Group #16955120 — subgroups

Subgroups — #16955120

Order 4352.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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×Dic₂₇₂	C₂	1	2	yes	maximal
2	#14531037	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
4	#14526966	C₂	1	2	yes	maximal
5	C₁₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₆	—	2	1	no	—
7	C₂×Dic₁₃₆	C₂²	1	2	yes	—
8	C₂×Dic₁₃₆	C₂²	1	1	yes	characteristic
9	#11763189	—	2	1	no	—
10	C₄×Dic₆₈	—	2	1	no	—
11	Dic₂₇₂	—	2	4	no	—
12	#11762635	C₂²	1	1	yes	characteristic
13	#11762614	—	2	1	no	—
14	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
15	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₈	—	2	1	no	—
17	C₂×Dic₆₈	—	2	2	no	—
18	C₂×Dic₆₈	—	4	1	no	—
19	#10587555	—	4	1	no	—
20	C₄×Dic₃₄	—	4	1	no	—
21	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
24	#10587489	—	2	1	no	—
25	#10587487	—	4	1	no	—
26	Dic₁₃₆	—	2	2	no	—
27	Dic₁₃₆	—	2	1	no	—
28	Dic₁₃₆	—	2	2	no	—
29	Dic₁₃₆	—	4	2	no	—
30	Dic₁₃₆	D₄	1	2	yes	—
31	C₁₇×C₃₂	—	2	2	no	—
32	C₂×Dic₃₄	—	4	1	no	—
33	C₂×Dic₃₄	—	4	2	no	—
34	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₁₇	D₈	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	4	1	no	—
38	C₄⋊Dic₁₇	—	8	1	no	—
39	C₄⋊Dic₁₇	—	8	1	no	—
40	#63225	—	8	1	no	—
41	C₄×Dic₁₇	—	8	1	no	—
42	Dic₆₈	—	2	2	no	—
43	Dic₆₈	—	4	2	no	—
44	Dic₆₈	—	4	1	no	—
45	Dic₆₈	—	4	2	no	—
46	Dic₆₈	—	8	2	no	—
47	C₁₆×C₁₇	—	2	1	no	—
48	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₁₇	#41	1	1	yes	characteristic
50	#13688	—	17	1	no	maximal, Sylow
51	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₁₇	—	8	1	no	—
55	C₂×Dic₁₇	—	8	2	no	—
56	C₂×Dic₁₇	—	8	1	no	—
57	Dic₃₄	—	4	2	no	—
58	Dic₃₄	—	8	1	no	—
59	Dic₃₄	—	8	2	no	—
60	Dic₃₄	—	8	2	no	—
61	C₈×C₁₇	—	2	1	no	—
62	C₈×C₁₇	#121	1	1	yes	characteristic
63	C₈×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Q₆₄	—	17	2	no	—
65	#2247	—	17	1	no	—
66	C₄×Q₃₂	—	17	1	no	—
67	C₄⋊C₃₂	D₁₇	1	1	yes	characteristic
68	#1416	—	17	2	no	—
69	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₁₇	—	2	1	no	—
71	C₄×C₁₇	Q₆₄	1	2	yes	complemented
72	C₄×C₁₇	#459	1	1	yes	characteristic
73	C₄×C₁₇	#509	1	1	yes	characteristic
74	Dic₁₇	—	8	2	no	—
75	Dic₁₇	—	16	2	no	—
76	Dic₁₇	—	16	1	no	—
77	C₂×Q₃₂	—	17	1	no	—
78	C₂×Q₃₂	—	17	2	no	—
79	C₂×Q₃₂	—	34	1	no	—
80	#494	—	34	1	no	—
81	C₄×Q₁₆	—	34	1	no	—
82	Q₆₄	—	34	4	no	—
83	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
84	#366	—	17	1	no	—
85	#358	—	34	1	no	—
86	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
87	C₂×C₁₇	#2206	1	1	yes	characteristic
88	C₂×C₁₇	C₂×Q₆₄	1	1	yes	characteristic
89	C₂×C₁₇	#2264	1	1	yes	characteristic
90	C₂×Q₁₆	—	34	1	no	—
91	C₂×Q₁₆	—	34	2	no	—
92	C₂×Q₁₆	—	68	1	no	—
93	C₄⋊Q₈	—	68	1	no	—
94	C₄×Q₈	—	68	1	no	—
95	Q₃₂	—	17	2	no	—
96	Q₃₂	—	34	2	no	—
97	Q₃₂	—	34	1	no	—
98	Q₃₂	—	34	2	no	—
99	Q₃₂	—	68	2	no	—
100	C₂×C₁₆	D₆₈	1	1	yes	characteristic
101	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
102	#107	—	34	1	no	—
103	#103	—	68	1	no	—
104	C₄×C₈	D₆₈	1	1	yes	characteristic
105	C₃₂	—	2	2	no	—
106	C₁₇	#13688	1	1	yes	characteristic, complemented, Sylow
107	C₂×Q₈	—	68	1	no	—
108	C₂×Q₈	—	68	2	no	—
109	Q₁₆	—	34	2	no	—
110	Q₁₆	—	68	2	no	—
111	Q₁₆	—	68	1	no	—
112	Q₁₆	—	68	2	no	—
113	Q₁₆	—	136	2	no	—
114	C₂×C₈	D₁₃₆	1	1	yes	characteristic
115	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
116	C₄⋊C₄	—	68	1	no	—
117	C₄⋊C₄	—	136	1	no	—
118	C₄⋊C₄	—	136	1	no	—
119	C₄⋊C₄	—	136	1	no	—
120	C₄²	—	136	1	no	—
121	C₄²	D₁₃₆	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	D₄×D₁₇	1	1	yes	characteristic
124	C₁₆	#63256	1	1	yes	characteristic
125	Q₈	—	68	2	no	—
126	Q₈	—	136	2	no	—
127	Q₈	—	136	2	no	—
128	Q₈	—	136	1	no	—
129	C₂×C₄	—	136	1	no	—
130	C₂×C₄	—	136	2	no	—
131	C₂×C₄	—	136	1	no	—
132	C₂×C₄	D₂₇₂	1	1	yes	characteristic
133	C₂×C₄	D₂₇₂	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#10587581	1	1	yes	characteristic
137	C₈	#10587594	1	1	yes	characteristic
138	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	272	2	no	—
141	C₄	—	272	1	no	—
142	C₄	—	136	2	no	—
143	C₄	Dic₂₇₂	1	2	yes	complemented
144	C₄	#11763005	1	1	yes	characteristic
145	C₄	#11763247	1	1	yes	characteristic
146	C₂	#14530820	1	1	yes	characteristic
147	C₂	C₂×Dic₂₇₂	1	1	yes	characteristic
148	C₂	#14531104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₇₂

