups

Subgroups — #16945993

Order 4352.

208 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 107 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	#14527540	C₂	1	1	yes	maximal, characteristic
2	#14527532	C₂	1	1	yes	maximal, characteristic
3	#14526850	C₂	1	4	yes	maximal, complemented
4	C₁₇×#2163	C₂	1	1	yes	maximal, characteristic
5	#11762862	—	2	1	no	—
6	#11762833	—	2	1	no	—
7	#11762562	—	2	1	no	—
8	#11762560	—	2	1	no	—
9	#11762560	C₂²	1	2	yes	—
10	#11762560	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₁₆	C₄	1	2	yes	—
13	C₁₇×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₁₇×#346	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₂×#63223	—	4	1	no	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	C₂²⋊#3622	—	4	1	no	—
25	C₄⋊#3622	—	2	2	no	—
26	C₄⋊#3622	—	2	2	no	—
27	C₄⋊#3622	—	4	1	no	—
28	C₄×#3622	—	4	1	no	—
29	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	—	4	1	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
40	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
41	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
42	#63223	—	8	2	no	—
43	C₂×#3622	—	4	2	no	—
44	C₂×#3622	—	4	2	no	—
45	C₁₆×C₁₇	—	2	4	no	—
46	#12029	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	#106	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	—	2	1	no	—
50	C₂×C₄×C₁₇	—	2	1	no	—
51	C₂×C₄×C₁₇	—	4	1	no	—
52	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₈×C₁₇	—	2	1	no	—
59	C₈×C₁₇	—	2	1	no	—
60	C₈×C₁₇	—	2	1	no	—
61	C₈×C₁₇	#131	1	2	yes	—
62	#3622	—	8	2	no	—
63	#3622	—	8	2	no	—
64	#2163	D₁₇	1	1	yes	characteristic
65	#1566	—	17	1	no	—
66	#1564	—	17	1	no	—
67	#1368	—	17	4	no	—
68	C₂²×C₁₇	—	2	1	no	—
69	C₂²×C₁₇	—	2	1	no	—
70	C₂²×C₁₇	—	2	1	no	—
71	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	—	2	1	no	—
74	C₄×C₁₇	—	2	1	no	—
75	C₄×C₁₇	—	2	1	no	—
76	C₄×C₁₇	#365	1	2	yes	—
77	C₄×C₁₇	#424	1	1	yes	characteristic
78	C₄×C₁₇	#430	1	1	yes	characteristic
79	#431	—	34	1	no	—
80	#423	—	34	1	no	—
81	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
82	C₄⋊C₁₆	Dic₁₇	1	2	yes	—
83	C₂²⋊C₁₆	Dic₁₇	1	2	yes	—
84	#346	C₂×D₁₇	1	2	yes	—
85	#335	—	34	1	no	—
86	#334	—	17	2	no	—
87	#334	—	17	2	no	—
88	#334	—	34	1	no	—
89	C₂×C₁₇	—	4	1	no	—
90	C₂×C₁₇	#1570	1	1	yes	characteristic
91	C₂×C₁₇	C₂×#365	1	1	yes	characteristic
92	C₂×C₁₇	#2155	1	1	yes	characteristic
93	C₂×M₁₆	—	68	1	no	—
94	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
96	C₂×C₄²	D₆₈	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
98	C₄⋊C₈	—	34	2	no	—
99	C₄⋊C₈	—	34	2	no	—
100	C₄⋊C₈	—	68	1	no	—
101	C₂²⋊C₈	—	68	1	no	—
102	C₄×C₈	—	68	1	no	—
103	C₄×C₈	Dic₃₄	1	1	yes	characteristic
104	C₄×C₈	D₆₈	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
107	C₁₇	#12029	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₄	#63219	1	1	yes	characteristic
109	C₂²×C₄	#63219	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
111	M₁₆	—	136	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	68	2	no	—
115	C₂×C₈	—	68	2	no	—
116	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
117	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
119	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
120	C₄²	#63219	1	1	yes	characteristic
121	C₄²	#63219	1	1	yes	characteristic
122	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
123	C₄²	C₂×D₆₈	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₂³	#10587488	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#10587488	1	1	yes	characteristic
131	C₂×C₄	#10587488	1	1	yes	characteristic
132	C₂×C₄	#10587488	1	1	yes	characteristic
133	C₂×C₄	C₂×#63219	1	1	yes	characteristic
134	C₂×C₄	C₂×#63219	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	136	2	no	—
140	C₈	—	136	2	no	—
141	C₈	#10587634	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	C₂×#10587488	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#11762632	1	2	yes	—
151	C₄	#11762836	1	1	yes	characteristic
152	C₄	#11762858	1	1	yes	characteristic
153	C₂	—	4	1	no	—
154	C₂	#14527555	1	1	yes	characteristic
155	C₂	C₂×#11762632	1	1	yes	characteristic
156	C₂	#14530557	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14527540

