Group #17981234 — subgroups

Subgroups — #17981234

Order 4352.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14545908	C₂	1	2	yes	maximal
2	C₂×C₁₆×#595	C₂	1	1	yes	maximal, characteristic
3	#11763922	—	2	2	no	—
4	#11763899	C₄	1	4	yes	—
5	#11763898	—	2	2	no	—
6	C₂×#10587529	C₄	1	1	yes	characteristic
7	C₂×C₈×#595	C₂²	1	1	yes	characteristic
8	C₁₆×#595	—	2	2	no	—
9	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄×#595	D₄	1	1	yes	characteristic
11	C₂×#63242	Q₈	1	1	yes	characteristic
12	C₂²⋊#3633	—	4	2	no	—
13	C₄⋊#3633	—	2	4	no	—
14	C₄×#3633	—	4	2	no	—
15	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#63216	—	2	1	no	—
17	C₈×#595	C₂×C₄	1	2	yes	—
18	C₈×#595	D₄	1	2	yes	—
19	#10587529	—	2	1	no	—
20	#10587529	C₈	1	2	yes	complemented
21	C₂×C₁₆×C₁₇	C₈	1	1	yes	characteristic, complemented
22	C₂×#63214	C₈	1	1	yes	characteristic, complemented
23	C₁₆×D₁₇	—	2	1	no	—
24	C₁₆×D₁₇	—	2	1	no	—
25	C₂²×#595	SD₁₆	1	2	yes	—
26	C₂×#3633	—	4	4	no	—
27	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×#3624	—	2	1	no	—
29	C₄×#595	C₂²⋊C₄	1	2	yes	—
30	C₄×#595	D₈	1	1	yes	characteristic
31	C₄×#595	Q₁₆	1	1	yes	characteristic
32	#63242	—	2	1	no	—
33	#63242	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×#3622	M₁₆	1	1	yes	characteristic
36	C₈×D₁₇	C₄²	1	1	yes	characteristic
37	C₈×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
39	C₈×D₁₇	M₁₆	1	1	yes	characteristic
40	#63216	—	2	2	no	—
41	C₁₆×C₁₇	—	2	1	no	—
42	#63214	—	2	1	no	—
43	#7155	—	17	1	no	maximal, Sylow
44	C₂²×D₁₇	#106	1	1	yes	characteristic
45	C₂×#595	—	2	4	no	—
46	C₂×#595	#102	1	2	yes	—
47	C₂×#595	#103	1	2	yes	—
48	#3633	—	8	4	no	—
49	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
50	C₂×Dic₁₇	#108	1	1	yes	characteristic
51	C₄×D₁₇	#95	1	1	yes	characteristic
52	C₄×D₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₄×D₁₇	#107	1	1	yes	characteristic
54	C₄×D₁₇	#108	1	1	yes	characteristic
55	#3624	—	2	2	no	—
56	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
57	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
58	#3622	#97	1	1	yes	characteristic
59	#3622	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₆	—	17	1	no	—
61	#1564	—	17	2	no	—
62	C₂²×C₁₇	#334	1	1	yes	characteristic
63	C₂×D₁₇	—	2	2	no	—
64	C₂×D₁₇	#334	1	1	yes	characteristic
65	C₂×D₁₇	#340	1	1	yes	characteristic
66	#595	—	2	2	no	—
67	#595	#359	1	4	yes	—
68	C₄×C₁₇	#335	1	1	yes	characteristic
69	C₄×C₁₇	#336	1	1	yes	characteristic
70	Dic₁₇	#335	1	1	yes	characteristic
71	Dic₁₇	#341	1	1	yes	characteristic
72	C₂²×C₁₆	—	17	1	no	—
73	C₂²×C₁₆	—	17	1	no	—
74	#431	—	34	2	no	—
75	C₂×C₄×C₈	—	17	1	no	—
76	C₄×C₁₆	—	34	2	no	—
77	#335	—	17	4	no	—
78	#334	—	34	2	no	—
79	C₂×C₁₇	—	2	1	no	—
80	C₂×C₁₇	#1277	1	1	yes	characteristic
81	D₁₇	—	2	1	no	—
82	D₁₇	#1382	1	2	yes	—
83	C₂²×C₈	—	17	1	no	—
84	C₂²×C₈	—	17	1	no	—
85	C₂×C₄²	—	17	1	no	—
86	C₂×C₁₆	—	17	2	no	—
87	C₂×C₁₆	—	17	1	no	—
88	C₂×C₁₆	—	34	1	no	—
89	C₂×C₁₆	—	34	1	no	—
90	C₂×C₁₆	—	34	1	no	—
91	C₂×C₁₆	—	34	1	no	—
92	C₂×C₁₆	#3633	1	1	yes	characteristic, complemented
93	C₄⋊C₈	—	34	4	no	—
94	C₂²⋊C₈	—	68	2	no	—
95	C₄×C₈	—	17	2	no	—
96	C₄×C₈	—	17	2	no	—
97	C₄×C₈	—	68	2	no	—
98	C₁₇	#7155	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	—	17	1	no	—
100	C₂²×C₄	—	17	2	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	17	1	no	—
104	C₂×C₈	—	17	1	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	2	no	—
107	C₂×C₈	—	34	1	no	—
108	C₂×C₈	—	34	1	no	—
109	C₂×C₈	—	68	4	no	—
110	C₂×C₈	C₂×#3633	1	1	yes	characteristic
111	C₄²	—	17	2	no	—
112	C₄²	—	17	1	no	—
113	C₄²	—	17	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	34	2	no	—
116	C₁₆	—	34	1	no	—
117	C₂³	—	17	1	no	—
118	C₂×C₄	—	17	2	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	17	1	no	—
124	C₂×C₄	—	17	2	no	—
125	C₂×C₄	—	34	4	no	—
126	C₂×C₄	C₄⋊#3633	1	1	yes	characteristic
127	C₈	—	17	1	no	—
128	C₈	—	17	1	no	—
129	C₈	—	34	2	no	—
130	C₈	—	136	4	no	—
131	C₈	C₄×#3633	1	1	yes	characteristic
132	C₈	C₂²⋊#3633	1	1	yes	characteristic
133	C₂²	—	17	1	no	—
134	C₂²	—	17	1	no	—
135	C₂²	—	34	2	no	—
136	C₂²	#11763898	1	1	yes	characteristic
137	C₄	—	17	4	no	—
138	C₄	—	17	1	no	—
139	C₄	—	17	1	no	—
140	C₄	—	34	2	no	—
141	C₄	#11763899	1	1	yes	characteristic
142	C₄	(C₂×#3633)⋊C₄	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	17	2	no	—
145	C₂	—	34	1	no	—
146	C₂	#14545790	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545908

