Group #21437071 — subgroups

Subgroups — #21437071

Order 7552.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16838074	C₂	1	2	yes	maximal, complemented
2	#16837779	C₂	1	3	yes	maximal, complemented
3	#16837598	C₂	1	6	yes	maximal, complemented
4	C₄×D₄₇₂	C₂	1	3	yes	maximal, complemented
5	C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11745689	—	2	2	no	—
7	C₂×D₄₇₂	C₂²	1	3	yes	—
8	C₂×D₄₇₂	C₂²	1	6	yes	—
9	#14273724	C₂²	1	6	yes	—
10	#14273702	—	2	6	no	—
11	C₄×D₂₃₆	C₂²	1	6	yes	—
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273649	C₂²	1	6	yes	—
14	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₉	C₂²	1	3	yes	—
16	C₄×C₈×C₅₉	C₂²	1	3	yes	—
17	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
18	#11745689	—	4	8	no	—
19	C₂×D₂₃₆	—	2	6	no	—
20	C₂×D₂₃₆	—	2	6	no	—
21	C₂×D₂₃₆	C₂³	1	6	yes	—
22	C₂×C₄×D₅₉	—	2	6	no	—
23	C₂×C₈×C₅₉	C₂³	1	3	yes	—
24	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₅₉	D₄	1	3	yes	—
26	C₄²×C₅₉	C₂³	1	3	yes	—
27	(C₂×D₅₉)⋊C₄	—	2	6	no	—
28	C₄⋊Dic₅₉	C₂³	1	2	yes	—
29	D₄₇₂	—	2	6	no	—
30	D₄₇₂	—	4	6	no	—
31	C₂²×D₅₉	—	2	6	no	—
32	Q₈×C₅₉	D₈	1	4	yes	complemented
33	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₅₉	C₂×D₄	1	3	yes	—
35	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
36	C₂×Dic₅₉	—	2	2	no	—
37	D₂₃₆	—	2	6	no	—
38	D₂₃₆	—	2	6	no	—
39	D₂₃₆	—	4	12	no	—
40	D₂₃₆	—	8	6	no	—
41	C₄×D₅₉	—	4	12	no	—
42	C₈×C₅₉	—	2	3	no	—
43	C₈×C₅₉	#41	1	2	yes	complemented
44	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
45	C₂×D₅₉	—	4	6	no	—
46	C₂×D₅₉	—	4	6	no	—
47	C₂×D₅₉	—	4	6	no	—
48	C₄×C₅₉	—	2	3	no	—
49	C₄×C₅₉	C₂×D₈	1	6	yes	—
50	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
51	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
52	Dic₅₉	—	4	2	no	—
53	#3391	—	59	1	no	maximal, Sylow
54	C₂×C₅₉	#550	1	1	yes	characteristic
55	C₂×C₅₉	C₂²×D₈	1	1	yes	characteristic
56	C₂×C₅₉	#576	1	1	yes	characteristic
57	D₅₉	—	8	6	no	—
58	#550	—	59	2	no	—
59	#493	—	59	3	no	—
60	#459	—	59	6	no	—
61	C₈×Q₈	D₅₉	1	1	yes	characteristic, complemented
62	C₄×D₈	—	59	3	no	—
63	C₅₉	#3391	1	1	yes	characteristic, complemented, Sylow
64	C₂×#41	—	118	2	no	—
65	C₂×D₈	—	59	3	no	—
66	C₂×D₈	—	59	6	no	—
67	#127	—	59	6	no	—
68	#121	—	118	6	no	—
69	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
70	C₄×D₄	—	59	6	no	—
71	#107	—	59	1	no	—
72	C₄⋊C₈	C₂×D₅₉	1	3	yes	—
73	#102	—	59	6	no	—
74	C₄×C₈	C₂×D₅₉	1	3	yes	—
75	#41	—	236	8	no	—
76	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
77	C₂×D₄	—	59	6	no	—
78	C₂×D₄	—	118	6	no	—
79	C₂×D₄	—	118	6	no	—
80	C₂²×C₄	—	118	6	no	—
81	D₈	—	118	6	no	—
82	D₈	—	236	6	no	—
83	C₂×C₈	C₂²×D₅₉	1	3	yes	—
84	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
85	C₄⋊C₄	—	59	2	no	—
86	C₄⋊C₄	D₂₃₆	1	3	yes	—
87	C₂²⋊C₄	—	118	6	no	—
88	C₄²	C₂²×D₅₉	1	3	yes	—
89	C₂³	—	118	6	no	—
90	Q₈	D₄₇₂	1	4	yes	complemented
91	D₄	—	118	6	no	—
92	D₄	—	118	6	no	—
93	D₄	—	472	6	no	—
94	D₄	—	236	12	no	—
95	C₂×C₄	—	118	2	no	—
96	C₂×C₄	—	236	12	no	—
97	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
98	C₂×C₄	C₂×D₂₃₆	1	3	yes	—
99	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
100	C₈	—	2	3	no	—
101	C₈	#11745689	1	2	yes	complemented
102	C₂²	—	236	6	no	—
103	C₂²	—	236	6	no	—
104	C₂²	—	236	6	no	—
105	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
106	C₄	—	2	3	no	—
107	C₄	—	236	2	no	—
108	C₄	C₂×D₄₇₂	1	6	yes	—
109	C₄	C₂×#11745689	1	1	yes	characteristic
110	C₄	#14273773	1	1	yes	characteristic
111	C₂	—	472	6	no	—
112	C₂	#16838074	1	1	yes	characteristic
113	C₂	C₂²×D₄₇₂	1	1	yes	characteristic
114	C₂	#16838207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16838074

