Group #21426178 — subgroups

Subgroups — #21426178

Order 7552.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16837845	C₂	1	2	yes	maximal
2	#16837782	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837520	C₂	1	1	yes	maximal, characteristic
4	#16837219	C₂	1	2	yes	maximal, complemented
5	C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273749	C₂²	1	1	yes	characteristic
7	C₂×Dic₂₃₆	—	2	1	no	—
8	C₂×Dic₂₃₆	C₂²	1	2	yes	—
9	#14273733	—	2	1	no	—
10	#14273727	—	2	1	no	—
11	Dic₄₇₂	—	2	2	no	—
12	#14273672	—	2	2	no	—
13	#14273663	C₂²	1	1	yes	characteristic
14	#14273655	—	2	1	no	—
15	M₃₂×C₅₉	C₂²	1	2	yes	—
16	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	#11745685	—	2	1	no	—
18	C₂×Dic₁₁₈	—	2	2	no	—
19	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
21	C₄²×C₅₉	D₄	1	1	yes	characteristic
22	C₄⋊Dic₅₉	—	4	2	no	—
23	#11745664	—	2	1	no	—
24	Dic₂₃₆	—	2	2	no	—
25	Dic₂₃₆	—	2	2	no	—
26	Dic₂₃₆	—	4	1	no	—
27	Dic₂₃₆	—	4	1	no	—
28	Dic₂₃₆	D₄	1	1	yes	characteristic
29	D₄₇₂	D₄	1	1	yes	characteristic
30	#11745660	—	2	1	no	—
31	#11745659	—	4	1	no	—
32	C₈×D₅₉	—	4	1	no	—
33	C₁₆×C₅₉	—	2	2	no	—
34	C₂×C₄×C₅₉	—	2	1	no	—
35	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂²⋊D₅₉	—	4	1	no	—
37	C₂×Dic₅₉	—	4	2	no	—
38	D₂₃₆	—	2	1	no	—
39	C₄×D₅₉	—	4	1	no	—
40	Dic₁₁₈	—	2	1	no	—
41	Dic₁₁₈	—	4	2	no	—
42	Dic₁₁₈	—	4	2	no	—
43	C₈×C₅₉	D₈	1	1	yes	characteristic
44	C₈×C₅₉	D₈	1	1	yes	characteristic
45	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₅₉	#41	1	1	yes	characteristic
47	C₅₉⋊C₈	—	4	1	no	—
48	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₂×D₅₉	—	4	1	no	—
50	C₄×C₅₉	—	2	2	no	—
51	C₄×C₅₉	#121	1	1	yes	characteristic
52	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
53	Dic₅₉	—	4	1	no	—
54	Dic₅₉	—	8	2	no	—
55	#2214	—	59	1	no	maximal, Sylow
56	C₂×C₅₉	—	2	1	no	—
57	C₂×C₅₉	#459	1	1	yes	characteristic
58	D₅₉	—	8	1	no	—
59	#510	—	59	2	no	—
60	#494	—	59	1	no	—
61	#443	—	59	1	no	—
62	#364	D₅₉	1	1	yes	characteristic, complemented
63	#362	—	59	2	no	—
64	C₅₉	#2214	1	1	yes	characteristic, complemented, Sylow
65	#135	—	59	1	no	—
66	C₂×Q₁₆	—	59	2	no	—
67	C₂×Q₁₆	—	118	1	no	—
68	#131	—	118	1	no	—
69	C₄⋊Q₈	—	118	1	no	—
70	Q₃₂	—	118	2	no	—
71	SD₃₂	—	118	2	no	—
72	M₃₂	C₂×D₅₉	1	2	yes	—
73	#108	—	59	1	no	—
74	#104	—	118	1	no	—
75	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
76	#41	—	118	1	no	—
77	C₂×Q₈	—	118	2	no	—
78	Q₁₆	—	59	1	no	—
79	Q₁₆	—	118	2	no	—
80	Q₁₆	—	118	2	no	—
81	Q₁₆	—	236	1	no	—
82	Q₁₆	—	236	1	no	—
83	SD₁₆	—	118	1	no	—
84	D₈	—	59	1	no	—
85	M₁₆	—	118	1	no	—
86	M₁₆	—	236	1	no	—
87	C₂×C₈	—	236	1	no	—
88	C₂×C₈	D₂₃₆	1	1	yes	characteristic
89	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
90	C₄⋊C₄	—	236	2	no	—
91	C₄²	D₂₃₆	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	Q₈	—	118	1	no	—
94	Q₈	—	236	2	no	—
95	Q₈	—	236	2	no	—
96	D₄	—	118	1	no	—
97	D₄	—	236	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	236	2	no	—
100	C₂×C₄	—	236	1	no	—
101	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
102	C₈	—	236	1	no	—
103	C₈	D₄₇₂	1	1	yes	characteristic
104	C₈	D₄₇₂	1	1	yes	characteristic
105	C₈	D₄×D₅₉	1	1	yes	characteristic
106	C₈	#11745689	1	1	yes	characteristic
107	C₂²	—	236	1	no	—
108	C₂²	#14273755	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	472	2	no	—
111	C₄	—	236	1	no	—
112	C₄	#14273702	1	1	yes	characteristic
113	C₄	C₂×D₄₇₂	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	472	1	no	—
116	C₂	#16837598	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837845

