Group #17948799 — subgroups

Subgroups — #17948799

Order 4352.

244 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 58 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	#14544668	C₂	1	2	yes	maximal
2	#14530397	C₂	1	1	yes	maximal, characteristic
3	D₁₇⋊#108	—	2	2	no	—
4	#11763795	—	2	2	no	—
5	#11763678	C₄	1	4	yes	—
6	C₂×#10587587	C₂²	1	1	yes	characteristic
7	C₂×#10587483	C₄	1	1	yes	characteristic
8	#11762588	—	2	2	no	—
9	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
10	C₂×#63252	D₄	1	1	yes	characteristic
11	D₁₇⋊M₁₆	—	2	4	no	—
12	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×#63223	Q₈	1	1	yes	characteristic
14	#10587587	—	4	1	no	—
15	#10587587	—	4	1	no	—
16	C₂×#63218	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₁₇	D₄	1	2	yes	—
18	#10587553	—	4	2	no	—
19	#10587545	—	4	2	no	—
20	#10587536	—	4	2	no	—
21	#10587535	—	4	2	no	—
22	C₂×C₁₆×C₁₇	—	2	1	no	—
23	C₂×C₁₆×C₁₇	—	4	1	no	—
24	#10587483	—	4	1	no	—
25	C₂×#63214	—	2	1	no	—
26	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₇	SD₁₆	1	2	yes	—
28	#63252	—	4	1	no	—
29	#63252	—	4	1	no	—
30	C₂×D₆₈	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×D₁₇	C₂²⋊C₄	1	2	yes	—
32	C₂×Dic₃₄	D₈	1	1	yes	characteristic
33	#63247	—	4	4	no	—
34	#63243	—	4	4	no	—
35	#63242	—	4	4	no	—
36	C₂×C₈×C₁₇	—	2	1	no	—
37	C₂×C₈×C₁₇	—	2	1	no	—
38	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	#63223	—	4	1	no	—
41	C₂×#3622	C₄⋊C₄	1	2	yes	—
42	#63218	—	4	2	no	—
43	C₈×D₁₇	—	2	2	no	—
44	C₈×D₁₇	—	2	2	no	—
45	C₁₆×C₁₇	—	4	1	no	—
46	#63214	—	4	1	no	—
47	#7229	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	#106	1	1	yes	characteristic
49	C₂²×D₁₇	#103	1	2	yes	—
50	C₂×C₄×C₁₇	—	2	1	no	—
51	C₂×C₄×C₁₇	—	2	1	no	—
52	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
54	C₂²⋊D₁₇	—	4	2	no	—
55	C₂×Dic₁₇	#102	1	2	yes	—
56	D₆₈	—	4	1	no	—
57	C₄×D₁₇	—	2	2	no	—
58	C₄×D₁₇	—	2	2	no	—
59	Dic₃₄	—	4	1	no	—
60	#3624	—	8	4	no	—
61	C₈×C₁₇	—	2	1	no	—
62	C₈×C₁₇	#99	1	1	yes	characteristic
63	C₈×C₁₇	#99	1	1	yes	characteristic
64	#3622	—	2	2	no	—
65	#2113	—	17	1	no	—
66	#1813	—	17	2	no	—
67	C₂²×C₁₇	—	2	1	no	—
68	C₂²×C₁₇	—	4	1	no	—
69	C₂²×C₁₇	#340	1	1	yes	characteristic
70	C₂×D₁₇	—	2	2	no	—
71	C₂×D₁₇	—	4	2	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	H(3,ℤ/4)	1	1	yes	characteristic
74	C₄×C₁₇	#342	1	1	yes	characteristic
75	Dic₁₇	—	2	2	no	—
76	C₂×#131	—	17	1	no	—
77	C₂×M₃₂	—	17	1	no	—
78	C₂²×C₁₆	#595	1	1	yes	characteristic
79	C₂×#108	—	34	2	no	—
80	#413	—	34	2	no	—
81	C₂²⋊C₁₆	—	34	2	no	—
82	#338	—	17	4	no	—
83	C₂×C₁₇	—	2	1	no	—
84	C₂×C₁₇	—	4	1	no	—
85	C₂×C₁₇	#1289	1	1	yes	characteristic
86	D₁₇	—	4	2	no	—
87	C₂×#41	—	17	1	no	—
88	#131	—	68	1	no	—
89	#131	—	68	1	no	—
90	C₂×M₁₆	—	17	1	no	—
91	C₂×M₁₆	—	17	2	no	—
92	C₂×M₁₆	—	34	4	no	—
93	C₂²×C₈	—	17	2	no	—
94	C₂²×C₈	C₂×#595	1	1	yes	characteristic
95	M₃₂	—	68	1	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	4	1	no	—
98	C₂×C₁₆	—	34	1	no	—
99	#108	—	68	2	no	—
100	#108	—	68	2	no	—
101	#101	—	68	2	no	—
102	#100	—	68	2	no	—
103	C₁₇	#7229	1	1	yes	characteristic, complemented, Sylow
104	#41	—	68	1	no	—
105	#41	—	68	1	no	—
106	C₂×Q₈	—	17	1	no	—
107	C₂×D₄	—	17	2	no	—
108	C₂×D₄	—	17	1	no	—
109	C₂²×C₄	—	17	2	no	—
110	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
111	M₁₆	—	68	2	no	—
112	M₁₆	—	68	1	no	—
113	M₁₆	—	68	4	no	—
114	M₁₆	—	68	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	17	2	no	—
118	C₂×C₈	—	34	2	no	—
119	C₂×C₈	—	34	2	no	—
120	C₂×C₈	—	68	4	no	—
121	C₂×C₈	C₄×#595	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
123	C₁₆	—	4	1	no	—
124	C₁₆	—	68	1	no	—
125	C₂³	—	17	2	no	—
126	C₂³	#10587533	1	1	yes	characteristic
127	Q₈	—	68	1	no	—
128	D₄	—	68	1	no	—
129	D₄	—	68	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	17	2	no	—
133	C₂×C₄	—	34	2	no	—
134	C₂×C₄	—	34	2	no	—
135	C₂×C₄	#10587534	1	1	yes	characteristic
136	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	34	2	no	—
139	C₈	—	136	4	no	—
140	C₈	#10587539	1	1	yes	characteristic
141	C₈	#10587551	1	1	yes	characteristic
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	34	2	no	—
145	C₂²	—	68	2	no	—
146	C₂²	#11763683	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	34	2	no	—
149	C₄	#11763677	1	1	yes	characteristic
150	C₄	#11763687	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	68	2	no	—
154	C₂	#14543647	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544668

