Group #17960221 — subgroups

Subgroups — #17960221

Order 4352.

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

normal subgroups only

This group has 212 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	#14545173	C₂	1	2	yes	maximal, complemented
2	#14543819	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543747	C₂	1	2	yes	maximal, complemented
4	#14543695	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530664	C₂	1	1	yes	maximal, characteristic
6	#11763868	—	2	2	no	—
7	#11763731	C₂²	1	2	yes	—
8	#11763708	—	2	2	no	—
9	#11763695	C₂²	1	1	yes	characteristic
10	C₈×#3624	C₂²	1	1	yes	characteristic
11	#11763241	C₄	1	1	yes	characteristic
12	#11762925	C₂²	1	2	yes	—
13	#11762615	C₄	1	2	yes	—
14	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#368	C₄	1	1	yes	characteristic
16	#10587593	—	2	2	no	—
17	#10587590	C₂×C₄	1	2	yes	—
18	C₄×#3624	—	2	1	no	—
19	#10587538	—	2	2	no	—
20	C₂×#63216	D₄	1	1	yes	characteristic
21	C₂×#63216	D₄	1	1	yes	characteristic
22	#10587530	—	4	2	no	—
23	#10587529	—	4	2	no	—
24	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₁₇×#108	C₂×C₄	1	2	yes	—
26	#10587496	—	2	2	no	—
27	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
28	C₂×#63214	C₂×C₄	1	1	yes	characteristic
29	Dic₁₃₆	C₈	1	1	yes	characteristic, complemented
30	#10587471	C₈	1	2	yes	complemented
31	D₂₇₂	C₈	1	1	yes	characteristic, complemented
32	#63252	—	2	2	no	—
33	C₂×#3624	—	2	2	no	—
34	M₁₆×C₁₇	—	2	2	no	—
35	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₁₇	#41	1	1	yes	characteristic
37	C₂×#3622	C₂×D₄	1	1	yes	characteristic
38	Dic₆₈	C₂×C₈	1	2	yes	—
39	D₁₃₆	C₂×C₈	1	2	yes	—
40	#63219	—	2	2	no	—
41	#63218	—	4	2	no	—
42	C₈×D₁₇	—	4	2	no	—
43	#63216	—	2	1	no	—
44	#63216	—	4	2	no	—
45	#63216	D₈	1	2	yes	—
46	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
47	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
48	#63214	—	2	1	no	—
49	#12047	—	17	1	no	maximal, Sylow
50	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₁₇	—	4	2	no	—
52	C₂×Dic₁₇	—	2	1	no	—
53	D₆₈	—	2	2	no	—
54	C₄×D₁₇	—	4	2	no	—
55	Dic₃₄	—	2	2	no	—
56	#3624	—	4	2	no	—
57	C₈×C₁₇	—	4	2	no	—
58	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₁₇	#131	1	1	yes	characteristic
60	#3622	C₂×D₈	1	1	yes	characteristic
61	#3622	#135	1	1	yes	characteristic
62	#2178	—	17	1	no	—
63	#2170	—	17	2	no	—
64	#1373	—	17	1	no	—
65	#1335	—	17	2	no	—
66	#1311	—	17	1	no	—
67	C₂²×C₁₇	#443	1	1	yes	characteristic
68	C₂×D₁₇	—	4	2	no	—
69	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
70	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
71	Dic₁₇	—	2	2	no	—
72	Dic₁₇	—	4	2	no	—
73	#508	—	17	1	no	—
74	#504	—	34	2	no	—
75	#443	—	17	2	no	—
76	#368	#595	1	1	yes	characteristic
77	#364	—	17	2	no	—
78	#359	—	17	2	no	—
79	#350	—	34	2	no	—
80	C₄×C₁₆	—	17	1	no	—
81	C₄×C₁₆	—	17	1	no	—
82	C₈²	—	17	1	no	—
83	C₂×C₁₇	—	2	1	no	—
84	C₂×C₁₇	C₈×D₈	1	1	yes	characteristic
85	D₁₇	—	8	2	no	—
86	#135	—	17	2	no	—
87	#131	—	34	2	no	—
88	Q₃₂	—	17	1	no	—
89	SD₃₂	—	17	2	no	—
90	D₁₆	—	17	1	no	—
91	M₃₂	—	34	2	no	—
92	M₃₂	—	68	2	no	—
93	C₂×C₁₆	—	17	1	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	68	2	no	—
97	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
98	#108	C₂×#595	1	2	yes	—
99	#104	—	34	2	no	—
100	C₄×C₈	—	17	1	no	—
101	C₄×C₈	—	34	1	no	—
102	C₁₇	#12047	1	1	yes	characteristic, complemented, Sylow
103	#41	—	34	2	no	—
104	Q₁₆	—	17	2	no	—
105	SD₁₆	—	34	2	no	—
106	D₈	—	17	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	68	2	no	—
109	C₂×C₈	—	17	1	no	—
110	C₂×C₈	—	34	2	no	—
111	C₂×C₈	—	68	2	no	—
112	C₂×C₈	C₂²×#595	1	1	yes	characteristic
113	C₄²	—	17	1	no	—
114	C₁₆	—	17	2	no	—
115	C₁₆	—	34	1	no	—
116	C₁₆	—	34	1	no	—
117	C₁₆	—	68	2	no	—
118	C₁₆	#63242	1	1	yes	characteristic
119	C₁₆	#63242	1	1	yes	characteristic
120	Q₈	—	34	2	no	—
121	D₄	—	34	2	no	—
122	D₄	—	68	2	no	—
123	C₂×C₄	—	34	1	no	—
124	C₂×C₄	—	68	2	no	—
125	C₂×C₄	D₄×#595	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	17	1	no	—
128	C₈	—	17	1	no	—
129	C₈	—	68	2	no	—
130	C₈	C₂×#63242	1	1	yes	characteristic
131	C₈	#10587680	1	1	yes	characteristic
132	C₂²	—	68	2	no	—
133	C₂²	#11763853	1	1	yes	characteristic
134	C₄	—	34	2	no	—
135	C₄	—	68	2	no	—
136	C₄	#11763837	1	1	yes	characteristic
137	C₄	D₈×#595	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	136	2	no	—
140	C₂	#14544365	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545173

