Group #17948556 — subgroups

Subgroups — #17948556

Order 4352.

210 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 96 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×#11763731	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587538	—	2	2	no	—
4	#11763731	—	2	2	no	—
5	#11763731	C₄	1	4	yes	—
6	C₂²×#63214	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
8	C₁₆×Dic₁₇	—	2	2	no	—
9	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
12	C₂²×#3622	D₄	1	1	yes	characteristic
13	#10587538	—	2	4	no	—
14	#10587538	—	4	2	no	—
15	C₂×#63216	—	4	2	no	—
16	C₂×C₁₆×C₁₇	—	2	1	no	—
17	C₂×C₁₆×C₁₇	—	2	1	no	—
18	C₂×C₁₆×C₁₇	C₈	1	2	yes	—
19	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
20	C₈×Dic₁₇	D₄	1	2	yes	—
21	C₂×#63214	—	2	1	no	—
22	C₂×#63214	—	2	1	no	—
23	C₂×#63214	C₈	1	2	yes	—
24	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
32	C₄×Dic₁₇	—	2	1	no	—
33	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
34	C₂×#3622	C₂²⋊C₄	1	2	yes	—
35	C₂×#3622	D₈	1	1	yes	characteristic
36	C₂×#3622	SD₁₆	1	2	yes	—
37	C₂×#3622	Q₁₆	1	1	yes	characteristic
38	#63216	—	4	4	no	—
39	C₁₆×C₁₇	—	2	2	no	—
40	#63214	—	2	2	no	—
41	#7160	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	#108	1	1	yes	characteristic
43	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#108	1	1	yes	characteristic
49	C₂×Dic₁₇	—	2	4	no	—
50	C₂×Dic₁₇	—	2	2	no	—
51	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
52	C₈×C₁₇	#97	1	1	yes	characteristic
53	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
55	#3622	#102	1	2	yes	—
56	#3622	#103	1	2	yes	—
57	C₂×#364	—	17	2	no	—
58	C₂×C₄×C₁₆	—	17	1	no	—
59	C₂²×C₁₇	—	2	2	no	—
60	C₂²×C₁₇	#334	1	1	yes	characteristic
61	C₂²×C₁₇	#335	1	1	yes	characteristic
62	C₂²×C₁₇	#341	1	1	yes	characteristic
63	C₄×C₁₇	#334	1	1	yes	characteristic
64	C₄×C₁₇	#335	1	1	yes	characteristic
65	C₄×C₁₇	#336	1	1	yes	characteristic
66	C₄×C₁₇	#340	1	1	yes	characteristic
67	Dic₁₇	—	2	4	no	—
68	C₂×M₃₂	—	34	2	no	—
69	C₂²×C₁₆	—	17	1	no	—
70	C₂²×C₁₆	#595	1	1	yes	characteristic
71	C₂×C₄×C₈	—	17	1	no	—
72	#364	—	17	4	no	—
73	#364	—	34	2	no	—
74	C₄×C₁₆	—	34	2	no	—
75	C₂×C₁₇	—	2	1	no	—
76	C₂×C₁₇	—	2	1	no	—
77	C₂×C₁₇	#1277	1	1	yes	characteristic
78	C₂×C₁₇	#1373	1	2	yes	—
79	C₂²×C₈	—	17	1	no	—
80	C₂²×C₈	C₂×#595	1	1	yes	characteristic
81	C₂×C₄²	—	17	1	no	—
82	M₃₂	—	34	4	no	—
83	M₃₂	—	68	2	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	17	2	no	—
87	C₂×C₁₆	—	34	1	no	—
88	C₂×C₁₆	—	34	1	no	—
89	C₂×C₁₆	—	68	2	no	—
90	C₂×C₁₆	#3624	1	2	yes	—
91	C₄×C₈	—	17	2	no	—
92	C₄×C₈	—	17	2	no	—
93	C₁₇	#7160	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₄	—	34	1	no	—
95	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
96	C₂×C₈	—	17	2	no	—
97	C₂×C₈	—	17	1	no	—
98	C₂×C₈	—	17	1	no	—
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	#63242	1	1	yes	characteristic
101	C₂×C₈	#63243	1	1	yes	characteristic
102	C₂×C₈	C₄×#595	1	1	yes	characteristic
103	C₂×C₈	C₂×#3624	1	1	yes	characteristic
104	C₂×C₈	#63247	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
106	C₄²	—	17	2	no	—
107	C₄²	—	34	1	no	—
108	C₁₆	—	2	2	no	—
109	C₁₆	—	34	2	no	—
110	C₁₆	—	68	4	no	—
111	C₂³	#10587536	1	1	yes	characteristic
112	C₂×C₄	—	34	4	no	—
113	C₂×C₄	—	34	2	no	—
114	C₂×C₄	#10587533	1	1	yes	characteristic
115	C₂×C₄	#10587534	1	1	yes	characteristic
116	C₂×C₄	#10587535	1	1	yes	characteristic
117	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
118	C₂×C₄	#10587544	1	1	yes	characteristic
119	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
120	C₈	—	17	2	no	—
121	C₈	—	17	2	no	—
122	C₈	C₄×#3624	1	1	yes	characteristic
123	C₈	#10587542	1	1	yes	characteristic
124	C₈	#10587543	1	1	yes	characteristic
125	C₈	C₂²⋊#3624	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	#11763671	1	1	yes	characteristic
128	C₂²	#11763672	1	1	yes	characteristic
129	C₂²	#11763685	1	1	yes	characteristic
130	C₄	—	34	4	no	—
131	C₄	#11763670	1	1	yes	characteristic
132	C₄	#11763673	1	1	yes	characteristic
133	C₄	#11763675	1	1	yes	characteristic
134	C₄	#11763683	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#14543614	1	1	yes	characteristic
138	C₂	#14543819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763731