C₂

yes

maximal

#16837782

C₂

yes

maximal, characteristic, complemented

#16837520

C₂

yes

maximal, characteristic

#16837219

C₂

yes

maximal, complemented

C₅₉×#364

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

C₂²

yes

—

#14273733

—

#14273727

—

Dic₄₇₂

—

#14273672

—

#14273663

C₂²

yes

characteristic

#14273655

—

M₃₂×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

characteristic

#11745685

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄⋊Dic₅₉

—

#11745664

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

D₄

yes

characteristic

D₄₇₂

D₄

yes

characteristic

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

D₈

yes

characteristic

C₈×C₅₉

D₈

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2214

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#459

yes

characteristic

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#362

—

C₅₉

#2214

yes

characteristic, complemented, Sylow

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

—

118

—

#131

—

118

—

C₄⋊Q₈

—

118

—

Q₃₂

—

118

—

SD₃₂

—

118

—

M₃₂

C₂×D₅₉

yes

—

#108

—

#104

—

118

—

C₄×C₈

C₂×D₅₉

yes

characteristic

#41

—

118

—

C₂×Q₈

—

118

—

Q₁₆

—

Q₁₆

—

118

—

Q₁₆

—

118

—

Q₁₆

—

236

—

Q₁₆

—

236

—

SD₁₆

—

118

—

D₈

—

M₁₆

—

118

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

236

—

C₄²

D₂₃₆

yes

characteristic

C₁₆

—

Q₈

—

118

—

Q₈

—

236

—

Q₈

—

236

—

D₄

—

118

—

D₄

—

236

—

C₂×C₄

—

C₂×C₄

—

236

—

100

C₂×C₄

—

236

—

101

C₂×C₄

C₂×D₂₃₆

yes

characteristic

102

C₈

—

236

—

103

C₈

D₄₇₂

yes

characteristic

104

C₈

D₄₇₂

yes

characteristic

105

C₈

D₄×D₅₉

yes

characteristic

106

C₈

#11745689

yes

characteristic

107

C₂²

—

236

—

108

C₂²

#14273755

yes

characteristic

109

C₄

—

110

C₄

—

472

—

111

C₄

—

236

—

112

C₄

#14273702

yes

characteristic

113

C₄

C₂×D₄₇₂

yes

characteristic

114

C₂

—

115

C₂

—

472

—

116

C₂

#16837598

yes

characteristic

	subgroup	G/H
center	C₂	#16837598
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#364	C₂
socle	C₂×C₅₉	#459
hypercenter	#364	D₅₉
hypocenter	C₅₉	#2214
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#16837598

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#364

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
2. #16837782	C₂	semidirect
4. #16837219	C₂	semidirect
5. C₅₉×#364	C₂	semidirect
62. #364	D₅₉	semidirect
64. C₅₉	#2214	semidirect

H ≅ G/N

type

2. #16837782

C₂

semidirect

4. #16837219

C₂

semidirect

5. C₅₉×#364

C₂

semidirect

62. #364

D₅₉

semidirect

64. C₅₉

#2214

semidirect

Subgroups — #21426178

Conjugacy classes of subgroups

Complements