Group #46832324 — subgroups

Subgroups — #1738×C₂₇₁ #46832324

Order 34688.

372 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 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	C₂²×M₁₆×C₂₇₁	C₂	1	2	yes	maximal
2	C₂²×M₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic
3	#341×C₂₇₁	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₂₇₁	—	2	4	no	—
6	C₂×M₁₆×C₂₇₁	—	2	4	no	—
7	C₂×M₁₆×C₂₇₁	—	2	2	no	—
8	C₂×M₁₆×C₂₇₁	C₄	1	8	yes	—
9	C₂×M₁₆×C₂₇₁	C₄	1	4	yes	—
10	C₂×M₁₆×C₂₇₁	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₇₁	—	2	2	no	—
12	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
13	C₂⁴×C₂₇₁	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₂₇₁	C₂×C₄	1	4	yes	—
15	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₂₇₁	D₄	1	4	yes	—
17	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
18	C₂²×C₄×C₂₇₁	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₇₁	—	2	8	no	—
21	M₁₆×C₂₇₁	—	2	8	no	—
22	M₁₆×C₂₇₁	—	2	4	no	—
23	M₁₆×C₂₇₁	C₂×C₄	1	8	yes	—
24	C₂×C₈×C₂₇₁	—	2	8	no	—
25	C₂×C₈×C₂₇₁	—	2	4	no	—
26	C₂×C₈×C₂₇₁	—	2	4	no	—
27	C₂×C₈×C₂₇₁	C₂×C₄	1	4	yes	—
28	C₂³×C₂₇₁	—	4	2	no	—
29	C₂³×C₂₇₁	C₄⋊C₄	1	2	yes	—
30	C₂³×C₂₇₁	C₄⋊C₄	1	4	yes	—
31	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₂₇₁	C₄²	1	4	yes	—
33	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	4	yes	—
36	C₂×C₄×C₂₇₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
42	C₈×C₂₇₁	—	2	8	no	—
43	C₈×C₂₇₁	—	2	4	no	—
44	C₂²×C₂₇₁	—	2	4	no	—
45	C₂²×C₂₇₁	—	2	2	no	—
46	C₂²×C₂₇₁	—	4	4	no	—
47	C₂²×C₂₇₁	#95	1	4	yes	—
48	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	2	yes	—
49	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₂₇₁	#95	1	4	yes	—
51	C₄×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	2	yes	—
54	C₂×C₂₇₁	—	2	4	no	—
55	C₂×C₂₇₁	—	2	2	no	—
56	C₂×C₂₇₁	C₂×#95	1	1	yes	characteristic
57	C₂×C₂₇₁	#430	1	2	yes	—
58	C₂₇₁	#1738	1	1	yes	characteristic, direct product factor, Sylow
59	#1738	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×M₁₆	C₂×C₂₇₁	1	2	yes	—
61	C₂²×M₁₆	C₂×C₂₇₁	1	1	yes	characteristic
62	#341	C₂×C₂₇₁	1	4	yes	complemented
63	C₂³×C₄	C₂²×C₂₇₁	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	2	4	no	—
67	C₂×M₁₆	C₄×C₂₇₁	1	8	yes	—
68	C₂×M₁₆	C₄×C₂₇₁	1	4	yes	—
69	C₂×M₁₆	C₂²×C₂₇₁	1	4	yes	—
70	C₂²×C₈	—	2	2	no	—
71	C₂²×C₈	C₂²×C₂₇₁	1	2	yes	—
72	C₂⁴	Q₈×C₂₇₁	1	1	yes	characteristic
73	C₂²×C₄	C₂×C₄×C₂₇₁	1	4	yes	—
74	C₂²×C₄	C₂×C₄×C₂₇₁	1	2	yes	—
75	C₂²×C₄	D₄×C₂₇₁	1	2	yes	—
76	C₂²×C₄	D₄×C₂₇₁	1	4	yes	—
77	C₂²×C₄	Q₈×C₂₇₁	1	1	yes	characteristic
78	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	2	8	no	—
81	M₁₆	—	2	4	no	—
82	M₁₆	C₂×C₄×C₂₇₁	1	8	yes	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	C₂×C₄×C₂₇₁	1	4	yes	—
87	C₂³	—	4	2	no	—
88	C₂³	C₄⋊C₄×C₂₇₁	1	2	yes	—
89	C₂³	C₄⋊C₄×C₂₇₁	1	4	yes	—
90	C₂³	C₂×Q₈×C₂₇₁	1	1	yes	characteristic
91	C₂×C₄	C₄²×C₂₇₁	1	4	yes	—
92	C₂×C₄	C₂²⋊C₄×C₂₇₁	1	4	yes	—
93	C₂×C₄	C₂²⋊C₄×C₂₇₁	1	4	yes	—
94	C₂×C₄	C₂²⋊C₄×C₂₇₁	1	4	yes	—
95	C₂×C₄	C₄⋊C₄×C₂₇₁	1	4	yes	—
96	C₂×C₄	C₄⋊C₄×C₂₇₁	1	2	yes	—
97	C₂×C₄	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₄×C₂₇₁	1	2	yes	—
99	C₂×C₄	C₂×D₄×C₂₇₁	1	1	yes	characteristic
100	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
101	C₈	—	2	8	no	—
102	C₈	—	2	4	no	—
103	C₂²	—	2	4	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	4	no	—
106	C₂²	#95×C₂₇₁	1	4	yes	—
107	C₂²	C₂×C₄⋊C₄×C₂₇₁	1	2	yes	—
108	C₂²	C₂×C₄⋊C₄×C₂₇₁	1	1	yes	characteristic
109	C₄	#95×C₂₇₁	1	4	yes	—
110	C₄	C₂×C₄²×C₂₇₁	1	1	yes	characteristic
111	C₄	C₂×C₂²⋊C₄×C₂₇₁	1	2	yes	—
112	C₄	C₂×C₂²⋊C₄×C₂₇₁	1	1	yes	characteristic
113	C₂	—	2	4	no	—
114	C₂	—	2	2	no	—
115	C₂	C₂×#95×C₂₇₁	1	1	yes	characteristic
116	C₂	#430×C₂₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₁₆×C₂₇₁

