Group #17958021 — subgroups

Subgroups — #17958021

Order 4352.

298 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 148 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	#14544990	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763650	C₂	1	1	yes	maximal, characteristic
3	#14543703	C₂	1	4	yes	maximal, complemented
4	M₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587538	C₄	1	2	yes	—
6	C₂×C₄×#3624	C₄	1	2	yes	—
7	#11763698	C₂²	1	2	yes	—
8	#11763695	C₄	1	2	yes	—
9	#11763650	C₄	1	2	yes	—
10	#11763650	C₂²	1	2	yes	—
11	C₂×#10587483	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
13	#11762584	C₂²	1	2	yes	—
14	C₁₆×Dic₁₇	C₄	1	2	yes	—
15	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
16	C₂²×#3624	—	2	2	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
20	C₄×#3624	—	2	2	no	—
21	C₄×#3624	C₂×C₄	1	4	yes	—
22	#10587538	C₈	1	8	yes	complemented
23	C₂×#63216	C₂×C₄	1	4	yes	—
24	C₁₇×M₃₂	—	2	2	no	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
26	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
30	#10587483	—	2	2	no	—
31	C₂×#63214	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
33	C₂²×Dic₁₇	M₁₆	1	2	yes	—
34	C₂×#3624	—	2	4	no	—
35	C₂×#3624	—	4	4	no	—
36	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
38	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
39	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
41	C₄×Dic₁₇	M₁₆	1	2	yes	—
42	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×#3622	C₄²	1	1	yes	characteristic
44	C₂×#3622	C₂×C₈	1	4	yes	—
45	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
46	#63216	C₂×C₈	1	8	yes	complemented
47	C₁₆×C₁₇	—	2	2	no	—
48	#63214	—	2	2	no	—
49	#11404	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	#97	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#97	1	2	yes	—
54	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
55	C₂×Dic₁₇	—	2	4	no	—
56	C₂×Dic₁₇	#97	1	2	yes	—
57	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
58	#3624	—	4	4	no	—
59	C₈×C₁₇	#97	1	2	yes	—
60	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	#3622	C₄×C₈	1	2	yes	—
63	#3622	C₂²×C₈	1	2	yes	—
64	C₄×M₃₂	—	17	1	no	—
65	C₄×M₃₂	—	17	1	no	—
66	C₂×#322	—	17	1	no	—
67	#1313	—	17	4	no	—
68	C₂²×C₁₇	—	2	2	no	—
69	C₂²×C₁₇	#322	1	2	yes	—
70	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
71	C₄×C₁₇	#322	1	2	yes	—
72	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
74	Dic₁₇	—	2	2	no	—
75	Dic₁₇	#347	1	4	yes	—
76	C₂×M₃₂	—	17	2	no	—
77	C₂×M₃₂	—	17	1	no	—
78	C₂×M₃₂	#595	1	1	yes	characteristic
79	C₂×C₄×C₈	—	17	1	no	—
80	C₂×C₄×C₈	—	17	2	no	—
81	#346	—	17	2	no	—
82	#346	—	17	2	no	—
83	C₄×C₁₆	—	17	2	no	—
84	C₄×C₁₆	—	17	2	no	—
85	#322	—	17	2	no	—
86	#322	—	17	2	no	—
87	C₂×C₁₇	—	2	2	no	—
88	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
89	C₂×C₁₇	C₂×#347	1	2	yes	—
90	C₂²×C₈	—	17	1	no	—
91	C₂²×C₈	—	34	2	no	—
92	C₂²×C₈	C₂×#595	1	1	yes	characteristic
93	C₂×C₄²	—	17	1	no	—
94	M₃₂	—	2	2	no	—
95	M₃₂	—	17	8	no	—
96	M₃₂	—	34	2	no	—
97	C₂×C₁₆	—	17	4	no	—
98	C₂×C₁₆	—	17	2	no	—
99	C₂×C₁₆	C₂×#595	1	2	yes	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	—	17	1	no	—
103	C₄×C₈	—	17	1	no	—
104	C₄×C₈	—	17	4	no	—
105	C₄×C₈	—	34	2	no	—
106	C₁₇	#11404	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₄	—	17	2	no	—
108	C₂²×C₄	C₄×#595	1	1	yes	characteristic
109	C₂×C₈	—	17	1	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	4	no	—
112	C₂×C₈	—	34	4	no	—
113	C₂×C₈	—	68	4	no	—
114	C₂×C₈	#63243	1	2	yes	—
115	C₂×C₈	C₄×#595	1	1	yes	characteristic
116	C₂×C₈	#63247	1	2	yes	—
117	C₂×C₈	C₂²×#595	1	1	yes	characteristic
118	C₄²	—	17	1	no	—
119	C₄²	—	17	1	no	—
120	C₄²	—	17	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	17	8	no	—
123	C₁₆	—	34	2	no	—
124	C₂³	#10587532	1	1	yes	characteristic
125	C₂×C₄	—	17	2	no	—
126	C₂×C₄	—	17	2	no	—
127	C₂×C₄	—	34	4	no	—
128	C₂×C₄	C₈×#595	1	2	yes	—
129	C₂×C₄	#10587532	1	1	yes	characteristic
130	C₂×C₄	#10587542	1	2	yes	—
131	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
132	C₈	—	17	2	no	—
133	C₈	—	17	2	no	—
134	C₈	—	68	4	no	—
135	C₈	#10587542	1	2	yes	—
136	C₈	C₂×#63243	1	1	yes	characteristic
137	C₈	C₂×#63247	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#11763651	1	2	yes	—
140	C₂²	C₂×#10587532	1	1	yes	characteristic
141	C₄	—	17	4	no	—
142	C₄	—	34	2	no	—
143	C₄	#11763651	1	2	yes	—
144	C₄	C₂×C₈×#595	1	1	yes	characteristic
145	C₄	C₂×#10587542	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	C₂×#11763651	1	1	yes	characteristic
148	C₂	#14544999	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544990

