Group #17958378 — subgroups

Subgroups — #17958378

Order 4352.

288 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 98 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	#14545145	C₂	1	2	yes	maximal
2	#14543722	C₂	1	4	yes	maximal
3	#14537719	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#63216	—	2	2	no	—
5	C₂²⋊#63216	C₂²	1	4	yes	—
6	#11763697	—	2	2	no	—
7	#11762940	—	2	2	no	—
8	Q₈×#3622	—	2	2	no	—
9	#11762836	C₂²	1	2	yes	—
10	#11762833	C₄	1	1	yes	characteristic
11	#11762765	C₄	1	2	yes	—
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	C₁₇×#533	C₄	1	1	yes	characteristic
14	C₁₇×C₄⋊Q₈	C₈	1	2	yes	—
15	C₁₇×#125	C₈	1	2	yes	—
16	C₁₇×C₂²⋊Q₈	—	2	2	no	—
17	C₄×Q₈×C₁₇	—	2	2	no	—
18	C₁₇×#117	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#63223	C₂×C₄	1	2	yes	—
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂³	1	1	yes	characteristic
23	C₂×#63216	—	2	4	no	—
24	C₄⋊#3622	—	2	4	no	—
25	C₄⋊#3622	—	2	2	no	—
26	#10587474	—	2	2	no	—
27	C₄×#3622	—	2	2	no	—
28	C₄×#3622	D₄	1	2	yes	—
29	C₄×#3622	D₄	1	2	yes	—
30	C₂×Q₈×C₁₇	—	2	2	no	—
31	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₇	—	2	4	no	—
34	C₄⋊C₄×C₁₇	—	2	2	no	—
35	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
36	C₂²⋊C₄×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	—	2	2	no	—
38	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
39	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
41	#63223	—	4	2	no	—
42	C₂×#3622	—	2	2	no	—
43	C₂×#3622	—	2	2	no	—
44	C₂×#3622	—	4	1	no	—
45	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#3622	C₂×D₄	1	2	yes	—
48	#63216	—	4	4	no	—
49	#11497	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
51	Q₈×C₁₇	—	4	4	no	—
52	C₂×C₄×C₁₇	—	2	1	no	—
53	C₂×C₄×C₁₇	—	2	1	no	—
54	C₂×C₄×C₁₇	—	2	4	no	—
55	C₂×C₄×C₁₇	—	4	2	no	—
56	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	#3622	—	2	2	no	—
61	#3622	—	2	1	no	—
62	#3622	—	4	2	no	—
63	#3622	#101	1	2	yes	—
64	#2960	—	17	1	no	—
65	#2163	—	17	2	no	—
66	#1323	—	17	4	no	—
67	C₂²×C₁₇	—	2	2	no	—
68	C₂²×C₁₇	—	2	1	no	—
69	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
70	C₄×C₁₇	—	2	2	no	—
71	C₄×C₁₇	—	2	2	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	—	4	4	no	—
74	C₄×C₁₇	#410	1	1	yes	characteristic
75	C₄×C₁₇	C₂×#101	1	1	yes	characteristic
76	#533	#595	1	1	yes	characteristic
77	#446	—	34	2	no	—
78	C₈×Q₈	—	34	2	no	—
79	#424	—	17	2	no	—
80	#423	—	17	1	no	—
81	#405	—	17	2	no	—
82	C₂×C₄×C₈	—	17	1	no	—
83	C₄⋊C₁₆	—	34	2	no	—
84	C₂²⋊C₁₆	—	17	4	no	—
85	#346	—	34	2	no	—
86	C₂×C₁₇	—	4	1	no	—
87	C₂×C₁₇	#1461	1	1	yes	characteristic
88	C₂×C₁₇	#2117	1	2	yes	—
89	C₂×M₁₆	—	17	2	no	—
90	C₂²×C₈	—	17	1	no	—
91	C₂²×C₈	—	17	1	no	—
92	C₄⋊Q₈	#3624	1	2	yes	—
93	#125	#3624	1	2	yes	—
94	C₂²⋊Q₈	—	2	2	no	—
95	C₄×Q₈	—	2	2	no	—
96	#117	C₂×#595	1	2	yes	—
97	C₂×C₄²	C₂×#595	1	1	yes	characteristic
98	C₂×C₁₆	—	34	4	no	—
99	C₄⋊C₈	—	34	4	no	—
100	C₄⋊C₈	—	34	2	no	—
101	#97	—	34	2	no	—
102	C₄×C₈	—	17	2	no	—
103	C₄×C₈	—	17	2	no	—
104	C₄×C₈	—	34	2	no	—
105	C₁₇	#11497	1	1	yes	characteristic, complemented, Sylow
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#63247	1	2	yes	—
108	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
109	M₁₆	—	68	2	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	2	no	—
113	C₂×C₈	—	34	2	no	—
114	C₂×C₈	—	34	2	no	—
115	C₂×C₈	—	68	1	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×#3624	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂×#3624	1	1	yes	characteristic
122	C₄²	C₂×#3624	1	1	yes	characteristic
123	C₄²	C₂²⋊#595	1	2	yes	—
124	C₁₆	—	68	4	no	—
125	C₂³	C₂×#63247	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊#3624	1	2	yes	—
132	C₂×C₄	C₂²⋊#3624	1	2	yes	—
133	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
135	C₈	—	17	2	no	—
136	C₈	—	34	2	no	—
137	C₈	—	34	1	no	—
138	C₈	—	68	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊#3624	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	#11763790	1	1	yes	characteristic
147	C₄	D₁₇⋊#101	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#14544027	1	1	yes	characteristic
150	C₂	#14545026	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14545145

