Group #21426357 — subgroups

Subgroups — #21426357

Order 7552.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14273672	C₂	1	2	yes	maximal, complemented
3	C₂×D₉₄₄	C₂	1	1	yes	maximal, characteristic
4	#16837786	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₃₆	C₂²	1	2	yes	—
7	C₂×#11745660	—	2	2	no	—
8	C₂×D₄₇₂	C₂²	1	2	yes	—
9	#14273715	—	2	2	no	—
10	Dic₄₇₂	—	2	2	no	—
11	#14273672	—	2	4	no	—
12	D₉₄₄	—	2	2	no	—
13	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₂₃₆	—	2	2	no	—
17	C₂×Dic₁₁₈	—	2	2	no	—
18	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
20	C₄²×C₅₉	D₄	1	1	yes	characteristic
21	(C₂×D₅₉)⋊C₄	—	4	4	no	—
22	Dic₂₃₆	—	2	2	no	—
23	Dic₂₃₆	—	2	2	no	—
24	D₄₇₂	—	2	2	no	—
25	D₄₇₂	—	2	2	no	—
26	#11745660	—	4	4	no	—
27	C₁₆×C₅₉	D₄	1	2	yes	—
28	C₁₆×C₅₉	D₄	1	2	yes	—
29	C₂²×D₅₉	—	4	2	no	—
30	C₂×C₄×C₅₉	D₈	1	2	yes	—
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₅₉	—	4	2	no	—
33	D₂₃₆	—	4	2	no	—
34	D₂₃₆	—	4	2	no	—
35	Dic₁₁₈	—	4	2	no	—
36	Dic₁₁₈	—	4	2	no	—
37	C₈×C₅₉	D₈	1	2	yes	—
38	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
41	C₂×D₅₉	—	8	2	no	—
42	C₂×D₅₉	—	8	4	no	—
43	C₄×C₅₉	—	2	2	no	—
44	C₄×C₅₉	#127	1	1	yes	characteristic
45	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
46	Dic₅₉	—	8	2	no	—
47	#2249	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	#493	1	1	yes	characteristic
49	C₂×C₅₉	#508	1	2	yes	—
50	D₅₉	—	16	2	no	—
51	C₂×Q₃₂	—	59	1	no	—
52	C₂×SD₃₂	—	59	2	no	—
53	C₂×D₁₆	—	59	1	no	—
54	#495	—	59	2	no	—
55	C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
56	C₅₉	#2249	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₁₆	—	59	2	no	—
58	C₂×SD₁₆	—	118	2	no	—
59	C₂×D₈	—	59	2	no	—
60	#124	—	118	2	no	—
61	Q₃₂	—	118	2	no	—
62	SD₃₂	—	118	4	no	—
63	D₁₆	—	118	2	no	—
64	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
65	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	118	2	no	—
68	C₂×D₄	—	118	2	no	—
69	Q₁₆	—	118	2	no	—
70	Q₁₆	—	118	2	no	—
71	SD₁₆	—	236	4	no	—
72	D₈	—	118	2	no	—
73	D₈	—	118	2	no	—
74	C₂×C₈	D₂₃₆	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
76	C₂²⋊C₄	—	236	4	no	—
77	C₄²	D₂₃₆	1	1	yes	characteristic
78	C₁₆	D₂₃₆	1	2	yes	—
79	C₁₆	D₂₃₆	1	2	yes	—
80	C₂³	—	236	2	no	—
81	Q₈	—	236	2	no	—
82	Q₈	—	236	2	no	—
83	D₄	—	236	2	no	—
84	D₄	—	236	2	no	—
85	C₂×C₄	—	236	2	no	—
86	C₂×C₄	D₄₇₂	1	2	yes	—
87	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
88	C₈	D₄₇₂	1	2	yes	—
89	C₈	C₂×D₂₃₆	1	1	yes	characteristic
90	C₈	C₂×D₂₃₆	1	1	yes	characteristic
91	C₂²	—	472	2	no	—
92	C₂²	—	472	4	no	—
93	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	472	2	no	—
96	C₄	#14273724	1	1	yes	characteristic
97	C₄	C₂×D₄₇₂	1	1	yes	characteristic
98	C₂	—	944	2	no	—
99	C₂	#16837779	1	1	yes	characteristic
100	C₂	#16837834	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₇₂

C₂

yes

maximal, characteristic, complemented

C₂×#14273672

C₂

yes

maximal, complemented

C₂×D₉₄₄

C₂

yes

maximal, characteristic

#16837786

C₂

yes

maximal, complemented

C₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

—

C₂×#11745660

—

C₂×D₄₇₂

C₂²

yes

—

#14273715

—

Dic₄₇₂

—

#14273672

—

D₉₄₄

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

(C₂×D₅₉)⋊C₄

—

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

—

#11745660

—

C₁₆×C₅₉

D₄

yes

—

C₁₆×C₅₉

D₄

yes

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

D₈

yes

—

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#127

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

C₄×C₁₆

D₅₉

yes

characteristic, complemented

C₅₉

#2249

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

118

—

C₂×D₈

—

#124

—

118

—

Q₃₂

—

118

—

SD₃₂

—

118

—

D₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₂×C₁₆

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

Q₁₆

—

118

—

Q₁₆

—

118

—

SD₁₆

—

236

—

D₈

—

118

—

D₈

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₂²⋊C₄

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

D₂₃₆

yes

—

C₁₆

D₂₃₆

yes

—

C₂³

—

236

—

Q₈

—

236

—

Q₈

—

236

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₈

D₄₇₂

yes

—

C₈

C₂×D₂₃₆

yes

characteristic

C₈

C₂×D₂₃₆

yes

characteristic

C₂²

—

472

—

C₂²

—

472

—

C₂²

C₂×D₄₇₂

yes

characteristic

C₄

—

C₄

—

472

—

C₄

#14273724

yes

characteristic

C₄

C₂×D₄₇₂

yes

characteristic

C₂

—

944

—

C₂

#16837779

yes

characteristic

100

C₂

#16837834

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₄×C₁₆	D₅₉
hypocenter	C₅₉	#2249
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₄×C₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₄×C₁₆

D₅₉

hypocenter

C₅₉

#2249

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. C₂×Dic₄₇₂	C₂	semidirect
2. C₂×#14273672	C₂	semidirect
4. #16837786	C₂	semidirect
5. C₄×C₁₆×C₅₉	C₂	semidirect
55. C₄×C₁₆	D₅₉	semidirect
56. C₅₉	#2249	semidirect

H ≅ G/N

type

1. C₂×Dic₄₇₂

C₂

semidirect

2. C₂×#14273672

C₂

semidirect

4. #16837786

C₂

semidirect

5. C₄×C₁₆×C₅₉

C₂

semidirect

55. C₄×C₁₆

D₅₉

semidirect

56. C₅₉

#2249

semidirect

Subgroups — #21426357

Conjugacy classes of subgroups

Complements