Group #17981224 — subgroups

Subgroups — #17981224

Order 4352.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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	#14545988	C₂	1	2	yes	maximal
2	C₄×#10587532	C₂	1	1	yes	maximal, characteristic
3	#11763912	C₄	1	4	yes	complemented
4	#11763906	—	2	2	no	—
5	#11763903	—	2	2	no	—
6	C₂×#10587532	—	2	2	no	—
7	C₄×#63242	C₂²	1	1	yes	characteristic
8	C₄²×#595	C₄	1	1	yes	characteristic
9	C₄×C₈×D₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄×#595	—	2	1	no	—
11	C₂×C₄×#595	—	2	1	no	—
12	C₂×C₄×#595	C₈	1	2	yes	—
13	C₂×#63242	D₄	1	2	yes	—
14	C₂×#63241	—	2	4	no	—
15	C₂×C₈×D₁₇	—	2	1	no	—
16	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄×#3624	C₂×C₄	1	2	yes	—
18	C₄×#3624	D₄	1	1	yes	characteristic
19	C₄×#3624	Q₈	1	1	yes	characteristic
20	#10587532	—	2	8	no	—
21	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
22	C₈×Dic₁₇	—	2	1	no	—
23	C₄×#3622	C₈	1	1	yes	characteristic
24	C₂²×#595	—	2	2	no	—
25	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
28	C₂×#3624	C₂²⋊C₄	1	2	yes	—
29	C₂×#3624	C₄⋊C₄	1	2	yes	—
30	C₄×#595	—	2	4	no	—
31	C₄×#595	—	4	2	no	—
32	C₄×#595	—	4	2	no	—
33	#63242	—	2	4	no	—
34	#63241	—	4	4	no	—
35	C₂×C₈×C₁₇	—	2	1	no	—
36	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
38	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
40	C₂×#3622	—	2	1	no	—
41	C₈×D₁₇	—	2	4	no	—
42	#7149	—	17	1	no	maximal, Sylow
43	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×#595	—	2	4	no	—
45	C₂×#595	—	4	2	no	—
46	C₂×#595	—	4	2	no	—
47	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
50	C₂×Dic₁₇	#95	1	1	yes	characteristic
51	C₂×Dic₁₇	#97	1	1	yes	characteristic
52	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₄×D₁₇	—	2	2	no	—
54	C₄×D₁₇	—	2	2	no	—
55	C₄×D₁₇	—	2	2	no	—
56	#3624	—	2	2	no	—
57	#3624	#104	1	4	yes	—
58	C₈×C₁₇	—	2	2	no	—
59	#3622	—	2	2	no	—
60	#2163	—	17	2	no	—
61	C₄×#97	—	17	1	no	—
62	C₂²×C₁₇	#336	1	1	yes	characteristic
63	C₂×D₁₇	—	2	1	no	—
64	C₂×D₁₇	—	2	1	no	—
65	C₂×D₁₇	#349	1	2	yes	—
66	#595	—	4	4	no	—
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	—	2	1	no	—
69	C₄×C₁₇	—	2	1	no	—
70	Dic₁₇	—	2	1	no	—
71	Dic₁₇	#339	1	2	yes	—
72	Dic₁₇	#347	1	2	yes	—
73	C₂×#97	—	34	2	no	—
74	C₂×C₄×C₈	—	17	1	no	—
75	C₂×C₄×C₈	—	17	1	no	—
76	C₄³	—	17	1	no	—
77	C₄⋊C₁₆	—	17	4	no	—
78	C₂²⋊C₁₆	—	34	2	no	—
79	#346	—	34	2	no	—
80	C₂×C₁₇	#1276	1	1	yes	characteristic
81	C₂×C₁₇	#1375	1	1	yes	characteristic
82	C₂×C₁₇	#1377	1	1	yes	characteristic
83	D₁₇	—	2	2	no	—
84	C₂²×C₈	—	17	2	no	—
85	C₂²×C₈	—	34	1	no	—
86	C₂×C₄²	—	17	1	no	—
87	C₂×C₄²	—	17	2	no	—
88	C₂×C₄²	—	34	1	no	—
89	C₂×C₄²	—	34	1	no	—
90	C₂×C₁₆	—	34	4	no	—
91	#97	—	34	8	no	—
92	C₄×C₈	—	17	1	no	—
93	C₄×C₈	—	17	2	no	—
94	C₄×C₈	—	17	1	no	—
95	C₄×C₈	—	17	1	no	—
96	C₄×C₈	—	34	1	no	—
97	C₄×C₈	#3633	1	1	yes	characteristic
98	C₁₇	#7149	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	—	17	1	no	—
100	C₂²×C₄	—	17	1	no	—
101	C₂²×C₄	—	17	1	no	—
102	C₂²×C₄	—	34	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	17	2	no	—
105	C₂×C₈	—	17	2	no	—
106	C₂×C₈	—	34	4	no	—
107	C₂×C₈	—	34	4	no	—
108	C₂×C₈	—	34	1	no	—
109	C₄²	—	17	1	no	—
110	C₄²	—	17	1	no	—
111	C₄²	—	17	1	no	—
112	C₄²	—	34	4	no	—
113	C₄²	—	68	2	no	—
114	C₄²	—	68	2	no	—
115	C₄²	C₂×#3633	1	1	yes	characteristic
116	C₁₆	—	68	4	no	—
117	C₂³	—	17	1	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	34	2	no	—
122	C₂×C₄	—	34	2	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	34	4	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	C₄×#3633	1	1	yes	characteristic
128	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	—	17	4	no	—
132	C₈	—	34	2	no	—
133	C₈	—	34	2	no	—
134	C₂²	—	17	2	no	—
135	C₂²	—	34	1	no	—
136	C₂²	—	34	1	no	—
137	C₂²	(C₂×#3633)⋊C₄	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	17	2	no	—
142	C₄	—	17	2	no	—
143	C₄	—	34	1	no	—
144	C₄	—	68	4	no	—
145	C₂	—	34	2	no	—
146	C₂	#14545789	1	1	yes	characteristic
147	C₂	#14545857	1	1	yes	characteristic
148	C₂	#14545861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545988

