Group #17960302 — subgroups

Subgroups — #17960302

Order 4352.

246 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 70 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	#14544382	C₂	1	2	yes	maximal, complemented
2	#14543809	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543743	C₂	1	2	yes	maximal, complemented
4	#14543700	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530659	C₂	1	1	yes	maximal, characteristic
6	#11763837	—	2	2	no	—
7	#11763698	C₂²	1	1	yes	characteristic
8	#11763670	C₂²	1	2	yes	—
9	#11763657	—	2	2	no	—
10	#11763650	C₂²	1	1	yes	characteristic
11	C₂×D₂₇₂	C₄	1	1	yes	characteristic
12	#11762891	C₂²	1	2	yes	—
13	#11762609	C₄	1	2	yes	—
14	#11762584	C₂²	1	1	yes	characteristic
15	C₁₇×#367	C₄	1	1	yes	characteristic
16	C₂×#63242	—	4	2	no	—
17	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
18	#10587579	—	2	2	no	—
19	#10587543	—	4	2	no	—
20	C₄⋊#3624	—	2	2	no	—
21	C₄×#3624	D₄	1	2	yes	—
22	C₂×#63216	—	2	1	no	—
23	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₁₇×#107	C₂×C₄	1	2	yes	—
25	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂×#63214	C₂×C₄	1	1	yes	characteristic
27	#10587480	—	2	2	no	—
28	D₂₇₂	C₈	1	4	yes	complemented
29	C₂×D₆₈	—	2	2	no	—
30	C₂×C₄×D₁₇	—	4	2	no	—
31	C₂×#3624	—	2	2	no	—
32	C₂×#3624	—	4	2	no	—
33	#63242	—	8	4	no	—
34	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₇	—	2	2	no	—
36	#63227	—	4	2	no	—
37	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×#3622	#41	1	1	yes	characteristic
39	D₁₃₆	—	2	2	no	—
40	D₁₃₆	C₂×C₈	1	4	yes	complemented
41	#63216	—	4	1	no	—
42	C₁₆×C₁₇	C₂×C₈	1	2	yes	complemented
43	#63214	—	2	1	no	—
44	#12067	—	17	1	no	maximal, Sylow
45	C₂²×D₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
48	C₂×Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
49	D₆₈	—	2	4	no	—
50	D₆₈	—	4	2	no	—
51	C₄×D₁₇	—	8	2	no	—
52	C₄×D₁₇	—	8	2	no	—
53	#3624	—	4	2	no	—
54	#3624	—	8	2	no	—
55	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₁₇	#131	1	1	yes	characteristic
57	#3622	—	2	1	no	—
58	#2177	—	17	1	no	—
59	#1580	—	17	2	no	—
60	#1368	—	17	1	no	—
61	#1333	—	17	2	no	—
62	#1313	—	17	1	no	—
63	C₂²×C₁₇	#442	1	1	yes	characteristic
64	C₂×D₁₇	—	4	4	no	—
65	C₂×D₁₇	—	8	2	no	—
66	C₂×D₁₇	—	8	2	no	—
67	C₄×C₁₇	—	8	2	no	—
68	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
69	C₄×C₁₇	#444	1	1	yes	characteristic
70	Dic₁₇	—	2	2	no	—
71	C₂×D₁₆	—	17	1	no	—
72	C₄×D₈	—	17	2	no	—
73	C₈×D₄	—	34	2	no	—
74	#367	#595	1	1	yes	characteristic
75	#357	—	17	2	no	—
76	#346	—	17	1	no	—
77	#346	—	17	1	no	—
78	#334	—	17	2	no	—
79	D₄⋊C₈	—	34	2	no	—
80	#322	—	17	1	no	—
81	C₂×C₁₇	#1580	1	1	yes	characteristic
82	C₂×C₁₇	AGL(1,ℤ/16)	1	2	yes	—
83	D₁₇	—	8	4	no	—
84	C₂×D₈	—	17	2	no	—
85	C₂²×C₈	—	68	2	no	—
86	C₄×D₄	—	34	2	no	—
87	D₁₆	—	17	4	no	—
88	C₂×C₁₆	—	17	1	no	—
89	C₂×C₁₆	—	34	1	no	—
90	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
91	#107	C₂×#595	1	2	yes	—
92	C₄⋊C₈	—	34	2	no	—
93	#102	—	34	2	no	—
94	C₂²⋊C₈	—	68	2	no	—
95	C₄×C₈	—	17	1	no	—
96	C₄×C₈	—	17	2	no	—
97	C₁₇	#12067	1	1	yes	characteristic, complemented, Sylow
98	C₂×D₄	—	34	2	no	—
99	C₂²×C₄	—	68	2	no	—
100	D₈	—	17	4	no	—
101	D₈	—	34	2	no	—
102	C₂×C₈	—	17	1	no	—
103	C₂×C₈	—	34	2	no	—
104	C₂×C₈	—	68	2	no	—
105	C₂×C₈	—	136	4	no	—
106	C₂×C₈	C₂²×#595	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	68	2	no	—
109	C₄²	—	17	1	no	—
110	C₁₆	—	34	1	no	—
111	C₁₆	—	68	1	no	—
112	C₁₆	#63242	1	2	yes	complemented
113	C₂³	—	68	2	no	—
114	D₄	—	34	4	no	—
115	D₄	—	68	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	17	2	no	—
118	C₂×C₄	—	136	2	no	—
119	C₂×C₄	—	136	2	no	—
120	C₂×C₄	D₄×#595	1	1	yes	characteristic
121	C₈	—	34	1	no	—
122	C₈	—	68	2	no	—
123	C₈	—	136	2	no	—
124	C₈	C₂×#63242	1	1	yes	characteristic
125	C₈	#10587680	1	1	yes	characteristic
126	C₂²	—	68	4	no	—
127	C₂²	—	136	2	no	—
128	C₂²	—	136	2	no	—
129	C₂²	#11763850	1	1	yes	characteristic
130	C₄	—	8	2	no	—
131	C₄	—	34	2	no	—
132	C₄	#11763837	1	1	yes	characteristic
133	C₄	#11763855	1	1	yes	characteristic
134	C₂	—	136	4	no	—
135	C₂	#14544382	1	1	yes	characteristic
136	C₂	#14545195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14544382

