Group #21426218 — subgroups

Subgroups — #21426218

Order 7552.

140 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16837818	C₂	1	1	yes	maximal, characteristic
2	#16837674	C₂	1	2	yes	maximal
3	#16837232	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837218	C₂	1	2	yes	maximal, complemented
5	C₂×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273758	—	2	2	no	—
7	#14273733	C₂²	1	1	yes	characteristic
8	#14273667	C₂²	1	1	yes	characteristic
9	#14273666	—	2	1	no	—
10	C₁₆×D₅₉	—	2	1	no	—
11	#14273663	C₂²	1	2	yes	—
12	#14273646	—	2	2	no	—
13	C₂×Q₁₆×C₅₉	C₂²	1	2	yes	—
14	Q₃₂×C₅₉	—	2	1	no	—
15	Q₃₂×C₅₉	—	2	1	no	—
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	—	2	2	no	—
18	#11745685	D₄	1	1	yes	characteristic
19	Q₁₆×C₅₉	—	2	2	no	—
20	Q₁₆×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	#11745672	—	4	2	no	—
23	#11745671	—	4	2	no	—
24	#11745664	—	2	2	no	—
25	#11745664	D₄	1	1	yes	characteristic
26	#11745659	—	2	1	no	—
27	C₈×D₅₉	—	2	1	no	—
28	C₁₆×C₅₉	D₄	1	1	yes	characteristic
29	C₁₆×C₅₉	D₄	1	1	yes	characteristic
30	C₅₉⋊C₁₆	—	2	1	no	—
31	Q₈×C₅₉	—	4	2	no	—
32	Q₈×C₅₉	—	4	2	no	—
33	C₂×C₄×C₅₉	—	4	2	no	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²⋊D₅₉	—	2	1	no	—
36	D₂₃₆	D₈	1	1	yes	characteristic
37	C₄×D₅₉	—	2	1	no	—
38	Dic₁₁₈	D₈	1	1	yes	characteristic
39	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₈×C₅₉	#41	1	1	yes	characteristic
41	C₅₉⋊C₈	—	2	1	no	—
42	C₅₉⋊C₈	—	4	2	no	—
43	C₂²×C₅₉	#135	1	1	yes	characteristic
44	C₂×D₅₉	—	2	1	no	—
45	C₄×C₅₉	—	8	2	no	—
46	C₄×C₅₉	#121	1	1	yes	characteristic
47	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
48	Dic₅₉	—	2	1	no	—
49	#2222	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	—	2	1	no	—
51	C₂×C₅₉	#466	1	1	yes	characteristic
52	D₅₉	—	4	1	no	—
53	C₂×Q₃₂	D₅₉	1	1	yes	characteristic, complemented
54	#504	—	59	1	no	—
55	#473	—	59	2	no	—
56	#368	—	59	1	no	—
57	#362	—	59	2	no	—
58	C₅₉	#2222	1	1	yes	characteristic, complemented, Sylow
59	#137	—	118	2	no	—
60	C₂×Q₁₆	C₂×D₅₉	1	2	yes	—
61	#131	—	59	1	no	—
62	Q₃₂	—	2	1	no	—
63	Q₃₂	—	2	1	no	—
64	M₃₂	—	59	1	no	—
65	M₃₂	—	118	1	no	—
66	C₂×C₁₆	—	118	1	no	—
67	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
68	#108	—	59	2	no	—
69	#101	—	118	2	no	—
70	#41	—	59	1	no	—
71	C₂×Q₈	—	2	2	no	—
72	Q₁₆	—	2	2	no	—
73	Q₁₆	—	2	2	no	—
74	Q₁₆	—	236	2	no	—
75	SD₁₆	—	236	2	no	—
76	M₁₆	—	59	1	no	—
77	M₁₆	—	118	1	no	—
78	M₁₆	—	118	2	no	—
79	C₂×C₈	—	118	1	no	—
80	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
81	C₁₆	—	118	1	no	—
82	C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
83	C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
84	Q₈	—	4	2	no	—
85	Q₈	—	4	2	no	—
86	Q₈	—	59	1	no	—
87	D₄	—	59	1	no	—
88	D₄	—	118	1	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	118	1	no	—
91	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
92	C₈	—	118	1	no	—
93	C₈	—	236	2	no	—
94	C₈	#11745687	1	1	yes	characteristic
95	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
96	C₂²	—	118	1	no	—
97	C₂²	#14273746	1	1	yes	characteristic
98	C₄	—	8	2	no	—
99	C₄	—	118	1	no	—
100	C₄	#14273698	1	1	yes	characteristic
101	C₄	D₈×D₅₉	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	236	1	no	—
104	C₂	#16837635	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837818

C₂

yes

maximal, characteristic

#16837674

C₂

yes

maximal

#16837232

C₂

yes

maximal, characteristic, complemented

#16837218

C₂

yes

maximal, complemented

C₂×Q₃₂×C₅₉

C₂

yes

maximal, characteristic, complemented

#14273758

—

#14273733

C₂²

yes

characteristic

#14273667

C₂²

yes

characteristic

#14273666

—

C₁₆×D₅₉

—

#14273663

C₂²

yes

—

#14273646

—

C₂×Q₁₆×C₅₉

C₂²

yes

—

Q₃₂×C₅₉

—

Q₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

#11745685

D₄

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

—

—

D₄

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

D₄

yes

characteristic

C₁₆×C₅₉

D₄

yes

characteristic

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

D₂₃₆

D₈

yes

characteristic

C₄×D₅₉

—

Dic₁₁₈

D₈

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

#135

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

#2222

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#466

yes

characteristic

D₅₉

—

C₂×Q₃₂

D₅₉

yes

characteristic, complemented

—

—

—

—

C₅₉

#2222

yes

characteristic, complemented, Sylow

#137

—

118

—

C₂×Q₁₆

C₂×D₅₉

yes

—

#131

—

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#108

—

#101

—

118

—

#41

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

236

—

SD₁₆

—

236

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₁₆

—

118

—

C₁₆

C₂²⋊D₅₉

yes

characteristic

C₁₆

C₂²⋊D₅₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

118

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₈

—

118

—

C₈

—

236

—

C₈

#11745687

yes

characteristic

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

C₂²

—

118

—

C₂²

#14273746

yes

characteristic

C₄

—

C₄

—

118

—

100

C₄

#14273698

yes

characteristic

101

C₄

D₈×D₅₉

yes

characteristic

102

C₂

—

103

C₂

—

236

—

104

C₂

#16837635

yes

characteristic

	subgroup	G/H
center	C₂	#16837635
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×Q₃₂×C₅₉	C₂
socle	C₂×C₅₉	#466
hypercenter	C₂×Q₃₂	D₅₉
hypocenter	C₅₉	#2222
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂

#16837635

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×Q₃₂×C₅₉

C₂

socle

C₂×C₅₉

#466

hypercenter

C₂×Q₃₂

D₅₉

hypocenter

C₅₉

#2222

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
3. #16837232	C₂	semidirect
4. #16837218	C₂	semidirect
5. C₂×Q₃₂×C₅₉	C₂	semidirect
53. C₂×Q₃₂	D₅₉	semidirect
58. C₅₉	#2222	semidirect

H ≅ G/N

type

3. #16837232

C₂

semidirect

4. #16837218

C₂

semidirect

5. C₂×Q₃₂×C₅₉

C₂

semidirect

53. C₂×Q₃₂

D₅₉

semidirect

58. C₅₉

#2222

semidirect

Subgroups — #21426218

Conjugacy classes of subgroups

Complements