Group #21422212 — subgroups

Subgroups — #21422212

Order 7552.

128 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14273667	C₂	1	1	yes	maximal, characteristic
2	C₂×#14273663	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#427	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	1	no	—
5	#14273667	—	2	1	no	—
6	#14273667	C₄	1	2	yes	complemented
7	C₂×C₅₉⋊C₁₆	—	2	1	no	—
8	#14273663	—	2	1	no	—
9	#14273663	C₄	1	2	yes	complemented
10	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	#117×C₅₉	—	2	1	no	—
12	#107×C₅₉	—	2	1	no	—
13	#106×C₅₉	C₄	1	2	yes	—
14	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	Q₈	1	1	yes	characteristic
21	C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₂²⋊C₄×C₅₉	—	4	1	no	—
23	C₄²×C₅₉	—	4	1	no	—
24	#11745664	—	2	2	no	—
25	#11745664	—	4	1	no	—
26	C₂×C₅₉⋊C₈	—	4	1	no	—
27	C₅₉⋊C₁₆	—	2	2	no	—
28	C₂³×C₅₉	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	4	2	no	—
30	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
34	C₂×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₉	Q₁₆	1	1	yes	characteristic
36	C₈×C₅₉	C₄²	1	1	yes	characteristic
37	C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
40	C₅₉⋊C₈	—	4	2	no	—
41	C₂²×C₅₉	—	4	1	no	—
42	C₂²×C₅₉	#102	1	1	yes	characteristic
43	C₂²×C₅₉	#103	1	1	yes	characteristic
44	C₂²×C₅₉	#106	1	1	yes	characteristic
45	C₄×C₅₉	—	8	2	no	—
46	C₄×C₅₉	#95	1	1	yes	characteristic
47	C₄×C₅₉	#102	1	1	yes	characteristic
48	C₄×C₅₉	#103	1	1	yes	characteristic
49	C₄×C₅₉	#107	1	1	yes	characteristic
50	#1389	—	59	1	no	maximal, Sylow
51	C₂×C₅₉	—	2	1	no	—
52	C₂×C₅₉	—	2	1	no	—
53	C₂×C₅₉	—	2	1	no	—
54	C₂×C₅₉	#340	1	1	yes	characteristic
55	C₂×M₃₂	—	59	1	no	—
56	C₂×#108	—	59	1	no	—
57	#427	D₅₉	1	1	yes	characteristic
58	C₅₉	#1389	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	118	1	no	—
60	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	M₃₂	—	59	2	no	—
63	M₃₂	—	118	1	no	—
64	C₂×C₁₆	—	118	1	no	—
65	#108	—	59	2	no	—
66	#108	—	118	1	no	—
67	#107	—	2	1	no	—
68	#106	Dic₅₉	1	2	yes	—
69	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
70	M₁₆	—	118	2	no	—
71	M₁₆	—	236	1	no	—
72	C₂×C₈	—	236	1	no	—
73	C₂×C₈	Dic₁₁₈	1	1	yes	characteristic
74	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
75	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
76	C₂×C₈	D₂₃₆	1	1	yes	characteristic
77	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₂²⋊C₄	—	4	1	no	—
81	C₄²	—	4	1	no	—
82	C₁₆	—	118	2	no	—
83	C₂³	#11745670	1	1	yes	characteristic
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	#11745669	1	1	yes	characteristic
88	C₂×C₄	#11745671	1	1	yes	characteristic
89	C₂×C₄	#11745672	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
91	C₈	—	236	2	no	—
92	C₈	C₄×Dic₅₉	1	1	yes	characteristic
93	C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
94	C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
95	C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
96	C₂²	—	4	1	no	—
97	C₂²	#14273648	1	1	yes	characteristic
98	C₂²	#14273650	1	1	yes	characteristic
99	C₂²	#14273658	1	1	yes	characteristic
100	C₄	—	8	2	no	—
101	C₄	C₅₉⋊#95	1	1	yes	characteristic
102	C₄	#14273647	1	1	yes	characteristic
103	C₄	#14273651	1	1	yes	characteristic
104	C₄	#14273660	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	—	2	1	no	—
108	C₂	#16837164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273667

C₂

yes

maximal, characteristic

C₂×#14273663

C₂

yes

maximal, characteristic

C₅₉×#427

C₂

yes

maximal, characteristic

C₂×#11745664

—

#14273667

—

#14273667

C₄

yes

complemented

C₂×C₅₉⋊C₁₆

—

#14273663

—

#14273663

C₄

yes

complemented

C₂²×C₈×C₅₉

C₂²

yes

characteristic

#117×C₅₉

—

#107×C₅₉

—

#106×C₅₉

C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

characteristic

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

characteristic

C₂×C₄×C₅₉

Q₁₆

yes

characteristic

C₈×C₅₉

C₄²

yes

characteristic

C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#340

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

#427

D₅₉

yes

characteristic

C₅₉

#1389

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

118

—

#108

—

#108

—

118

—

#107

—

#106

Dic₅₉

yes

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₂×C₈

Dic₁₁₈

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

118

—

C₂³

#11745670

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

C₈

—

236

—

C₈

C₄×Dic₅₉

yes

characteristic

C₈

Dic₅₉⋊C₄

yes

characteristic

C₈

C₄⋊Dic₅₉

yes

characteristic

C₈

(C₂×D₅₉)⋊C₄

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

101

C₄

C₅₉⋊#95

yes

characteristic

102

C₄

#14273647

yes

characteristic

103

C₄

#14273651

yes

characteristic

104

C₄

#14273660

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

108

C₂

#16837164

yes

characteristic

	subgroup	G/H
center	C₄	#14273647
commutator	C₈×C₅₉	C₄²
Frattini	C₂²×C₈	C₂×D₅₉
Fitting	C₅₉×#427	C₂
socle	C₂×C₅₉	#340
hypercenter	#427	D₅₉
hypocenter	C₅₉	#1389
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₄

#14273647

commutator

C₈×C₅₉

C₄²

Frattini

C₂²×C₈

C₂×D₅₉

Fitting

C₅₉×#427

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
6. #14273667	C₄	semidirect
9. #14273663	C₄	semidirect
58. C₅₉	#1389	semidirect

H ≅ G/N

type

6. #14273667

C₄

semidirect

9. #14273663

C₄

semidirect

58. C₅₉

#1389

semidirect

Subgroups — #21422212

Conjugacy classes of subgroups

Complements