Group #21422080 — subgroups

Subgroups — #21422080

Order 7552.

114 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 31 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	#16837655	C₂	1	1	yes	maximal, characteristic
2	#16837184	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₇₂	C₄	1	1	yes	characteristic, complemented
5	#14273701	—	2	1	no	—
6	#14273667	—	2	1	no	—
7	#14273659	C₄	1	1	yes	characteristic, complemented
8	#14273649	—	2	1	no	—
9	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
10	#117×C₅₉	—	2	1	no	—
11	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₉	—	4	1	no	—
13	C₂×D₂₃₆	—	2	1	no	—
14	M₁₆×C₅₉	—	4	1	no	—
15	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
17	C₄⋊C₄×C₅₉	—	4	1	no	—
18	C₂²⋊C₄×C₅₉	—	4	1	no	—
19	C₄²×C₅₉	—	2	2	no	—
20	(C₂×D₅₉)⋊C₄	—	4	1	no	—
21	C₄⋊Dic₅₉	—	2	1	no	—
22	D₄₇₂	—	4	1	no	—
23	C₅₉⋊C₁₆	—	4	1	no	—
24	C₂³×C₅₉	D₈	1	1	yes	characteristic
25	C₂²×D₅₉	—	4	1	no	—
26	C₂×C₄×C₅₉	—	2	1	no	—
27	C₂×C₄×C₅₉	—	2	1	no	—
28	C₂×C₄×C₅₉	—	4	2	no	—
29	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
31	C₂²⋊D₅₉	—	8	2	no	—
32	C₂×Dic₅₉	—	4	1	no	—
33	D₂₃₆	—	4	1	no	—
34	D₂₃₆	—	4	1	no	—
35	C₈×C₅₉	—	2	1	no	—
36	C₈×C₅₉	—	2	1	no	—
37	C₂²×C₅₉	—	2	1	no	—
38	C₂²×C₅₉	—	4	1	no	—
39	C₂²×C₅₉	#102	1	1	yes	characteristic
40	C₂×D₅₉	—	8	1	no	—
41	C₂×D₅₉	—	8	1	no	—
42	C₂×D₅₉	—	8	1	no	—
43	C₄×C₅₉	—	2	1	no	—
44	C₄×C₅₉	—	8	2	no	—
45	C₄×C₅₉	#99	1	1	yes	characteristic
46	C₄×C₅₉	#104	1	1	yes	characteristic
47	Dic₅₉	—	8	1	no	—
48	#1339	—	59	1	no	maximal, Sylow
49	C₂×C₅₉	—	2	1	no	—
50	C₂×C₅₉	—	4	1	no	—
51	C₂×C₅₉	#327	1	1	yes	characteristic
52	D₅₉	—	16	1	no	—
53	#469	—	59	1	no	—
54	#349	—	59	1	no	—
55	H(3,ℤ/4)	D₅₉	1	1	yes	characteristic, complemented
56	C₅₉	#1339	1	1	yes	characteristic, complemented, Sylow
57	C₂×D₈	—	59	1	no	—
58	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
59	#121	—	118	1	no	—
60	#117	—	2	1	no	—
61	M₃₂	—	118	1	no	—
62	#106	—	59	1	no	—
63	#102	—	118	1	no	—
64	C₂×D₄	—	118	1	no	—
65	C₂×D₄	—	236	1	no	—
66	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
67	D₈	—	236	1	no	—
68	M₁₆	—	4	1	no	—
69	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
70	C₂×C₈	D₂₃₆	1	1	yes	characteristic
71	C₄⋊C₄	—	4	1	no	—
72	C₄⋊C₄	—	118	1	no	—
73	C₂²⋊C₄	—	4	1	no	—
74	C₂²⋊C₄	—	236	1	no	—
75	C₄²	—	2	2	no	—
76	C₁₆	—	236	1	no	—
77	C₂³	—	236	1	no	—
78	C₂³	#11745669	1	1	yes	characteristic
79	D₄	—	472	2	no	—
80	D₄	—	236	1	no	—
81	D₄	—	236	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	236	1	no	—
86	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	#11745671	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	472	1	no	—
93	C₂²	—	472	1	no	—
94	C₂²	—	472	1	no	—
95	C₂²	#14273647	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	8	2	no	—
98	C₄	—	472	1	no	—
99	C₄	#14273641	1	1	yes	characteristic
100	C₄	#14273655	1	1	yes	characteristic
101	C₂	—	2	1	no	—
102	C₂	—	4	1	no	—
103	C₂	—	944	1	no	—
104	C₂	#16837135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837655

C₂

yes

maximal, characteristic

#16837184

C₂

yes

maximal, characteristic, complemented

C₅₉×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₄

yes

characteristic, complemented

#14273701

—

#14273667

—

#14273659

C₄

yes

characteristic, complemented

#14273649

—

C₂×M₁₆×C₅₉

C₂²

yes

characteristic

#117×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×C₂²⋊D₅₉

—

C₂×D₂₃₆

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

D₄₇₂

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

D₈

yes

characteristic

C₂²×D₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₈×C₅₉

—

C₈×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#102

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

#1339

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#327

yes

characteristic

D₅₉

—

#469

—

#349

—

H(3,ℤ/4)

D₅₉

yes

characteristic, complemented

C₅₉

#1339

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

C₂×D₅₉

yes

characteristic

#121

—

118

—

#117

—

M₃₂

—

118

—

#106

—

#102

—

118

—

C₂×D₄

—

118

—

C₂×D₄

—

236

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

D₈

—

236

—

M₁₆

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

C₂²⋊C₄

—

236

—

C₄²

—

C₁₆

—

236

—

C₂³

—

236

—

C₂³

#11745669

yes

characteristic

D₄

—

472

—

D₄

—

236

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂×C₄

#11745671

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

472

—

C₂²

—

472

—

C₂²

—

472

—

C₂²

#14273647

yes

characteristic

C₄

—

C₄

—

C₄

—

472

—

C₄

#14273641

yes

characteristic

100

C₄

#14273655

yes

characteristic

101

C₂

—

102

C₂

—

103

C₂

—

944

—

104

C₂

#16837135

yes

characteristic

	subgroup	G/H
center	C₂	#16837135
commutator	C₂×C₈×C₅₉	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₅₉
Fitting	C₅₉×H(3,ℤ/4)	C₂
socle	C₂×C₅₉	#327
hypercenter	H(3,ℤ/4)	D₅₉
hypocenter	C₅₉	#1339
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₂

#16837135

commutator

C₂×C₈×C₅₉

C₂×C₄

Frattini

C₂×M₁₆

C₂×D₅₉

Fitting

C₅₉×H(3,ℤ/4)

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #16837184	C₂	semidirect
3. C₅₉×H(3,ℤ/4)	C₂	semidirect
4. C₂×D₄₇₂	C₄	semidirect
7. #14273659	C₄	semidirect
55. H(3,ℤ/4)	D₅₉	semidirect
56. C₅₉	#1339	semidirect

H ≅ G/N

type

2. #16837184

C₂

semidirect

3. C₅₉×H(3,ℤ/4)

C₂

semidirect

4. C₂×D₄₇₂

C₄

semidirect

7. #14273659

C₄

semidirect

55. H(3,ℤ/4)

D₅₉

semidirect

56. C₅₉

#1339

semidirect

Subgroups — #21422080

Conjugacy classes of subgroups

Complements