C₂

yes

maximal

C₂×C₁₆×#595

C₂

yes

maximal, characteristic

#11763922

—

#11763899

C₄

yes

—

#11763898

—

C₂×#10587529

C₄

yes

characteristic

C₂×C₈×#595

C₂²

yes

characteristic

C₁₆×#595

—

C₂×C₁₆×D₁₇

C₄

yes

characteristic

C₂×C₄×#595

D₄

yes

characteristic

C₂×#63242

Q₈

yes

characteristic

C₂²⋊#3633

—

C₄⋊#3633

—

C₄×#3633

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×#63216

—

C₈×#595

C₂×C₄

yes

—

C₈×#595

D₄

yes

—

#10587529

—

#10587529

C₈

yes

complemented

C₂×C₁₆×C₁₇

C₈

yes

characteristic, complemented

C₂×#63214

C₈

yes

characteristic, complemented

C₁₆×D₁₇

—

C₁₆×D₁₇

—

C₂²×#595

SD₁₆

yes

—

C₂×#3633

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×#3624

—

C₄×#595

C₂²⋊C₄

yes

—

C₄×#595

D₈

yes

characteristic

C₄×#595

Q₁₆

yes

characteristic

#63242

—

#63242

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×#3622

M₁₆

yes

characteristic

C₈×D₁₇

C₄²

yes

characteristic

C₈×D₁₇

C₂²⋊C₄

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

#63216

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

—

yes

—

C₂×#595

#103

yes

—

#3633

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

C₂×Dic₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

C₄⋊C₈

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

C₈×C₁₇

C₄×C₈

yes

characteristic

C₈×C₁₇

C₂²⋊C₈

yes

yes

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₆

—

#1564

—

C₂²×C₁₇

#334

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

yes

C₂×D₁₇

yes

—

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

yes

Dic₁₇

yes

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

C₄×C₁₆

—

#335

—

#334

—

C₂×C₁₇

—

C₂×C₁₇

#1277

yes

characteristic

D₁₇

—

D₁₇

#1382

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#3633

yes

characteristic, complemented

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#7155

yes

characteristic, complemented, Sylow

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂×#3633

yes

characteristic

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₁₆

—

115

C₁₆

—

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

C₄⋊#3633

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

136

—

131

C₈

C₄×#3633

yes

characteristic

132

C₈

C₂²⋊#3633

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#11763898

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#11763899

yes

characteristic

142

C₄

(C₂×#3633)⋊C₄

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#14545790

yes

characteristic

	subgroup	G/H
center	C₄	#11763899
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₂×C₈	C₂×#3633
Fitting	C₂×C₁₆×C₁₇	C₈
socle	C₂×C₁₇	#1277
prime radical	C₂²×D₁₇	#106
cosocle	C₂×C₈×#595	C₂²
hypercenter	C₂×C₁₆	#3633
hypocenter	C₁₇	#7155
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#11763899

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₂×C₈

C₂×#3633

Fitting

C₂×C₁₆×C₁₇

C₈

C₂×C₁₇

C₂²×D₁₇

C₂×C₈×#595

C₂²

C₂×C₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
20. #10587529	C₈	semidirect
21. C₂×C₁₆×C₁₇	C₈	semidirect
22. C₂×#63214	C₈	semidirect
92. C₂×C₁₆	#3633	semidirect
98. C₁₇	#7155	semidirect

H ≅ G/N

type

20. #10587529

C₈

semidirect

21. C₂×C₁₆×C₁₇

C₈

semidirect

22. C₂×#63214

C₈

semidirect

92. C₂×C₁₆

#3633

semidirect

98. C₁₇

#7155

semidirect

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

subgroup	index
#14545908	2
C₂×C₁₆×#595	2
#7155	17

subgroup

index

#14545908

C₂×C₁₆×#595

#7155

Subgroups — #17981234

Conjugacy classes of subgroups

Complements

Maximal subgroups