C₂

yes

maximal, complemented

#14543809

C₂

yes

maximal, characteristic, complemented

#14543743

C₂

yes

maximal, complemented

#14543700

C₂

yes

maximal, characteristic, complemented

#14530659

C₂

yes

maximal, characteristic

#11763837

—

#11763698

C₂²

yes

characteristic

#11763670

C₂²

yes

—

#11763657

—

#11763650

C₂²

yes

characteristic

C₂×D₂₇₂

C₄

yes

characteristic

#11762891

C₂²

yes

—

#11762609

C₄

yes

—

#11762584

C₂²

yes

characteristic

C₁₇×#367

C₄

yes

characteristic

C₂×#63242

—

C₂×D₁₃₆

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂×#63216

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#107

C₂×C₄

yes

—

C₈×Dic₁₇

C₂³

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

#10587480

—

D₂₇₂

C₈

yes

complemented

C₂×D₆₈

—

C₂×C₄×D₁₇

—

C₂×#3624

—

C₂×#3624

—

#63242

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₁₇

—

#63227

—

C₄×Dic₁₇

C₂×D₄

yes

yes

D₁₃₆

—

D₁₃₆

C₂×C₈

yes

complemented

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

—

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

AGL(1,ℤ/8)

yes

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

#3624

—

#3624

—

C₈×C₁₇

C₂²×C₈

yes

C₈×C₁₇

yes

—

—

—

—

—

—

C₂²×C₁₇

#442

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

#444

yes

characteristic

Dic₁₇

—

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

yes

—

—

—

—

D₄⋊C₈

—

#322

—

C₂×C₁₇

#1580

yes

characteristic

C₂×C₁₇

AGL(1,ℤ/16)

yes

—

D₁₇

—

C₂×D₈

—

C₂²×C₈

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12067

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂²×C₄

—

100

D₈

—

101

D₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

136

—

106

C₂×C₈

C₂²×#595

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

#63242

yes

complemented

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

136

—

119

C₂×C₄

—

136

—

120

C₂×C₄

D₄×#595

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

136

—

124

C₈

C₂×#63242

yes

characteristic

125

C₈

#10587680

yes

characteristic

126

C₂²

—

127

C₂²

—

136

—

128

C₂²

—

136

—

129

C₂²

#11763850

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#11763837

yes

characteristic

133

C₄

#11763855

yes

characteristic

134

C₂

—

136

—

135

C₂

#14544382

yes

characteristic

136

C₂

#14545195

yes

—

	subgroup	G/H
center	C₂²	#11763850
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#367	C₄
socle	C₂²×C₁₇	#442
prime radical	C₂×D₂₇₂	C₄
cosocle	C₈×Dic₁₇	C₂³
hypercenter	#367	#595
hypocenter	C₁₇	#12067
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#11763850

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#367

C₄

socle

C₂²×C₁₇

#442

prime radical

C₂×D₂₇₂

C₄

cosocle

C₈×Dic₁₇

C₂³

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. #14544382	C₂	semidirect
2. #14543809	C₂	semidirect
3. #14543743	C₂	semidirect
4. #14543700	C₂	semidirect
28. D₂₇₂	C₈	semidirect
40. D₁₃₆	C₂×C₈	semidirect
42. C₁₆×C₁₇	C₂×C₈	semidirect
97. C₁₇	#12067	semidirect
112. C₁₆	#63242	semidirect

H ≅ G/N

type

1. #14544382

C₂

semidirect

2. #14543809

C₂

semidirect

3. #14543743

C₂

semidirect

4. #14543700

C₂

semidirect

28. D₂₇₂

C₈

semidirect

40. D₁₃₆

C₂×C₈

semidirect

42. C₁₆×C₁₇

C₂×C₈

semidirect

97. C₁₇

#12067

semidirect

112. C₁₆

#63242

semidirect

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

subgroup	index
#14544382	2
#14543809	2
#14543743	2
#14543700	2
#14530659	2
#12067	17

subgroup

index

Subgroups — #17960302

Conjugacy classes of subgroups

Complements

Maximal subgroups