Group #16928128 — subgroups

Subgroups — #16928128

Order 4352.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	2	yes	maximal
2	#14527543	C₂	1	4	yes	maximal
3	C₂×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587490	—	2	4	no	—
5	#11762833	C₂²	1	4	yes	—
6	#11762562	—	2	8	no	—
7	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
9	C₈²×C₁₇	C₄	1	1	yes	characteristic
10	C₈²×C₁₇	C₄	1	2	yes	—
11	C₈²×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	D₄	1	2	yes	—
13	C₂²×C₈×C₁₇	Q₈	1	4	yes	—
14	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
15	C₂×#63223	—	2	4	no	—
16	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₇	D₄	1	4	yes	—
21	C₄×C₈×C₁₇	Q₈	1	2	yes	—
22	#10587490	—	4	8	no	—
23	C₄⋊#3622	—	2	8	no	—
24	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
26	C₂×C₈×C₁₇	—	2	8	no	—
27	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
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	8	no	—
37	C₂×#3622	—	4	4	no	—
38	#8823	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	4	no	—
41	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
46	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
47	C₈×C₁₇	—	2	4	no	—
48	C₈×C₁₇	#107	1	4	yes	—
49	C₈×C₁₇	#107	1	4	yes	—
50	C₈×C₁₇	#108	1	8	yes	complemented
51	#3622	—	8	4	no	—
52	#1848	—	17	2	no	—
53	#1567	—	17	4	no	—
54	C₂×C₈²	D₁₇	1	1	yes	characteristic
55	C₂²×C₁₇	—	2	2	no	—
56	C₂²×C₁₇	#384	1	1	yes	characteristic
57	C₂²×C₁₇	#500	1	2	yes	—
58	C₄×C₁₇	—	2	2	no	—
59	C₄×C₁₇	C₂×#107	1	2	yes	—
60	C₄×C₁₇	C₂×#108	1	4	yes	—
61	C₄×C₁₇	#493	1	1	yes	characteristic
62	C₄×C₁₇	#494	1	1	yes	characteristic
63	C₂×#108	—	34	4	no	—
64	#423	—	17	4	no	—
65	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
67	#335	—	34	8	no	—
68	C₈²	Dic₁₇	1	1	yes	characteristic
69	C₈²	Dic₁₇	1	2	yes	—
70	C₈²	Dic₁₇	1	1	yes	characteristic
71	C₂×C₁₇	—	2	2	no	—
72	C₂×C₁₇	#1845	1	1	yes	characteristic
73	C₂×C₁₇	#1848	1	2	yes	—
74	C₂×M₁₆	—	34	4	no	—
75	C₂²×C₈	Dic₃₄	1	4	yes	—
76	C₂²×C₈	D₆₈	1	2	yes	—
77	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
78	#108	—	68	8	no	—
79	C₄⋊C₈	—	34	8	no	—
80	C₄×C₈	Dic₃₄	1	2	yes	—
81	C₄×C₈	D₆₈	1	4	yes	—
82	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
83	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
84	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
86	C₁₇	#8823	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₄	C₂×Dic₃₄	1	2	yes	—
88	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
89	M₁₆	—	68	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	68	4	no	—
92	C₂×C₈	D₁₃₆	1	4	yes	—
93	C₂×C₈	Dic₆₈	1	4	yes	—
94	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
95	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
97	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
98	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
99	C₄²	C₂×D₆₈	1	2	yes	—
100	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
101	C₂³	#10587555	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	#10587555	1	2	yes	—
104	C₂×C₄	#10587560	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
106	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	136	4	no	—
111	C₈	#10587489	1	4	yes	—
112	C₈	#10587489	1	4	yes	—
113	C₈	#10587490	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#11762679	1	1	yes	characteristic
116	C₂²	#11763214	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#10587489	1	2	yes	—
119	C₄	C₂×#10587490	1	4	yes	—
120	C₄	#11763186	1	1	yes	characteristic
121	C₄	#11763189	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#14528944	1	1	yes	characteristic
124	C₂	#14528956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528956

