Group #21426245 — subgroups

Subgroups — #21426245

Order 7552.

130 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#16837227	C₂	1	2	yes	maximal
3	C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#16837205	C₂	1	2	yes	maximal
5	C₅₉×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₁₈	—	2	1	no	—
8	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
9	#14273661	C₂²	1	1	yes	characteristic
10	#14273652	—	2	1	no	—
11	C₄⋊Q₈×C₅₉	—	2	1	no	—
12	#107×C₅₉	—	2	1	no	—
13	#107×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₉	—	4	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	C₄⋊C₄×C₅₉	—	2	2	no	—
20	C₄⋊C₄×C₅₉	—	4	1	no	—
21	C₄²×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊Dic₅₉	—	2	1	no	—
23	Dic₅₉⋊C₄	—	4	1	no	—
24	C₄×Dic₅₉	—	4	1	no	—
25	Dic₂₃₆	—	2	1	no	—
26	Dic₂₃₆	Q₈	1	2	yes	complemented
27	C₅₉⋊C₁₆	—	2	2	no	—
28	Q₈×C₅₉	—	8	2	no	—
29	C₂×C₄×C₅₉	—	4	2	no	—
30	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
31	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅₉	—	4	1	no	—
34	C₂×Dic₅₉	—	4	1	no	—
35	Dic₁₁₈	—	2	2	no	—
36	Dic₁₁₈	—	4	1	no	—
37	C₈×C₅₉	—	2	1	no	—
38	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₅₉	#41	1	1	yes	characteristic
40	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
41	C₄×C₅₉	—	2	1	no	—
42	C₄×C₅₉	—	8	2	no	—
43	C₄×C₅₉	Q₃₂	1	2	yes	—
44	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
45	C₄×C₅₉	#137	1	1	yes	characteristic
46	Dic₅₉	—	4	2	no	—
47	Dic₅₉	—	8	1	no	—
48	#2227	—	59	1	no	maximal, Sylow
49	C₂×C₅₉	#474	1	1	yes	characteristic
50	C₂×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
51	C₂×C₅₉	#509	1	1	yes	characteristic
52	#500	D₅₉	1	1	yes	characteristic
53	C₄×Q₁₆	—	59	1	no	—
54	#366	—	59	2	no	—
55	C₄⋊C₁₆	—	59	1	no	—
56	#358	—	59	2	no	—
57	C₅₉	#2227	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	59	1	no	—
59	C₄⋊Q₈	—	2	1	no	—
60	C₄×Q₈	—	118	1	no	—
61	C₂×C₁₆	—	59	2	no	—
62	#107	—	2	1	no	—
63	#107	—	59	1	no	—
64	#107	C₂×D₅₉	1	2	yes	—
65	#103	—	118	1	no	—
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	118	1	no	—
69	Q₁₆	—	59	2	no	—
70	Q₁₆	—	118	1	no	—
71	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	118	1	no	—
76	C₄⋊C₄	—	236	1	no	—
77	C₄²	—	236	1	no	—
78	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
79	C₁₆	—	118	2	no	—
80	Q₈	—	8	2	no	—
81	Q₈	—	118	2	no	—
82	Q₈	—	236	1	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	236	1	no	—
85	C₂×C₄	—	236	1	no	—
86	C₂×C₄	#11745669	1	1	yes	characteristic
87	C₂×C₄	#11745669	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	Q₈×D₅₉	1	1	yes	characteristic
91	C₈	#11745689	1	1	yes	characteristic
92	C₂²	C₂×#11745669	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	8	2	no	—
95	C₄	—	472	1	no	—
96	C₄	—	236	2	no	—
97	C₄	#14273673	1	2	yes	—
98	C₄	#14273703	1	1	yes	characteristic
99	C₄	#14273758	1	1	yes	characteristic
100	C₂	#16837679	1	1	yes	characteristic
101	C₂	C₂×#14273673	1	1	yes	characteristic
102	C₂	#16837838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837227

C₂

yes

maximal

C₄⋊(C₅₉⋊C₁₆)

C₂

yes

maximal, characteristic

#16837205

C₂

yes

maximal

C₅₉×#500

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

C₄×Dic₁₁₈

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273652

—

C₄⋊Q₈×C₅₉

—

#107×C₅₉

—

#107×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

Q₈

yes

complemented

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

Q₃₂

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×Q₃₂

yes

C₂×C₅₉

yes

D₅₉

yes

characteristic

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

—

#358

—

C₅₉

#2227

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

118

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×D₅₉

yes

—

#103

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

118

—

Q₁₆

—

Q₁₆

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

—

236

—

C₄²

C₂²⋊D₅₉

yes

characteristic

C₁₆

—

118

—

Q₈

—

Q₈

—

118

—

Q₈

—

236

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

Q₈×D₅₉

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

472

—

C₄

—

236

—

C₄

#14273673

yes

—

C₄

yes

C₄

yes

100

C₂

#16837679

yes

characteristic

101

C₂

C₂×#14273673

yes

characteristic

102

C₂

#16837838

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#500	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	#500	D₅₉
hypocenter	C₅₉	#2227
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#500

C₂

socle

C₂²×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
26. Dic₂₃₆	Q₈	semidirect
57. C₅₉	#2227	semidirect

H ≅ G/N

type

26. Dic₂₃₆

Q₈

semidirect

57. C₅₉

#2227

semidirect

Subgroups — #21426245

Conjugacy classes of subgroups

Complements