C₂

yes

maximal

#14530397

C₂

yes

maximal, characteristic

D₁₇⋊#108

—

#11763795

—

#11763678

C₄

yes

—

C₂×#10587587

C₂²

yes

characteristic

C₂×#10587483

C₄

yes

characteristic

#11762588

—

C₂²×C₁₆×C₁₇

C₄

yes

characteristic

C₂×#63252

D₄

yes

characteristic

D₁₇⋊M₁₆

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×#63223

Q₈

yes

—

—

C₂×C₄

yes

—

C₂×C₈×D₁₇

D₄

yes

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

#10587483

—

C₂×#63214

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₂²⋊D₁₇

SD₁₆

yes

—

#63252

—

#63252

—

C₂×D₆₈

Q₁₆

yes

characteristic

C₂×C₄×D₁₇

C₂²⋊C₄

yes

—

C₂×Dic₃₄

D₈

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

#63223

—

C₂×#3622

C₄⋊C₄

yes

—

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂²×D₁₇

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂²⋊D₁₇

—

C₂×Dic₁₇

#102

yes

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

yes

—

—

C₂²⋊C₁₆

—

#338

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1289

yes

characteristic

D₁₇

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

100

#108

—

101

#101

—

102

#100

—

103

C₁₇

#7229

yes

characteristic, complemented, Sylow

104

#41

—

105

#41

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄⋊#595

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

C₄×#595

yes

characteristic

122

C₂×C₈

C₂²⋊#595

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₂³

—

126

C₂³

#10587533

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

#10587534

yes

characteristic

136

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

136

—

140

C₈

#10587539

yes

characteristic

141

C₈

#10587551

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#11763683

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#11763677

yes

characteristic

150

C₄

#11763687

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#14543647

yes

characteristic

	subgroup	G/H
center	C₄	#11763677
commutator	C₂×C₈×C₁₇	C₄²
Frattini	C₂²×C₈	C₂×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂×C₁₇	#1289
prime radical	C₂×#63252	D₄
cosocle	C₂×#10587587	C₂²
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#7229
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₄

#11763677

commutator

C₂×C₈×C₁₇

C₄²

Frattini

C₂²×C₈

C₂×#595

Fitting

C₂²×C₁₆×C₁₇

C₄

C₂×C₁₇

D₄

C₂²

C₂²×C₁₆

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
103. C₁₇	#7229	semidirect

H ≅ G/N

type

103. C₁₇

#7229

semidirect

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

subgroup	index
#14544668	2
#14530397	2
#7229	17

subgroup

index

#14544668

#14530397

#7229

Subgroups — #17948799

Conjugacy classes of subgroups

Complements

Maximal subgroups