Group #17960390 — subgroups

Subgroups — #17960390

Order 4352.

206 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14544424	C₂	1	2	yes	maximal
2	#14543817	C₂	1	1	yes	maximal, characteristic
3	#14543740	C₂	1	2	yes	maximal
4	#14543695	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763860	—	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	#10587544	—	4	2	no	—
19	#10587542	—	4	2	no	—
20	C₄⋊#3624	—	2	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	—	2	2	no	—
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	2	yes	—
34	Q₁₆×C₁₇	—	2	2	no	—
35	Q₁₆×C₁₇	—	2	2	no	—
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	2	2	no	—
38	C₄⋊Dic₁₇	—	4	2	no	—
39	C₄×Dic₁₇	—	4	2	no	—
40	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×#3622	#41	1	1	yes	characteristic
42	#63216	—	2	2	no	—
43	C₁₆×C₁₇	M₁₆	1	2	yes	—
44	#63214	—	2	1	no	—
45	#12085	—	17	1	no	maximal, Sylow
46	Q₈×C₁₇	—	4	2	no	—
47	Q₈×C₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₁₇	—	4	2	no	—
51	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
52	C₂×Dic₁₇	#135	1	1	yes	characteristic
53	#3624	—	2	2	no	—
54	#3624	—	4	1	no	—
55	#3624	—	8	2	no	—
56	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
57	C₈×C₁₇	#131	1	1	yes	characteristic
58	#3622	—	2	1	no	—
59	C₄×Q₃₂	—	17	1	no	—
60	#1591	—	17	2	no	—
61	#1372	—	17	1	no	—
62	#1332	—	17	2	no	—
63	#1311	—	17	1	no	—
64	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
65	C₄×C₁₇	—	8	2	no	—
66	C₄×C₁₇	#436	1	1	yes	characteristic
67	C₄×C₁₇	#443	1	1	yes	characteristic
68	Dic₁₇	—	2	1	no	—
69	Dic₁₇	—	8	2	no	—
70	Dic₁₇	#510	1	2	yes	—
71	C₂×Q₃₂	#595	1	1	yes	characteristic
72	#446	—	34	2	no	—
73	C₄×Q₁₆	—	17	2	no	—
74	#366	—	17	1	no	—
75	#358	—	17	2	no	—
76	C₄×C₁₆	—	17	1	no	—
77	C₄×C₁₆	—	17	1	no	—
78	#335	—	17	2	no	—
79	Q₈⋊C₈	—	34	2	no	—
80	C₈²	—	17	1	no	—
81	C₂×C₁₇	#1589	1	1	yes	characteristic
82	C₂×C₁₇	#2176	1	1	yes	characteristic
83	C₂×C₁₇	#2179	1	1	yes	characteristic
84	C₂×Q₁₆	C₂×#595	1	2	yes	—
85	C₄×Q₈	—	34	2	no	—
86	Q₃₂	—	2	2	no	—
87	C₂×C₁₆	—	17	1	no	—
88	C₂×C₁₆	—	34	1	no	—
89	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
90	#107	—	17	2	no	—
91	C₄⋊C₈	—	34	2	no	—
92	C₄⋊C₈	—	68	2	no	—
93	#103	—	34	2	no	—
94	#97	—	68	2	no	—
95	C₄×C₈	—	17	1	no	—
96	C₄×C₈	—	17	1	no	—
97	C₄×C₈	—	17	1	no	—
98	C₁₇	#12085	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	2	2	no	—
102	C₂×C₈	—	17	2	no	—
103	C₂×C₈	—	17	1	no	—
104	C₂×C₈	—	34	1	no	—
105	C₂×C₈	—	68	2	no	—
106	C₂×C₈	C₂²×#595	1	1	yes	characteristic
107	C₄⋊C₄	—	34	2	no	—
108	C₄⋊C₄	—	68	2	no	—
109	C₄²	—	17	1	no	—
110	C₄²	—	68	2	no	—
111	C₁₆	—	34	2	no	—
112	C₁₆	—	34	1	no	—
113	C₁₆	#63247	1	2	yes	—
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	17	1	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	D₄×#595	1	1	yes	characteristic
121	C₈	—	34	2	no	—
122	C₈	—	34	1	no	—
123	C₈	—	68	1	no	—
124	C₈	—	136	2	no	—
125	C₈	#10587678	1	1	yes	characteristic
126	C₈	C₂×#63247	1	1	yes	characteristic
127	C₂²	D₈×#595	1	1	yes	characteristic
128	C₄	—	8	2	no	—
129	C₄	—	17	2	no	—
130	C₄	—	34	1	no	—
131	C₄	—	136	2	no	—
132	C₄	#11763844	1	1	yes	characteristic
133	C₄	#11763851	1	1	yes	characteristic
134	C₂	#14544417	1	1	yes	characteristic
135	C₂	#14545181	1	1	yes	characteristic
136	C₂	#14545187	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544424

C₂

yes

maximal

#14543817

C₂

yes

maximal, characteristic

#14543740

C₂

yes

maximal

#14543695

C₂

yes

maximal, characteristic

Q₃₂×Dic₁₇

C₂

yes

maximal, characteristic

#11763860

—

#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

—

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

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×D₄

yes

yes

—

C₁₆×C₁₇

M₁₆

yes

—

#63214

—

#12085

—

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

—

—

—

C₈×C₁₇

C₂×M₁₆

yes

C₈×C₁₇

yes

—

C₄×Q₃₂

—

—

—

—

—

C₂²×C₁₇

C₄×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#510

yes

—

C₂×Q₃₂

#595

yes

characteristic

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#595

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12085

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂²×#595

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#63247

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×#595

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

136

—

125

C₈

#10587678

yes

characteristic

126

C₈

C₂×#63247

yes

characteristic

127

C₂²

D₈×#595

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

136

—

132

C₄

#11763844

yes

characteristic

133

C₄

#11763851

yes

characteristic

134

C₂

#14544417

yes

characteristic

135

C₂

#14545181

yes

characteristic

136

C₂

#14545187

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#595
commutator	C₂×C₈×C₁₇	C₂²×C₄
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	#12085
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#595

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₂×C₁₇×Q₃₂

C₄

socle, prime radical

C₂²×C₁₇

C₄×D₈

cosocle

C₈×Dic₁₇

C₂³

C₂×Q₃₂

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
98. C₁₇	#12085	semidirect

H ≅ G/N

type

98. C₁₇

#12085

semidirect

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

subgroup	index
#14544424	2
#14543817	2
#14543740	2
#14543695	2
Q₃₂×Dic₁₇	2
#12085	17

subgroup

index

Q₃₂×Dic₁₇

Subgroups — #17960390

Conjugacy classes of subgroups

Complements

Maximal subgroups