Group #14545152 — subgroups

Subgroups — #14545152

Order 2176.

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₄⋊#63216	C₂	1	2	yes	maximal, complemented
2	#11763704	C₂	1	2	yes	maximal
3	#11763698	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763695	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762766	C₂	1	1	yes	maximal, characteristic
6	C₂×#63218	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
8	#10587558	C₄	1	1	yes	characteristic
9	C₂×#63216	C₂²	1	2	yes	—
10	C₂×#63216	C₂²	1	2	yes	—
11	C₁₇×#97	C₄	1	1	yes	characteristic
12	#10587486	C₄	1	1	yes	characteristic
13	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄×#3622	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	#63227	C₈	1	2	yes	—
20	#63225	C₈	1	2	yes	—
21	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂×#3622	C₂³	1	1	yes	characteristic
24	#63218	—	2	2	no	—
25	C₈×D₁₇	—	2	2	no	—
26	#63216	—	2	2	no	—
27	#63216	—	2	2	no	—
28	C₂²×D₁₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×Dic₁₇	C₂×C₈	1	2	yes	—
33	C₄×D₁₇	—	2	1	no	—
34	C₄×D₁₇	—	2	1	no	—
35	C₈×C₁₇	—	2	1	no	—
36	C₈×C₁₇	M₁₆	1	2	yes	—
37	#3622	—	2	1	no	—
38	#3622	#41	1	2	yes	—
39	#2164	—	17	1	no	maximal, Sylow
40	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×D₁₇	—	2	1	no	—
42	C₂×D₁₇	—	2	2	no	—
43	C₄×C₁₇	—	2	2	no	—
44	C₄×C₁₇	#117	1	1	yes	characteristic
45	C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
46	Dic₁₇	—	2	1	no	—
47	Dic₁₇	—	2	2	no	—
48	#405	—	17	1	no	—
49	C₄⋊C₁₆	—	17	2	no	—
50	C₂²⋊C₁₆	—	17	2	no	—
51	#346	—	17	1	no	—
52	C₄×C₁₆	—	17	1	no	—
53	C₂×C₁₇	#431	1	1	yes	characteristic
54	C₂×C₁₇	#504	1	2	yes	—
55	D₁₇	—	4	1	no	—
56	C₂×M₁₆	—	17	1	no	—
57	C₂²×C₈	—	17	1	no	—
58	#117	—	17	1	no	—
59	C₂×C₁₆	—	17	2	no	—
60	C₂×C₁₆	—	17	2	no	—
61	#97	—	17	1	no	—
62	#97	#595	1	1	yes	characteristic
63	C₄×C₈	—	17	1	no	—
64	C₄×C₈	—	17	1	no	—
65	C₁₇	#2164	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	17	1	no	—
67	M₁₆	—	34	2	no	—
68	C₂×C₈	—	17	1	no	—
69	C₂×C₈	—	17	1	no	—
70	C₂×C₈	—	34	2	no	—
71	C₂×C₈	C₂×#595	1	1	yes	characteristic
72	C₂×C₈	C₂×#595	1	1	yes	characteristic
73	C₄⋊C₄	—	17	2	no	—
74	C₂²⋊C₄	—	17	2	no	—
75	C₄²	—	17	1	no	—
76	C₄²	C₂×#595	1	1	yes	characteristic
77	C₁₆	—	34	2	no	—
78	C₁₆	—	34	2	no	—
79	C₂³	—	17	1	no	—
80	C₂×C₄	—	17	2	no	—
81	C₂×C₄	—	17	1	no	—
82	C₂×C₄	—	34	1	no	—
83	C₂×C₄	—	34	1	no	—
84	C₂×C₄	#63242	1	2	yes	—
85	C₂×C₄	C₂²×#595	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	17	2	no	—
88	C₈	—	34	1	no	—
89	C₈	#63243	1	2	yes	—
90	C₂²	—	34	1	no	—
91	C₂²	—	34	2	no	—
92	C₂²	C₂×#63242	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	34	1	no	—
95	C₄	—	34	2	no	—
96	C₄	C₂×#63243	1	1	yes	characteristic
97	C₄	#10587677	1	1	yes	characteristic
98	C₂	—	68	1	no	—
99	C₂	#11763828	1	1	yes	characteristic
100	C₂	#11763868	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#63216

C₂

yes

maximal, complemented

#11763704

C₂

yes

maximal

#11763698

C₂

yes

maximal, characteristic, complemented

#11763695

C₂

yes

maximal, characteristic, complemented

#11762766

C₂

yes

maximal, characteristic

C₂×#63218

C₄

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

characteristic

#10587558

C₄

yes

characteristic

C₂×#63216

C₂²

yes

—

C₂×#63216

C₂²

yes

—

C₁₇×#97

C₄

yes

characteristic

#10587486

C₄

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

#63227

C₈

yes

—

#63225

C₈

yes

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

C₂³

yes

characteristic

#63218

—

C₈×D₁₇

—

#63216

—

#63216

—

C₂²×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₂×C₈

yes

characteristic

C₂×Dic₁₇

C₂×C₈

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

M₁₆

yes

—

#3622

—

#3622

#41

yes

—

#2164

—

maximal, Sylow

C₂²×C₁₇

C₂²×C₈

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#117

yes

characteristic

C₄×C₁₇

C₂×M₁₆

yes

characteristic

Dic₁₇

—

Dic₁₇

—

#405

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂×C₁₇

#431

yes

characteristic

C₂×C₁₇

#504

yes

—

D₁₇

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₁₆

—

C₂×C₁₆

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₁₇

#2164

yes

characteristic, complemented, Sylow

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#63242

yes

—

C₂×C₄

C₂²×#595

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#63243

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#63242

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#11763828

yes

characteristic

100

C₂

#11763868

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂²×C₈
Fitting	C₁₇×#97	C₄
prime radical	C₂²×D₁₇	C₂×C₈
cosocle	C₂×#3622	C₂³
hypercenter	#97	#595
hypocenter	C₁₇	#2164
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#595

commutator, socle

C₂²×C₁₇

C₂²×C₈

Fitting

C₁₇×#97

C₄

prime radical

C₂²×D₁₇

C₂×C₈

C₂³

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
1. C₄⋊#63216	C₂	semidirect
3. #11763698	C₂	semidirect
4. #11763695	C₂	semidirect
65. C₁₇	#2164	semidirect

H ≅ G/N

type

1. C₄⋊#63216

C₂

semidirect

3. #11763698

C₂

semidirect

4. #11763695

C₂

semidirect

65. C₁₇

#2164

semidirect

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

subgroup	index
C₄⋊#63216	2
#11763704	2
#11763698	2
#11763695	2
#11762766	2
#2164	17

subgroup

index

Subgroups — #14545152

Conjugacy classes of subgroups

Complements

Maximal subgroups