Group #21426266 — subgroups

Subgroups — #21426266

Order 7552.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#16837799	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837518	C₂	1	1	yes	maximal, characteristic
3	#16837226	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837222	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837213	C₂	1	1	yes	maximal, characteristic
6	#16837211	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#365	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273749	C₂²	1	1	yes	characteristic
9	#14273735	—	2	1	no	—
10	#14273726	—	2	1	no	—
11	#14273667	C₂²	1	1	yes	characteristic
12	#14273659	C₂²	1	1	yes	characteristic
13	#14273658	—	2	1	no	—
14	#14273653	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
17	#108×C₅₉	C₂²	1	1	yes	characteristic
18	#106×C₅₉	C₂²	1	1	yes	characteristic
19	#11745685	—	2	1	no	—
20	C₂×Dic₁₁₈	—	4	1	no	—
21	M₁₆×C₅₉	—	2	1	no	—
22	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₉	—	2	1	no	—
24	C₄⋊Dic₅₉	—	2	1	no	—
25	Dic₅₉⋊C₄	—	4	1	no	—
26	C₄×Dic₅₉	D₄	1	1	yes	characteristic
27	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
28	Dic₂₃₆	Q₈	1	1	yes	characteristic, complemented
29	D₄₇₂	Q₈	1	1	yes	characteristic, complemented
30	#11745660	—	2	1	no	—
31	#11745659	—	4	1	no	—
32	C₈×D₅₉	—	4	1	no	—
33	C₁₆×C₅₉	—	2	1	no	—
34	C₅₉⋊C₁₆	—	2	1	no	—
35	C₂×C₄×C₅₉	—	4	1	no	—
36	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₅₉	—	4	1	no	—
38	C₂×Dic₅₉	—	2	1	no	—
39	C₂×Dic₅₉	—	4	1	no	—
40	D₂₃₆	—	2	1	no	—
41	C₄×D₅₉	—	4	1	no	—
42	Dic₁₁₈	—	2	1	no	—
43	Dic₁₁₈	—	8	2	no	—
44	C₈×C₅₉	—	4	1	no	—
45	C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₅₉	#41	1	1	yes	characteristic
47	C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
48	C₅₉⋊C₈	SD₁₆	1	1	yes	characteristic
49	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₅₉	—	4	1	no	—
51	C₄×C₅₉	—	8	1	no	—
52	C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₅₉	—	2	2	no	—
55	Dic₅₉	—	4	1	no	—
56	Dic₅₉	—	8	1	no	—
57	#2230	—	59	1	no	maximal, Sylow
58	C₂×C₅₉	—	2	1	no	—
59	C₂×C₅₉	#476	1	1	yes	characteristic
60	D₅₉	—	8	1	no	—
61	#498	—	59	1	no	—
62	#443	—	59	1	no	—
63	#365	—	59	1	no	—
64	#365	D₅₉	1	1	yes	characteristic, complemented
65	#364	—	59	1	no	—
66	#360	—	59	1	no	—
67	#360	—	59	1	no	—
68	C₅₉	#2230	1	1	yes	characteristic, complemented, Sylow
69	#135	—	59	1	no	—
70	#131	—	118	1	no	—
71	C₄⋊Q₈	—	118	1	no	—
72	M₃₂	—	59	1	no	—
73	M₃₂	C₂×D₅₉	1	1	yes	characteristic
74	#108	C₂×D₅₉	1	1	yes	characteristic
75	#106	—	59	1	no	—
76	#106	—	118	1	no	—
77	#106	C₂×D₅₉	1	1	yes	characteristic
78	#104	—	118	1	no	—
79	C₄×C₈	—	59	1	no	—
80	#41	—	118	1	no	—
81	C₂×Q₈	—	236	1	no	—
82	Q₁₆	—	59	1	no	—
83	SD₁₆	—	118	1	no	—
84	D₈	—	59	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	236	1	no	—
87	C₂×C₈	—	59	1	no	—
88	C₂×C₈	—	236	1	no	—
89	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	118	1	no	—
92	C₄⋊C₄	—	236	1	no	—
93	C₄²	—	59	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	118	1	no	—
96	Q₈	—	118	1	no	—
97	Q₈	—	472	2	no	—
98	D₄	—	118	1	no	—
99	D₄	—	236	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	118	1	no	—
102	C₂×C₄	—	236	1	no	—
103	C₂×C₄	—	236	1	no	—
104	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	59	1	no	—
107	C₈	—	59	1	no	—
108	C₈	#11745685	1	1	yes	characteristic
109	C₈	Q₈×D₅₉	1	1	yes	characteristic
110	C₂²	—	236	1	no	—
111	C₂²	#14273754	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	118	2	no	—
114	C₄	—	472	1	no	—
115	C₄	—	236	1	no	—
116	C₄	#14273705	1	1	yes	characteristic
117	C₄	SD₁₆×D₅₉	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	472	1	no	—
120	C₂	#16837691	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837799

