Group #27167998 — subgroups

Subgroups — C₁₀₁×#2644 #27167998

Order 12928.

420 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 184 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	#397×C₁₀₁	C₂	1	6	yes	maximal, complemented
2	#396×C₁₀₁	C₂	1	2	yes	maximal, complemented
3	#390×C₁₀₁	C₂	1	3	yes	maximal
4	#389×C₁₀₁	C₂	1	3	yes	maximal, complemented
5	#376×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	3	yes	—
8	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	6	yes	—
9	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	6	yes	—
10	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	6	yes	—
11	C₂×C₄²×C₁₀₁	C₂²	1	3	yes	—
12	#95×C₁₀₁	C₂²	1	1	yes	characteristic
13	#95×C₁₀₁	C₂²	1	3	yes	—
14	#95×C₁₀₁	C₂²	1	6	yes	—
15	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
16	C₂×D₄×C₁₀₁	—	2	6	no	—
17	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
18	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
19	C₂²×C₄×C₁₀₁	C₂³	1	6	yes	—
20	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₁	—	2	6	no	—
22	C₄⋊C₄×C₁₀₁	—	4	6	no	—
23	C₂²⋊C₄×C₁₀₁	—	4	6	no	—
24	C₂²⋊C₄×C₁₀₁	—	4	6	no	—
25	C₄²×C₁₀₁	—	2	6	no	—
26	C₂³×C₁₀₁	—	2	6	no	—
27	C₂³×C₁₀₁	—	2	6	no	—
28	C₂³×C₁₀₁	—	2	2	no	—
29	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
30	D₄×C₁₀₁	—	4	4	no	—
31	C₂×C₄×C₁₀₁	—	2	3	no	—
32	C₂×C₄×C₁₀₁	—	2	3	no	—
33	C₂×C₄×C₁₀₁	—	2	6	no	—
34	C₂×C₄×C₁₀₁	—	2	3	no	—
35	C₂×C₄×C₁₀₁	—	2	3	no	—
36	C₂×C₄×C₁₀₁	—	2	6	no	—
37	C₂×C₄×C₁₀₁	—	2	6	no	—
38	C₂×C₄×C₁₀₁	—	2	3	no	—
39	C₂×C₄×C₁₀₁	—	2	3	no	—
40	C₂×C₄×C₁₀₁	#41	1	6	yes	—
41	C₂²×C₁₀₁	—	4	2	no	—
42	C₂²×C₁₀₁	—	4	6	no	—
43	C₂²×C₁₀₁	—	4	6	no	—
44	C₂²×C₁₀₁	C₂×#41	1	3	yes	—
45	C₂²×C₁₀₁	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
47	C₄×C₁₀₁	—	4	3	no	—
48	C₄×C₁₀₁	—	4	6	no	—
49	C₄×C₁₀₁	—	4	3	no	—
50	C₄×C₁₀₁	#126	1	4	yes	—
51	C₂×C₁₀₁	—	8	2	no	—
52	C₂×C₁₀₁	C₂×#126	1	1	yes	characteristic
53	C₂×C₁₀₁	#540	1	3	yes	—
54	C₂×C₁₀₁	#542	1	3	yes	—
55	#2644	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₀₁	#2644	1	1	yes	characteristic, direct product factor, Sylow
57	#397	C₂×C₁₀₁	1	6	yes	complemented
58	#396	C₂×C₁₀₁	1	2	yes	complemented
59	#390	C₂×C₁₀₁	1	3	yes	—
60	#389	C₂×C₁₀₁	1	3	yes	complemented
61	#376	C₂×C₁₀₁	1	1	yes	characteristic, complemented
62	C₂²×D₄	C₂²×C₁₀₁	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂²×C₁₀₁	1	3	yes	—
64	C₂×C₄⋊C₄	C₂²×C₁₀₁	1	6	yes	—
65	C₂×C₂²⋊C₄	C₂²×C₁₀₁	1	6	yes	—
66	C₂×C₂²⋊C₄	C₂²×C₁₀₁	1	6	yes	—
67	C₂×C₄²	C₂²×C₁₀₁	1	3	yes	—
68	#95	C₂²×C₁₀₁	1	1	yes	characteristic
69	#95	C₂²×C₁₀₁	1	3	yes	—
70	#95	C₂²×C₁₀₁	1	6	yes	—
71	C₂⁴	C₂³×C₁₀₁	1	2	yes	—
72	C₂×D₄	—	2	6	no	—
73	C₂²×C₄	C₂³×C₁₀₁	1	3	yes	—
74	C₂²×C₄	C₂³×C₁₀₁	1	3	yes	—
75	C₂²×C₄	C₂³×C₁₀₁	1	6	yes	—
76	C₂²×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
77	C₄⋊C₄	—	2	6	no	—
78	C₄⋊C₄	—	4	6	no	—
79	C₂²⋊C₄	—	4	6	no	—
80	C₂²⋊C₄	—	4	6	no	—
81	C₄²	—	2	6	no	—
82	C₂³	—	2	6	no	—
83	C₂³	—	2	6	no	—
84	C₂³	—	2	2	no	—
85	C₂³	C₂⁴×C₁₀₁	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	#41×C₁₀₁	1	6	yes	—
97	C₂²	—	4	2	no	—
98	C₂²	—	4	6	no	—
99	C₂²	—	4	6	no	—
100	C₂²	C₂×#41×C₁₀₁	1	3	yes	—
101	C₂²	H(5,𝔽₂)×C₁₀₁	1	3	yes	—
102	C₂²	2₋¹⁺⁴×C₁₀₁	1	1	yes	characteristic
103	C₄	—	4	3	no	—
104	C₄	—	4	6	no	—
105	C₄	—	4	3	no	—
106	C₄	#126×C₁₀₁	1	4	yes	—
107	C₂	—	8	2	no	—
108	C₂	C₂×#126×C₁₀₁	1	1	yes	characteristic
109	C₂	#540×C₁₀₁	1	3	yes	—
110	C₂	#542×C₁₀₁	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#397×C₁₀₁

