Group #16955355 — subgroups

Subgroups — #16955355

Order 4352.

150 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 75 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	#14531063	C₂	1	1	yes	maximal, characteristic
2	#14531059	C₂	1	1	yes	maximal, characteristic
3	#14526986	C₂	1	2	yes	maximal
4	#14526984	C₂	1	2	yes	maximal
5	C₁₇×#1397	C₂	1	1	yes	maximal, characteristic
6	#11763214	—	2	1	no	—
7	#11763211	—	2	1	no	—
8	#11762637	—	2	1	no	—
9	#11762635	—	2	1	no	—
10	#11762635	C₂²	1	1	yes	characteristic
11	#11762635	C₂²	1	2	yes	—
12	#11762635	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
14	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	#10587556	—	4	1	no	—
16	#10587555	—	4	1	no	—
17	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
19	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
20	#10587489	—	2	1	no	—
21	#10587489	—	2	1	no	—
22	#10587489	—	2	2	no	—
23	#10587489	—	4	1	no	—
24	#10587488	—	4	1	no	—
25	C₁₇×C₃₂	Q₈	1	4	yes	complemented
26	C₂×Dic₃₄	—	8	1	no	—
27	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
28	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₄²×C₁₇	D₈	1	1	yes	characteristic
30	C₄⋊Dic₁₇	—	4	1	no	—
31	C₄⋊Dic₁₇	—	4	1	no	—
32	C₄⋊Dic₁₇	—	4	2	no	—
33	C₄⋊Dic₁₇	—	8	1	no	—
34	#63225	—	8	1	no	—
35	#63225	—	8	1	no	—
36	C₁₆×C₁₇	Q₁₆	1	2	yes	—
37	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
38	#13734	—	17	1	no	maximal, Sylow
39	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₁₇	—	8	2	no	—
43	C₂×Dic₁₇	—	8	1	no	—
44	C₂×Dic₁₇	—	8	1	no	—
45	Dic₃₄	—	16	2	no	—
46	C₈×C₁₇	Q₃₂	1	2	yes	—
47	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
49	#2253	—	17	1	no	—
50	#2252	—	17	1	no	—
51	#1424	—	17	2	no	—
52	#1423	—	17	2	no	—
53	#1397	D₁₇	1	1	yes	characteristic
54	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
55	C₄×C₁₇	—	2	1	no	—
56	C₄×C₁₇	—	2	1	no	—
57	C₄×C₁₇	#500	1	1	yes	characteristic
58	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
59	Dic₁₇	—	16	2	no	—
60	Dic₁₇	—	16	1	no	—
61	Dic₁₇	—	16	1	no	—
62	#500	—	34	1	no	—
63	#499	—	34	1	no	—
64	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
65	#367	—	34	1	no	—
66	#366	—	17	1	no	—
67	#366	—	17	2	no	—
68	#366	—	17	1	no	—
69	#366	—	34	1	no	—
70	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
71	C₂×C₁₇	#2252	1	1	yes	characteristic
72	C₂×C₁₇	#2263	1	1	yes	characteristic
73	C₂×C₁₇	#2264	1	1	yes	characteristic
74	C₄⋊Q₈	—	68	1	no	—
75	#125	—	68	1	no	—
76	C₂×C₁₆	D₆₈	1	1	yes	characteristic
77	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
78	#107	—	34	2	no	—
79	#107	—	34	1	no	—
80	#107	—	34	1	no	—
81	#107	—	68	1	no	—
82	#106	—	68	1	no	—
83	C₄×C₈	D₆₈	1	1	yes	characteristic
84	C₃₂	Dic₃₄	1	4	yes	complemented
85	C₁₇	#13734	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	136	1	no	—
87	C₂×C₈	D₁₃₆	1	1	yes	characteristic
88	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
89	C₄⋊C₄	—	68	1	no	—
90	C₄⋊C₄	—	68	1	no	—
91	C₄⋊C₄	—	68	2	no	—
92	C₄⋊C₄	—	136	1	no	—
93	C₄⋊C₄	—	136	1	no	—
94	C₄⋊C₄	—	136	1	no	—
95	C₄²	D₁₃₆	1	1	yes	characteristic
96	C₁₆	Dic₆₈	1	2	yes	—
97	C₁₆	C₂×Dic₃₄	1	2	yes	—
98	Q₈	—	272	2	no	—
99	C₂×C₄	—	136	1	no	—
100	C₂×C₄	—	136	1	no	—
101	C₂×C₄	—	136	2	no	—
102	C₂×C₄	D₂₇₂	1	1	yes	characteristic
103	C₂×C₄	D₂₇₂	1	1	yes	characteristic
104	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
105	C₈	Dic₁₃₆	1	2	yes	—
106	C₈	#10587555	1	1	yes	characteristic
107	C₈	C₂×Dic₆₈	1	1	yes	characteristic
108	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	272	2	no	—
112	C₄	—	272	1	no	—
113	C₄	—	272	1	no	—
114	C₄	#11763214	1	1	yes	characteristic
115	C₄	C₂×Dic₁₃₆	1	1	yes	characteristic
116	C₂	#14531059	1	1	yes	characteristic
117	C₂	#14531099	1	1	yes	characteristic
118	C₂	#14531104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531063

