Group #17948832 — subgroups

Subgroups — #17948832

Order 4352.

206 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 74 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	#14544341	C₂	1	2	yes	maximal
2	C₂×#11762627	C₂	1	1	yes	maximal, characteristic
3	#11763812	—	2	2	no	—
4	#11763673	C₄	1	4	yes	—
5	#11763672	—	2	2	no	—
6	C₂²×#63214	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
8	#11762627	—	2	2	no	—
9	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
10	C₂×#63247	—	4	2	no	—
11	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
13	C₂²×#3622	Q₈	1	1	yes	characteristic
14	#10587544	—	4	2	no	—
15	C₄⋊#3624	—	2	4	no	—
16	C₂×C₁₆×C₁₇	—	2	1	no	—
17	C₂×C₁₆×C₁₇	—	4	1	no	—
18	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
19	C₈×Dic₁₇	D₄	1	2	yes	—
20	C₂×#63214	—	2	1	no	—
21	C₂×#63214	—	4	1	no	—
22	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
24	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
25	#63247	—	8	2	no	—
26	#63247	—	8	2	no	—
27	C₂×#3624	—	4	4	no	—
28	C₂×C₈×C₁₇	—	2	1	no	—
29	C₂×C₈×C₁₇	—	2	1	no	—
30	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×Dic₁₇	C₂²⋊C₄	1	2	yes	—
33	C₄×Dic₁₇	SD₁₆	1	2	yes	—
34	C₂×#3622	—	4	1	no	—
35	C₂×#3622	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₁₇	—	4	1	no	—
37	#63214	—	4	1	no	—
38	#7241	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	#107	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
43	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
44	C₂×Dic₁₇	—	4	1	no	—
45	C₂×Dic₁₇	#102	1	2	yes	—
46	C₂×Dic₁₇	#103	1	2	yes	—
47	C₂×Dic₁₇	D₁₆	1	1	yes	characteristic
48	C₂×Dic₁₇	SD₃₂	1	2	yes	—
49	C₂×Dic₁₇	Q₃₂	1	1	yes	characteristic
50	#3624	—	8	4	no	—
51	C₈×C₁₇	—	2	1	no	—
52	C₈×C₁₇	#100	1	1	yes	characteristic
53	C₈×C₁₇	#101	1	1	yes	characteristic
54	#3622	—	2	2	no	—
55	C₂×C₄⋊C₁₆	—	17	1	no	—
56	#1567	—	17	2	no	—
57	C₂²×C₁₇	—	2	1	no	—
58	C₂²×C₁₇	—	2	1	no	—
59	C₂²×C₁₇	#340	1	1	yes	characteristic
60	C₂²×C₁₇	#366	1	1	yes	characteristic
61	C₂²×C₁₇	#367	1	1	yes	characteristic
62	C₄×C₁₇	—	2	1	no	—
63	C₄×C₁₇	#338	1	1	yes	characteristic
64	C₄×C₁₇	#343	1	1	yes	characteristic
65	Dic₁₇	—	2	2	no	—
66	Dic₁₇	#357	1	2	yes	—
67	Dic₁₇	#358	1	2	yes	—
68	C₂²×C₁₆	—	17	1	no	—
69	C₂²×C₁₆	#595	1	1	yes	characteristic
70	#423	—	34	2	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₄⋊C₁₆	—	34	2	no	—
73	#335	—	17	4	no	—
74	#335	—	34	2	no	—
75	C₂×C₁₇	—	2	1	no	—
76	C₂×C₁₇	—	2	1	no	—
77	C₂×C₁₇	#1292	1	1	yes	characteristic
78	C₂×C₁₇	#1380	1	1	yes	characteristic
79	C₂×C₁₇	#1385	1	1	yes	characteristic
80	C₂×M₁₆	—	68	2	no	—
81	C₂²×C₈	—	17	1	no	—
82	C₂²×C₈	C₂×#595	1	1	yes	characteristic
83	C₂×C₄²	—	17	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	4	1	no	—
86	C₂×C₁₆	—	34	1	no	—
87	C₂×C₁₆	—	68	1	no	—
88	C₄⋊C₈	—	34	4	no	—
89	C₄⋊C₈	—	68	2	no	—
90	C₄×C₈	—	17	2	no	—
91	C₄×C₈	—	17	2	no	—
92	C₁₇	#7241	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	17	1	no	—
94	C₂²×C₄	—	17	1	no	—
95	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
96	M₁₆	—	136	2	no	—
97	M₁₆	—	136	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	—	68	1	no	—
102	C₂×C₈	—	68	4	no	—
103	C₂×C₈	C₄×#595	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
105	C₄²	—	17	2	no	—
106	C₄²	—	17	2	no	—
107	C₁₆	—	4	1	no	—
108	C₁₆	—	68	1	no	—
109	C₂³	#10587534	1	1	yes	characteristic
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	17	1	no	—
113	C₂×C₄	—	17	1	no	—
114	C₂×C₄	—	17	2	no	—
115	C₂×C₄	—	17	2	no	—
116	C₂×C₄	—	17	2	no	—
117	C₂×C₄	—	68	1	no	—
118	C₂×C₄	#10587533	1	1	yes	characteristic
119	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	34	2	no	—
122	C₈	—	136	4	no	—
123	C₈	#10587545	1	1	yes	characteristic
124	C₈	#10587553	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#11763683	1	1	yes	characteristic
128	C₂²	#11763735	1	1	yes	characteristic
129	C₂²	#11763736	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	17	2	no	—
132	C₄	—	17	2	no	—
133	C₄	—	34	2	no	—
134	C₄	#11763678	1	1	yes	characteristic
135	C₄	#11763689	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	#14543654	1	1	yes	characteristic
139	C₂	#14543842	1	1	yes	characteristic
140	C₂	#14543854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544341

