Group #17948523 — subgroups

Subgroups — #17948523

Order 4352.

302 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 104 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	#14545141	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#63216	C₄	1	4	yes	complemented
4	C₂²⋊#63216	—	2	2	no	—
5	C₄×#63216	—	2	2	no	—
6	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₇	—	2	2	no	—
8	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	—	2	1	no	—
11	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₁₇	—	2	1	no	—
13	C₂×C₄×Dic₁₇	—	2	1	no	—
14	C₂×C₄×Dic₁₇	C₈	1	2	yes	—
15	C₂²×#3622	D₄	1	2	yes	—
16	C₂×#63216	—	2	4	no	—
17	C₄×C₈×C₁₇	—	2	1	no	—
18	C₄×C₈×C₁₇	C₈	1	2	yes	—
19	C₈×Dic₁₇	—	2	8	no	—
20	C₄×#3622	C₂×C₄	1	2	yes	—
21	C₄×#3622	D₄	1	1	yes	characteristic
22	C₄×#3622	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
26	C₂²×Dic₁₇	—	2	2	no	—
27	C₂×C₈×C₁₇	—	2	4	no	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₄²×C₁₇	C₄²	1	1	yes	characteristic
30	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
33	C₄×Dic₁₇	—	2	4	no	—
34	C₄×Dic₁₇	—	2	4	no	—
35	C₄×Dic₁₇	—	2	2	no	—
36	C₄×Dic₁₇	C₁₆	1	4	yes	complemented
37	C₂×#3622	—	2	4	no	—
38	C₂×#3622	C₂²⋊C₄	1	2	yes	—
39	C₂×#3622	C₄⋊C₄	1	2	yes	—
40	#63216	—	4	4	no	—
41	#7148	—	17	1	no	maximal, Sylow
42	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	—	2	2	no	—
45	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#97	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×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
52	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
53	C₂×Dic₁₇	—	2	4	no	—
54	C₂×Dic₁₇	—	2	4	no	—
55	C₂×Dic₁₇	—	2	4	no	—
56	C₈×C₁₇	—	2	4	no	—
57	#3622	—	2	2	no	—
58	#3622	#104	1	4	yes	—
59	#2162	—	17	2	no	—
60	C₄²×C₈	—	17	1	no	—
61	C₂²×C₁₇	—	2	1	no	—
62	C₂²×C₁₇	—	2	1	no	—
63	C₂²×C₁₇	#336	1	1	yes	characteristic
64	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	#339	1	2	yes	—
68	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
69	C₄×C₁₇	#346	1	1	yes	characteristic
70	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
71	C₄×C₁₇	#364	1	2	yes	—
72	Dic₁₇	—	2	8	no	—
73	C₂×C₄×C₈	—	17	1	no	—
74	C₂×C₄×C₈	—	34	2	no	—
75	C₂×C₄×C₈	#595	1	1	yes	characteristic
76	C₄³	—	17	1	no	—
77	C₄⋊C₁₆	—	17	4	no	—
78	C₂²⋊C₁₆	—	34	2	no	—
79	C₄×C₁₆	—	34	2	no	—
80	C₂×C₁₇	—	2	2	no	—
81	C₂×C₁₇	#1276	1	1	yes	characteristic
82	C₂×C₁₇	#1374	1	1	yes	characteristic
83	C₂×C₁₇	#1376	1	1	yes	characteristic
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	17	2	no	—
86	C₂×C₄²	—	17	2	no	—
87	C₂×C₄²	—	34	1	no	—
88	C₂×C₄²	—	34	1	no	—
89	C₂×C₄²	C₂×#595	1	1	yes	characteristic
90	C₂×C₁₆	—	34	4	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	17	2	no	—
93	C₄×C₈	—	17	1	no	—
94	C₄×C₈	—	17	1	no	—
95	C₄×C₈	—	34	8	no	—
96	C₄×C₈	#3624	1	2	yes	—
97	C₁₇	#7148	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₄	—	34	2	no	—
99	C₂²×C₄	#63242	1	1	yes	characteristic
100	C₂²×C₄	#63243	1	1	yes	characteristic
101	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	17	2	no	—
105	C₂×C₈	—	17	2	no	—
106	C₂×C₈	—	34	4	no	—
107	C₄²	—	17	4	no	—
108	C₄²	—	34	4	no	—
109	C₄²	—	34	4	no	—
110	C₄²	—	34	2	no	—
111	C₄²	C₄×#595	1	1	yes	characteristic
112	C₄²	C₄⋊#595	1	1	yes	characteristic
113	C₄²	C₂×#3624	1	1	yes	characteristic
114	C₄²	#63247	1	1	yes	characteristic
115	C₁₆	—	68	4	no	—
116	C₂³	#10587543	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	34	4	no	—
120	C₂×C₄	—	34	4	no	—
121	C₂×C₄	—	34	4	no	—
122	C₂×C₄	#10587529	1	2	yes	—
123	C₂×C₄	#10587530	1	2	yes	—
124	C₂×C₄	C₄×#3624	1	1	yes	characteristic
125	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
126	C₂×C₄	#10587542	1	1	yes	characteristic
127	C₂×C₄	#10587544	1	1	yes	characteristic
128	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₈	—	17	4	no	—
132	C₈	—	34	2	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#11763675	1	1	yes	characteristic
136	C₂²	#11763704	1	2	yes	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	34	8	no	—
140	C₄	#11763680	1	2	yes	—
141	C₄	#11763695	1	1	yes	characteristic
142	C₄	#11763698	1	1	yes	characteristic
143	C₄	#11763730	1	2	yes	—
144	C₄	#11763731	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	#14543612	1	1	yes	characteristic
147	C₂	#14543823	1	1	yes	characteristic
148	C₂	#14543829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545141

