Group #21423989 — subgroups

Subgroups — C₄×#14273661 #21423989

Order 7552.

280 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14273661	C₂	1	2	yes	maximal
2	C₄×C₄⋊Dic₅₉	C₂	1	2	yes	maximal
3	#16837165	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₅₉	C₂²	1	4	yes	—
6	C₂×C₄×Dic₅₉	—	2	2	no	—
7	#14273661	C₄	1	8	yes	direct product factor
8	C₅₉⋊#95	—	2	2	no	—
9	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₅₉	C₄	1	4	yes	complemented
13	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₉	—	2	4	no	—
17	C₂×C₈×C₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₉	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₄²×C₅₉	C₂×C₄	1	2	yes	—
22	C₄²×C₅₉	Q₈	1	2	yes	—
23	C₄⋊Dic₅₉	—	2	4	no	—
24	C₄⋊Dic₅₉	C₂×C₄	1	8	yes	—
25	C₄×Dic₅₉	—	4	4	no	—
26	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₉	—	2	2	no	—
28	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₅₉	D₈	1	2	yes	—
31	C₂×C₄×C₅₉	Q₁₆	1	2	yes	—
32	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₅₉	#41	1	2	yes	—
36	C₂×Dic₅₉	—	2	8	no	—
37	C₂×Dic₅₉	—	4	4	no	—
38	C₂×Dic₅₉	—	4	4	no	—
39	C₈×C₅₉	—	2	2	no	—
40	C₈×C₅₉	C₄²	1	4	yes	complemented
41	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₅₉	C₄×D₄	1	2	yes	—
43	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
44	C₂²×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
45	C₂²×C₅₉	#135	1	2	yes	—
46	C₄×C₅₉	—	2	2	no	—
47	C₄×C₅₉	#107	1	4	yes	complemented
48	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₅₉	#117	1	1	yes	characteristic
50	C₄×C₅₉	C₄×Q₈	1	2	yes	—
51	Dic₅₉	—	4	8	no	—
52	C₄×#107	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₅₉	C₂×#107	1	1	yes	characteristic
55	C₂×C₅₉	#427	1	1	yes	characteristic
56	C₂×C₅₉	C₄×D₈	1	2	yes	—
57	C₂×C₅₉	C₄×Q₁₆	1	2	yes	—
58	C₂×#107	—	59	2	no	—
59	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
60	C₄×C₄⋊C₄	—	59	2	no	—
61	#340	—	59	2	no	—
62	C₅₉	C₄×#107	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
65	C₂×C₄⋊C₄	—	59	4	no	—
66	C₂×C₄²	—	118	2	no	—
67	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
68	#107	—	59	8	no	—
69	C₄×C₈	Dic₅₉	1	4	yes	complemented
70	#95	—	118	2	no	—
71	C₂²×C₄	—	118	4	no	—
72	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
73	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
74	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	C₄×D₅₉	1	4	yes	—
77	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
78	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
79	C₄⋊C₄	—	59	8	no	—
80	C₄⋊C₄	—	118	4	no	—
81	C₄²	—	236	4	no	—
82	C₄²	Dic₁₁₈	1	2	yes	—
83	C₄²	C₂×Dic₅₉	1	2	yes	—
84	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	118	8	no	—
87	C₂×C₄	—	236	4	no	—
88	C₂×C₄	—	236	4	no	—
89	C₂×C₄	D₄₇₂	1	2	yes	—
90	C₂×C₄	Dic₂₃₆	1	2	yes	—
91	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
92	C₂×C₄	C₄⋊Dic₅₉	1	2	yes	—
93	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
95	C₂×C₄	#11745685	1	2	yes	—
96	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	C₄×Dic₅₉	1	4	yes	complemented
99	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
100	C₂²	C₄×D₂₃₆	1	2	yes	—
101	C₂²	C₂×D₄₇₂	1	1	yes	characteristic
102	C₂²	C₂×Dic₂₃₆	1	1	yes	characteristic
103	C₂²	#14273749	1	2	yes	—
104	C₄	—	2	2	no	—
105	C₄	—	236	8	no	—
106	C₄	#14273661	1	4	yes	direct product factor
107	C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
108	C₄	#14273686	1	1	yes	characteristic
109	C₄	C₄×Dic₁₁₈	1	2	yes	—
110	C₂	C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
111	C₂	C₂×#14273661	1	1	yes	characteristic
112	C₂	#16837440	1	1	yes	characteristic
113	C₂	C₄×D₄₇₂	1	2	yes	—
114	C₂	C₄×Dic₂₃₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273661

