Group #21426249 — subgroups

Subgroups — #21426249

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	#16837807	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#16837228	C₂	1	2	yes	maximal
4	#16837207	C₂	1	2	yes	maximal
5	C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	#14273727	—	2	1	no	—
8	C₄×Dic₁₁₈	—	2	1	no	—
9	#14273661	—	2	1	no	—
10	#14273661	C₂²	1	2	yes	—
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273652	—	2	1	no	—
13	C₂×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₁₈	—	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	1	no	—
21	C₄⋊Dic₅₉	—	2	2	no	—
22	C₄⋊Dic₅₉	—	4	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	C₁₆×C₅₉	—	2	2	no	—
28	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₉	—	4	1	no	—
32	C₂×Dic₅₉	—	4	1	no	—
33	C₂×Dic₅₉	—	4	2	no	—
34	Dic₁₁₈	—	2	2	no	—
35	Dic₁₁₈	—	4	1	no	—
36	Dic₁₁₈	—	8	2	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	Q₃₂	1	2	yes	—
43	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
44	C₄×C₅₉	#137	1	1	yes	characteristic
45	Dic₅₉	—	4	2	no	—
46	Dic₅₉	—	8	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	—	59	1	no	—
53	C₄×Q₁₆	—	59	1	no	—
54	#366	—	59	2	no	—
55	C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
56	#358	—	59	2	no	—
57	C₅₉	#2227	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	59	1	no	—
59	C₄⋊Q₈	—	118	1	no	—
60	C₄×Q₈	—	118	1	no	—
61	C₂×C₁₆	C₂×D₅₉	1	2	yes	—
62	#107	—	59	2	no	—
63	#107	—	59	1	no	—
64	#107	—	118	1	no	—
65	#103	—	118	1	no	—
66	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
67	C₂×Q₈	—	118	1	no	—
68	C₂×Q₈	—	236	1	no	—
69	Q₁₆	—	59	2	no	—
70	Q₁₆	—	118	1	no	—
71	C₂×C₈	D₂₃₆	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
73	C₄⋊C₄	—	118	1	no	—
74	C₄⋊C₄	—	118	2	no	—
75	C₄⋊C₄	—	236	1	no	—
76	C₄⋊C₄	—	236	1	no	—
77	C₄²	—	236	1	no	—
78	C₄²	D₂₃₆	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	Q₈	—	118	2	no	—
81	Q₈	—	472	2	no	—
82	Q₈	—	236	1	no	—
83	C₂×C₄	—	236	1	no	—
84	C₂×C₄	—	236	2	no	—
85	C₂×C₄	—	236	1	no	—
86	C₂×C₄	D₄₇₂	1	1	yes	characteristic
87	C₂×C₄	D₄₇₂	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	#11745687	1	1	yes	characteristic
91	C₈	Q₈×D₅₉	1	1	yes	characteristic
92	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	472	2	no	—
95	C₄	—	472	1	no	—
96	C₄	—	236	2	no	—
97	C₄	Dic₄₇₂	1	2	yes	—
98	C₄	#14273707	1	1	yes	characteristic
99	C₄	#14273760	1	1	yes	characteristic
100	C₂	#16837683	1	1	yes	characteristic
101	C₂	C₂×Dic₄₇₂	1	1	yes	characteristic
102	C₂	#16837840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic

C₄×Dic₂₃₆

C₂

yes

maximal, characteristic

#16837228

C₂

yes

maximal

#16837207

C₂

yes

maximal

C₅₉×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₃₆

C₂²

yes

characteristic

#14273727

—

C₄×Dic₁₁₈

—

#14273661

—

#14273661

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273652

—

C₂×C₁₆×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

Q₈

yes

complemented

C₁₆×C₅₉

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

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

Q₃₂

yes

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

#137

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂×Q₃₂

yes

C₂×C₅₉

yes

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

D₅₉

yes

characteristic

#358

—

C₅₉

#2227

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

118

—

C₄×Q₈

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

—

#107

—

#107

—

#107

—

118

—

#103

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×Q₈

—

236

—

Q₁₆

—

Q₁₆

—

118

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄⋊C₄

—

236

—

C₄²

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

Q₈

—

118

—

Q₈

—

472

—

Q₈

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

characteristic

C₂×C₄

D₄₇₂

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₈

—

C₈

#11745687

yes

characteristic

C₈

Q₈×D₅₉

yes

characteristic

C₂²

C₂×D₄₇₂

yes

characteristic

C₄

—

C₄

—

472

—

C₄

—

472

—

C₄

—

236

—

C₄

Dic₄₇₂

yes

—

C₄

yes

C₄

yes

100

C₂

#16837683

yes

characteristic

101

C₂

C₂×Dic₄₇₂

yes

characteristic

102

C₂

#16837840

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#2227
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×C₄⋊C₁₆

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₅₉

hypocenter

C₅₉

#2227

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

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 — #21426249

Conjugacy classes of subgroups

Complements