Group #16928136 — subgroups

Subgroups — #16928136

Order 4352.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	#14528956	C₂	1	1	yes	maximal, characteristic
2	#14527542	C₂	1	4	yes	maximal
3	C₂×#11762530	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587490	—	2	2	no	—
6	#11762833	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₂²	1	2	yes	—
8	#11762561	—	2	8	no	—
9	#11762530	C₄	1	4	yes	—
10	C₂×C₁₇×#108	—	2	2	no	—
11	C₁₇×#423	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₇	—	2	2	no	—
14	C₂²×C₈×C₁₇	D₄	1	2	yes	—
15	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×#63223	—	2	2	no	—
17	C₂²×#3622	Q₈	1	4	yes	—
18	C₁₇×#108	—	4	4	no	—
19	C₁₇×C₄⋊C₈	—	2	4	no	—
20	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₇	Q₈	1	2	yes	—
22	#10587490	—	4	4	no	—
23	C₄⋊#3622	—	2	4	no	—
24	C₄×#3622	C₂×C₄	1	4	yes	—
25	C₄×#3622	D₄	1	4	yes	—
26	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
28	M₁₆×C₁₇	—	4	4	no	—
29	C₂×C₈×C₁₇	—	4	2	no	—
30	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₁₇	D₈	1	4	yes	—
32	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
33	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₁₇	C₂×D₄	1	2	yes	—
35	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
36	#63223	—	4	4	no	—
37	C₂×#3622	—	4	4	no	—
38	C₂×#3622	—	4	2	no	—
39	C₂×#3622	C₄⋊C₄	1	8	yes	—
40	#8826	—	17	1	no	maximal, Sylow
41	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇	—	2	4	no	—
43	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
47	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
49	C₈×C₁₇	—	8	2	no	—
50	C₈×C₁₇	#107	1	8	yes	—
51	#3622	—	2	8	no	—
52	#3622	—	8	2	no	—
53	#1848	—	17	1	no	—
54	#1848	D₁₇	1	1	yes	characteristic
55	#1567	—	17	4	no	—
56	C₂×#322	—	17	1	no	—
57	C₂²×C₁₇	—	2	2	no	—
58	C₂²×C₁₇	#384	1	1	yes	characteristic
59	C₂²×C₁₇	#500	1	2	yes	—
60	C₄×C₁₇	—	2	2	no	—
61	C₄×C₁₇	C₂×#107	1	2	yes	—
62	C₄×C₁₇	#430	1	4	yes	—
63	C₄×C₁₇	#493	1	1	yes	characteristic
64	C₄×C₁₇	#494	1	1	yes	characteristic
65	C₂×#108	—	2	2	no	—
66	C₂×#108	—	34	2	no	—
67	#423	—	17	2	no	—
68	#423	C₂×D₁₇	1	2	yes	—
69	C₂×C₄×C₈	—	17	2	no	—
70	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
71	#335	—	34	8	no	—
72	#322	—	17	4	no	—
73	C₂×C₁₇	—	2	2	no	—
74	C₂×C₁₇	#1845	1	1	yes	characteristic
75	C₂×C₁₇	#1849	1	2	yes	—
76	C₂×M₁₆	—	2	2	no	—
77	C₂×M₁₆	—	34	2	no	—
78	C₂²×C₈	—	17	4	no	—
79	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
80	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
81	#108	—	4	4	no	—
82	#108	—	68	4	no	—
83	C₄⋊C₈	—	2	4	no	—
84	C₄⋊C₈	—	34	4	no	—
85	C₄×C₈	—	17	4	no	—
86	C₄×C₈	—	17	4	no	—
87	C₄×C₈	Dic₃₄	1	2	yes	—
88	C₄×C₈	C₄×D₁₇	1	2	yes	—
89	C₁₇	#8826	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₄	Q₈×D₁₇	1	2	yes	—
91	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
92	M₁₆	—	4	4	no	—
93	M₁₆	—	68	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	17	8	no	—
96	C₂×C₈	—	68	4	no	—
97	C₂×C₈	—	68	2	no	—
98	C₂×C₈	#63225	1	4	yes	—
99	C₂×C₈	#63228	1	4	yes	—
100	C₂×C₈	#63231	1	4	yes	—
101	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
102	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
103	C₄²	D₄×D₁₇	1	2	yes	—
104	C₂³	#10587630	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	#10587574	1	2	yes	—
107	C₂×C₄	C₄⋊C₄×D₁₇	1	2	yes	—
108	C₂×C₄	C₂×#63225	1	1	yes	characteristic
109	C₂×C₄	C₂×#63228	1	2	yes	—
110	C₂×C₄	#10587626	1	1	yes	characteristic
111	C₂×C₄	C₂×#63231	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	34	8	no	—
114	C₈	—	136	2	no	—
115	C₈	#10587478	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#11762678	1	1	yes	characteristic
118	C₂²	#11763212	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#10587478	1	2	yes	—
121	C₄	#11762856	1	4	yes	—
122	C₄	#11763185	1	1	yes	characteristic
123	C₄	#11763188	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#14528943	1	1	yes	characteristic
126	C₂	#14528957	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528956