C₂

yes

maximal

C₄×C₄⋊Dic₅₉

C₂

yes

maximal

#16837165

C₂

yes

maximal

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

C₂×C₄×Dic₅₉

—

#14273661

C₄

yes

direct product factor

C₅₉⋊#95

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₄

yes

complemented

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

D₈

yes

—

C₂×C₄×C₅₉

Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄²

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×D₄

yes

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₂²×C₅₉

C₂×Q₁₆

yes

characteristic

C₂²×C₅₉

#135

yes

—

C₄×C₅₉

—

C₄×C₅₉

#107

yes

complemented

C₄×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×Q₈

yes

—

Dic₅₉

—

C₄×#107

—

maximal, Sylow

C₂×C₅₉

C₄×C₄⋊C₄

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

C₄×D₈

yes

—

C₂×C₅₉

C₄×Q₁₆

yes

—

C₂×#107

—

C₂×C₄×C₈

D₅₉

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₅₉

C₄×#107

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

118

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

C₄×C₈

Dic₅₉

yes

complemented

#95

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄²

—

236

—

C₄²

Dic₁₁₈

yes

—

C₄²

C₂×Dic₅₉

yes

—

C₂³

C₂×D₂₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

—

236

—

C₂×C₄

D₄₇₂

yes

—

C₂×C₄

Dic₂₃₆

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

—

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₄

#11745685

yes

—

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

C₈

C₄×Dic₅₉

yes

complemented

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

100

C₂²

C₄×D₂₃₆

yes

—

101

C₂²

C₂×D₄₇₂

yes

characteristic

102

C₂²

C₂×Dic₂₃₆

yes

characteristic

103

C₂²

#14273749

yes

—

104

C₄

—

105

C₄

—

236

—

106

C₄

#14273661

yes

direct product factor

107

C₄

C₂×C₄×Dic₅₉

yes

characteristic

108

C₄

#14273686

yes

characteristic

109

C₄

C₄×Dic₁₁₈

yes

—

110

C₂

C₄×C₄⋊Dic₅₉

yes

characteristic

111

C₂

C₂×#14273661

yes

characteristic

112

C₂

#16837440

yes

characteristic

113

C₂

C₄×D₄₇₂

yes

—

114

C₂

C₄×Dic₂₃₆

yes

—

	subgroup	G/H
center	C₂²×C₄	D₂₃₆
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	C₄×#107
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₂₃₆

commutator

C₄×C₅₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₅₉

hypocenter

C₅₉

C₄×#107

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #14273661	C₄	direct (N × H)
12. C₄×C₈×C₅₉	C₄	semidirect
40. C₈×C₅₉	C₄²	semidirect
47. C₄×C₅₉	#107	semidirect
62. C₅₉	C₄×#107	semidirect
69. C₄×C₈	Dic₅₉	semidirect
98. C₈	C₄×Dic₅₉	semidirect
106. C₄	#14273661	direct (N × H)

H ≅ G/N

type

7. #14273661

C₄

direct (N × H)

12. C₄×C₈×C₅₉

C₄

semidirect

40. C₈×C₅₉

C₄²

semidirect

47. C₄×C₅₉

#107

semidirect

62. C₅₉

C₄×#107

semidirect

69. C₄×C₈

Dic₅₉

semidirect

98. C₈

C₄×Dic₅₉

semidirect

106. C₄

#14273661

direct (N × H)

Subgroups — C₄×#14273661 #21423989

Conjugacy classes of subgroups

Complements

Subgroups — C4×#14273661 #21423989

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14273661 #21423989