C₂

yes

maximal

C₂×C₁₆×Dic₁₇

C₂

yes

maximal, characteristic

C₂×#10587538

—

#11763731

—

#11763731

C₄

yes

—

C₂²×#63214

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

C₁₆×Dic₁₇

—

C₂²×C₁₆×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

Q₈

yes

characteristic

C₂²×#3622

D₄

yes

—

—

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₂×#63214

—

C₂×#63214

—

C₂×#63214

C₈

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

—

C₂×#3622

C₂²⋊C₄

yes

—

C₂×#3622

D₈

yes

characteristic

C₂×#3622

SD₁₆

yes

—

C₂×#3622

Q₁₆

yes

characteristic

#63216

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

C₄×C₈

yes

characteristic

C₈×C₁₇

#97

yes

characteristic

C₈×C₁₇

C₂²⋊C₈

yes

characteristic

C₈×C₁₇

C₂²⋊C₈

yes

characteristic

#3622

#102

yes

—

#3622

#103

yes

—

C₂×#364

—

C₂×C₄×C₁₆

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

#595

yes

characteristic

C₂×C₄×C₈

—

#364

—

#364

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1277

yes

characteristic

C₂×C₁₇

#1373

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#3624

yes

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#7160

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄⋊#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#63242

yes

characteristic

101

C₂×C₈

#63243

yes

characteristic

102

C₂×C₈

C₄×#595

yes

characteristic

103

C₂×C₈

C₂×#3624

yes

characteristic

104

C₂×C₈

#63247

yes

characteristic

105

C₂×C₈

C₂²⋊#595

yes

characteristic

106

C₄²

—

107

C₄²

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

111

C₂³

#10587536

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#10587533

yes

characteristic

115

C₂×C₄

#10587534

yes

characteristic

116

C₂×C₄

#10587535

yes

characteristic

117

C₂×C₄

C₄⋊#3624

yes

characteristic

118

C₂×C₄

#10587544

yes

characteristic

119

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

C₄×#3624

yes

characteristic

123

C₈

#10587542

yes

characteristic

124

C₈

#10587543

yes

characteristic

125

C₈

C₂²⋊#3624

yes

characteristic

126

C₂²

—

127

C₂²

#11763671

yes

characteristic

128

C₂²

#11763672

yes

characteristic

129

C₂²

#11763685

yes

characteristic

130

C₄

—

131

C₄

#11763670

yes

characteristic

132

C₄

#11763673

yes

characteristic

133

C₄

#11763675

yes

characteristic

134

C₄

#11763683

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#14543614

yes

characteristic

138

C₂

#14543819

yes

—

	subgroup	G/H
center	C₂×C₄	#10587534
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₂²×C₈	C₂×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#335
prime radical	C₂³×C₁₇	#108
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#7160
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

#10587534

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₂²×C₈

C₂×#595

Fitting

C₂²×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₈×Dic₁₇

C₂²

C₂²×C₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
93. C₁₇	#7160	semidirect

H ≅ G/N

type

93. C₁₇

#7160

semidirect

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

subgroup	index
C₂×#11763731	2
C₂×C₁₆×Dic₁₇	2
#7160	17

subgroup

index

C₂×#11763731

C₂×C₁₆×Dic₁₇

#7160

Subgroups — #17948556

Conjugacy classes of subgroups

Complements

Maximal subgroups