C₂

yes

maximal, characteristic

#14531059

C₂

yes

maximal, characteristic

#14526986

C₂

yes

maximal

#14526984

C₂

yes

maximal

C₁₇×#1397

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

characteristic

#11762635

C₂²

yes

—

#11762635

C₂²

yes

characteristic

C₂×C₁₇×C₃₂

C₂²

yes

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

#10587556

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

—

C₁₇×C₃₂

Q₈

yes

complemented

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

D₈

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

C₁₆×C₁₇

Q₁₆

yes

—

C₁₆×C₁₇

C₂×Q₈

yes

—

#13734

—

maximal, Sylow

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

D₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

Q₃₂

yes

—

C₈×C₁₇

C₄⋊Q₈

yes

characteristic

C₈×C₁₇

C₂×Q₁₆

yes

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#500

yes

characteristic

C₄×C₁₇

C₂×Q₃₂

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

#500

—

#499

—

C₂×C₃₂

C₂×D₁₇

yes

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₄⋊Q₈

—

#125

—

C₂×C₁₆

D₆₈

yes

characteristic

C₂×C₁₆

C₂²×D₁₇

yes

—

—

—

—

—

C₄×C₈

D₆₈

yes

characteristic

C₃₂

Dic₃₄

yes

complemented

C₁₇

#13734

yes

characteristic, complemented, Sylow

C₂×Q₈

—

136

—

C₂×C₈

D₁₃₆

yes

characteristic

C₂×C₈

C₂×D₆₈

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

136

—

C₄⋊C₄

—

136

—

C₄⋊C₄

—

136

—

C₄²

D₁₃₆

yes

characteristic

C₁₆

Dic₆₈

yes

—

C₁₆

C₂×Dic₃₄

yes

—

Q₈

—

272

—

C₂×C₄

—

136

—

100

C₂×C₄

—

136

—

101

C₂×C₄

—

136

—

102

C₂×C₄

D₂₇₂

yes

characteristic

103

C₂×C₄

D₂₇₂

yes

characteristic

104

C₂×C₄

C₂×D₁₃₆

yes

characteristic

105

C₈

Dic₁₃₆

yes

—

106

C₈

#10587555

yes

characteristic

107

C₈

C₂×Dic₆₈

yes

characteristic

108

C₂²

C₂×D₂₇₂

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

272

—

112

C₄

—

272

—

113

C₄

—

272

—

114

C₄

#11763214

yes

characteristic

115

C₄

C₂×Dic₁₃₆

yes

characteristic

116

C₂

#14531059

yes

characteristic

117

C₂

#14531099

yes

characteristic

118

C₂

#14531104

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₇₂
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×#1397	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	#1397	D₁₇
hypocenter	C₁₇	#13734
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₂₇₂

commutator, cosocle

C₂×C₁₆×C₁₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₁₇×#1397

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₁₆

D₁₇

C₁₇

G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
25. C₁₇×C₃₂	Q₈	semidirect
84. C₃₂	Dic₃₄	semidirect
85. C₁₇	#13734	semidirect

H ≅ G/N

type

25. C₁₇×C₃₂

Q₈

semidirect

84. C₃₂

Dic₃₄

semidirect

85. C₁₇

#13734

semidirect

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

subgroup	index
#14531063	2
#14531059	2
#14526986	2
#14526984	2
C₁₇×#1397	2
#13734	17

subgroup

index

C₁₇×#1397

Subgroups — #16955355

Conjugacy classes of subgroups

Complements

Maximal subgroups