C₂

yes

maximal, characteristic

C₂×#11763650

C₂

yes

maximal, characteristic

#14543703

C₂

yes

maximal, complemented

M₃₂×Dic₁₇

C₂

yes

maximal, characteristic

C₂×#10587538

C₄

yes

—

C₂×C₄×#3624

C₄

yes

—

#11763698

C₂²

yes

—

#11763695

C₄

yes

—

#11763650

C₄

yes

—

#11763650

C₂²

yes

—

C₂×#10587483

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762584

C₂²

yes

—

C₁₆×Dic₁₇

C₄

yes

—

C₂×C₁₇×M₃₂

C₄

yes

characteristic

C₂²×#3624

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₄×#3624

—

C₄×#3624

C₂×C₄

yes

—

#10587538

C₈

yes

complemented

C₂×#63216

C₂×C₄

yes

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂³

yes

characteristic

#10587483

—

C₂×#63214

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₄²

yes

characteristic

C₂²×Dic₁₇

M₁₆

yes

—

C₂×#3624

—

C₂×#3624

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

—

C₄×Dic₁₇

C₂²×C₄

yes

characteristic

C₂×#3622

C₄²

yes

characteristic

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

C₂²×C₄

yes

characteristic

#63216

C₂×C₈

yes

complemented

C₁₆×C₁₇

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

C₄×C₈

yes

—

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×C₄×C₁₇

#97

yes

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#97

yes

—

C₂×Dic₁₇

C₂×M₁₆

yes

—

#3624

—

C₈×C₁₇

#97

yes

—

C₈×C₁₇

C₂×M₁₆

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

characteristic

#3622

C₄×C₈

yes

—

#3622

C₂²×C₈

yes

—

C₄×M₃₂

—

C₄×M₃₂

—

C₂×#322

—

#1313

—

C₂²×C₁₇

—

C₂²×C₁₇

#322

yes

—

C₂²×C₁₇

C₂×#97

yes

characteristic

C₄×C₁₇

#322

yes

—

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

C₄×C₁₇

C₂×#97

yes

characteristic

Dic₁₇

—

Dic₁₇

#347

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

#595

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

#322

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#322

yes

characteristic

C₂×C₁₇

C₂×#347

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₁₇

#11404

yes

characteristic, complemented, Sylow

107

C₂²×C₄

—

108

C₂²×C₄

C₄×#595

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

#63243

yes

—

115

C₂×C₈

C₄×#595

yes

characteristic

116

C₂×C₈

#63247

yes

—

117

C₂×C₈

C₂²×#595

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#10587532

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×#595

yes

—

129

C₂×C₄

#10587532

yes

characteristic

130

C₂×C₄

#10587542

yes

—

131

C₂×C₄

C₂×C₄×#595

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#10587542

yes

—

136

C₈

C₂×#63243

yes

characteristic

137

C₈

C₂×#63247

yes

characteristic

138

C₂²

—

139

C₂²

#11763651

yes

—

140

C₂²

C₂×#10587532

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#11763651

yes

—

144

C₄

C₂×C₈×#595

yes

characteristic

145

C₄

C₂×#10587542

yes

characteristic

146

C₂

—

147

C₂

C₂×#11763651

yes

characteristic

148

C₂

#14544999

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂×C₁₇×M₃₂	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	#97
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂×M₃₂	#595
hypocenter	C₁₇	#11404
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

subgroup

G/H

center

C₂×C₄

C₂×C₄×#595

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₂×C₁₇×M₃₂

C₄

socle

C₂²×C₁₇

C₂×#97

prime radical

C₂³×C₁₇

#97

cosocle

C₈×Dic₁₇

C₂³

C₂×M₃₂

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #14543703	C₂	semidirect
22. #10587538	C₈	semidirect
46. #63216	C₂×C₈	semidirect
106. C₁₇	#11404	semidirect

H ≅ G/N

type

3. #14543703

C₂

semidirect

22. #10587538

C₈

semidirect

46. #63216

C₂×C₈

semidirect

106. C₁₇

#11404

semidirect

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

subgroup	index
#14544990	2
C₂×#11763650	2
#14543703	2
M₃₂×Dic₁₇	2
#11404	17

subgroup

index

M₃₂×Dic₁₇

Subgroups — #17958021

Conjugacy classes of subgroups

Complements

Maximal subgroups