C₂

yes

maximal

C₄×C₈×Dic₁₇

C₂

yes

maximal, characteristic

C₄⋊#63216

C₄

yes

complemented

C₂²⋊#63216

—

C₄×#63216

—

C₂×C₄×#3622

C₂²

yes

characteristic

C₂×C₈×Dic₁₇

—

C₄²×Dic₁₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

C₈

yes

—

C₂²×#3622

D₄

yes

—

C₂×#63216

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₈

yes

—

C₈×Dic₁₇

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

D₄

yes

characteristic

C₄×#3622

Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₁₆

yes

complemented

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

#63216

—

#7148

—

maximal, Sylow

C₂³×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

M₃₂

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#3622

—

#3622

#104

yes

—

#2162

—

C₄²×C₈

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#336

yes

characteristic

C₂²×C₁₇

C₂²⋊C₁₆

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#339

yes

—

C₄×C₁₇

C₄×C₁₆

yes

characteristic

C₄×C₁₇

#346

yes

characteristic

C₄×C₁₇

C₄⋊C₁₆

yes

—

C₄×C₁₇

#364

yes

—

Dic₁₇

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#595

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#595

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3624

yes

—

C₁₇

#7148

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

#63242

yes

characteristic

100

C₂²×C₄

#63243

yes

characteristic

101

C₂²×C₄

C₂²⋊#595

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×#595

yes

characteristic

112

C₄²

C₄⋊#595

yes

characteristic

113

C₄²

C₂×#3624

yes

characteristic

114

C₄²

#63247

yes

characteristic

115

C₁₆

—

116

C₂³

#10587543

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#10587529

yes

—

123

C₂×C₄

#10587530

yes

—

124

C₂×C₄

C₄×#3624

yes

characteristic

125

C₂×C₄

C₄⋊#3624

yes

characteristic

126

C₂×C₄

#10587542

yes

characteristic

127

C₂×C₄

#10587544

yes

characteristic

128

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

129

C₂×C₄

C₂²⋊#3624

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

#11763675

yes

characteristic

136

C₂²

#11763704

yes

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#11763680

yes

—

141

C₄

#11763695

yes

characteristic

142

C₄

#11763698

yes

characteristic

143

C₄

#11763730

yes

—

144

C₄

#11763731

yes

—

145

C₂

—

146

C₂

#14543612

yes

characteristic

147

C₂

#14543823

yes

characteristic

148

C₂

#14543829

yes

characteristic

	subgroup	G/H
center	C₄²	C₄⋊#595
commutator	C₄×C₁₇	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7148
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₄⋊#595

commutator

C₄×C₁₇

C₄×C₁₆

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₄×C₈×C₁₇

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂³×C₁₇

C₂²⋊C₈

cosocle

C₂×C₄×#3622

C₂²

C₂×C₄×C₈

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. C₄⋊#63216	C₄	semidirect
36. C₄×Dic₁₇	C₁₆	semidirect
97. C₁₇	#7148	semidirect

H ≅ G/N

type

3. C₄⋊#63216

C₄

semidirect

36. C₄×Dic₁₇

C₁₆

semidirect

97. C₁₇

#7148

semidirect

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

subgroup	index
#14545141	2
C₄×C₈×Dic₁₇	2
#7148	17

subgroup

index

#14545141

C₄×C₈×Dic₁₇

#7148

Subgroups — #17948523

Conjugacy classes of subgroups

Complements

Maximal subgroups