Group #46832800 — subgroups

Subgroups — #2214×C₂₇₁ #46832800

Order 34688.

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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	#510×C₂₇₁	C₂	1	2	yes	maximal
2	#494×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#443×C₂₇₁	C₂	1	1	yes	maximal, characteristic
4	#364×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#362×C₂₇₁	C₂	1	2	yes	maximal, complemented
6	#135×C₂₇₁	C₂²	1	1	yes	characteristic
7	C₂×Q₁₆×C₂₇₁	—	2	1	no	—
8	C₂×Q₁₆×C₂₇₁	C₂²	1	2	yes	—
9	#131×C₂₇₁	—	2	1	no	—
10	C₄⋊Q₈×C₂₇₁	—	2	1	no	—
11	Q₃₂×C₂₇₁	—	2	2	no	—
12	SD₃₂×C₂₇₁	—	2	2	no	—
13	M₃₂×C₂₇₁	C₂²	1	2	yes	—
14	#108×C₂₇₁	C₂²	1	1	yes	characteristic
15	#104×C₂₇₁	—	2	1	no	—
16	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
17	#41×C₂₇₁	—	2	1	no	—
18	C₂×Q₈×C₂₇₁	—	2	2	no	—
19	Q₁₆×C₂₇₁	—	2	2	no	—
20	Q₁₆×C₂₇₁	—	2	2	no	—
21	Q₁₆×C₂₇₁	—	4	1	no	—
22	Q₁₆×C₂₇₁	—	4	1	no	—
23	Q₁₆×C₂₇₁	D₄	1	1	yes	characteristic
24	SD₁₆×C₂₇₁	—	2	1	no	—
25	D₈×C₂₇₁	D₄	1	1	yes	characteristic
26	M₁₆×C₂₇₁	—	2	1	no	—
27	M₁₆×C₂₇₁	—	4	1	no	—
28	C₂×C₈×C₂₇₁	—	4	1	no	—
29	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₇₁	—	4	2	no	—
32	C₄²×C₂₇₁	D₄	1	1	yes	characteristic
33	C₁₆×C₂₇₁	—	2	2	no	—
34	Q₈×C₂₇₁	—	2	1	no	—
35	Q₈×C₂₇₁	—	4	2	no	—
36	Q₈×C₂₇₁	—	4	2	no	—
37	D₄×C₂₇₁	—	2	1	no	—
38	D₄×C₂₇₁	—	4	1	no	—
39	C₂×C₄×C₂₇₁	—	2	1	no	—
40	C₂×C₄×C₂₇₁	—	4	1	no	—
41	C₂×C₄×C₂₇₁	—	4	2	no	—
42	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
43	C₈×C₂₇₁	—	4	1	no	—
44	C₈×C₂₇₁	D₈	1	1	yes	characteristic
45	C₈×C₂₇₁	D₈	1	1	yes	characteristic
46	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₇₁	#41	1	1	yes	characteristic
48	C₂²×C₂₇₁	—	4	1	no	—
49	C₂²×C₂₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₄×C₂₇₁	—	2	2	no	—
51	C₄×C₂₇₁	—	4	1	no	—
52	C₄×C₂₇₁	—	8	2	no	—
53	C₄×C₂₇₁	#121	1	1	yes	characteristic
54	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₂₇₁	—	2	1	no	—
56	C₂×C₂₇₁	—	8	1	no	—
57	C₂×C₂₇₁	#459	1	1	yes	characteristic
58	C₂₇₁	#2214	1	1	yes	characteristic, direct product factor, Sylow
59	#2214	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	#510	C₂×C₂₇₁	1	2	yes	—
61	#494	C₂×C₂₇₁	1	1	yes	characteristic, complemented
62	#443	C₂×C₂₇₁	1	1	yes	characteristic
63	#364	C₂×C₂₇₁	1	1	yes	characteristic, complemented
64	#362	C₂×C₂₇₁	1	2	yes	complemented
65	#135	C₂²×C₂₇₁	1	1	yes	characteristic
66	C₂×Q₁₆	—	2	1	no	—
67	C₂×Q₁₆	C₂²×C₂₇₁	1	2	yes	—
68	#131	—	2	1	no	—
69	C₄⋊Q₈	—	2	1	no	—
70	Q₃₂	—	2	2	no	—
71	SD₃₂	—	2	2	no	—
72	M₃₂	C₂²×C₂₇₁	1	2	yes	—
73	#108	C₂²×C₂₇₁	1	1	yes	characteristic
74	#104	—	2	1	no	—
75	C₄×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
76	#41	—	2	1	no	—
77	C₂×Q₈	—	2	2	no	—
78	Q₁₆	—	2	2	no	—
79	Q₁₆	—	2	2	no	—
80	Q₁₆	—	4	1	no	—
81	Q₁₆	—	4	1	no	—
82	Q₁₆	D₄×C₂₇₁	1	1	yes	characteristic
83	SD₁₆	—	2	1	no	—
84	D₈	D₄×C₂₇₁	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	D₄×C₂₇₁	1	1	yes	characteristic
89	C₂×C₈	C₂³×C₂₇₁	1	1	yes	characteristic
90	C₄⋊C₄	—	4	2	no	—
91	C₄²	D₄×C₂₇₁	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	Q₈	—	2	1	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	2	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×D₄×C₂₇₁	1	1	yes	characteristic
102	C₈	—	4	1	no	—
103	C₈	D₈×C₂₇₁	1	1	yes	characteristic
104	C₈	D₈×C₂₇₁	1	1	yes	characteristic
105	C₈	C₂×D₄×C₂₇₁	1	1	yes	characteristic
106	C₈	#41×C₂₇₁	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	AGL(1,ℤ/8)×C₂₇₁	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	4	1	no	—
111	C₄	—	8	2	no	—
112	C₄	#121×C₂₇₁	1	1	yes	characteristic
113	C₄	C₂×D₈×C₂₇₁	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	8	1	no	—
116	C₂	#459×C₂₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#510×C₂₇₁