C₂

yes

maximal

C₂²×M₁₆×C₂₇₁

C₂

yes

maximal, characteristic

#341×C₂₇₁

C₂

yes

maximal, complemented

C₂³×C₄×C₂₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₇₁

—

C₂×M₁₆×C₂₇₁

—

C₂×M₁₆×C₂₇₁

—

C₂×M₁₆×C₂₇₁

C₄

yes

—

C₂×M₁₆×C₂₇₁

C₄

yes

—

C₂×M₁₆×C₂₇₁

C₂²

yes

—

C₂²×C₈×C₂₇₁

—

C₂²×C₈×C₂₇₁

C₂²

yes

—

C₂⁴×C₂₇₁

Q₈

yes

characteristic

C₂²×C₄×C₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

C₂×C₄

yes

—

C₂²×C₄×C₂₇₁

D₄

yes

—

C₂²×C₄×C₂₇₁

D₄

yes

—

C₂²×C₄×C₂₇₁

Q₈

yes

characteristic

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

—

C₂³×C₂₇₁

—

C₂³×C₂₇₁

C₄⋊C₄

yes

—

C₂³×C₂₇₁

C₄⋊C₄

yes

—

C₂³×C₂₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₁

C₄²

yes

—

C₂×C₄×C₂₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂²×C₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