C₂

yes

maximal, characteristic

#14527542

C₂

yes

maximal

C₂×#11762530

C₂

yes

maximal, characteristic

C₁₇×#1848

C₂

yes

maximal, characteristic

C₂×#10587490

—

#11762833

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

—

#11762561

—

#11762530

C₄

yes

—

C₂×C₁₇×#108

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₂²×#3622

Q₈

yes

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

#10587490

—

C₄⋊#3622

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

C₄⋊C₄

yes

—

#8826

—

maximal, Sylow

C₂³×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

#107

yes

—

—

—

—

D₁₇

yes

characteristic

#1567

—

C₂×#322

—

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

characteristic

C₂²×C₁₇

#500

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#107

yes

—

C₄×C₁₇

#430

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

—

—

C₂×D₁₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

#335

—

#322

—

C₂×C₁₇

—

C₂×C₁₇

#1845

yes

characteristic

C₂×C₁₇

#1849

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₁₇

yes

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

C₄×D₁₇

yes

—

C₁₇

#8826

yes

characteristic, complemented, Sylow

C₂²×C₄

Q₈×D₁₇

yes

—

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#63225

yes

—

C₂×C₈

#63228

yes

—

100

C₂×C₈

#63231

yes

—

101

C₄²

C₂×Dic₃₄

yes

characteristic

102

C₄²

C₂×C₄×D₁₇

yes

characteristic

103

C₄²

D₄×D₁₇

yes

—

104

C₂³

#10587630

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

#10587574

yes

—

107

C₂×C₄

C₄⋊C₄×D₁₇

yes

—

108

C₂×C₄

C₂×#63225

yes

characteristic

109

C₂×C₄

C₂×#63228

yes

—

110

C₂×C₄

#10587626

yes

characteristic

111

C₂×C₄

C₂×#63231

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

136

—

115

C₈

#10587478

yes

—

116

C₂²

—

117

C₂²

#11762678

yes

characteristic

118

C₂²

#11763212

yes

—

119

C₄

—

120

C₄

C₂×#10587478

yes

—

121

C₄

#11762856

yes

—

122

C₄

#11763185

yes

characteristic

123

C₄

#11763188

yes

characteristic

124

C₂

—

125

C₂

#14528943

yes

characteristic

126

C₂

#14528957

yes

—

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#8826
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1848

C₂

socle

C₂²×C₁₇

#384

prime radical

C₂³×C₁₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
89. C₁₇	#8826	semidirect

H ≅ G/N

type

89. C₁₇

#8826

semidirect

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

subgroup	index
#14528956	2
#14527542	2
C₂×#11762530	2
C₁₇×#1848	2
#8826	17

subgroup

index

C₁₇×#1848

Subgroups — #16928136

Conjugacy classes of subgroups

Complements

Maximal subgroups