Group #17960203 — subgroups

Subgroups — Q₃₂×#3624 #17960203

Order 4352.

226 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 84 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	Q₁₆×#3624	C₂	1	2	yes	maximal
2	#14543815	C₂	1	1	yes	maximal, characteristic
3	#14543740	C₂	1	2	yes	maximal
4	C₁₆×#3624	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3624	—	2	2	no	—
7	#11763695	C₂²	1	1	yes	characteristic
8	#11763672	C₂²	1	2	yes	—
9	Q₈⋊#3624	—	2	2	no	—
10	C₈×#3624	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₁₇	C₂²	1	2	yes	—
12	#11762635	C₄	1	1	yes	characteristic
13	#11762613	C₄	1	2	yes	—
14	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×Q₃₂	C₄	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
17	Q₈×Dic₁₇	—	2	2	no	—
18	C₄⋊#3624	—	2	2	no	—
19	C₄⋊#3624	—	4	2	no	—
20	C₄×#3624	—	4	2	no	—
21	C₄×#3624	D₄	1	1	yes	characteristic
22	C₄×#3624	D₄	1	1	yes	characteristic
23	C₂×#63216	—	2	1	no	—
24	C₁₇×Q₃₂	C₈	1	4	yes	complemented
25	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
26	#10587506	—	2	2	no	—
27	#10587489	C₂×C₄	1	2	yes	—
28	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
29	C₂×#63214	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₁₇	—	2	2	no	—
31	C₂×#3624	—	2	1	no	—
32	C₂×#3624	—	4	2	no	—
33	C₂×#3624	D₈	1	1	yes	characteristic
34	C₂×#3624	D₈	1	1	yes	characteristic
35	Q₁₆×C₁₇	—	2	2	no	—
36	Q₁₆×C₁₇	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊Dic₁₇	—	2	2	no	—
39	C₄⋊Dic₁₇	—	4	2	no	—
40	C₄×Dic₁₇	—	4	2	no	—
41	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×#3622	#41	1	1	yes	characteristic
43	#63216	—	2	2	no	—
44	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
45	#63214	—	2	1	no	—
46	C₈×Q₃₂	—	17	1	no	maximal, Sylow
47	Q₈×C₁₇	—	2	4	no	—
48	Q₈×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	4	2	no	—
50	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₁₇	#135	1	1	yes	characteristic
54	#3624	—	2	2	no	—
55	#3624	—	2	1	no	—
56	#3624	—	8	2	no	—
57	#3624	Q₃₂	1	2	yes	direct product factor
58	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₁₇	#131	1	1	yes	characteristic
60	#3622	—	2	1	no	—
61	C₄×Q₃₂	—	17	1	no	—
62	C₈×Q₁₆	—	17	2	no	—
63	#1371	—	17	1	no	—
64	#1332	—	17	2	no	—
65	C₈×C₁₆	—	17	1	no	—
66	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
67	C₄×C₁₇	—	4	4	no	—
68	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
69	C₄×C₁₇	#443	1	1	yes	characteristic
70	Dic₁₇	—	2	1	no	—
71	Dic₁₇	—	8	2	no	—
72	Dic₁₇	C₂×Q₃₂	1	1	yes	characteristic
73	Dic₁₇	#508	1	1	yes	characteristic
74	C₂×Q₃₂	#595	1	1	yes	characteristic
75	C₈×Q₈	—	34	2	no	—
76	C₄×Q₁₆	—	17	2	no	—
77	#366	—	17	1	no	—
78	#358	—	17	2	no	—
79	C₄×C₁₆	—	17	1	no	—
80	C₄×C₁₆	—	17	1	no	—
81	#335	—	17	2	no	—
82	Q₈⋊C₈	—	34	2	no	—
83	C₈²	—	17	1	no	—
84	C₂×C₁₇	C₈×D₈	1	1	yes	characteristic
85	C₂×C₁₇	C₄×Q₃₂	1	1	yes	characteristic
86	C₂×C₁₇	#2178	1	1	yes	characteristic
87	C₂×Q₁₆	C₂×#595	1	2	yes	—
88	C₄×Q₈	—	34	2	no	—
89	Q₃₂	#3624	1	4	yes	direct product factor
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	34	1	no	—
92	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
93	#107	—	17	2	no	—
94	C₄⋊C₈	—	34	2	no	—
95	C₄⋊C₈	—	68	2	no	—
96	#103	—	34	2	no	—
97	C₄×C₈	—	17	1	no	—
98	C₄×C₈	—	17	1	no	—
99	C₄×C₈	—	17	1	no	—
100	C₄×C₈	—	68	2	no	—
101	C₁₇	C₈×Q₃₂	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	C₂×#3624	1	4	yes	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	17	1	no	—
107	C₂×C₈	—	17	1	no	—
108	C₂×C₈	—	34	1	no	—
109	C₂×C₈	—	68	2	no	—
110	C₂×C₈	C₂²×#595	1	1	yes	characteristic
111	C₄⋊C₄	—	34	2	no	—
112	C₄⋊C₄	—	68	2	no	—
113	C₄²	—	17	1	no	—
114	C₄²	—	68	2	no	—
115	C₁₆	—	34	2	no	—
116	C₁₆	—	34	1	no	—
117	C₁₆	C₂×#3624	1	2	yes	—
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	68	2	no	—
124	C₂×C₄	D₄×#595	1	1	yes	characteristic
125	C₈	—	17	2	no	—
126	C₈	—	34	1	no	—
127	C₈	—	34	2	no	—
128	C₈	—	34	1	no	—
129	C₈	—	136	2	no	—
130	C₈	#10587678	1	1	yes	characteristic
131	C₈	C₂²×#3624	1	1	yes	characteristic
132	C₂²	D₈×#595	1	1	yes	characteristic
133	C₄	—	4	4	no	—
134	C₄	—	17	1	no	—
135	C₄	—	17	1	no	—
136	C₄	—	34	1	no	—
137	C₄	—	136	2	no	—
138	C₄	D₄×#3624	1	1	yes	characteristic
139	C₄	#11763851	1	1	yes	characteristic
140	C₂	D₈×#3624	1	1	yes	characteristic
141	C₂	Q₃₂×#595	1	1	yes	characteristic
142	C₂	#14545185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#3624

