Group #17981398 — subgroups

Subgroups — #17981398

Order 4352.

124 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 41 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	C₂²⋊#10587528	C₂	1	2	yes	maximal
2	#14545121	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587528	—	2	2	no	—
4	C₂×#10587538	C₄	1	1	yes	characteristic
5	C₂²×#63216	C₂²	1	1	yes	characteristic
6	#11763730	—	2	2	no	—
7	#11762767	C₄	1	1	yes	characteristic
8	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
9	#10587612	C₈	1	1	yes	characteristic
10	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
11	#10587538	—	4	1	no	—
12	C₂×#63216	—	2	1	no	—
13	C₂×#63216	—	4	1	no	—
14	C₂×#63216	D₄	1	2	yes	—
15	#10587528	—	4	2	no	—
16	#10587486	—	2	1	no	—
17	C₈×Dic₁₇	—	2	1	no	—
18	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
19	M₁₆×C₁₇	—	4	1	no	—
20	C₂×C₈×C₁₇	—	2	1	no	—
21	C₂²⋊Dic₁₇	—	2	1	no	—
22	C₄⋊Dic₁₇	C₁₆	1	2	yes	—
23	C₄×Dic₁₇	—	2	1	no	—
24	C₂×#3622	—	2	1	no	—
25	C₂×#3622	—	2	1	no	—
26	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×#3622	M₁₆	1	1	yes	characteristic
28	#63216	—	2	2	no	—
29	#63216	—	4	1	no	—
30	#7363	—	17	1	no	maximal, Sylow
31	C₂³×C₁₇	M₃₂	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	1	no	—
33	C₂×C₄×C₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
36	C₂×Dic₁₇	—	2	2	no	—
37	C₈×C₁₇	—	4	1	no	—
38	#3622	—	2	1	no	—
39	#3622	#99	1	1	yes	characteristic
40	#3622	#101	1	1	yes	characteristic
41	#2151	—	17	1	no	—
42	C₂²⋊C₃₂	—	17	2	no	—
43	C₂²×C₁₇	—	2	1	no	—
44	C₂²×C₁₇	—	2	2	no	—
45	C₂²×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
46	C₄×C₁₇	—	2	1	no	—
47	C₄×C₁₇	#324	1	1	yes	characteristic
48	C₄×C₁₇	#349	1	1	yes	characteristic
49	Dic₁₇	—	4	2	no	—
50	C₂×M₃₂	—	17	1	no	—
51	C₂²×C₁₆	—	17	1	no	—
52	#405	—	17	1	no	—
53	C₂×C₃₂	—	34	2	no	—
54	C₄⋊C₁₆	—	34	2	no	—
55	C₂×C₁₇	—	4	1	no	—
56	C₂×C₁₇	#1322	1	1	yes	characteristic
57	C₂×C₁₇	#1401	1	2	yes	—
58	C₂×M₁₆	#3633	1	1	yes	characteristic
59	C₂²×C₈	—	17	1	no	—
60	#117	—	17	1	no	—
61	M₃₂	—	68	1	no	—
62	C₂×C₁₆	—	17	2	no	—
63	C₂×C₁₆	—	34	1	no	—
64	C₂×C₁₆	—	68	1	no	—
65	#97	—	34	1	no	—
66	C₄×C₈	—	34	1	no	—
67	C₃₂	—	68	2	no	—
68	C₁₇	#7363	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
70	M₁₆	—	4	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	17	1	no	—
73	C₂×C₈	—	17	1	no	—
74	C₂×C₈	—	34	1	no	—
75	C₂×C₈	—	34	1	no	—
76	C₄⋊C₄	—	17	2	no	—
77	C₂²⋊C₄	—	34	1	no	—
78	C₄²	—	34	1	no	—
79	C₁₆	—	34	2	no	—
80	C₁₆	—	68	1	no	—
81	C₂³	#10587666	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	34	2	no	—
85	C₂×C₄	#10587665	1	1	yes	characteristic
86	C₂×C₄	C₂²⋊#3633	1	1	yes	characteristic
87	C₈	—	4	1	no	—
88	C₈	—	17	1	no	—
89	C₈	—	17	1	no	—
90	C₈	—	34	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#11763906	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	68	2	no	—
96	C₄	#11763895	1	1	yes	characteristic
97	C₄	#11763907	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#14545818	1	1	yes	characteristic
100	C₂	#14545881	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊#10587528

C₂

yes

maximal

#14545121

C₂

yes

maximal, characteristic

C₂×#10587528

—

C₂×#10587538

C₄

yes

characteristic

C₂²×#63216

C₂²

yes

characteristic

#11763730

—

#11762767

C₄

yes

characteristic

C₂×M₁₆×C₁₇

C₈

yes

characteristic

#10587612

C₈

yes

characteristic

C₂²×#3622

C₂×C₄

yes

—

—

—

D₄

yes

—

#10587528

—

#10587486

—

C₈×Dic₁₇

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

C₁₆

yes

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

M₁₆

yes

—

—

—

C₂³×C₁₇

M₃₂

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×Dic₁₇

—

C₈×C₁₇

—

—

yes

yes

—

C₂²⋊C₃₂

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²⋊C₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1322

yes

characteristic

C₂×C₁₇

#1401

yes

—

C₂×M₁₆

#3633

yes

characteristic

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₃₂

—

C₁₇

#7363

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×#3633

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

#10587666

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10587665

yes

characteristic

C₂×C₄

C₂²⋊#3633

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

#11763906

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#14545818

yes

characteristic

100

C₂

#14545881

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#3633
commutator	C₂×C₄×C₁₇	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#3633
Fitting	C₂×M₁₆×C₁₇	C₈
socle	C₂²×C₁₇	C₂²⋊C₁₆
prime radical	C₂³×C₁₇	M₃₂
cosocle	C₂²×#63216	C₂²
hypercenter	C₂×M₁₆	#3633
hypocenter	C₁₇	#7363
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²⋊#3633

commutator

C₂×C₄×C₁₇

C₂×C₁₆

Frattini

C₂²×C₄

C₂×#3633

Fitting

C₂×M₁₆×C₁₇

C₈

socle

C₂²×C₁₇

C₂²⋊C₁₆

prime radical

C₂³×C₁₇

M₃₂

cosocle

C₂²×#63216

C₂²

C₂×M₁₆

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
68. C₁₇	#7363	semidirect

H ≅ G/N

type

68. C₁₇

#7363

semidirect

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

subgroup	index
C₂²⋊#10587528	2
#14545121	2
#7363	17

subgroup

index

C₂²⋊#10587528

#14545121

#7363

Subgroups — #17981398

Conjugacy classes of subgroups

Complements

Maximal subgroups