Group #17981222 — subgroups

Subgroups — #17981222

Order 4352.

302 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 81 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	#14545984	C₂	1	2	yes	maximal
2	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#63241	C₄	1	4	yes	complemented
4	#11763906	—	2	2	no	—
5	C₄×#63241	—	2	2	no	—
6	C₄×#63242	C₂²	1	1	yes	characteristic
7	C₂×C₈×#595	—	2	2	no	—
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	C₈×#595	—	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	2	no	—
31	C₄×#595	—	2	2	no	—
32	C₄×#595	—	2	4	no	—
33	C₄×#595	—	2	2	no	—
34	C₄×#595	C₁₆	1	4	yes	complemented
35	#63242	—	2	4	no	—
36	#63241	—	4	4	no	—
37	C₂×C₈×C₁₇	—	2	1	no	—
38	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
39	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
40	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
42	C₂×#3622	—	2	1	no	—
43	C₈×D₁₇	—	2	4	no	—
44	#7148	—	17	1	no	maximal, Sylow
45	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×#595	—	2	4	no	—
47	C₂×#595	—	2	4	no	—
48	C₂×#595	—	2	4	no	—
49	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
52	C₂×Dic₁₇	#95	1	1	yes	characteristic
53	C₂×Dic₁₇	#97	1	1	yes	characteristic
54	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
55	C₄×D₁₇	—	2	1	no	—
56	C₄×D₁₇	—	2	1	no	—
57	C₄×D₁₇	—	2	2	no	—
58	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
59	C₄×D₁₇	M₃₂	1	2	yes	—
60	#3624	—	2	2	no	—
61	#3624	#104	1	4	yes	—
62	C₈×C₁₇	—	2	2	no	—
63	#3622	—	2	2	no	—
64	#2162	—	17	2	no	—
65	C₄²×C₈	—	17	1	no	—
66	C₂²×C₁₇	#336	1	1	yes	characteristic
67	C₂×D₁₇	—	2	1	no	—
68	C₂×D₁₇	—	2	1	no	—
69	C₂×D₁₇	C₂²⋊C₁₆	1	2	yes	—
70	#595	—	2	8	no	—
71	C₄×C₁₇	—	2	1	no	—
72	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
73	C₄×C₁₇	#346	1	1	yes	characteristic
74	C₄×C₁₇	#364	1	1	yes	characteristic
75	C₄×C₁₇	#364	1	1	yes	characteristic
76	Dic₁₇	—	2	1	no	—
77	Dic₁₇	#339	1	2	yes	—
78	Dic₁₇	C₄⋊C₁₆	1	2	yes	—
79	C₂×C₄×C₈	—	17	1	no	—
80	C₂×C₄×C₈	—	17	1	no	—
81	C₂×C₄×C₈	—	34	2	no	—
82	C₄³	—	17	1	no	—
83	C₄⋊C₁₆	—	17	4	no	—
84	C₂²⋊C₁₆	—	34	2	no	—
85	C₄×C₁₆	—	34	2	no	—
86	C₂×C₁₇	#1276	1	1	yes	characteristic
87	C₂×C₁₇	#1374	1	1	yes	characteristic
88	C₂×C₁₇	#1376	1	1	yes	characteristic
89	D₁₇	—	2	2	no	—
90	C₂²×C₈	—	17	2	no	—
91	C₂²×C₈	—	34	1	no	—
92	C₂×C₄²	—	17	1	no	—
93	C₂×C₄²	—	17	2	no	—
94	C₂×C₄²	—	34	1	no	—
95	C₂×C₄²	—	34	1	no	—
96	C₂×C₁₆	—	34	4	no	—
97	C₄×C₈	—	17	1	no	—
98	C₄×C₈	—	17	2	no	—
99	C₄×C₈	—	17	1	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	34	1	no	—
102	C₄×C₈	—	34	8	no	—
103	C₄×C₈	#3633	1	1	yes	characteristic
104	C₁₇	#7148	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	—	17	1	no	—
106	C₂²×C₄	—	17	1	no	—
107	C₂²×C₄	—	17	1	no	—
108	C₂²×C₄	—	34	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	17	2	no	—
111	C₂×C₈	—	17	2	no	—
112	C₂×C₈	—	34	4	no	—
113	C₂×C₈	—	34	4	no	—
114	C₂×C₈	—	34	1	no	—
115	C₄²	—	17	1	no	—
116	C₄²	—	17	1	no	—
117	C₄²	—	17	1	no	—
118	C₄²	—	17	4	no	—
119	C₄²	—	34	2	no	—
120	C₄²	—	34	4	no	—
121	C₄²	—	34	2	no	—
122	C₄²	—	34	2	no	—
123	C₄²	C₂×#3633	1	1	yes	characteristic
124	C₁₆	—	68	4	no	—
125	C₂³	—	17	1	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	17	1	no	—
128	C₂×C₄	—	17	2	no	—
129	C₂×C₄	—	17	2	no	—
130	C₂×C₄	—	17	1	no	—
131	C₂×C₄	—	34	2	no	—
132	C₂×C₄	—	34	4	no	—
133	C₂×C₄	—	34	4	no	—
134	C₂×C₄	—	34	4	no	—
135	C₂×C₄	—	34	1	no	—
136	C₂×C₄	—	34	1	no	—
137	C₂×C₄	C₄×#3633	1	1	yes	characteristic
138	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	17	4	no	—
142	C₈	—	34	2	no	—
143	C₈	—	34	2	no	—
144	C₂²	—	17	2	no	—
145	C₂²	—	34	1	no	—
146	C₂²	—	34	1	no	—
147	C₂²	(C₂×#3633)⋊C₄	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	17	2	no	—
150	C₄	—	17	2	no	—
151	C₄	—	34	8	no	—
152	C₄	—	34	1	no	—
153	C₄	#11763902	1	1	yes	characteristic
154	C₄	#11763904	1	1	yes	characteristic
155	C₄	#11763913	1	1	yes	characteristic
156	C₄	#11763914	1	1	yes	characteristic
157	C₂	—	34	2	no	—
158	C₂	#14545789	1	1	yes	characteristic
159	C₂	#14545855	1	1	yes	characteristic
160	C₂	#14545859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545984