C₂

yes

maximal

#14531037

C₂

yes

maximal, characteristic

C₄×Dic₁₃₆

C₂

yes

maximal, characteristic

#14526966

C₂

yes

maximal

C₁₇×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

—

C₂×Dic₁₃₆

C₂²

yes

—

C₂×Dic₁₃₆

C₂²

yes

characteristic

#11763189

—

C₄×Dic₆₈

—

Dic₂₇₂

—

#11762635

C₂²

yes

characteristic

#11762614

—

C₂×C₁₇×C₃₂

C₂²

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

C₄×Dic₃₄

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

#10587489

—

#10587487

—

Dic₁₃₆

—

Dic₁₃₆

—

Dic₁₃₆

—

Dic₁₃₆

—

Dic₁₃₆

D₄

yes

—

C₁₇×C₃₂

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

D₈

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

C₁₆×C₁₇

C₂×D₄

yes

C₁₆×C₁₇

yes

—

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

—

C₄⋊C₃₂

D₁₇

yes

characteristic

#1416

—

C₂²×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

Q₆₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

C₂×D₁₇

yes

—

#366

—

#358

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₂×C₁₇

#2206

yes

characteristic

C₂×C₁₇

C₂×Q₆₄

yes

characteristic

C₂×C₁₇

#2264

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

100

C₂×C₁₆

D₆₈

yes

characteristic

101

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

102

#107

—

103

#103

—

104

C₄×C₈

D₆₈

yes

characteristic

105

C₃₂

—

106

C₁₇

#13688

yes

characteristic, complemented, Sylow

107

C₂×Q₈

—

108

C₂×Q₈

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

112

Q₁₆

—

113

Q₁₆

—

136

—

114

C₂×C₈

D₁₃₆

yes

characteristic

115

C₂×C₈

C₂×D₆₈

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

136

—

118

C₄⋊C₄

—

136

—

119

C₄⋊C₄

—

136

—

120

C₄²

—

136

—

121

C₄²

D₁₃₆

yes

characteristic

122

C₁₆

—

123

C₁₆

D₄×D₁₇

yes

characteristic

124

C₁₆

#63256

yes

characteristic

125

Q₈

—

126

Q₈

—

136

—

127

Q₈

—

136

—

128

Q₈

—

136

—

129

C₂×C₄

—

136

—

130

C₂×C₄

—

136

—

131

C₂×C₄

—

136

—

132

C₂×C₄

D₂₇₂

yes

characteristic

133

C₂×C₄

D₂₇₂

yes

characteristic

134

C₂×C₄

C₂×D₁₃₆

yes

characteristic

135

C₈

—

136

C₈

#10587581

yes

characteristic

137

C₈

#10587594

yes

characteristic

138

C₂²

C₂×D₂₇₂

yes

characteristic

139

C₄

—

140

C₄

—

272

—

141

C₄

—

272

—

142

C₄

—

136

—

143

C₄

Dic₂₇₂

yes

complemented

144

C₄

#11763005

yes

characteristic

145

C₄

#11763247

yes

characteristic

146

C₂

#14530820

yes

characteristic

147

C₂

C₂×Dic₂₇₂

yes

characteristic

148

C₂

#14531104

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₇₂
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	C₄⋊C₃₂	D₁₇
hypocenter	C₁₇	#13688
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center

C₂²

C₂×D₂₇₂

commutator, cosocle

C₂×C₁₆×C₁₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₁₇×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₁₆

hypercenter

C₄⋊C₃₂

D₁₇

hypocenter

C₁₇

#13688

LCS

G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
71. C₄×C₁₇	Q₆₄	semidirect
106. C₁₇	#13688	semidirect
143. C₄	Dic₂₇₂	semidirect

H ≅ G/N

type

71. C₄×C₁₇

Q₆₄

semidirect

106. C₁₇

#13688

semidirect

143. C₄

Dic₂₇₂

semidirect

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

subgroup	index
C₂×Dic₂₇₂	2
#14531037	2
C₄×Dic₁₃₆	2
#14526966	2
C₁₇×C₄⋊C₃₂	2
#13688	17

subgroup

index

C₂×Dic₂₇₂

#14531037

C₄×Dic₁₃₆

#14526966

C₁₇×C₄⋊C₃₂

#13688

Subgroups — #16955120

Conjugacy classes of subgroups

Complements

Maximal subgroups