C₂

yes

maximal

C₂×#11762627

C₂

yes

maximal, characteristic

#11763812

—

#11763673

C₄

yes

—

#11763672

—

C₂²×#63214

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762627

—

C₂²×C₁₆×C₁₇

C₄

yes

characteristic

C₂×#63247

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

Q₈

yes

characteristic

#10587544

—

C₄⋊#3624

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

D₄

yes

—

C₂×#63214

—

C₂×#63214

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₇

D₈

yes

characteristic

C₂²×Dic₁₇

Q₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

—

C₄×Dic₁₇

SD₁₆

yes

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

#102

yes

—

C₂×Dic₁₇

#103

yes

—

C₂×Dic₁₇

D₁₆

yes

characteristic

C₂×Dic₁₇

SD₃₂

yes

—

C₂×Dic₁₇

Q₃₂

yes

characteristic

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

C₂×C₄⋊C₁₆

—

#1567

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

#357

yes

—

Dic₁₇

#358

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

#595

yes

characteristic

#423

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

#335

—

#335

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#7241

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#595

yes

characteristic

M₁₆

—

136

—

M₁₆

—

136

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄×#595

yes

characteristic

104

C₂×C₈

C₂²⋊#595

yes

characteristic

105

C₄²

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

109

C₂³

#10587534

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#10587533

yes

characteristic

119

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

136

—

123

C₈

#10587545

yes

characteristic

124

C₈

#10587553

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

#11763683

yes

characteristic

128

C₂²

#11763735

yes

characteristic

129

C₂²

#11763736

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#11763678

yes

characteristic

135

C₄

#11763689

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#14543654

yes

characteristic

139

C₂

#14543842

yes

characteristic

140

C₂

#14543854

yes

characteristic

	subgroup	G/H
center	C₂²	#11763683
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂²×C₈	C₂×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂³×C₁₇	#107
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#7241
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#11763683

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₂²×C₈

C₂×#595

Fitting

C₂²×C₁₆×C₁₇

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₈×Dic₁₇

C₂²

C₂²×C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
92. C₁₇	#7241	semidirect

H ≅ G/N

type

92. C₁₇

#7241

semidirect

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

subgroup	index
#14544341	2
C₂×#11762627	2
#7241	17

subgroup

index

#14544341

C₂×#11762627

#7241

Subgroups — #17948832

Conjugacy classes of subgroups

Complements

Maximal subgroups