Group #21425816 — subgroups

Subgroups — #21425816

Order 7552.

220 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#14273667	C₂	1	1	yes	maximal, characteristic
2	#16837440	C₂	1	1	yes	maximal, characteristic
3	#16837206	C₂	1	4	yes	maximal, complemented
4	C₂²×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	#14273686	—	2	1	no	—
6	#14273667	—	2	2	no	—
7	C₂×C₅₉⋊C₁₆	C₂²	1	2	yes	—
8	#14273661	C₂²	1	2	yes	—
9	#14273659	—	2	1	no	—
10	C₂²×Q₈×C₅₉	—	2	1	no	—
11	C₂×Q₁₆×C₅₉	—	2	1	no	—
12	C₂×Q₁₆×C₅₉	—	2	2	no	—
13	C₂×Q₁₆×C₅₉	C₄	1	4	yes	complemented
14	C₂×Q₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₉	—	2	4	no	—
17	C₂×Q₈×C₅₉	—	2	2	no	—
18	C₂×Q₈×C₅₉	—	4	1	no	—
19	C₂×Q₈×C₅₉	—	4	2	no	—
20	C₂²×C₄×C₅₉	—	4	1	no	—
21	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
22	Q₁₆×C₅₉	—	2	4	no	—
23	Q₁₆×C₅₉	—	2	2	no	—
24	Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
25	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₅₉	D₄	1	2	yes	—
28	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂²⋊Dic₅₉	—	4	1	no	—
30	C₄⋊Dic₅₉	—	2	2	no	—
31	C₄×Dic₅₉	—	4	1	no	—
32	C₅₉⋊C₁₆	—	2	2	no	—
33	C₂³×C₅₉	D₈	1	1	yes	characteristic
34	Q₈×C₅₉	—	2	4	no	—
35	Q₈×C₅₉	—	4	4	no	—
36	Q₈×C₅₉	—	4	2	no	—
37	C₂×C₄×C₅₉	—	4	4	no	—
38	C₂×C₄×C₅₉	—	4	2	no	—
39	C₂×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₉	D₈	1	1	yes	characteristic
41	C₂×C₄×C₅₉	SD₁₆	1	2	yes	—
42	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅₉	—	4	2	no	—
44	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
45	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₅₉	—	2	2	no	—
48	C₂²×C₅₉	#102	1	2	yes	—
49	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
50	C₄×C₅₉	—	4	4	no	—
51	C₄×C₅₉	#102	1	2	yes	—
52	C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₅₉	—	8	2	no	—
55	#2141	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	2	no	—
57	C₂×C₅₉	C₂×#102	1	1	yes	characteristic
58	C₂×C₅₉	#510	1	2	yes	—
59	C₂²×Q₁₆	D₅₉	1	1	yes	characteristic
60	C₂×M₃₂	—	59	1	no	—
61	#427	—	59	1	no	—
62	#358	—	59	4	no	—
63	C₅₉	#2141	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	2	1	no	—
65	C₂×Q₁₆	—	2	1	no	—
66	C₂×Q₁₆	—	2	2	no	—
67	C₂×Q₁₆	Dic₅₉	1	4	yes	complemented
68	C₂×Q₁₆	C₂×D₅₉	1	2	yes	—
69	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
70	#117	—	118	1	no	—
71	M₃₂	—	118	2	no	—
72	C₂×C₁₆	—	59	2	no	—
73	#107	—	59	2	no	—
74	#106	—	118	1	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂²×C₄	—	4	1	no	—
80	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	2	2	no	—
83	Q₁₆	C₂×Dic₅₉	1	4	yes	—
84	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
85	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
87	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
88	C₄⋊C₄	—	118	2	no	—
89	C₂²⋊C₄	—	236	1	no	—
90	C₄²	—	236	1	no	—
91	C₁₆	—	118	2	no	—
92	C₂³	#11745669	1	1	yes	characteristic
93	Q₈	—	2	4	no	—
94	Q₈	—	4	4	no	—
95	Q₈	—	4	2	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	236	2	no	—
99	C₂×C₄	#11745669	1	1	yes	characteristic
100	C₂×C₄	#11745670	1	2	yes	—
101	C₂×C₄	C₂²⋊Dic₅₉	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
103	C₈	C₂²⋊Dic₅₉	1	2	yes	—
104	C₈	C₂²×Dic₅₉	1	1	yes	characteristic
105	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#14273648	1	2	yes	—
108	C₂²	C₂×#11745669	1	1	yes	characteristic
109	C₄	—	4	4	no	—
110	C₄	—	472	2	no	—
111	C₄	#14273648	1	2	yes	—
112	C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
113	C₄	C₂×#11745670	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#14273648	1	1	yes	characteristic
116	C₂	#16837843	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273667

C₂

yes

maximal, characteristic

#16837440

C₂

yes

maximal, characteristic

#16837206

C₂

yes

maximal, complemented

C₂²×Q₁₆×C₅₉

C₂

yes

maximal, characteristic

#14273686

—

#14273667

—

C₂×C₅₉⋊C₁₆

C₂²

yes

—

#14273661

C₂²

yes

—

#14273659

—

C₂²×Q₈×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

—

C₂×Q₁₆×C₅₉

C₄

yes

complemented

C₂×Q₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

characteristic

Q₁₆×C₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

D₈

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₂²×C₅₉

—

C₂²×C₅₉

#102

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#102

yes

—

C₄×C₅₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

#2141

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₂×#102

yes

characteristic

C₂×C₅₉

#510

yes

—

C₂²×Q₁₆

D₅₉

yes

characteristic

C₂×M₃₂

—

#427

—

#358

—

C₅₉

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₅₉

yes

complemented

C₂×Q₁₆

C₂×D₅₉

yes

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

118

—

M₃₂

—

118

—

C₂×C₁₆

—

#107

—

#106

—

118

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₉

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₂²⋊C₄

—

236

—

C₄²

—

236

—

C₁₆

—

118

—

C₂³

#11745669

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

#11745669

yes

characteristic

100

C₂×C₄

#11745670

yes

—

101

C₂×C₄

C₂²⋊Dic₅₉

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

103

C₈

C₂²⋊Dic₅₉

yes

—

104

C₈

C₂²×Dic₅₉

yes

characteristic

105

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

106

C₂²

—

107

C₂²

#14273648

yes

—

108

C₂²

C₂×#11745669

yes

characteristic

109

C₄

—

110

C₄

—

472

—

111

C₄

#14273648

yes

—

112

C₄

C₂×C₂²⋊Dic₅₉

yes

characteristic

113

C₄

C₂×#11745670

yes

characteristic

114

C₂

—

115

C₂

C₂×#14273648

yes

characteristic

116

C₂

#16837843

yes

—

	subgroup	G/H
center	C₂²	C₂×#11745669
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂²×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂²×Q₁₆	D₅₉
hypocenter	C₅₉	#2141
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×#11745669

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂²×Q₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂²×Q₁₆

D₅₉

hypocenter

C₅₉

#2141

LCS

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
3. #16837206	C₂	semidirect
13. C₂×Q₁₆×C₅₉	C₄	semidirect
63. C₅₉	#2141	semidirect
67. C₂×Q₁₆	Dic₅₉	semidirect

H ≅ G/N

type

3. #16837206

C₂

semidirect

13. C₂×Q₁₆×C₅₉

C₄

semidirect

63. C₅₉

#2141

semidirect

67. C₂×Q₁₆

Dic₅₉

semidirect

Subgroups — #21425816

Conjugacy classes of subgroups

Complements