C₂

yes

maximal, characteristic

#14527532

C₂

yes

maximal, characteristic

#14526850

C₂

yes

maximal, complemented

C₁₇×#2163

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#11762560

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊C₁₆

C₄

yes

—

C₁₇×C₂²⋊C₁₆

C₄

yes

—

C₁₇×#346

C₂²

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

Q₈

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×#63223

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

Q₈

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

C₂²⋊#3622

—

—

—

—

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

SD₁₆

yes

characteristic

C₂²×C₄×C₁₇

SD₁₆

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

SD₁₆

yes

characteristic

C₄²×C₁₇

SD₁₆

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

—

—

C₁₆×C₁₇

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#131

yes

—

#3622

—

#3622

—

#2163

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#106

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#365

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₄⋊C₁₆

Dic₁₇

yes

—

C₂²⋊C₁₆

Dic₁₇

yes

—

#346

C₂×D₁₇

yes

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂²×C₈

Dic₃₄

yes

characteristic

C₂²×C₈

C₂×Dic₁₇

yes

characteristic

C₂×C₄²

D₆₈

yes

characteristic

C₂×C₁₆

C₂×Dic₁₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

Dic₃₄

yes

characteristic

104

C₄×C₈

D₆₈

yes

characteristic

105

C₄×C₈

C₂×Dic₁₇

yes

characteristic

106

C₄×C₈

C₂²×D₁₇

yes

characteristic

107

C₁₇

#12029

yes

characteristic, complemented, Sylow

108

C₂²×C₄

#63219

yes

characteristic

109

C₂²×C₄

#63219

yes

characteristic

110

C₂²×C₄

C₄⋊Dic₁₇

yes

characteristic

111

M₁₆

—

136

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄⋊Dic₁₇

yes

characteristic

117

C₂×C₈

C₄⋊Dic₁₇

yes

characteristic

118

C₂×C₈

C₂×Dic₃₄

yes

characteristic

119

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

120

C₄²

#63219

yes

characteristic

121

C₄²

#63219

yes

characteristic

122

C₄²

C₄⋊Dic₁₇

yes

characteristic

123

C₄²

C₂×D₆₈

yes

characteristic

124

C₁₆

—

125

C₂³

#10587488

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10587488

yes

characteristic

131

C₂×C₄

#10587488

yes

characteristic

132

C₂×C₄

#10587488

yes

characteristic

133

C₂×C₄

C₂×#63219

yes

characteristic

134

C₂×C₄

C₂×#63219

yes

characteristic

135

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

136

—

140

C₈

—

136

—

141

C₈

#10587634

yes

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×#10587488

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#11762632

yes

—

151

C₄

#11762836

yes

characteristic

152

C₄

#11762858

yes

characteristic

153

C₂

—

154

C₂

#14527555

yes

characteristic

155

C₂

C₂×#11762632

yes

characteristic

156

C₂

#14530557

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#10587488
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#2163	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	#106
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#2163	D₁₇
hypocenter	C₁₇	#12029
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

C₂×#10587488

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#2163

C₂

C₂²×C₁₇

C₂³×C₁₇

C₄×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
3. #14526850	C₂	semidirect
107. C₁₇	#12029	semidirect

H ≅ G/N

type

3. #14526850

C₂

semidirect

107. C₁₇

#12029

semidirect

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

subgroup	index
#14527540	2
#14527532	2
#14526850	2
C₁₇×#2163	2
#12029	17

subgroup

index

C₁₇×#2163

Subgroups — #16945993

Conjugacy classes of subgroups

Complements

Maximal subgroups