C₂

yes

maximal

#14527543

C₂

yes

maximal

C₂×C₈²×C₁₇

C₂

yes

maximal, characteristic

C₂×#10587490

—

#11762833

C₂²

yes

—

#11762562

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₄

yes

characteristic

C₈²×C₁₇

C₄

yes

—

C₈²×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

Q₈

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

#10587490

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

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

—

—

—

C₂³×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

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

—

C₈×C₁₇

#107

yes

—

C₈×C₁₇

yes

—

—

—

C₂×C₈²

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

characteristic

C₂²×C₁₇

#500

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#107

yes

—

C₄×C₁₇

C₂×#108

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

—

#335

—

C₈²

Dic₁₇

yes

characteristic

C₈²

Dic₁₇

yes

—

C₈²

Dic₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#1845

yes

characteristic

C₂×C₁₇

#1848

yes

—

C₂×M₁₆

—

C₂²×C₈

Dic₃₄

yes

—

C₂²×C₈

D₆₈

yes

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₃₄

yes

—

C₄×C₈

D₆₈

yes

—

C₄×C₈

C₂×Dic₁₇

yes

—

C₄×C₈

C₂×Dic₁₇

yes

characteristic

C₄×C₈

C₂×Dic₁₇

yes

characteristic

C₄×C₈

C₂×Dic₁₇

yes

—

C₁₇

#8823

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₃₄

yes

—

C₂²×C₄

C₂×D₆₈

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₃₆

yes

—

C₂×C₈

Dic₆₈

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₂×C₈

C₄⋊Dic₁₇

yes

—

C₄²

C₂×Dic₃₄

yes

characteristic

C₄²

C₂×D₆₈

yes

—

100

C₄²

C₂²×Dic₁₇

yes

characteristic

101

C₂³

#10587555

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

#10587555

yes

—

104

C₂×C₄

#10587560

yes

characteristic

105

C₂×C₄

C₂×D₁₃₆

yes

—

106

C₂×C₄

C₂×Dic₆₈

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

108

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

—

109

C₈

—

110

C₈

—

136

—

111

C₈

#10587489

yes

—

112

C₈

#10587489

yes

—

113

C₈

#10587490

yes

complemented

114

C₂²

—

115

C₂²

#11762679

yes

characteristic

116

C₂²

#11763214

yes

—

117

C₄

—

118

C₄

C₂×#10587489

yes

—

119

C₄

C₂×#10587490

yes

—

120

C₄

#11763186

yes

characteristic

121

C₄

#11763189

yes

characteristic

122

C₂

—

123

C₂

#14528944

yes

characteristic

124

C₂

#14528956

yes

—

	subgroup	G/H
center	C₂×C₄	#10587560
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₈²×C₁₇	C₂
socle	C₂²×C₁₇	#384
prime radical	C₂³×C₁₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×C₈²	D₁₇
hypocenter	C₁₇	#8823
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#10587560

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₈²×C₁₇

C₂

socle

C₂²×C₁₇

#384

prime radical

C₂³×C₁₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₇

C₂³

hypercenter

C₂×C₈²

D₁₇

hypocenter

C₁₇

#8823

LCS

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
50. C₈×C₁₇	#108	semidirect
86. C₁₇	#8823	semidirect
113. C₈	#10587490	semidirect

H ≅ G/N

type

50. C₈×C₁₇

#108

semidirect

86. C₁₇

#8823

semidirect

113. C₈

#10587490

semidirect

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

subgroup	index
#14528956	2
#14527543	2
C₂×C₈²×C₁₇	2
#8823	17

subgroup

index

#14528956

#14527543

C₂×C₈²×C₁₇

#8823

Subgroups — #16928128

Conjugacy classes of subgroups

Complements

Maximal subgroups