Group #21426268 — subgroups

Subgroups — #21426268

Order 7552.

126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16837800	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837520	C₂	1	1	yes	maximal, characteristic
3	#16837225	C₂	1	2	yes	maximal, complemented
4	#16837213	C₂	1	2	yes	maximal
5	C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273749	C₂²	1	1	yes	characteristic
7	#14273733	—	2	1	no	—
8	#14273727	—	2	1	no	—
9	#14273663	C₂²	1	1	yes	characteristic
10	#14273659	—	2	1	no	—
11	#14273659	C₂²	1	2	yes	—
12	#14273655	—	2	1	no	—
13	M₃₂×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	#11745685	—	2	1	no	—
16	C₂×Dic₁₁₈	—	4	1	no	—
17	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
19	C₄²×C₅₉	D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₉	—	2	2	no	—
21	C₄⋊Dic₅₉	—	4	1	no	—
22	#11745664	—	2	1	no	—
23	Dic₂₃₆	Q₈	1	1	yes	characteristic, complemented
24	D₄₇₂	Q₈	1	1	yes	characteristic, complemented
25	#11745660	—	2	1	no	—
26	#11745659	—	4	1	no	—
27	C₈×D₅₉	—	4	1	no	—
28	C₁₆×C₅₉	—	2	2	no	—
29	C₂×C₄×C₅₉	—	2	1	no	—
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂²⋊D₅₉	—	4	1	no	—
32	C₂×Dic₅₉	—	4	2	no	—
33	D₂₃₆	—	2	1	no	—
34	C₄×D₅₉	—	4	1	no	—
35	Dic₁₁₈	—	2	1	no	—
36	Dic₁₁₈	—	8	2	no	—
37	C₈×C₅₉	SD₁₆	1	1	yes	characteristic
38	C₈×C₅₉	SD₁₆	1	1	yes	characteristic
39	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₅₉	#41	1	1	yes	characteristic
41	C₅₉⋊C₈	—	4	1	no	—
42	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂×D₅₉	—	4	1	no	—
44	C₄×C₅₉	—	2	2	no	—
45	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
46	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
47	Dic₅₉	—	4	1	no	—
48	Dic₅₉	—	8	2	no	—
49	#2230	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	—	2	1	no	—
51	C₂×C₅₉	#476	1	1	yes	characteristic
52	D₅₉	—	8	1	no	—
53	#498	—	59	1	no	—
54	#443	—	59	1	no	—
55	#365	—	59	2	no	—
56	#364	D₅₉	1	1	yes	characteristic, complemented
57	#360	—	59	2	no	—
58	C₅₉	#2230	1	1	yes	characteristic, complemented, Sylow
59	#135	—	59	1	no	—
60	#131	—	118	1	no	—
61	C₄⋊Q₈	—	118	1	no	—
62	M₃₂	C₂×D₅₉	1	2	yes	—
63	#108	—	59	1	no	—
64	#106	—	59	2	no	—
65	#106	—	118	1	no	—
66	#104	—	118	1	no	—
67	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
68	#41	—	118	1	no	—
69	C₂×Q₈	—	236	1	no	—
70	Q₁₆	—	59	1	no	—
71	SD₁₆	—	118	1	no	—
72	D₈	—	59	1	no	—
73	M₁₆	—	118	1	no	—
74	M₁₆	—	236	1	no	—
75	C₂×C₈	—	236	1	no	—
76	C₂×C₈	D₂₃₆	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
78	C₄⋊C₄	—	118	2	no	—
79	C₄⋊C₄	—	236	1	no	—
80	C₄²	D₂₃₆	1	1	yes	characteristic
81	C₁₆	—	2	2	no	—
82	Q₈	—	118	1	no	—
83	Q₈	—	472	2	no	—
84	D₄	—	118	1	no	—
85	D₄	—	236	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	236	2	no	—
88	C₂×C₄	—	236	1	no	—
89	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
90	C₈	—	236	1	no	—
91	C₈	#11745660	1	1	yes	characteristic
92	C₈	#11745660	1	1	yes	characteristic
93	C₈	#11745687	1	1	yes	characteristic
94	C₈	Q₈×D₅₉	1	1	yes	characteristic
95	C₂²	—	236	1	no	—
96	C₂²	#14273755	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	472	2	no	—
99	C₄	—	236	1	no	—
100	C₄	#14273707	1	1	yes	characteristic
101	C₄	C₂×#11745660	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	472	1	no	—
104	C₂	#16837693	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837800

C₂

yes

maximal, characteristic, complemented

#16837520

C₂

yes

maximal, characteristic

#16837225

C₂

yes

maximal, complemented

#16837213

C₂

yes

maximal

C₅₉×#364

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

—

—

C₂²

yes

characteristic

#14273659

—

#14273659

C₂²

yes

—

#14273655

—

M₃₂×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

#11745685

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

#11745664

—

Dic₂₃₆

Q₈

yes

characteristic, complemented

D₄₇₂

Q₈

yes

characteristic, complemented

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

SD₁₆

yes

characteristic

C₈×C₅₉

SD₁₆

yes

characteristic

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2230

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#476

yes

characteristic

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#360

—

C₅₉

#2230

yes

characteristic, complemented, Sylow

#135

—

#131

—

118

—

C₄⋊Q₈

—

118

—

M₃₂

C₂×D₅₉

yes

—

#108

—

#106

—

#106

—

118

—

#104

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

#41

—

118

—

C₂×Q₈

—

236

—

Q₁₆

—

SD₁₆

—

118

—

D₈

—

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

Q₈

—

118

—

Q₈

—

472

—

D₄

—

118

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₈

—

236

—

C₈

yes

C₈

yes

C₈

yes

C₈

Q₈×D₅₉

yes

characteristic

C₂²

—

236

—

C₂²

#14273755

yes

characteristic

C₄

—

C₄

—

472

—

C₄

—

236

—

100

C₄

#14273707

yes

characteristic

101

C₄

C₂×#11745660

yes

characteristic

102

C₂

—

103

C₂

—

472

—

104

C₂

#16837693

yes

characteristic

	subgroup	G/H
center	C₂	#16837693
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#364	C₂
socle	C₂×C₅₉	#476
hypercenter	#364	D₅₉
hypocenter	C₅₉	#2230
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#16837693

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#364

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #16837800	C₂	semidirect
3. #16837225	C₂	semidirect
5. C₅₉×#364	C₂	semidirect
23. Dic₂₃₆	Q₈	semidirect
24. D₄₇₂	Q₈	semidirect
56. #364	D₅₉	semidirect
58. C₅₉	#2230	semidirect

H ≅ G/N

type

1. #16837800

C₂

semidirect

3. #16837225

C₂

semidirect

5. C₅₉×#364

C₂

semidirect

23. Dic₂₃₆

Q₈

semidirect

24. D₄₇₂

Q₈

semidirect

56. #364

D₅₉

semidirect

58. C₅₉

#2230

semidirect

Subgroups — #21426268

Conjugacy classes of subgroups

Complements