Group #17960161 — subgroups

Subgroups — #17960161

Order 4352.

180 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 68 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	#14545159	C₂	1	2	yes	maximal
2	#14543820	C₂	1	2	yes	maximal, complemented
3	#14543812	C₂	1	2	yes	maximal, complemented
4	#14530382	C₂	1	1	yes	maximal, characteristic
5	#11763731	C₂²	1	4	yes	—
6	#11763706	—	2	2	no	—
7	#11763701	—	2	2	no	—
8	C₂×#10587469	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₁₇	C₄	1	1	yes	characteristic
10	#11762763	C₂²	1	1	yes	characteristic
11	#11762584	C₂²	1	1	yes	characteristic
12	C₄×#63214	C₄	1	1	yes	characteristic
13	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#346	C₄	1	1	yes	characteristic
15	C₂×#63218	D₄	1	1	yes	characteristic
16	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
17	#10587558	Q₈	1	1	yes	characteristic
18	#10587538	—	2	4	no	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	#10587486	Q₈	1	1	yes	characteristic
23	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63214	C₂×C₄	1	1	yes	characteristic
25	C₂×#63214	C₂×C₄	1	1	yes	characteristic
26	#10587474	D₄	1	1	yes	characteristic
27	#10587469	—	2	2	no	—
28	C₁₆×D₁₇	—	2	2	no	—
29	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
33	#63227	—	2	1	no	—
34	#63225	—	2	1	no	—
35	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#3622	C₂×D₄	1	1	yes	characteristic
38	#63218	—	2	2	no	—
39	C₈×D₁₇	—	2	1	no	—
40	C₈×D₁₇	—	2	1	no	—
41	#63216	—	4	4	no	—
42	C₁₆×C₁₇	—	2	1	no	—
43	C₁₆×C₁₇	—	2	1	no	—
44	#63214	M₁₆	1	2	yes	—
45	#63214	M₁₆	1	2	yes	—
46	#12027	—	17	1	no	maximal, Sylow
47	C₂²×D₁₇	#108	1	1	yes	characteristic
48	C₂×C₄×C₁₇	—	2	1	no	—
49	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	1	no	—
51	C₂×Dic₁₇	#108	1	1	yes	characteristic
52	C₄×D₁₇	—	2	1	no	—
53	C₄×D₁₇	—	2	1	no	—
54	C₈×C₁₇	—	2	1	no	—
55	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
57	#3622	—	2	1	no	—
58	#3622	#135	1	2	yes	—
59	#2166	—	17	2	no	—
60	#2108	—	17	1	no	—
61	#1373	—	17	2	no	—
62	#1369	—	17	2	no	—
63	C₂²×C₁₇	C₂×#108	1	1	yes	characteristic
64	C₂×D₁₇	—	2	1	no	—
65	C₂×D₁₇	—	4	1	no	—
66	C₄×C₁₇	—	4	1	no	—
67	C₄×C₁₇	#423	1	1	yes	characteristic
68	C₄×C₁₇	#427	1	1	yes	characteristic
69	Dic₁₇	—	2	1	no	—
70	Dic₁₇	—	4	1	no	—
71	C₂×M₃₂	—	17	1	no	—
72	C₂²×C₁₆	—	17	1	no	—
73	#405	—	17	1	no	—
74	#364	—	17	4	no	—
75	#349	—	34	2	no	—
76	#347	—	34	2	no	—
77	#346	—	17	1	no	—
78	#346	#595	1	1	yes	characteristic
79	C₄×C₁₆	—	17	1	no	—
80	C₄×C₁₆	—	17	1	no	—
81	C₂×C₁₇	—	2	1	no	—
82	C₂×C₁₇	#1568	1	1	yes	characteristic
83	D₁₇	—	4	1	no	—
84	C₂×M₁₆	—	17	1	no	—
85	C₂²×C₈	—	17	1	no	—
86	#117	—	17	1	no	—
87	M₃₂	—	34	2	no	—
88	M₃₂	—	34	4	no	—
89	C₂×C₁₆	—	17	1	no	—
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	34	2	no	—
92	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
93	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
94	#97	—	17	1	no	—
95	#97	—	17	1	no	—
96	C₄×C₈	—	17	1	no	—
97	C₄×C₈	C₂×#595	1	1	yes	characteristic
98	C₁₇	#12027	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	—	17	1	no	—
100	M₁₆	—	34	2	no	—
101	C₂×C₈	—	17	1	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	34	1	no	—
104	C₂×C₈	—	34	1	no	—
105	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
106	C₂×C₈	C₂²×#595	1	1	yes	characteristic
107	C₄⋊C₄	—	34	1	no	—
108	C₂²⋊C₄	—	34	1	no	—
109	C₄²	—	17	1	no	—
110	C₄²	C₄⋊#595	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	17	2	no	—
114	C₁₆	—	17	2	no	—
115	C₁₆	—	68	4	no	—
116	C₂³	—	17	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	17	1	no	—
119	C₂×C₄	—	34	1	no	—
120	C₂×C₄	—	34	1	no	—
121	C₂×C₄	—	34	1	no	—
122	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	17	2	no	—
125	C₈	—	34	1	no	—
126	C₈	D₁₇⋊M₁₆	1	1	yes	characteristic
127	C₈	D₁₇⋊M₁₆	1	1	yes	characteristic
128	C₂²	—	34	1	no	—
129	C₂²	—	68	1	no	—
130	C₂²	D₁₇⋊#108	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	34	1	no	—
133	C₄	—	68	1	no	—
134	C₄	#11763814	1	1	yes	characteristic
135	C₄	#11763819	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	68	1	no	—
138	C₂	#14544347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545159

