Group #17960218 — subgroups

Subgroups — #17960218

Order 4352.

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

normal subgroups only

This group has 216 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	C₁₆×#3624	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	1	yes	characteristic
46	#63216	D₈	1	1	yes	characteristic
47	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
49	#63214	—	2	1	no	—
50	#12046	—	17	1	no	maximal, Sylow
51	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
52	C₂²⋊D₁₇	—	4	2	no	—
53	C₂×Dic₁₇	—	2	1	no	—
54	D₆₈	—	2	2	no	—
55	C₄×D₁₇	—	4	2	no	—
56	Dic₃₄	—	2	2	no	—
57	#3624	—	2	4	no	—
58	C₈×C₁₇	—	4	2	no	—
59	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₁₇	#131	1	1	yes	characteristic
61	#3622	C₂×D₈	1	1	yes	characteristic
62	#3622	#135	1	1	yes	characteristic
63	#2178	—	17	1	no	—
64	#2170	—	17	2	no	—
65	#1373	—	17	1	no	—
66	#1335	—	17	2	no	—
67	C₈×C₁₆	—	17	1	no	—
68	C₂²×C₁₇	#443	1	1	yes	characteristic
69	C₂×D₁₇	—	4	2	no	—
70	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
71	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
72	Dic₁₇	—	2	2	no	—
73	Dic₁₇	—	4	2	no	—
74	#508	—	17	1	no	—
75	#504	—	34	2	no	—
76	#443	—	17	2	no	—
77	#368	#595	1	1	yes	characteristic
78	#364	—	17	2	no	—
79	#359	—	17	2	no	—
80	#350	—	34	2	no	—
81	C₄×C₁₆	—	17	1	no	—
82	C₄×C₁₆	—	17	1	no	—
83	C₈²	—	17	1	no	—
84	C₂×C₁₇	—	2	1	no	—
85	C₂×C₁₇	C₈×D₈	1	1	yes	characteristic
86	D₁₇	—	8	2	no	—
87	#135	—	17	2	no	—
88	#131	—	34	2	no	—
89	Q₃₂	—	17	1	no	—
90	SD₃₂	—	17	2	no	—
91	D₁₆	—	17	1	no	—
92	M₃₂	—	34	2	no	—
93	M₃₂	—	68	2	no	—
94	C₂×C₁₆	—	17	1	no	—
95	C₂×C₁₆	—	17	1	no	—
96	C₂×C₁₆	—	17	1	no	—
97	C₂×C₁₆	—	68	2	no	—
98	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
99	#108	C₂×#595	1	2	yes	—
100	#104	—	34	2	no	—
101	C₄×C₈	—	17	1	no	—
102	C₄×C₈	—	34	1	no	—
103	C₁₇	#12046	1	1	yes	characteristic, complemented, Sylow
104	#41	—	34	2	no	—
105	Q₁₆	—	17	2	no	—
106	SD₁₆	—	34	2	no	—
107	D₈	—	17	2	no	—
108	M₁₆	—	2	2	no	—
109	M₁₆	—	68	2	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	34	2	no	—
112	C₂×C₈	—	68	2	no	—
113	C₂×C₈	C₂²×#595	1	1	yes	characteristic
114	C₄²	—	17	1	no	—
115	C₁₆	—	17	1	no	—
116	C₁₆	—	17	1	no	—
117	C₁₆	—	34	1	no	—
118	C₁₆	—	34	1	no	—
119	C₁₆	—	68	2	no	—
120	C₁₆	#63242	1	1	yes	characteristic
121	C₁₆	#63242	1	1	yes	characteristic
122	Q₈	—	34	2	no	—
123	D₄	—	34	2	no	—
124	D₄	—	68	2	no	—
125	C₂×C₄	—	34	1	no	—
126	C₂×C₄	—	68	2	no	—
127	C₂×C₄	D₄×#595	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	17	1	no	—
130	C₈	—	17	1	no	—
131	C₈	—	34	4	no	—
132	C₈	C₂×#63242	1	1	yes	characteristic
133	C₈	#10587680	1	1	yes	characteristic
134	C₂²	—	68	2	no	—
135	C₂²	#11763853	1	1	yes	characteristic
136	C₄	—	34	2	no	—
137	C₄	—	68	2	no	—
138	C₄	#11763837	1	1	yes	characteristic
139	C₄	D₈×#595	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	136	2	no	—
142	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

C₁₆×#3624

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

characteristic

#63216

D₈

yes

characteristic

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

—

—

—

—

C₈×C₁₆

—

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

—

100

#104

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₁₇

#12046

yes

characteristic, complemented, Sylow

104

#41

—

105

Q₁₆

—

106

SD₁₆

—

107

D₈

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×#595

yes

characteristic

114

C₄²

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

#63242

yes

characteristic

121

C₁₆

#63242

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

D₄×#595

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

C₂×#63242

yes

characteristic

133

C₈

#10587680

yes

characteristic

134

C₂²

—

135

C₂²

#11763853

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#11763837

yes

characteristic

139

C₄

D₈×#595

yes

characteristic

140

C₂

—

141

C₂

—

136

—

142

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	#12046
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. C₁₆×#3624	C₂	semidirect
29. Dic₁₃₆	C₈	semidirect
30. #10587471	C₈	semidirect
31. D₂₇₂	C₈	semidirect
103. C₁₇	#12046	semidirect

H ≅ G/N

type

1. #14545173

C₂

semidirect

2. #14543819

C₂

semidirect

3. #14543747

C₂

semidirect

4. C₁₆×#3624

C₂

semidirect

29. Dic₁₃₆

C₈

semidirect

30. #10587471

C₈

semidirect

31. D₂₇₂

C₈

semidirect

103. C₁₇

#12046

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
C₁₆×#3624	2
#14530664	2
#12046	17

subgroup

index

C₁₆×#3624

Subgroups — #17960218

Conjugacy classes of subgroups

Complements

Maximal subgroups