C₂

yes

maximal

#494×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#443×C₂₇₁

C₂

yes

maximal, characteristic

#364×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#362×C₂₇₁

C₂

yes

maximal, complemented

#135×C₂₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₂₇₁

—

C₂×Q₁₆×C₂₇₁

C₂²

yes

—

#131×C₂₇₁

—

C₄⋊Q₈×C₂₇₁

—

Q₃₂×C₂₇₁

—

SD₃₂×C₂₇₁

—

M₃₂×C₂₇₁

C₂²

yes

—

#108×C₂₇₁

C₂²

yes

characteristic

#104×C₂₇₁

—

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

#41×C₂₇₁

—

C₂×Q₈×C₂₇₁

—

Q₁₆×C₂₇₁

—

Q₁₆×C₂₇₁

—

Q₁₆×C₂₇₁

—

Q₁₆×C₂₇₁

—

Q₁₆×C₂₇₁

D₄

yes

characteristic

SD₁₆×C₂₇₁

—

D₈×C₂₇₁

D₄

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

D₄

yes

characteristic

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

D₄

yes

characteristic

C₁₆×C₂₇₁

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

—

D₄×C₂₇₁

—

D₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

—

C₈×C₂₇₁

D₈

yes

characteristic

C₈×C₂₇₁

D₈

yes

characteristic

C₈×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

#41

yes

characteristic

C₂²×C₂₇₁

—

C₂²×C₂₇₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

#121

yes

characteristic

C₄×C₂₇₁

C₂×D₈

yes

characteristic

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#459

yes

characteristic

C₂₇₁

#2214

yes

characteristic, direct product factor, Sylow

#2214

C₂₇₁

yes

maximal, characteristic, direct product factor, Sylow

#510

C₂×C₂₇₁

yes

—

#494

C₂×C₂₇₁

yes

characteristic, complemented

#443

C₂×C₂₇₁

yes

characteristic

#364

C₂×C₂₇₁

yes

characteristic, complemented

#362

C₂×C₂₇₁

yes

complemented

#135

C₂²×C₂₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₂²×C₂₇₁

yes

—

#131

—

C₄⋊Q₈

—

Q₃₂

—

SD₃₂

—

M₃₂

C₂²×C₂₇₁

yes

—

#108

C₂²×C₂₇₁

yes

characteristic

#104

—

C₄×C₈

C₂²×C₂₇₁

yes

characteristic

#41

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

D₄×C₂₇₁

yes

characteristic

SD₁₆

—

D₈

D₄×C₂₇₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

D₄×C₂₇₁

yes

characteristic

C₂×C₈

C₂³×C₂₇₁

yes

characteristic

C₄⋊C₄

—

C₄²

D₄×C₂₇₁

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×D₄×C₂₇₁

yes

characteristic

102

C₈

—

103

C₈

D₈×C₂₇₁

yes

characteristic

104

C₈

D₈×C₂₇₁

yes

characteristic

105

C₈

C₂×D₄×C₂₇₁

yes

characteristic

106

C₈

#41×C₂₇₁

yes

characteristic

107

C₂²

—

108

C₂²

AGL(1,ℤ/8)×C₂₇₁

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#121×C₂₇₁

yes

characteristic

113

C₄

C₂×D₈×C₂₇₁

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#459×C₂₇₁

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₂₇₁	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₇₁
prime radical	#135×C₂₇₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₇₁ ◁ C₂×C₄×C₂₇₁ ◁ C₄×C₈×C₂₇₁ ◁ G

subgroup

G/H

center, socle

C₂×C₂₇₁

#459

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₂₇₁

prime radical

#135×C₂₇₁

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₂₇₁ ◁ C₂×C₄×C₂₇₁ ◁ C₄×C₈×C₂₇₁ ◁ G

N	H ≅ G/N	type
2. #494×C₂₇₁	C₂	semidirect
4. #364×C₂₇₁	C₂	semidirect
5. #362×C₂₇₁	C₂	semidirect
58. C₂₇₁	#2214	direct (N × H)
59. #2214	C₂₇₁	direct (N × H)
61. #494	C₂×C₂₇₁	semidirect
63. #364	C₂×C₂₇₁	semidirect
64. #362	C₂×C₂₇₁	semidirect

H ≅ G/N

type

2. #494×C₂₇₁

C₂

semidirect

4. #364×C₂₇₁

C₂

semidirect

5. #362×C₂₇₁

C₂

semidirect

58. C₂₇₁

#2214

direct (N × H)

59. #2214

C₂₇₁

direct (N × H)

61. #494

C₂×C₂₇₁

semidirect

63. #364

C₂×C₂₇₁

semidirect

64. #362

C₂×C₂₇₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#510×C₂₇₁	2
#494×C₂₇₁	2
#443×C₂₇₁	2
#364×C₂₇₁	2
#362×C₂₇₁	2
#2214	271

subgroup

index

#510×C₂₇₁

#494×C₂₇₁

#443×C₂₇₁

#364×C₂₇₁

#362×C₂₇₁

#2214

271

Subgroups — #2214×C₂₇₁ #46832800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2214×C271 #46832800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2214×C₂₇₁ #46832800