—

C₈×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

#95

yes

—

C₂²×C₂₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₁

#95

yes

—

C₄×C₂₇₁

C₂×C₄²

yes

characteristic

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₇₁

C₂×C₂²⋊C₄

yes

—

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

C₂×#95

yes

characteristic

C₂×C₂₇₁

#430

yes

—

C₂₇₁

#1738

yes

characteristic, direct product factor, Sylow

#1738

C₂₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×M₁₆

C₂×C₂₇₁

yes

—

C₂²×M₁₆

C₂×C₂₇₁

yes

characteristic

#341

C₂×C₂₇₁

yes

complemented

C₂³×C₄

C₂²×C₂₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₄×C₂₇₁

yes

—

C₂×M₁₆

C₄×C₂₇₁

yes

—

C₂×M₁₆

C₂²×C₂₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₂₇₁

yes

—

C₂⁴

Q₈×C₂₇₁

yes

characteristic

C₂²×C₄

C₂×C₄×C₂₇₁

yes

—

C₂²×C₄

C₂×C₄×C₂₇₁

yes

—

C₂²×C₄

D₄×C₂₇₁

yes

—

C₂²×C₄

D₄×C₂₇₁

yes

—

C₂²×C₄

Q₈×C₂₇₁

yes

characteristic

C₂²×C₄

C₂³×C₂₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₄×C₂₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₇₁

yes

—

C₂³

—

C₂³

C₄⋊C₄×C₂₇₁

yes

—

C₂³

C₄⋊C₄×C₂₇₁

yes

—

C₂³

C₂×Q₈×C₂₇₁

yes

characteristic

C₂×C₄

C₄²×C₂₇₁

yes

—

C₂×C₄

C₂²⋊C₄×C₂₇₁

yes

—

C₂×C₄

C₂²⋊C₄×C₂₇₁

yes

—

C₂×C₄

C₂²⋊C₄×C₂₇₁

yes

—

C₂×C₄

C₄⋊C₄×C₂₇₁

yes

—

C₂×C₄

C₄⋊C₄×C₂₇₁

yes

—

C₂×C₄

C₂²×C₄×C₂₇₁

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₇₁

yes

—

C₂×C₄

C₂×D₄×C₂₇₁

yes

characteristic

100

C₂×C₄

C₂×D₄×C₂₇₁

yes

—

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#95×C₂₇₁

yes

—

107

C₂²

C₂×C₄⋊C₄×C₂₇₁

yes

—

108

C₂²

C₂×C₄⋊C₄×C₂₇₁

yes

characteristic

109

C₄

#95×C₂₇₁

yes

—

110

C₄

C₂×C₄²×C₂₇₁

yes

characteristic

111

C₄

C₂×C₂²⋊C₄×C₂₇₁

yes

—

112

C₄

C₂×C₂²⋊C₄×C₂₇₁

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×#95×C₂₇₁

yes

characteristic

116

C₂

#430×C₂₇₁

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₇₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₂₇₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₇₁
socle	C₂²×C₂₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₇₁	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₇₁ ◁ C₂³×C₄×C₂₇₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₇₁

C₂×D₄

commutator

C₄

C₂×C₄²×C₂₇₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₇₁

socle

C₂²×C₂₇₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₇₁

Q₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₇₁ ◁ C₂³×C₄×C₂₇₁ ◁ G

N	H ≅ G/N	type
3. #341×C₂₇₁	C₂	semidirect
58. C₂₇₁	#1738	direct (N × H)
59. #1738	C₂₇₁	direct (N × H)
62. #341	C₂×C₂₇₁	semidirect

H ≅ G/N

type

3. #341×C₂₇₁

C₂

semidirect

58. C₂₇₁

#1738

direct (N × H)

59. #1738

C₂₇₁

direct (N × H)

62. #341

C₂×C₂₇₁

semidirect

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

subgroup	index
C₂²×M₁₆×C₂₇₁	2
#341×C₂₇₁	2
#1738	271

subgroup

index

C₂²×M₁₆×C₂₇₁

#341×C₂₇₁

#1738

271

Subgroups — #1738×C₂₇₁ #46832324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1738×C271 #46832324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1738×C₂₇₁ #46832324