Group #21422982 — subgroups

Subgroups — #21422982

Order 7552.

180 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
2	#16837481	C₂	1	2	yes	maximal, complemented
3	#16837155	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837131	C₂	1	2	yes	maximal, complemented
5	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₇₂	C₄	1	1	yes	characteristic
7	C₂×#11745659	—	2	2	no	—
8	C₄×D₂₃₆	C₂²	1	2	yes	—
9	#14273661	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	#14273649	C₄	1	2	yes	—
12	(C₂×D₅₉)⋊C₈	—	2	2	no	—
13	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×D₂₃₆	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	—	2	1	no	—
19	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
23	C₄²×C₅₉	C₂³	1	1	yes	characteristic
24	(C₂×D₅₉)⋊C₄	—	2	2	no	—
25	C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	D₄₇₂	—	2	2	no	—
28	#11745659	—	4	4	no	—
29	C₂²×D₅₉	—	2	2	no	—
30	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
33	C₂×Dic₅₉	—	2	2	no	—
34	D₂₃₆	—	2	2	no	—
35	D₂₃₆	—	2	2	no	—
36	C₄×D₅₉	—	4	2	no	—
37	C₄×D₅₉	—	4	2	no	—
38	C₈×C₅₉	—	2	2	no	—
39	C₈×C₅₉	—	2	1	no	—
40	C₈×C₅₉	—	2	1	no	—
41	C₈×C₅₉	M₁₆	1	2	yes	complemented
42	C₈×C₅₉	D₈	1	2	yes	complemented
43	C₅₉⋊C₈	—	4	2	no	—
44	C₂²×C₅₉	C₄×D₄	1	1	yes	characteristic
45	C₂×D₅₉	—	4	2	no	—
46	C₂×D₅₉	—	4	2	no	—
47	C₂×D₅₉	—	4	2	no	—
48	C₄×C₅₉	—	2	1	no	—
49	C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₅₉	#131	1	1	yes	characteristic
51	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
52	C₄×C₅₉	#135	1	1	yes	characteristic
53	Dic₅₉	—	4	2	no	—
54	#1589	—	59	1	no	maximal, Sylow
55	C₂×C₅₉	#436	1	1	yes	characteristic
56	C₂×C₅₉	C₄×D₈	1	1	yes	characteristic
57	C₂×C₅₉	#443	1	1	yes	characteristic
58	D₅₉	—	8	2	no	—
59	C₄×D₈	—	59	1	no	—
60	#436	—	59	2	no	—
61	#335	—	59	1	no	—
62	D₄⋊C₈	—	59	2	no	—
63	C₈²	D₅₉	1	1	yes	characteristic, complemented
64	C₅₉	#1589	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₈	—	59	1	no	—
66	C₂×M₁₆	—	118	2	no	—
67	C₄×D₄	—	59	2	no	—
68	#107	—	59	1	no	—
69	C₄⋊C₈	—	59	2	no	—
70	#102	—	59	2	no	—
71	C₂²⋊C₈	—	118	2	no	—
72	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
73	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
74	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
75	C₂×D₄	—	59	2	no	—
76	C₂²×C₄	—	118	2	no	—
77	D₈	—	118	2	no	—
78	M₁₆	—	236	4	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	118	2	no	—
81	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
82	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
83	C₂×C₈	D₂₃₆	1	1	yes	characteristic
84	C₂×C₈	D₂₃₆	1	1	yes	characteristic
85	C₄⋊C₄	—	59	2	no	—
86	C₂²⋊C₄	—	118	2	no	—
87	C₄²	C₂²×D₅₉	1	1	yes	characteristic
88	C₂³	—	118	2	no	—
89	D₄	—	118	2	no	—
90	D₄	—	118	2	no	—
91	C₂×C₄	—	118	2	no	—
92	C₂×C₄	—	236	2	no	—
93	C₂×C₄	—	236	2	no	—
94	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
95	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
96	C₂×C₄	#11745685	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₈	—	236	2	no	—
101	C₈	#11745659	1	2	yes	complemented
102	C₈	D₄₇₂	1	2	yes	complemented
103	C₂²	—	236	2	no	—
104	C₂²	—	236	2	no	—
105	C₂²	—	236	2	no	—
106	C₂²	C₄×D₂₃₆	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	236	2	no	—
109	C₄	C₂×#11745659	1	1	yes	characteristic
110	C₄	#14273733	1	1	yes	characteristic
111	C₄	C₂×D₄₇₂	1	1	yes	characteristic
112	C₄	#14273749	1	1	yes	characteristic
113	C₂	—	472	2	no	—
114	C₂	#16837481	1	1	yes	characteristic
115	C₂	C₄×D₄₇₂	1	1	yes	characteristic
116	C₂	#16837520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₇₂