C₂

yes

maximal, characteristic, complemented

#16837518

C₂

yes

maximal, characteristic

#16837226

C₂

yes

maximal, characteristic, complemented

#16837222

C₂

yes

maximal, characteristic, complemented

#16837213

C₂

yes

maximal, characteristic

#16837211

C₂

yes

maximal, characteristic

C₅₉×#365

C₂

yes

maximal, characteristic, complemented

#14273749

C₂²

yes

—

—

C₂²

yes

characteristic

#14273659

C₂²

yes

characteristic

#14273658

—

#14273653

—

C₈×Dic₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

C₂²

yes

characteristic

#108×C₅₉

C₂²

yes

characteristic

#106×C₅₉

C₂²

yes

characteristic

#11745685

—

C₂×Dic₁₁₈

—

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

Dic₂₃₆

Q₈

yes

characteristic, complemented

D₄₇₂

Q₈

yes

characteristic, complemented

#11745660

—

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

C₄×D₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

SD₁₆

yes

characteristic

C₅₉⋊C₈

SD₁₆

yes

characteristic

C₂²×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂²⋊Q₈

yes

characteristic

C₄×C₅₉

C₂×SD₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2230

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#476

yes

characteristic

D₅₉

—

—

—

—

D₅₉

yes

characteristic, complemented

#364

—

#360

—

#360

—

C₅₉

#2230

yes

characteristic, complemented, Sylow

#135

—

#131

—

118

—

C₄⋊Q₈

—

118

—

M₃₂

—

M₃₂

C₂×D₅₉

yes

characteristic

#108

C₂×D₅₉

yes

characteristic

#106

—

#106

—

118

—

#106

C₂×D₅₉

yes

characteristic

#104

—

118

—

C₄×C₈

—

#41

—

118

—

C₂×Q₈

—

236

—

Q₁₆

—

SD₁₆

—

118

—

D₈

—

M₁₆

—

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

236

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

C₄²

—

C₁₆

—

C₁₆

—

118

—

Q₈

—

118

—

Q₈

—

472

—

D₄

—

118

—

D₄

—

236

—

100

C₂×C₄

—

101

C₂×C₄

—

118

—

102

C₂×C₄

—

236

—

103

C₂×C₄

—

236

—

104

C₂×C₄

D₄×D₅₉

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#11745685

yes

characteristic

109

C₈

Q₈×D₅₉

yes

characteristic

110

C₂²

—

236

—

111

C₂²

#14273754

yes

characteristic

112

C₄

—

113

C₄

—

118

—

114

C₄

—

472

—

115

C₄

—

236

—

116

C₄

#14273705

yes

characteristic

117

C₄

SD₁₆×D₅₉

yes

characteristic

118

C₂

—

119

C₂

—

472

—

120

C₂

#16837691

yes

characteristic

	subgroup	G/H
center	C₂	#16837691
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#365	C₂
socle	C₂×C₅₉	#476
hypercenter	#365	D₅₉
hypocenter	C₅₉	#2230
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16837691

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₅₉×#365

C₂

C₂×C₅₉

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #16837799	C₂	semidirect
3. #16837226	C₂	semidirect
4. #16837222	C₂	semidirect
7. C₅₉×#365	C₂	semidirect
28. Dic₂₃₆	Q₈	semidirect
29. D₄₇₂	Q₈	semidirect
64. #365	D₅₉	semidirect
68. C₅₉	#2230	semidirect

H ≅ G/N

type

1. #16837799

C₂

semidirect

3. #16837226

C₂

semidirect

4. #16837222

C₂

semidirect

7. C₅₉×#365

C₂

semidirect

28. Dic₂₃₆

Q₈

semidirect

29. D₄₇₂

Q₈

semidirect

64. #365

D₅₉

semidirect

68. C₅₉

#2230

semidirect

Subgroups — #21426266

Conjugacy classes of subgroups

Complements