Group #17948343 — subgroups

Subgroups — #17948343

Order 4352.

200 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 50 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	#14543720	C₂	1	2	yes	maximal
2	#14528935	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#63216	—	2	2	no	—
4	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
5	C₂×#10587485	—	2	2	no	—
6	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
7	C₂×C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	—	2	1	no	—
9	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇	—	2	1	no	—
11	C₂×C₄×Dic₁₇	—	2	1	no	—
12	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
14	C₂²×#3622	D₄	1	2	yes	—
15	C₂×#63216	—	4	2	no	—
16	C₁₇×C₄⋊C₈	—	4	1	no	—
17	#10587485	—	4	4	no	—
18	C₄⋊#3622	—	4	1	no	—
19	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	1	no	—
23	C₂²×Dic₁₇	—	2	1	no	—
24	C₂×C₈×C₁₇	—	4	2	no	—
25	C₂×C₈×C₁₇	—	4	1	no	—
26	C₄²×C₁₇	—	2	1	no	—
27	C₄²×C₁₇	—	2	1	no	—
28	C₄×Dic₁₇	—	2	2	no	—
29	C₄×Dic₁₇	—	2	2	no	—
30	C₄×Dic₁₇	—	2	1	no	—
31	C₄×Dic₁₇	—	4	2	no	—
32	C₄×Dic₁₇	—	4	1	no	—
33	C₄×Dic₁₇	C₁₆	1	2	yes	complemented
34	C₂×#3622	—	2	2	no	—
35	C₂×#3622	—	4	2	no	—
36	#63216	—	8	2	no	—
37	#7063	—	17	1	no	maximal, Sylow
38	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₇	—	2	1	no	—
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	—	2	1	no	—
44	C₂×C₄×C₁₇	—	4	1	no	—
45	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#101	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	2	no	—
50	C₂×Dic₁₇	—	2	2	no	—
51	C₂×Dic₁₇	—	2	2	no	—
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	—	4	1	no	—
54	C₂×Dic₁₇	—	4	1	no	—
55	C₈×C₁₇	—	8	1	no	—
56	#3622	—	4	2	no	—
57	#1842	—	17	1	no	—
58	#1322	—	17	2	no	—
59	C₂²×C₁₇	—	2	1	no	—
60	C₂²×C₁₇	—	2	1	no	—
61	C₂²×C₁₇	#324	1	1	yes	characteristic
62	C₂²×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₂²×C₁₇	#349	1	1	yes	characteristic
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	—	4	1	no	—
68	C₄×C₁₇	#353	1	1	yes	characteristic
69	C₄×C₁₇	#356	1	1	yes	characteristic
70	Dic₁₇	—	2	4	no	—
71	Dic₁₇	—	4	2	no	—
72	C₂×C₄⋊C₈	—	17	1	no	—
73	C₂×C₄⋊C₈	—	34	2	no	—
74	C₂×C₄⋊C₈	#595	1	1	yes	characteristic
75	C₄³	—	17	1	no	—
76	C₂²⋊C₁₆	—	34	2	no	—
77	C₂×C₁₇	—	2	1	no	—
78	C₂×C₁₇	—	2	1	no	—
79	C₂×C₁₇	#1322	1	1	yes	characteristic
80	C₂×C₁₇	#1325	1	1	yes	characteristic
81	C₂×C₁₇	#1327	1	1	yes	characteristic
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	17	2	no	—
84	C₂×C₄²	—	17	1	no	—
85	C₂×C₄²	—	17	1	no	—
86	C₂×C₄²	—	34	1	no	—
87	C₂×C₄²	—	34	1	no	—
88	C₂×C₄²	C₂×#595	1	1	yes	characteristic
89	C₂×C₁₆	—	68	2	no	—
90	C₄⋊C₈	—	4	1	no	—
91	C₄⋊C₈	—	68	1	no	—
92	C₄⋊C₈	—	68	4	no	—
93	C₁₇	#7063	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₄	—	34	1	no	—
95	C₂²×C₄	—	34	1	no	—
96	C₂²×C₄	#63242	1	1	yes	characteristic
97	C₂²×C₄	#63243	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	34	2	no	—
102	C₂×C₈	—	68	2	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	17	2	no	—
106	C₄²	—	34	2	no	—
107	C₄²	—	34	2	no	—
108	C₄²	—	34	1	no	—
109	C₄²	—	68	1	no	—
110	C₄²	—	68	2	no	—
111	C₁₆	—	136	2	no	—
112	C₂³	#10587543	1	1	yes	characteristic
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	34	2	no	—
120	C₂×C₄	—	34	2	no	—
121	C₂×C₄	—	34	2	no	—
122	C₂×C₄	—	34	2	no	—
123	C₂×C₄	—	68	1	no	—
124	C₂×C₄	—	68	1	no	—
125	C₂×C₄	#10587529	1	1	yes	characteristic
126	C₂×C₄	#10587530	1	1	yes	characteristic
127	C₂×C₄	#10587545	1	1	yes	characteristic
128	C₂×C₄	#10587551	1	1	yes	characteristic
129	C₈	—	8	1	no	—
130	C₈	—	68	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#11763655	1	1	yes	characteristic
134	C₂²	#11763704	1	1	yes	characteristic
135	C₂²	#11763706	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	1	no	—
140	C₄	—	34	4	no	—
141	C₄	—	68	2	no	—
142	C₄	#11763713	1	1	yes	characteristic
143	C₄	#11763719	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#14543721	1	1	yes	characteristic
147	C₂	#14543727	1	1	yes	characteristic
148	C₂	#14543731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14543720