C₂

yes

maximal

#14543815

C₂

yes

maximal, characteristic

#14543740

C₂

yes

maximal

C₁₆×#3624

C₂

yes

maximal, characteristic

Q₃₂×Dic₁₇

C₂

yes

maximal, characteristic

Q₈×#3624

—

#11763695

C₂²

yes

characteristic

#11763672

C₂²

yes

—

Q₈⋊#3624

—

C₈×#3624

C₂²

yes

characteristic

Q₁₆×Dic₁₇

C₂²

yes

—

#11762635

C₄

yes

characteristic

#11762613

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

characteristic

C₂×C₁₇×Q₃₂

C₄

yes

characteristic

C₂×Q₁₆×C₁₇

C₂×C₄

yes

—

Q₈×Dic₁₇

—

—

—

—

D₄

yes

characteristic

C₄×#3624

D₄

yes

characteristic

C₂×#63216

—

C₁₇×Q₃₂

C₈

yes

complemented

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

#10587506

—

#10587489

C₂×C₄

yes

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×#3624

—

C₂×#3624

—

C₂×#3624

D₈

yes

characteristic

C₂×#3624

D₈

yes

characteristic

Q₁₆×C₁₇

—

Q₁₆×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×D₄

yes

yes

—

C₁₆×C₁₇

C₂×C₈

yes

—

#63214

—

C₈×Q₃₂

—

maximal, Sylow

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₈

yes

C₂×Dic₁₇

yes

—

—

—

Q₃₂

yes

direct product factor

C₈×C₁₇

C₂²×C₈

yes

C₈×C₁₇

yes

—

C₄×Q₃₂

—

C₈×Q₁₆

—

#1371

—

#1332

—

C₈×C₁₆

—

C₂²×C₁₇

C₄×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

#443

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₂×Q₃₂

yes

Dic₁₇

yes

C₂×Q₃₂

yes

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

C₈×D₈

yes

characteristic

C₂×C₁₇

C₄×Q₃₂

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

—

C₄×Q₈

—

Q₃₂

#3624

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₁₇

C₈×Q₃₂

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

Q₁₆

—

104

Q₁₆

C₂×#3624

yes

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×#595

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

C₂×#3624

yes

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₄×#595

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

136

—

130

C₈

#10587678

yes

characteristic

131

C₈

C₂²×#3624

yes

characteristic

132

C₂²

D₈×#595

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

136

—

138

C₄

D₄×#3624

yes

characteristic

139

C₄

#11763851

yes

characteristic

140

C₂

D₈×#3624

yes

characteristic

141

C₂

Q₃₂×#595

yes

characteristic

142

C₂

#14545185

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#595
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂×C₁₇×Q₃₂	C₄
socle, prime radical	C₂²×C₁₇	C₄×D₈
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂×Q₃₂	#595
hypocenter	C₁₇	C₈×Q₃₂
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#595

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₂×C₁₇×Q₃₂

C₄

socle, prime radical

C₂²×C₁₇

C₄×D₈

cosocle

C₈×Dic₁₇

C₂³

hypercenter

C₂×Q₃₂

#595

hypocenter

C₁₇

C₈×Q₃₂

LCS

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
24. C₁₇×Q₃₂	C₈	semidirect
57. #3624	Q₃₂	direct (N × H)
89. Q₃₂	#3624	direct (N × H)
101. C₁₇	C₈×Q₃₂	semidirect

H ≅ G/N

type

24. C₁₇×Q₃₂

C₈

semidirect

57. #3624

Q₃₂

direct (N × H)

89. Q₃₂

#3624

direct (N × H)

101. C₁₇

C₈×Q₃₂

semidirect

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

subgroup	index
Q₁₆×#3624	2
#14543815	2
#14543740	2
C₁₆×#3624	2
Q₃₂×Dic₁₇	2
C₈×Q₃₂	17

subgroup

index

Q₁₆×#3624

#14543815

#14543740

C₁₆×#3624

Q₃₂×Dic₁₇

C₈×Q₃₂

Subgroups — Q₃₂×#3624 #17960203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q32×#3624 #17960203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₃₂×#3624 #17960203