Group #16946348 — subgroups

Subgroups — #16946348

Order 4352.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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	#14530661	C₂	1	1	yes	maximal, characteristic
2	#14527697	C₂	1	2	yes	maximal, complemented
3	#14526750	C₂	1	2	yes	maximal, complemented
4	#14526704	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₇₂	C₄	1	1	yes	characteristic
7	C₄×D₁₃₆	C₂²	1	2	yes	—
8	#11762886	—	2	2	no	—
9	#11762637	C₄	1	1	yes	characteristic
10	#11762611	C₄	1	2	yes	—
11	#11762585	C₂²	1	1	yes	characteristic
12	#11762536	—	2	2	no	—
13	#11762530	C₂²	1	1	yes	characteristic
14	C₁₇×#346	C₂²	1	1	yes	characteristic
15	C₁₇×#334	C₂²	1	2	yes	—
16	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
17	C₂×#63218	—	4	2	no	—
18	C₄×D₆₈	—	2	2	no	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₁₇×C₄⋊C₈	—	2	2	no	—
22	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	#10587492	—	2	2	no	—
24	#10587491	—	4	2	no	—
25	#10587489	C₂×C₄	1	2	yes	—
26	C₂×#63214	—	2	1	no	—
27	C₄×#3622	D₄	1	2	yes	—
28	D₂₇₂	—	2	2	no	—
29	C₂×D₆₈	—	2	2	no	—
30	C₂×C₄×D₁₇	—	4	2	no	—
31	C₂×C₈×C₁₇	—	4	2	no	—
32	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
34	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
35	#63227	—	4	2	no	—
36	C₄⋊Dic₁₇	—	2	2	no	—
37	C₂×#3622	—	2	2	no	—
38	D₁₃₆	—	2	2	no	—
39	D₁₃₆	—	2	2	no	—
40	#63218	—	8	4	no	—
41	C₁₆×C₁₇	—	2	1	no	—
42	C₁₆×C₁₇	M₁₆	1	2	yes	complemented
43	#63214	—	4	1	no	—
44	#12089	—	17	1	no	maximal, Sylow
45	C₂²×D₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
48	C₂×Dic₁₇	—	4	2	no	—
49	D₆₈	—	4	2	no	—
50	D₆₈	—	4	2	no	—
51	C₄×D₁₇	—	8	2	no	—
52	C₄×D₁₇	—	8	2	no	—
53	C₈×C₁₇	—	2	1	no	—
54	C₈×C₁₇	—	8	2	no	—
55	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₁₇	#131	1	1	yes	characteristic
57	#3622	—	4	2	no	—
58	#2177	—	17	1	no	—
59	#1594	—	17	2	no	—
60	#1368	D₁₇	1	1	yes	characteristic, complemented
61	#1333	—	17	2	no	—
62	#1313	—	17	1	no	—
63	C₂²×C₁₇	#442	1	1	yes	characteristic
64	C₂×D₁₇	—	8	2	no	—
65	C₂×D₁₇	—	8	2	no	—
66	C₂×D₁₇	—	8	2	no	—
67	C₄×C₁₇	—	2	2	no	—
68	C₄×C₁₇	#436	1	1	yes	characteristic
69	C₄×C₁₇	#444	1	1	yes	characteristic
70	Dic₁₇	—	8	2	no	—
71	C₂×D₁₆	—	17	1	no	—
72	C₄×D₈	—	17	2	no	—
73	#436	—	34	2	no	—
74	#367	—	17	1	no	—
75	#357	—	17	2	no	—
76	#346	—	17	1	no	—
77	#346	C₂×D₁₇	1	1	yes	characteristic
78	#334	C₂×D₁₇	1	2	yes	—
79	D₄⋊C₈	—	34	2	no	—
80	#322	—	17	1	no	—
81	C₂×C₁₇	#1594	1	1	yes	characteristic
82	C₂×C₁₇	AGL(1,ℤ/16)	1	2	yes	—
83	D₁₇	—	16	2	no	—
84	C₂×D₈	—	17	2	no	—
85	C₂×M₁₆	—	68	2	no	—
86	C₄×D₄	—	34	2	no	—
87	D₁₆	—	34	2	no	—
88	C₂×C₁₆	—	34	1	no	—
89	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
91	#107	—	17	2	no	—
92	C₄⋊C₈	—	2	2	no	—
93	#102	—	34	2	no	—
94	C₂²⋊C₈	—	68	2	no	—
95	C₄×C₈	—	17	2	no	—
96	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
97	C₁₇	#12089	1	1	yes	characteristic, complemented, Sylow
98	C₂×D₄	—	34	2	no	—
99	C₂²×C₄	—	68	2	no	—
100	D₈	—	34	2	no	—
101	D₈	—	34	2	no	—
102	M₁₆	—	136	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	34	2	no	—
105	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
106	C₂×C₈	#63256	1	1	yes	characteristic
107	C₄⋊C₄	—	34	2	no	—
108	C₂²⋊C₄	—	68	2	no	—
109	C₄²	D₄×D₁₇	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	68	1	no	—
112	C₁₆	#63218	1	2	yes	complemented
113	C₂³	—	68	2	no	—
114	D₄	—	68	2	no	—
115	D₄	—	68	2	no	—
116	C₂×C₄	—	68	2	no	—
117	C₂×C₄	—	136	2	no	—
118	C₂×C₄	—	136	2	no	—
119	C₂×C₄	#10587579	1	1	yes	characteristic
120	C₂×C₄	#10587600	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	68	2	no	—
124	C₈	C₂×#63218	1	1	yes	characteristic
125	C₈	#10587593	1	1	yes	characteristic
126	C₂²	—	136	2	no	—
127	C₂²	—	136	2	no	—
128	C₂²	—	136	2	no	—
129	C₂²	#11762920	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	136	2	no	—
132	C₄	#11762886	1	1	yes	characteristic
133	C₄	#11762930	1	1	yes	characteristic
134	C₂	—	272	2	no	—
135	C₂	#14527697	1	1	yes	characteristic
136	C₂	#14530681	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14530661