C₂

yes

maximal

#14528935

C₂

yes

maximal, characteristic

C₂²⋊#63216

—

C₂×C₄⋊#3622

C₂²

yes

characteristic

C₂×#10587485

—

C₄²×Dic₁₇

C₄

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₄

yes

characteristic

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

C₈

yes

characteristic

C₂×C₄×Dic₁₇

C₈

yes

characteristic

C₂²×#3622

D₄

yes

—

C₂×#63216

—

C₁₇×C₄⋊C₈

—

#10587485

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₁₆

yes

—

—

—

—

C₂³×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

M₃₂

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#3622

—

#1842

—

#1322

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#324

yes

characteristic

C₂²×C₁₇

C₂²⋊C₁₆

yes

characteristic

C₂²×C₁₇

#349

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#595

yes

characteristic

C₄³

—

C₂²⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#595

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₁₇

#7063

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²⋊#595

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₁₆

—

136

—

112

C₂³

#10587543

yes

characteristic

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

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#10587529

yes

characteristic

126

C₂×C₄

#10587530

yes

characteristic

127

C₂×C₄

#10587545

yes

characteristic

128

C₂×C₄

#10587551

yes

characteristic

129

C₈

—

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

#11763655

yes

characteristic

134

C₂²

#11763704

yes

characteristic

135

C₂²

#11763706

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#11763713

yes

characteristic

143

C₄

#11763719

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#14543721

yes

characteristic

147

C₂

#14543727

yes

characteristic

148

C₂

#14543731

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587551
commutator	C₂×C₄×C₁₇	C₂×C₁₆
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₁₇×C₄⋊C₈	C₄
socle	C₂²×C₁₇	#324
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₄⋊#3622	C₂²
hypercenter	C₂×C₄⋊C₈	#595
hypocenter	C₁₇	#7063
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#10587551

commutator

C₂×C₄×C₁₇

C₂×C₁₆

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₁₇×C₄⋊C₈

C₄

socle

C₂²×C₁₇

#324

prime radical

C₂³×C₁₇

C₂²⋊C₈

cosocle

C₂×C₄⋊#3622

C₂²

C₂×C₄⋊C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
33. C₄×Dic₁₇	C₁₆	semidirect
93. C₁₇	#7063	semidirect

H ≅ G/N

type

33. C₄×Dic₁₇

C₁₆

semidirect

93. C₁₇

#7063

semidirect

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

subgroup	index
#14543720	2
#14528935	2
#7063	17

subgroup

index

#14543720

#14528935

#7063

Subgroups — #17948343

Conjugacy classes of subgroups

Complements

Maximal subgroups