C₂

yes

maximal

C₄×#10587532

C₂

yes

maximal, characteristic

#11763912

C₄

yes

—

—

—

C₂²

yes

characteristic

C₄²×#595

C₄

yes

characteristic

C₄×C₈×D₁₇

C₄

yes

characteristic

C₂×C₄×#595

—

C₂×C₄×#595

—

C₂×C₄×#595

C₈

yes

—

C₂×#63242

D₄

yes

—

C₂×#63241

—

C₂×C₈×D₁₇

—

C₄²×D₁₇

C₂×C₄

yes

characteristic

C₄×#3624

C₂×C₄

yes

—

C₄×#3624

D₄

yes

characteristic

C₄×#3624

Q₈

yes

characteristic

#10587532

—

C₄×C₈×C₁₇

C₈

yes

characteristic

C₈×Dic₁₇

—

C₄×#3622

C₈

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

characteristic

C₂×#3624

C₂²⋊C₄

yes

—

C₂×#3624

C₄⋊C₄

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

—

C₈×D₁₇

—

#7149

—

maximal, Sylow

C₂²×D₁₇

C₂²⋊C₈

yes

—

—

—

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

#104

yes

—

C₈×C₁₇

—

#3622

—

#2163

—

C₄×#97

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#349

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

#339

yes

—

Dic₁₇

#347

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3633

yes

characteristic

C₁₇

#7149

yes

characteristic, complemented, Sylow

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂×#3633

yes

characteristic

116

C₁₆

—

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₄×#3633

yes

characteristic

128

C₂×C₄

C₄⋊#3633

yes

characteristic

129

C₂×C₄

C₂²⋊#3633

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

(C₂×#3633)⋊C₄

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

#14545789

yes

characteristic

147

C₂

#14545857

yes

characteristic

148

C₂

#14545861

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#3633)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₄²	C₂×#3633
Fitting	C₄×C₈×C₁₇	C₈
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×#63242	C₂²
hypercenter	C₄×C₈	#3633
hypocenter	C₁₇	#7149
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

(C₂×#3633)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₄²

C₂×#3633

Fitting

C₄×C₈×C₁₇

C₈

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₂²⋊C₈

C₂²

C₄×C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. #11763912	C₄	semidirect
98. C₁₇	#7149	semidirect

H ≅ G/N

type

3. #11763912

C₄

semidirect

98. C₁₇

#7149

semidirect

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

subgroup	index
#14545988	2
C₄×#10587532	2
#7149	17

subgroup

index

#14545988

C₄×#10587532

#7149

Subgroups — #17981224

Conjugacy classes of subgroups

Complements

Maximal subgroups