C₂

yes

maximal

C₄×C₈×#595

C₂

yes

maximal, characteristic

C₄⋊#63241

C₄

yes

—

—

C₂²

yes

characteristic

C₂×C₈×#595

—

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

C₈×#595

—

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

—

—

—

—

—

C₁₆

yes

complemented

#63242

—

#63241

—

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

—

#7148

—

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

—

C₄×D₁₇

C₂×C₁₆

yes

—

C₄×D₁₇

M₃₂

yes

—

#3624

—

#3624

#104

yes

—

C₈×C₁₇

—

#3622

—

#2162

—

C₄²×C₈

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊C₁₆

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₄×C₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

#339

yes

—

Dic₁₇

C₄⋊C₁₆

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

#3633

yes

characteristic

104

C₁₇

#7148

yes

characteristic, complemented, Sylow

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₂×#3633

yes

characteristic

124

C₁₆

—

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄×#3633

yes

characteristic

138

C₂×C₄

C₄⋊#3633

yes

characteristic

139

C₂×C₄

C₂²⋊#3633

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

(C₂×#3633)⋊C₄

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#11763902

yes

characteristic

154

C₄

#11763904

yes

characteristic

155

C₄

#11763913

yes

characteristic

156

C₄

#11763914

yes

characteristic

157

C₂

—

158

C₂

#14545789

yes

characteristic

159

C₂

#14545855

yes

characteristic

160

C₂

#14545859

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#3633
commutator	C₄×C₁₇	C₄×C₁₆
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	#7148
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#3633

commutator

C₄×C₁₇

C₄×C₁₆

Frattini

C₄²

C₂×#3633

Fitting

C₄×C₈×C₁₇

C₈

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₂²⋊C₈

C₂²

C₄×C₈

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. C₄⋊#63241	C₄	semidirect
34. C₄×#595	C₁₆	semidirect
104. C₁₇	#7148	semidirect

H ≅ G/N

type

3. C₄⋊#63241

C₄

semidirect

34. C₄×#595

C₁₆

semidirect

104. C₁₇

#7148

semidirect

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

subgroup	index
#14545984	2
C₄×C₈×#595	2
#7148	17

subgroup

index

#14545984

C₄×C₈×#595

#7148

Subgroups — #17981222

Conjugacy classes of subgroups

Complements

Maximal subgroups