C₂

yes

maximal

#14543820

C₂

yes

maximal, complemented

#14543812

C₂

yes

maximal, complemented

#14530382

C₂

yes

maximal, characteristic

#11763731

C₂²

yes

—

#11763706

—

#11763701

—

C₂×#10587469

C₄

yes

characteristic

C₂×C₁₆×D₁₇

C₄

yes

characteristic

#11762763

C₂²

yes

characteristic

#11762584

C₂²

yes

characteristic

C₄×#63214

C₄

yes

characteristic

C₁₆×Dic₁₇

C₂²

yes

characteristic

C₁₇×#346

C₄

yes

characteristic

C₂×#63218

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

#10587558

Q₈

yes

characteristic

#10587538

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587486

Q₈

yes

characteristic

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

#10587474

D₄

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

#63227

—

#63225

—

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×D₄

yes

characteristic

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

#63216

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

#63214

M₁₆

yes

—

#63214

M₁₆

yes

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#108

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×M₁₆

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

—

yes

—

—

—

—

—

C₂²×C₁₇

C₂×#108

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1568

yes

characteristic

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#12027

yes

characteristic, complemented, Sylow

C₂²×C₄

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₄⋊#595

yes

characteristic

106

C₂×C₈

C₂²×#595

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

C₄⋊#595

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

D₁₇⋊M₁₆

yes

characteristic

127

C₈

D₁₇⋊M₁₆

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

D₁₇⋊#108

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#11763814

yes

characteristic

135

C₄

#11763819

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#14544347

yes

characteristic

	subgroup	G/H
center	C₄	#11763819
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂×C₁₇	#1568
prime radical	C₂²×D₁₇	#108
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12027
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

#11763819

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#346

C₄

C₂×C₁₇

C₂²×D₁₇

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #14543820	C₂	semidirect
3. #14543812	C₂	semidirect
98. C₁₇	#12027	semidirect

H ≅ G/N

type

2. #14543820

C₂

semidirect

3. #14543812

C₂

semidirect

98. C₁₇

#12027

semidirect

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

subgroup	index
#14545159	2
#14543820	2
#14543812	2
#14530382	2
#12027	17

subgroup

index

Subgroups — #17960161

Conjugacy classes of subgroups

Complements

Maximal subgroups