C₂

yes

maximal, characteristic

#14527697

C₂

yes

maximal, complemented

#14526750

C₂

yes

maximal, complemented

#14526704

C₂

yes

maximal, characteristic, complemented

C₁₇×#1368

C₂

yes

maximal, characteristic, complemented

C₂×D₂₇₂

C₄

yes

characteristic

C₄×D₁₃₆

C₂²

yes

—

#11762886

—

#11762637

C₄

yes

characteristic

#11762611

C₄

yes

—

#11762585

C₂²

yes

characteristic

#11762536

—

#11762530

C₂²

yes

characteristic

C₁₇×#346

C₂²

yes

characteristic

C₁₇×#334

C₂²

yes

—

C₂×D₁₃₆

C₂×C₄

yes

—

C₂×#63218

—

C₄×D₆₈

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂³

yes

—

—

C₂×C₄

yes

—

C₂×#63214

—

C₄×#3622

D₄

yes

—

D₂₇₂

—

C₂×D₆₈

—

C₂×C₄×D₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

#63227

—

C₄⋊Dic₁₇

—

C₂×#3622

—

D₁₃₆

—

D₁₃₆

—

#63218

—

C₁₆×C₁₇

—

C₁₆×C₁₇

M₁₆

yes

—

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×M₁₆

yes

C₈×C₁₇

yes

—

—

—

D₁₇

yes

characteristic, complemented

#1333

—

#1313

—

C₂²×C₁₇

#442

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×D₁₆

—

C₄×D₈

—

—

—

—

—

C₂×D₁₇

yes

characteristic

#334

C₂×D₁₇

yes

—

D₄⋊C₈

—

#322

—

C₂×C₁₇

#1594

yes

characteristic

C₂×C₁₇

AGL(1,ℤ/16)

yes

—

D₁₇

—

C₂×D₈

—

C₂×M₁₆

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₇

yes

characteristic

C₂×C₁₆

C₄×D₁₇

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12089

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂²×C₄

—

100

D₈

—

101

D₈

—

102

M₁₆

—

136

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

106

C₂×C₈

#63256

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

D₄×D₁₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

#63218

yes

complemented

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

136

—

118

C₂×C₄

—

136

—

119

C₂×C₄

#10587579

yes

characteristic

120

C₂×C₄

#10587600

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#63218

yes

characteristic

125

C₈

#10587593

yes

characteristic

126

C₂²

—

136

—

127

C₂²

—

136

—

128

C₂²

—

136

—

129

C₂²

#11762920

yes

characteristic

130

C₄

—

131

C₄

—

136

—

132

C₄

#11762886

yes

characteristic

133

C₄

#11762930

yes

characteristic

134

C₂

—

272

—

135

C₂

#14527697

yes

characteristic

136

C₂

#14530681

yes

—

	subgroup	G/H
center	C₂²	#11762920
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1368	C₂
socle	C₂²×C₁₇	#442
prime radical	C₂×D₂₇₂	C₄
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1368	D₁₇
hypocenter	C₁₇	#12089
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#11762920

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1368

C₂

socle

C₂²×C₁₇

#442

prime radical

C₂×D₂₇₂

C₄

cosocle

C₄×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
2. #14527697	C₂	semidirect
3. #14526750	C₂	semidirect
4. #14526704	C₂	semidirect
5. C₁₇×#1368	C₂	semidirect
42. C₁₆×C₁₇	M₁₆	semidirect
60. #1368	D₁₇	semidirect
97. C₁₇	#12089	semidirect
112. C₁₆	#63218	semidirect

H ≅ G/N

type

2. #14527697

C₂

semidirect

3. #14526750

C₂

semidirect

4. #14526704

C₂

semidirect

5. C₁₇×#1368

C₂

semidirect

42. C₁₆×C₁₇

M₁₆

semidirect

60. #1368

D₁₇

semidirect

97. C₁₇

#12089

semidirect

112. C₁₆

#63218

semidirect

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

subgroup	index
#14530661	2
#14527697	2
#14526750	2
#14526704	2
C₁₇×#1368	2
#12089	17

subgroup

index

C₁₇×#1368

Subgroups — #16946348

Conjugacy classes of subgroups

Complements

Maximal subgroups