C₂

yes

maximal, characteristic

#16837481

C₂

yes

maximal, complemented

#16837155

C₂

yes

maximal, characteristic, complemented

#16837131

C₂

yes

maximal, complemented

C₈²×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄₇₂

C₄

yes

characteristic

C₂×#11745659

—

C₄×D₂₃₆

C₂²

yes

—

#14273661

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

#14273649

C₄

yes

—

(C₂×D₅₉)⋊C₈

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

characteristic

C₂×D₂₃₆

C₂×C₄

yes

—

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

D₄₇₂

—

#11745659

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

complemented

C₈×C₅₉

D₈

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

C₄×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×M₁₆

yes

characteristic

C₄×C₅₉

#131

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×D₈

yes

characteristic

C₂×C₅₉

#443

yes

characteristic

D₅₉

—

C₄×D₈

—

#436

—

#335

—

D₄⋊C₈

—

C₈²

D₅₉

yes

characteristic, complemented

C₅₉

#1589

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

118

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

118

—

D₈

—

118

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

—

118

—

D₄

—

118

—

D₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

#11745685

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

236

—

101

C₈

#11745659

yes

complemented

102

C₈

D₄₇₂

yes

complemented

103

C₂²

—

236

—

104

C₂²

—

236

—

105

C₂²

—

236

—

106

C₂²

C₄×D₂₃₆

yes

characteristic

107

C₄

—

108

C₄

—

236

—

109

C₄

C₂×#11745659

yes

characteristic

110

C₄

#14273733

yes

characteristic

111

C₄

C₂×D₄₇₂

yes

characteristic

112

C₄

#14273749

yes

characteristic

113

C₂

—

472

—

114

C₂

#16837481

yes

characteristic

115

C₂

C₄×D₄₇₂

yes

characteristic

116

C₂

#16837520

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₈²×C₅₉	C₂
socle	C₂²×C₅₉	C₄×D₄
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1589
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₈²×C₅₉

C₂

socle

C₂²×C₅₉

C₄×D₄

hypercenter

C₈²

D₅₉

hypocenter

C₅₉

#1589

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
2. #16837481	C₂	semidirect
3. #16837155	C₂	semidirect
4. #16837131	C₂	semidirect
5. C₈²×C₅₉	C₂	semidirect
41. C₈×C₅₉	M₁₆	semidirect
42. C₈×C₅₉	D₈	semidirect
63. C₈²	D₅₉	semidirect
64. C₅₉	#1589	semidirect
101. C₈	#11745659	semidirect
102. C₈	D₄₇₂	semidirect

H ≅ G/N

type

2. #16837481

C₂

semidirect

3. #16837155

C₂

semidirect

4. #16837131

C₂

semidirect

5. C₈²×C₅₉

C₂

semidirect

41. C₈×C₅₉

M₁₆

semidirect

42. C₈×C₅₉

D₈

semidirect

63. C₈²

D₅₉

semidirect

64. C₅₉

#1589

semidirect

101. C₈

#11745659

semidirect

102. C₈

D₄₇₂

semidirect

Subgroups — #21422982

Conjugacy classes of subgroups

Complements