C₂

yes

maximal, complemented

#14543819

C₂

yes

maximal, characteristic, complemented

#14543747

C₂

yes

maximal, complemented

#14543695

C₂

yes

maximal, characteristic, complemented

#14530664

C₂

yes

maximal, characteristic

#11763868

—

#11763731

C₂²

yes

—

#11763708

—

#11763695

C₂²

yes

characteristic

C₈×#3624

C₂²

yes

characteristic

#11763241

C₄

yes

characteristic

#11762925

C₂²

yes

—

#11762615

C₄

yes

—

C₁₆×Dic₁₇

C₂²

yes

characteristic

C₁₇×#368

C₄

yes

characteristic

#10587593

—

#10587590

C₂×C₄

yes

—

C₄×#3624

—

#10587538

—

C₂×#63216

D₄

yes

characteristic

C₂×#63216

D₄

yes

characteristic

#10587530

—

#10587529

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#108

C₂×C₄

yes

—

#10587496

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

Dic₁₃₆

C₈

yes

characteristic, complemented

#10587471

C₈

yes

complemented

D₂₇₂

C₈

yes

characteristic, complemented

#63252

—

C₂×#3624

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

C₄×Dic₁₇

yes

C₂×D₄

yes

characteristic

Dic₆₈

C₂×C₈

yes

—

D₁₃₆

C₂×C₈

yes

—

#63219

—

#63218

—

C₈×D₁₇

—

#63216

—

#63216

—

#63216

D₈

yes

—

C₁₆×C₁₇

C₂×C₈

yes

characteristic

C₁₆×C₁₇

C₂×C₈

yes

—

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

C₂²×C₈

yes

C₈×C₁₇

yes

C₂×D₈

yes

yes

—

—

—

—

—

C₂²×C₁₇

#443

yes

characteristic

C₂×D₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

C₄×D₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

yes

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

C₈×D₈

yes

characteristic

D₁₇

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

#104

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₁₇

#12047

yes

characteristic, complemented, Sylow

103

#41

—

104

Q₁₆

—

105

SD₁₆

—

106

D₈

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×#595

yes

characteristic

113

C₄²

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

#63242

yes

characteristic

119

C₁₆

#63242

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

D₄×#595

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

C₂×#63242

yes

characteristic

131

C₈

#10587680

yes

characteristic

132

C₂²

—

133

C₂²

#11763853

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

#11763837

yes

characteristic

137

C₄

D₈×#595

yes

characteristic

138

C₂

—

139

C₂

—

136

—

140

C₂

#14544365

yes

characteristic

	subgroup	G/H
center	C₄	D₈×#595
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#368	C₄
socle	C₂×C₁₇	C₈×D₈
prime radical	#11763241	C₄
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#368	#595
hypocenter	C₁₇	#12047
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×#595

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#368

C₄

socle

C₂×C₁₇

C₈×D₈

prime radical

#11763241

C₄

cosocle

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #14545173	C₂	semidirect
2. #14543819	C₂	semidirect
3. #14543747	C₂	semidirect
4. #14543695	C₂	semidirect
29. Dic₁₃₆	C₈	semidirect
30. #10587471	C₈	semidirect
31. D₂₇₂	C₈	semidirect
102. C₁₇	#12047	semidirect

H ≅ G/N

type

1. #14545173

C₂

semidirect

2. #14543819

C₂

semidirect

3. #14543747

C₂

semidirect

4. #14543695

C₂

semidirect

29. Dic₁₃₆

C₈

semidirect

30. #10587471

C₈

semidirect

31. D₂₇₂

C₈

semidirect

102. C₁₇

#12047

semidirect

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

subgroup	index
#14545173	2
#14543819	2
#14543747	2
#14543695	2
#14530664	2
#12047	17

subgroup

index

Subgroups — #17960221

Conjugacy classes of subgroups

Complements

Maximal subgroups