Group #21424262 — subgroups

Subgroups — #21424262

Order 7552.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14273660	C₂	1	2	yes	maximal
2	C₂×#14273658	C₂	1	2	yes	maximal
3	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
4	#16837268	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
9	#14273660	—	2	4	no	—
10	#14273658	—	2	4	no	—
11	#14273638	C₄	1	4	yes	complemented
12	C₅₉⋊#95	—	2	2	no	—
13	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
14	C₂×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅₉	—	2	2	no	—
17	C₂²×C₄×C₅₉	D₄	1	2	yes	—
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₉	—	2	1	no	—
20	C₂²×Dic₅₉	—	2	1	no	—
21	C₄⋊C₄×C₅₉	—	2	2	no	—
22	C₄⋊C₄×C₅₉	—	2	2	no	—
23	C₄⋊C₄×C₅₉	—	4	4	no	—
24	C₄²×C₅₉	C₂×C₄	1	2	yes	—
25	C₄²×C₅₉	Q₈	1	2	yes	—
26	C₄⋊Dic₅₉	—	2	2	no	—
27	C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₅₉⋊C₈	D₄	1	4	yes	—
30	C₂×C₅₉⋊C₈	Q₈	1	4	yes	—
31	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	2	2	no	—
33	C₂×C₄×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	4	2	no	—
35	C₂×C₄×C₅₉	—	4	4	no	—
36	C₂×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
41	C₂×Dic₅₉	—	4	1	no	—
42	C₂×Dic₅₉	—	4	1	no	—
43	C₂×Dic₅₉	—	4	2	no	—
44	C₂×Dic₅₉	—	4	2	no	—
45	C₅₉⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₅₉	#127	1	1	yes	characteristic
48	C₂²×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₅₉	#137	1	2	yes	—
51	C₄×C₅₉	—	2	4	no	—
52	C₄×C₅₉	—	8	2	no	—
53	C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
55	Dic₅₉	—	8	1	no	—
56	Dic₅₉	—	8	1	no	—
57	#1847	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	#384	1	1	yes	characteristic
59	C₂×C₅₉	#428	1	2	yes	—
60	C₂×C₅₉	#496	1	1	yes	characteristic
61	C₂×C₅₉	#497	1	1	yes	characteristic
62	C₂×C₅₉	#501	1	2	yes	—
63	C₂×#107	—	59	2	no	—
64	C₂×#106	—	59	2	no	—
65	C₂×#97	—	59	1	no	—
66	#384	D₅₉	1	1	yes	characteristic
67	#382	—	59	1	no	—
68	C₅₉	#1847	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	59	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	59	1	no	—
72	C₂×C₄⋊C₄	—	59	1	no	—
73	C₂×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
74	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
75	#107	—	118	4	no	—
76	#106	—	118	4	no	—
77	#97	—	59	4	no	—
78	#95	—	118	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	118	1	no	—
81	C₂²×C₄	—	118	1	no	—
82	C₂²×C₄	C₂²⋊D₅₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
84	C₂×C₈	—	59	4	no	—
85	C₂×C₈	—	59	4	no	—
86	C₂×C₈	—	59	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	118	2	no	—
91	C₄⋊C₄	—	118	2	no	—
92	C₄²	Dic₁₁₈	1	2	yes	—
93	C₄²	C₄×D₅₉	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	236	1	no	—
100	C₂×C₄	—	236	1	no	—
101	C₂×C₄	—	236	2	no	—
102	C₂×C₄	—	236	2	no	—
103	C₂×C₄	Dic₅₉⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
106	C₂×C₄	D₄×D₅₉	1	2	yes	—
107	C₂×C₄	Q₈×D₅₉	1	2	yes	—
108	C₈	—	59	8	no	—
109	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
110	C₂²	#14273723	1	1	yes	characteristic
111	C₂²	#14273725	1	1	yes	characteristic
112	C₂²	#14273753	1	2	yes	—
113	C₂²	#14273758	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	472	1	no	—
117	C₄	—	472	1	no	—
118	C₄	C₄⋊C₄×D₅₉	1	2	yes	—
119	C₄	#14273726	1	2	yes	—
120	C₂	#16837271	1	1	yes	characteristic
121	C₂	#16837442	1	2	yes	—
122	C₂	#16837789	1	1	yes	characteristic
123	C₂	#16837794	1	1	yes	characteristic
124	C₂	#16837808	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273660

C₂

yes

maximal

C₂×#14273658

C₂

yes

maximal

C₂×#14273638

C₂

yes

maximal, characteristic

#16837268

C₂

yes

maximal, characteristic

C₅₉×#384

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₅₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₉

C₂²

yes

—

—

C₄

yes

complemented

C₅₉⋊#95

—

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

Q₈

yes

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

D₄

yes

—

C₂×C₅₉⋊C₈

Q₈

yes

—

C₂³×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#127

yes

characteristic

C₂²×C₅₉

C₄⋊Q₈

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

—

C₂²×C₅₉

#137

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₄×C₅₉

C₄⋊Q₈

yes

—

Dic₅₉

—

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₅₉

yes

characteristic

#382

—

C₅₉

#1847

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

#107

—

118

—

#106

—

118

—

#97

—

#95

—

118

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄²

Dic₁₁₈

yes

—

C₄²

C₄×D₅₉

yes

—

C₂³

C₂×C₂²⋊D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

236

—

100

C₂×C₄

—

236

—

101

C₂×C₄

—

236

—

102

C₂×C₄

—

236

—

103

C₂×C₄

Dic₅₉⋊C₄

yes

—

104

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

105

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

106

C₂×C₄

D₄×D₅₉

yes

—

107

C₂×C₄

Q₈×D₅₉

yes

—

108

C₈

—

109

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

110

C₂²

#14273723

yes

characteristic

111

C₂²

#14273725

yes

characteristic

112

C₂²

#14273753

yes

—

113

C₂²

#14273758

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

472

—

117

C₄

—

472

—

118

C₄

C₄⋊C₄×D₅₉

yes

—

119

C₄

#14273726

yes

—

120

C₂

#16837271

yes

characteristic

121

C₂

#16837442

yes

—

122

C₂

#16837789

yes

characteristic

123

C₂

#16837794

yes

characteristic

124

C₂

#16837808

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#384	C₂
socle	C₂³×C₅₉	C₂×D₄
hypercenter	#384	D₅₉
hypocenter	C₅₉	#1847
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#384

C₂

socle

C₂³×C₅₉

C₂×D₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #14273638	C₄	semidirect
45. C₅₉⋊C₈	C₄⋊C₄	semidirect
68. C₅₉	#1847	semidirect

H ≅ G/N

type

11. #14273638

C₄

semidirect

45. C₅₉⋊C₈

C₄⋊C₄

semidirect

68. C₅₉

#1847

semidirect

Subgroups — #21424262

Conjugacy classes of subgroups

Complements