Group #16949062 — subgroups

Subgroups — #16949062

Order 4352.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14531064	C₂	1	2	yes	maximal
2	#14531057	C₂	1	2	yes	maximal
3	#14528394	C₂	1	1	yes	maximal, characteristic
4	C₈×#63214	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1710	C₂	1	1	yes	maximal, characteristic
6	#11763214	C₂²	1	2	yes	—
7	#11763160	—	2	2	no	—
8	#11762636	—	2	4	no	—
9	#11762634	—	2	4	no	—
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₄×#63214	C₂²	1	1	yes	characteristic
12	C₁₇×#500	C₂²	1	2	yes	—
13	C₁₇×#487	—	2	2	no	—
14	C₈²×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×C₄⋊Q₈	—	2	2	no	—
16	#10587555	—	2	2	no	—
17	C₁₇×#107	—	2	2	no	—
18	C₁₇×#107	—	2	2	no	—
19	C₁₇×#103	—	4	4	no	—
20	C₄×C₈×C₁₇	D₄	1	2	yes	—
21	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	#10587489	—	2	2	no	—
23	#10587489	—	2	2	no	—
24	#10587487	—	4	4	no	—
25	C₂×#63214	D₄	1	2	yes	—
26	C₂×#63214	D₄	1	2	yes	—
27	C₂×Q₈×C₁₇	—	4	2	no	—
28	C₂×Dic₃₄	—	4	2	no	—
29	C₂×C₈×C₁₇	D₈	1	4	yes	—
30	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₇	—	4	2	no	—
33	C₄⋊C₄×C₁₇	—	4	2	no	—
34	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₁₇	—	4	2	no	—
36	C₄⋊Dic₁₇	—	4	2	no	—
37	#63214	Q₁₆	1	4	yes	—
38	#63214	Q₁₆	1	4	yes	—
39	#12522	—	17	1	no	maximal, Sylow
40	Q₈×C₁₇	—	8	4	no	—
41	C₂×C₄×C₁₇	—	8	2	no	—
42	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
44	C₂×Dic₁₇	—	8	2	no	—
45	Dic₃₄	—	8	4	no	—
46	C₈×C₁₇	—	2	4	no	—
47	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
48	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
49	#2254	—	17	2	no	—
50	#2252	—	17	2	no	—
51	#1710	—	17	1	no	—
52	#1710	D₁₇	1	1	yes	characteristic
53	C₈×C₁₆	—	17	1	no	—
54	C₂²×C₁₇	#493	1	1	yes	characteristic
55	C₄×C₁₇	—	16	2	no	—
56	C₄×C₁₇	#494	1	1	yes	characteristic
57	C₄×C₁₇	#494	1	1	yes	characteristic
58	C₄×C₁₇	#508	1	4	yes	—
59	Dic₁₇	—	16	2	no	—
60	#500	—	17	2	no	—
61	#500	C₂×D₁₇	1	2	yes	—
62	#487	—	2	2	no	—
63	#487	—	34	2	no	—
64	#367	—	34	4	no	—
65	#366	—	34	4	no	—
66	C₄×C₁₆	—	17	1	no	—
67	C₄×C₁₆	—	17	1	no	—
68	C₈²	C₂×D₁₇	1	1	yes	characteristic
69	C₂×C₁₇	#1713	1	1	yes	characteristic
70	C₂×C₁₇	#2249	1	2	yes	—
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	—	34	2	no	—
73	C₂×C₁₆	—	17	2	no	—
74	C₂×C₁₆	—	17	2	no	—
75	#107	—	2	2	no	—
76	#107	—	2	2	no	—
77	#107	—	34	2	no	—
78	#107	—	34	2	no	—
79	#103	—	4	4	no	—
80	#103	—	68	4	no	—
81	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
82	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
83	C₁₇	#12522	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	68	2	no	—
86	C₂×C₈	#63228	1	4	yes	—
87	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
88	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	68	2	no	—
92	C₄⋊C₄	—	68	2	no	—
93	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
94	C₁₆	—	17	4	no	—
95	C₁₆	—	17	4	no	—
96	Q₈	—	8	4	no	—
97	Q₈	—	136	4	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	136	2	no	—
100	C₂×C₄	C₂×#63228	1	2	yes	—
101	C₂×C₄	#10587626	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	Q₁₆×D₁₇	1	2	yes	—
104	C₈	Q₁₆×D₁₇	1	2	yes	—
105	C₂²	#11763185	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	272	2	no	—
108	C₄	#11763187	1	1	yes	characteristic
109	C₄	#11763187	1	1	yes	characteristic
110	C₄	#11763239	1	4	yes	—
111	C₂	#14528400	1	1	yes	characteristic
112	C₂	#14531043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14531064

C₂

yes

maximal

#14531057

C₂

yes

maximal

#14528394

C₂

yes

maximal, characteristic

C₈×#63214

C₂

yes

maximal, characteristic

C₁₇×#1710

C₂

yes

maximal, characteristic

#11763214

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#63214

C₂²

yes

characteristic

C₁₇×#500

C₂²

yes

—

C₁₇×#487

—

C₈²×C₁₇

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

—

#10587555

—

C₁₇×#107

—

C₁₇×#107

—

C₁₇×#103

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

C₂³

yes

—

—

—

D₄

yes

—

C₂×#63214

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

Q₁₆

yes

—

#63214

Q₁₆

yes

—

#12522

—

maximal, Sylow

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#127

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

—

C₂×Dic₁₇

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

—

C₈×C₁₇

C₂×Q₁₆

yes

—

—

—

—

D₁₇

yes

characteristic

C₈×C₁₆

—

C₂²×C₁₇

#493

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

#500

—

#500

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₁₇

yes

—

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12522

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

#63228

yes

—

C₂×C₈

D₄×D₁₇

yes

characteristic

C₂×C₈

D₄×D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

136

—

C₂×C₄

—

C₂×C₄

—

136

—

100

C₂×C₄

C₂×#63228

yes

—

101

C₂×C₄

#10587626

yes

characteristic

102

C₈

—

103

C₈

Q₁₆×D₁₇

yes

—

104

C₈

Q₁₆×D₁₇

yes

—

105

C₂²

#11763185

yes

characteristic

106

C₄

—

107

C₄

—

272

—

108

C₄

#11763187

yes

characteristic

109

C₄

#11763187

yes

characteristic

110

C₄

#11763239

yes

—

111

C₂

#14528400

yes

characteristic

112

C₂

#14531043

yes

—

	subgroup	G/H
center	C₂²	#11763185
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1710	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	#1710	D₁₇
hypocenter	C₁₇	#12522
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#11763185

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1710

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
83. C₁₇	#12522	semidirect

H ≅ G/N

type

83. C₁₇

#12522

semidirect

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

subgroup	index
#14531064	2
#14531057	2
#14528394	2
C₈×#63214	2
C₁₇×#1710	2
#12522	17

subgroup

index

C₁₇×#1710

Subgroups — #16949062

Conjugacy classes of subgroups

Complements

Maximal subgroups