C₂

yes

maximal, complemented

#16837779

C₂

yes

maximal, complemented

#16837598

C₂

yes

maximal, complemented

C₄×D₄₇₂

C₂

yes

maximal, complemented

C₈×Q₈×C₅₉

C₂

yes

maximal, characteristic, complemented

C₂×#11745689

—

C₂×D₄₇₂

C₂²

yes

—

C₂×D₄₇₂

C₂²

yes

—

#14273724

C₂²

yes

—

#14273702

—

C₄×D₂₃₆

C₂²

yes

—

#14273661

C₂²

yes

characteristic

#14273649

C₂²

yes

—

C₄×Q₈×C₅₉

C₂²

yes

characteristic

C₄⋊C₈×C₅₉

C₂²

yes

—

C₄×C₈×C₅₉

C₂²

yes

—

C₂×Q₈×C₅₉

D₄

yes

characteristic

#11745689

—

C₂×D₂₃₆

—

C₂×D₂₃₆

—

C₂×D₂₃₆

C₂³

yes

—

C₂×C₄×D₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₄⋊C₄×C₅₉

D₄

yes

—

C₄²×C₅₉

C₂³

yes

—

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

C₂³

yes

—

D₄₇₂

—

D₄₇₂

—

C₂²×D₅₉

—

Q₈×C₅₉

D₈

yes

complemented

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₂×C₄×C₅₉

C₂⁴

yes

characteristic

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

D₂₃₆

—

C₄×D₅₉

—

C₈×C₅₉

—

C₈×C₅₉

#41

yes

complemented

C₂²×C₅₉

C₂²×D₄

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×D₈

yes

—

C₄×C₅₉

C₂×#41

yes

characteristic

C₄×C₅₉

H(5,𝔽₂)

yes

characteristic

Dic₅₉

—

—

C₂×C₅₉

yes

C₂×C₅₉

C₂²×D₈

yes

characteristic

C₂×C₅₉

#576

yes

characteristic

D₅₉

—

#550

—

#493

—

#459

—

C₈×Q₈

D₅₉

yes

characteristic, complemented

C₄×D₈

—

C₅₉

#3391

yes

characteristic, complemented, Sylow

C₂×#41

—

118

—

C₂×D₈

—

C₂×D₈

—

#127

—

#121

—

118

—

C₄×Q₈

C₂×D₅₉

yes

characteristic

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₅₉

yes

—

#102

—

C₄×C₈

C₂×D₅₉

yes

—

#41

—

236

—

C₂×Q₈

D₂₃₆

yes

characteristic

C₂×D₄

—

C₂×D₄

—

118

—

C₂×D₄

—

118

—

C₂²×C₄

—

118

—

D₈

—

118

—

D₈

—

236

—

C₂×C₈

C₂²×D₅₉

yes

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₂₃₆

yes

—

C₂²⋊C₄

—

118

—

C₄²

C₂²×D₅₉

yes

—

C₂³

—

118

—

Q₈

D₄₇₂

yes

complemented

D₄

—

118

—

D₄

—

118

—

D₄

—

472

—

D₄

—

236

—

C₂×C₄

—

118

—

C₂×C₄

—

236

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂×D₂₃₆

yes

—

C₂×C₄

C₂³×D₅₉

yes

characteristic

100

C₈

—

101

C₈

#11745689

yes

complemented

102

C₂²

—

236

—

103

C₂²

—

236

—

104

C₂²

—

236

—

105

C₂²

C₂²×D₂₃₆

yes

characteristic

106

C₄

—

107

C₄

—

236

—

108

C₄

C₂×D₄₇₂

yes

—

109

C₄

C₂×#11745689

yes

characteristic

110

C₄

#14273773

yes

characteristic

111

C₂

—

472

—

112

C₂

#16838074

yes

characteristic

113

C₂

C₂²×D₄₇₂

yes

characteristic

114

C₂

#16838207

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₈×Q₈	D₅₉
hypocenter	C₅₉	#3391
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₉

Fitting

C₈×Q₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×D₄

hypercenter

C₈×Q₈

D₅₉

hypocenter

C₅₉

#3391

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16838074	C₂	semidirect
2. #16837779	C₂	semidirect
3. #16837598	C₂	semidirect
4. C₄×D₄₇₂	C₂	semidirect
5. C₈×Q₈×C₅₉	C₂	semidirect
32. Q₈×C₅₉	D₈	semidirect
43. C₈×C₅₉	#41	semidirect
61. C₈×Q₈	D₅₉	semidirect
63. C₅₉	#3391	semidirect
90. Q₈	D₄₇₂	semidirect
101. C₈	#11745689	semidirect

H ≅ G/N

type

1. #16838074

C₂

semidirect

2. #16837779

C₂

semidirect

3. #16837598

C₂

semidirect

4. C₄×D₄₇₂

C₂

semidirect

5. C₈×Q₈×C₅₉

C₂

semidirect

32. Q₈×C₅₉

D₈

semidirect

43. C₈×C₅₉

#41

semidirect

61. C₈×Q₈

D₅₉

semidirect

63. C₅₉

#3391

semidirect

90. Q₈

D₄₇₂

semidirect

101. C₈

#11745689

semidirect

Subgroups — #21437071

Conjugacy classes of subgroups

Complements