C₂

yes

maximal, complemented

#396×C₁₀₁

C₂

yes

maximal, complemented

#390×C₁₀₁

C₂

yes

maximal

#389×C₁₀₁

C₂

yes

maximal, complemented

#376×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

#95×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

C₂²

yes

—

#95×C₁₀₁

C₂²

yes

—

C₂⁴×C₁₀₁

C₂³

yes

—

C₂×D₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×#41

yes

—

C₂²×C₁₀₁

H(5,𝔽₂)

yes

—

C₂²×C₁₀₁

2₋¹⁺⁴

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#126

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#126

yes

characteristic

C₂×C₁₀₁

#540

yes

—

C₂×C₁₀₁

#542

yes

—

#2644

C₁₀₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₁

#2644

yes

characteristic, direct product factor, Sylow

#397

C₂×C₁₀₁

yes

complemented

#396

C₂×C₁₀₁

yes

complemented

#390

C₂×C₁₀₁

yes

—

#389

C₂×C₁₀₁

yes

complemented

#376

C₂×C₁₀₁

yes

characteristic, complemented

C₂²×D₄

C₂²×C₁₀₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₀₁

yes

—

C₂×C₄²

C₂²×C₁₀₁

yes

—

#95

C₂²×C₁₀₁

yes

characteristic

#95

C₂²×C₁₀₁

yes

—

#95

C₂²×C₁₀₁

yes

—

C₂⁴

C₂³×C₁₀₁

yes

—

C₂×D₄

—

C₂²×C₄

C₂³×C₁₀₁

yes

—

C₂²×C₄

C₂³×C₁₀₁

yes

—

C₂²×C₄

C₂³×C₁₀₁

yes

—

C₂²×C₄

C₂³×C₁₀₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₁₀₁

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#41×C₁₀₁

yes

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×#41×C₁₀₁

yes

—

101

C₂²

H(5,𝔽₂)×C₁₀₁

yes

—

102

C₂²

2₋¹⁺⁴×C₁₀₁

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#126×C₁₀₁

yes

—

107

C₂

—

108

C₂

C₂×#126×C₁₀₁

yes

characteristic

109

C₂

#540×C₁₀₁

yes

—

110

C₂

#542×C₁₀₁

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₀₁
prime radical	C₂²×D₄×C₁₀₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₁

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₁₀₁

prime radical

C₂²×D₄×C₁₀₁

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₁ ◁ G

N	H ≅ G/N	type
1. #397×C₁₀₁	C₂	semidirect
2. #396×C₁₀₁	C₂	semidirect
4. #389×C₁₀₁	C₂	semidirect
5. #376×C₁₀₁	C₂	semidirect
55. #2644	C₁₀₁	direct (N × H)
56. C₁₀₁	#2644	direct (N × H)
57. #397	C₂×C₁₀₁	semidirect
58. #396	C₂×C₁₀₁	semidirect
60. #389	C₂×C₁₀₁	semidirect
61. #376	C₂×C₁₀₁	semidirect

H ≅ G/N

type

1. #397×C₁₀₁

C₂

semidirect

2. #396×C₁₀₁

C₂

semidirect

4. #389×C₁₀₁

C₂

semidirect

5. #376×C₁₀₁

C₂

semidirect

55. #2644

C₁₀₁

direct (N × H)

56. C₁₀₁

#2644

direct (N × H)

57. #397

C₂×C₁₀₁

semidirect

58. #396

C₂×C₁₀₁

semidirect

60. #389

C₂×C₁₀₁

semidirect

61. #376

C₂×C₁₀₁

semidirect

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

subgroup	index
#397×C₁₀₁	2
#396×C₁₀₁	2
#390×C₁₀₁	2
#389×C₁₀₁	2
#376×C₁₀₁	2
#2644	101

subgroup

index

#397×C₁₀₁

#396×C₁₀₁

#390×C₁₀₁

#389×C₁₀₁

#376×C₁₀₁

#2644

101

Subgroups — C₁₀₁×#2644 #27167998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C101×#2644 #27167998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#2644 #27167998