C₂

yes

maximal

#14543722

C₂

yes

maximal

#14537719

C₂

yes

maximal, characteristic

C₄⋊#63216

—

C₂²⋊#63216

C₂²

yes

—

—

—

—

C₂²

yes

—

#11762833

C₄

yes

characteristic

#11762765

C₄

yes

—

C₂×C₄×#3622

C₂²

yes

characteristic

C₁₇×#533

C₄

yes

characteristic

C₁₇×C₄⋊Q₈

C₈

yes

—

C₁₇×#125

C₈

yes

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×#63223

C₂×C₄

yes

—

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂³

yes

—

—

—

—

—

D₄

yes

—

C₄×#3622

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₈

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₂×D₄

yes

—

#63216

—

#11497

—

maximal, Sylow

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

—

—

—

yes

—

#2960

—

#2163

—

#1323

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

yes

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂×C₁₇

—

C₂×C₁₇

#1461

yes

characteristic

C₂×C₁₇

#2117

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

#3624

yes

—

#125

#3624

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×#595

yes

—

C₂×C₄²

C₂×#595

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#97

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₁₇

#11497

yes

characteristic, complemented, Sylow

106

C₂×Q₈

—

107

C₂²×C₄

#63247

yes

—

108

C₂²×C₄

C₂²×#595

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×#3624

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂×#3624

yes

characteristic

122

C₄²

C₂×#3624

yes

characteristic

123

C₄²

C₂²⋊#595

yes

—

124

C₁₆

—

125

C₂³

C₂×#63247

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊#3624

yes

—

132

C₂×C₄

C₂²⋊#3624

yes

—

133

C₂×C₄

C₂²×#3624

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊#595

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#3624

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#11763790

yes

characteristic

147

C₄

D₁₇⋊#101

yes

characteristic

148

C₂

—

149

C₂

#14544027

yes

characteristic

150

C₂

#14545026

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#533	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×#3622	C₂³
hypercenter	#533	#595
hypocenter	C₁₇	#11497
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₁₇×#533

C₄

socle

C₂²×C₁₇

C₂×C₂²⋊C₈

prime radical

C₂³×C₁₇

C₂×M₁₆

cosocle

C₂²×#3622

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
105. C₁₇	#11497	semidirect

H ≅ G/N

type

105. C₁₇

#11497

semidirect

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

subgroup	index
#14545145	2
#14543722	2
#14537719	2
#11497	17

subgroup

index

Subgroups — #17958378

Conjugacy classes of subgroups

Complements

Maximal subgroups