Group #21423490 — subgroups

Subgroups — #21423490

Order 7552.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16837612	C₂	1	2	yes	maximal
3	#16837155	C₂	1	1	yes	maximal, characteristic
4	#16837147	C₂	1	2	yes	maximal
5	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745672	—	2	2	no	—
7	#14273727	C₂²	1	2	yes	—
8	#14273661	—	2	1	no	—
9	#14273661	—	2	1	no	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	#14273650	—	2	2	no	—
12	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₉	—	2	1	no	—
14	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	—	2	1	no	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	C₂×Dic₁₁₈	—	2	2	no	—
19	C₂×C₈×C₅₉	—	2	1	no	—
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₅₉	—	2	1	no	—
23	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	—	2	1	no	—
25	C₄²×C₅₉	C₂³	1	1	yes	characteristic
26	#11745672	—	4	4	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₅₉⋊C₈	—	2	2	no	—
30	Q₈×C₅₉	—	2	1	no	—
31	Q₈×C₅₉	D₈	1	2	yes	—
32	C₂×C₄×C₅₉	—	2	1	no	—
33	C₂×C₄×C₅₉	—	2	1	no	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
37	C₂×Dic₅₉	—	4	2	no	—
38	Dic₁₁₈	—	4	2	no	—
39	Dic₁₁₈	—	4	2	no	—
40	C₈×C₅₉	—	2	1	no	—
41	C₈×C₅₉	—	4	1	no	—
42	C₈×C₅₉	Q₁₆	1	2	yes	complemented
43	C₅₉⋊C₈	—	4	2	no	—
44	C₂²×C₅₉	#121	1	1	yes	characteristic
45	C₄×C₅₉	—	2	2	no	—
46	C₄×C₅₉	—	2	1	no	—
47	C₄×C₅₉	—	4	1	no	—
48	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
49	C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
50	C₄×C₅₉	#135	1	1	yes	characteristic
51	C₄×C₅₉	#137	1	1	yes	characteristic
52	Dic₅₉	—	8	2	no	—
53	#1674	—	59	1	no	maximal, Sylow
54	C₂×C₅₉	#462	1	1	yes	characteristic
55	C₂×C₅₉	#466	1	1	yes	characteristic
56	C₂×C₅₉	#473	1	1	yes	characteristic
57	#500	—	59	1	no	—
58	#462	—	59	2	no	—
59	C₈×Q₈	D₅₉	1	1	yes	characteristic
60	#335	—	59	1	no	—
61	#332	—	59	2	no	—
62	C₅₉	#1674	1	1	yes	characteristic, complemented, Sylow
63	C₂×Q₁₆	—	118	2	no	—
64	C₄⋊Q₈	—	59	2	no	—
65	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
66	#107	—	118	1	no	—
67	#107	—	118	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	59	2	no	—
70	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
71	#103	—	118	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
74	C₂×Q₈	—	118	2	no	—
75	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
76	Q₁₆	—	236	4	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	118	2	no	—
79	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	118	2	no	—
83	C₄⋊C₄	—	118	2	no	—
84	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
85	C₄²	—	2	1	no	—
86	C₄²	C₂²×D₅₉	1	1	yes	characteristic
87	Q₈	—	2	1	no	—
88	Q₈	—	236	2	no	—
89	Q₈	—	236	2	no	—
90	Q₈	D₄₇₂	1	2	yes	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	236	2	no	—
94	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
95	C₂×C₄	#11745685	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	4	1	no	—
99	C₈	—	236	2	no	—
100	C₈	#11745672	1	2	yes	complemented
101	C₂²	#14273701	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	472	2	no	—
106	C₄	C₂×D₄₇₂	1	1	yes	characteristic
107	C₄	C₂×#11745672	1	1	yes	characteristic
108	C₄	#14273749	1	1	yes	characteristic
109	C₄	#14273757	1	1	yes	characteristic
110	C₂	#16837612	1	1	yes	characteristic
111	C₂	#16837638	1	1	yes	characteristic
112	C₂	#16837677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837807

C₂

yes

maximal, characteristic

#16837612

C₂

yes

maximal

#16837155

C₂

yes

maximal, characteristic

#16837147

C₂

yes

maximal

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745672

—

#14273727

C₂²

yes

—

#14273661

—

#14273661

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273650

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₂³

yes

characteristic

#11745672

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

—

Q₈×C₅₉

—

Q₈×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

#121

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

C₂×Q₁₆

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

C₈×Q₈

D₅₉

yes

characteristic

#335

—

#332

—

C₅₉

#1674

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

#107

—

118

—

#107

—

118

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₉

yes

characteristic

#103

—

118

—

C₄×C₈

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

C₂×Q₈

D₂₃₆

yes

characteristic

Q₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

D₂₃₆

yes

characteristic

C₄²

—

C₄²

C₂²×D₅₉

yes

characteristic

Q₈

—

Q₈

—

236

—

Q₈

—

236

—

Q₈

D₄₇₂

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745685

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

C₈

—

236

—

100

C₈

#11745672

yes

complemented

101

C₂²

#14273701

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

472

—

106

C₄

C₂×D₄₇₂

yes

characteristic

107

C₄

C₂×#11745672

yes

characteristic

108

C₄

#14273749

yes

characteristic

109

C₄

#14273757

yes

characteristic

110

C₂

#16837612

yes

characteristic

111

C₂

#16837638

yes

characteristic

112

C₂

#16837677

yes

characteristic

	subgroup	G/H
center	C₂²	#14273701
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	#121
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#1674
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14273701

commutator

C₄²×C₅₉

C₂³

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

#121

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#1674

LCS

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
42. C₈×C₅₉	Q₁₆	semidirect
62. C₅₉	#1674	semidirect
100. C₈	#11745672	semidirect

H ≅ G/N

type

42. C₈×C₅₉

Q₁₆

semidirect

62. C₅₉

#1674

semidirect

100. C₈

#11745672

semidirect

Subgroups — #21423490

Conjugacy classes of subgroups

Complements