Group #21426328 — subgroups

Subgroups — #21426328

Order 7552.

138 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16837804	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837786	C₂	1	1	yes	maximal, characteristic
3	#16837207	C₂	1	2	yes	maximal, complemented
4	#16837204	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×#11745660	—	2	1	no	—
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	#14273719	—	2	1	no	—
10	#14273715	—	2	1	no	—
11	#14273661	C₂²	1	2	yes	—
12	#14273659	—	2	1	no	—
13	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×D₂₃₆	—	2	1	no	—
16	C₂×Dic₁₁₈	—	2	1	no	—
17	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
19	C₄²×C₅₉	D₄	1	1	yes	characteristic
20	(C₂×D₅₉)⋊C₄	—	4	2	no	—
21	C₄⋊Dic₅₉	—	2	2	no	—
22	Dic₅₉⋊C₄	—	4	2	no	—
23	Dic₂₃₆	—	2	2	no	—
24	D₄₇₂	—	2	2	no	—
25	#11745660	—	4	2	no	—
26	C₁₆×C₅₉	—	2	2	no	—
27	C₂²×D₅₉	—	4	1	no	—
28	C₂×C₄×C₅₉	D₈	1	2	yes	—
29	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₅₉	—	4	2	no	—
31	C₂×Dic₅₉	—	4	1	no	—
32	D₂₃₆	—	4	1	no	—
33	D₂₃₆	—	4	1	no	—
34	Dic₁₁₈	—	4	1	no	—
35	Dic₁₁₈	—	4	1	no	—
36	C₈×C₅₉	SD₁₆	1	2	yes	—
37	C₈×C₅₉	#41	1	2	yes	—
38	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
39	C₂×D₅₉	—	8	1	no	—
40	C₂×D₅₉	—	8	2	no	—
41	C₄×C₅₉	—	2	2	no	—
42	C₄×C₅₉	#124	1	1	yes	characteristic
43	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
44	Dic₅₉	—	8	1	no	—
45	Dic₅₉	—	8	2	no	—
46	#2242	—	59	1	no	maximal, Sylow
47	C₂×C₅₉	#486	1	1	yes	characteristic
48	C₂×C₅₉	#508	1	2	yes	—
49	D₅₉	—	16	1	no	—
50	#499	—	59	1	no	—
51	#495	—	59	1	no	—
52	#358	—	59	2	no	—
53	#357	—	59	2	no	—
54	C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
55	C₅₉	#2242	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	—	59	1	no	—
57	C₂×SD₁₆	—	118	1	no	—
58	C₂×D₈	—	59	1	no	—
59	#125	—	118	1	no	—
60	#124	—	118	1	no	—
61	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
62	#107	—	59	2	no	—
63	#106	—	118	1	no	—
64	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
65	C₂×Q₈	—	118	1	no	—
66	C₂×D₄	—	118	1	no	—
67	Q₁₆	—	118	2	no	—
68	SD₁₆	—	236	2	no	—
69	D₈	—	118	2	no	—
70	C₂×C₈	D₂₃₆	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
72	C₄⋊C₄	—	118	2	no	—
73	C₄⋊C₄	—	236	2	no	—
74	C₂²⋊C₄	—	236	2	no	—
75	C₄²	D₂₃₆	1	1	yes	characteristic
76	C₁₆	—	2	2	no	—
77	C₂³	—	236	1	no	—
78	Q₈	—	236	1	no	—
79	Q₈	—	236	1	no	—
80	D₄	—	236	1	no	—
81	D₄	—	236	1	no	—
82	C₂×C₄	—	236	1	no	—
83	C₂×C₄	—	236	2	no	—
84	C₂×C₄	D₄₇₂	1	2	yes	—
85	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
86	C₈	#11745660	1	2	yes	—
87	C₈	#11745685	1	2	yes	—
88	C₂²	—	472	1	no	—
89	C₂²	—	472	2	no	—
90	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	472	1	no	—
93	C₄	—	472	2	no	—
94	C₄	#14273715	1	1	yes	characteristic
95	C₄	C₂×#11745660	1	1	yes	characteristic
96	C₂	—	944	1	no	—
97	C₂	#16837749	1	1	yes	characteristic
98	C₂	#16837834	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837804

C₂

yes

maximal, characteristic, complemented

#16837786

C₂

yes

maximal, characteristic

#16837207

C₂

yes

maximal, complemented

#16837204

C₂

yes

maximal

C₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₂×#11745660

—

C₂×D₄₇₂

C₂²

yes

—

—

C₂²

yes

—

#14273659

—

C₂×C₁₆×C₅₉

C₂²

yes

—

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

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

Dic₂₃₆

—

D₄₇₂

—

#11745660

—

C₁₆×C₅₉

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

SD₁₆

yes

—

C₈×C₅₉

#41

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#124

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

D₅₉

—

—

—

—

—

C₄×C₁₆

D₅₉

yes

characteristic, complemented

C₅₉

#2242

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

118

—

C₂×D₈

—

#125

—

118

—

#124

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

—

#107

—

#106

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×D₄

—

118

—

Q₁₆

—

118

—

SD₁₆

—

236

—

D₈

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₂²⋊C₄

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

C₂³

—

236

—

Q₈

—

236

—

Q₈

—

236

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₈

#11745660

yes

—

C₈

#11745685

yes

—

C₂²

—

472

—

C₂²

—

472

—

C₂²

C₂×D₄₇₂

yes

characteristic

C₄

—

C₄

—

472

—

C₄

—

472

—

C₄

yes

C₄

yes

C₂

—

944

—

C₂

#16837749

yes

characteristic

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	#2242
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

#2242

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. #16837804	C₂	semidirect
3. #16837207	C₂	semidirect
5. C₄×C₁₆×C₅₉	C₂	semidirect
54. C₄×C₁₆	D₅₉	semidirect
55. C₅₉	#2242	semidirect

H ≅ G/N

type

1. #16837804

C₂

semidirect

3. #16837207

C₂

semidirect

5. C₄×C₁₆×C₅₉

C₂

semidirect

54. C₄×C₁₆

D₅₉

semidirect

55. C₅₉

#2242

semidirect

Subgroups — #21426328

Conjugacy classes of subgroups

Complements