Group #21422994 — subgroups

Subgroups — #21422994

Order 7552.

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

normal subgroups only

This group has 152 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	#16837532	C₂	1	2	yes	maximal
2	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#16837155	C₂	1	1	yes	maximal, characteristic
4	#16837134	C₂	1	2	yes	maximal
5	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₄	1	1	yes	characteristic
7	C₄×Dic₁₁₈	C₂²	1	2	yes	—
8	#14273661	C₄	1	1	yes	characteristic
9	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	Dic₅₉⋊C₈	—	2	2	no	—
11	#14273652	C₄	1	2	yes	—
12	#14273637	—	2	2	no	—
13	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₁₈	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₉	—	2	1	no	—
18	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄²×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
24	Dic₅₉⋊C₄	—	2	2	no	—
25	C₄×Dic₅₉	—	2	2	no	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	Dic₂₃₆	—	2	2	no	—
28	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
31	C₂×Dic₅₉	—	2	2	no	—
32	C₂×Dic₅₉	—	2	2	no	—
33	Dic₁₁₈	—	2	2	no	—
34	Dic₁₁₈	—	2	2	no	—
35	C₈×C₅₉	—	2	2	no	—
36	C₈×C₅₉	—	2	1	no	—
37	C₈×C₅₉	—	2	1	no	—
38	C₈×C₅₉	M₁₆	1	2	yes	—
39	C₈×C₅₉	Q₁₆	1	2	yes	complemented
40	C₅₉⋊C₈	—	4	2	no	—
41	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
42	C₄×C₅₉	—	2	1	no	—
43	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₅₉	#131	1	1	yes	characteristic
45	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₅₉	#135	1	1	yes	characteristic
47	Dic₅₉	—	4	2	no	—
48	Dic₅₉	—	4	2	no	—
49	#1591	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	#436	1	1	yes	characteristic
51	C₂×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
52	C₂×C₅₉	#443	1	1	yes	characteristic
53	#446	—	59	2	no	—
54	C₄×Q₁₆	—	59	1	no	—
55	#335	—	59	1	no	—
56	Q₈⋊C₈	—	59	2	no	—
57	C₈²	D₅₉	1	1	yes	characteristic
58	C₅₉	#1591	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	59	1	no	—
60	C₄×Q₈	—	59	2	no	—
61	#107	—	59	1	no	—
62	C₄⋊C₈	—	59	2	no	—
63	C₄⋊C₈	—	118	2	no	—
64	#103	—	59	2	no	—
65	#97	—	118	2	no	—
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
68	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
69	C₂×Q₈	—	59	2	no	—
70	Q₁₆	—	118	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	118	2	no	—
73	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
74	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
75	C₂×C₈	D₂₃₆	1	1	yes	characteristic
76	C₂×C₈	D₂₃₆	1	1	yes	characteristic
77	C₄⋊C₄	—	59	2	no	—
78	C₄⋊C₄	—	118	2	no	—
79	C₄²	—	118	2	no	—
80	C₄²	C₂²×D₅₉	1	1	yes	characteristic
81	Q₈	—	118	2	no	—
82	Q₈	—	118	2	no	—
83	C₂×C₄	—	118	2	no	—
84	C₂×C₄	—	118	2	no	—
85	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
87	C₂×C₄	#11745685	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	236	2	no	—
92	C₈	#11745659	1	2	yes	—
93	C₈	Dic₂₃₆	1	2	yes	complemented
94	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	236	2	no	—
97	C₄	—	236	2	no	—
98	C₄	C₂×#11745659	1	1	yes	characteristic
99	C₄	#14273733	1	1	yes	characteristic
100	C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
101	C₄	#14273749	1	1	yes	characteristic
102	C₂	#16837481	1	1	yes	characteristic
103	C₂	C₄×Dic₂₃₆	1	1	yes	characteristic
104	C₂	#16837520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837532

C₂

yes

maximal

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837155

C₂

yes

maximal, characteristic

#16837134

C₂

yes

maximal

C₈²×C₅₉

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₄

yes

characteristic

C₄×Dic₁₁₈

C₂²

yes

—

#14273661

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

Dic₅₉⋊C₈

—

#14273652

C₄

yes

—

#14273637

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

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

C₄²×C₅₉

C₂³

yes

characteristic

C₄⋊Dic₅₉

C₂×C₄

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

—

C₈×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×Q₁₆

yes

C₂×C₅₉

yes

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

D₅₉

yes

characteristic

C₅₉

#1591

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#103

—

#97

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄²

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

118

—

Q₈

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745685

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

236

—

C₈

#11745659

yes

—

C₈

Dic₂₃₆

yes

complemented

C₂²

C₄×D₂₃₆

yes

characteristic

C₄

—

C₄

—

236

—

C₄

—

236

—

C₄

yes

C₄

yes

100

C₄

C₂×Dic₂₃₆

yes

characteristic

101

C₄

#14273749

yes

characteristic

102

C₂

#16837481

yes

characteristic

103

C₂

C₄×Dic₂₃₆

yes

characteristic

104

C₂

#16837520

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1591
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈²×C₅₉

C₂

socle

C₂²×C₅₉

C₄×D₄

hypercenter

C₈²

D₅₉

hypocenter

C₅₉

#1591

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
39. C₈×C₅₉	Q₁₆	semidirect
58. C₅₉	#1591	semidirect
93. C₈	Dic₂₃₆	semidirect

H ≅ G/N

type

39. C₈×C₅₉

Q₁₆

semidirect

58. C₅₉

#1591

semidirect

93. C₈

Dic₂₃₆

semidirect

Subgroups — #21422994

Conjugacy classes of subgroups

Complements