Group #20251988 — subgroups

Subgroups — #20251988

Order 6016.

476 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 293 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	#15689606	C₂	1	2	yes	maximal
2	#15689165	C₂	1	6	yes	maximal, complemented
3	#15689161	C₂	1	3	yes	maximal, complemented
4	C₂×#13093958	C₂	1	3	yes	maximal, complemented
5	C₂×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₄₇	—	2	2	no	—
7	D₄×Dic₄₇	—	2	6	no	—
8	#13093983	C₂²	1	6	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	6	yes	—
10	#13093958	C₂²	1	3	yes	—
11	#13093958	C₂²	1	6	yes	—
12	#13093958	C₂²	1	1	yes	characteristic
13	#13093956	C₂²	1	6	yes	—
14	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
15	#131×C₄₇	C₄	1	8	yes	complemented
16	C₂×M₁₆×C₄₇	C₂²	1	3	yes	—
17	C₂²×C₈×C₄₇	C₂²	1	3	yes	—
18	#41×C₄₇	C₂×C₄	1	4	yes	—
19	#41×C₄₇	Q₈	1	4	yes	—
20	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
21	C₂×D₄×C₄₇	D₄	1	3	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
23	C₂²×Dic₄₇	—	2	6	no	—
24	M₁₆×C₄₇	C₂×C₄	1	12	yes	complemented
25	C₂×C₈×C₄₇	C₂×C₄	1	12	yes	complemented
26	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₄₇	C₂³	1	3	yes	—
28	C₂²⋊Dic₄₇	—	2	6	no	—
29	C₄⋊Dic₄₇	—	2	6	no	—
30	C₄⋊Dic₄₇	C₂³	1	2	yes	—
31	C₄⋊Dic₄₇	C₂³	1	6	yes	—
32	C₄×Dic₄₇	—	2	6	no	—
33	C₂³×C₄₇	C₂×D₄	1	3	yes	—
34	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
35	D₄×C₄₇	C₄⋊C₄	1	12	yes	—
36	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
38	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
39	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₄₇	—	2	2	no	—
41	C₂×Dic₄₇	—	2	6	no	—
42	C₂×Dic₄₇	—	8	6	no	—
43	C₈×C₄₇	C₂²×C₄	1	2	yes	—
44	C₈×C₄₇	C₂²×C₄	1	6	yes	—
45	C₂²×C₄₇	—	2	3	no	—
46	C₂²×C₄₇	—	2	3	no	—
47	C₂²×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
48	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
50	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
51	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
52	Dic₄₇	—	4	6	no	—
53	Dic₄₇	—	4	2	no	—
54	#2913	—	47	1	no	maximal, Sylow
55	C₂×C₄₇	—	2	6	no	—
56	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄₇	#576	1	1	yes	characteristic
58	C₂×C₄₇	#578	1	1	yes	characteristic
59	C₂×#131	D₄₇	1	1	yes	characteristic
60	#520	—	47	2	no	—
61	#428	—	47	6	no	—
62	#427	—	47	3	no	—
63	C₂×#107	—	47	3	no	—
64	C₄₇	#2913	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
66	#131	Dic₄₇	1	8	yes	complemented
67	C₂×M₁₆	C₂×D₄₇	1	3	yes	—
68	C₂²×C₈	C₂×D₄₇	1	3	yes	—
69	C₄×Q₈	—	94	2	no	—
70	C₄×D₄	—	94	6	no	—
71	#117	—	47	6	no	—
72	C₂×C₄⋊C₄	—	47	6	no	—
73	#107	—	47	3	no	—
74	#107	—	47	6	no	—
75	#107	—	47	1	no	—
76	#106	—	47	6	no	—
77	#41	Dic₉₄	1	4	yes	—
78	#41	C₂×Dic₄₇	1	4	yes	—
79	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
80	C₂×D₄	D₁₈₈	1	3	yes	—
81	C₂²×C₄	—	94	6	no	—
82	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
83	M₁₆	C₂×Dic₄₇	1	12	yes	complemented
84	C₂×C₈	C₂×Dic₄₇	1	12	yes	complemented
85	C₂×C₈	C₂²×D₄₇	1	3	yes	—
86	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
87	C₄⋊C₄	—	47	2	no	—
88	C₄⋊C₄	—	47	6	no	—
89	C₄⋊C₄	—	94	6	no	—
90	C₂²⋊C₄	—	94	6	no	—
91	C₄²	—	94	6	no	—
92	C₂³	C₂×D₁₈₈	1	3	yes	—
93	Q₈	C₄⋊Dic₄₇	1	4	yes	—
94	D₄	C₄⋊Dic₄₇	1	12	yes	—
95	C₂×C₄	—	94	2	no	—
96	C₂×C₄	—	94	6	no	—
97	C₂×C₄	—	376	6	no	—
98	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
99	C₂×C₄	C₂×D₁₈₈	1	3	yes	—
100	C₂×C₄	C₂²×Dic₄₇	1	6	yes	—
101	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
102	C₈	C₂²×Dic₄₇	1	2	yes	—
103	C₈	C₂²×Dic₄₇	1	6	yes	—
104	C₂²	—	2	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	C₂×C₄⋊Dic₄₇	1	6	yes	—
107	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
108	C₄	—	188	6	no	—
109	C₄	—	188	2	no	—
110	C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
111	C₄	C₂³×Dic₄₇	1	1	yes	characteristic
112	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
113	C₂	—	2	6	no	—
114	C₂	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
115	C₂	#15689928	1	1	yes	characteristic
116	C₂	#15689940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689606

C₂

yes

maximal

#15689165

C₂

yes

maximal, complemented

#15689161

C₂

yes

maximal, complemented

C₂×#13093958

C₂

yes

maximal, complemented

C₂×#131×C₄₇

C₂

yes

maximal, characteristic

Q₈×Dic₄₇

—

D₄×Dic₄₇

—

#13093983

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093958

C₂²

yes

—

#13093958

C₂²

yes

—

#13093958

C₂²

yes

characteristic

#13093956

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

C₄

yes

complemented

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

#41×C₄₇

C₂×C₄

yes

—

#41×C₄₇

Q₈

yes

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

—

M₁₆×C₄₇

C₂×C₄

yes

complemented

C₂×C₈×C₄₇

C₂×C₄

yes

complemented

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

Q₈×C₄₇

C₄⋊C₄

yes

—

D₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

C₂²×C₄

yes

—

C₈×C₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2913

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

—

C₄₇

#2913

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

#131

Dic₄₇

yes

complemented

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

C₂×D₄₇

yes

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×C₄⋊C₄

—

—

—

—

—

Dic₉₄

yes

—

#41

C₂×Dic₄₇

yes

—

C₂×Q₈

D₁₈₈

yes

characteristic

C₂×D₄

D₁₈₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

M₁₆

C₂×Dic₄₇

yes

complemented

C₂×C₈

C₂×Dic₄₇

yes

complemented

C₂×C₈

C₂²×D₄₇

yes

—

C₂×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

C₂×D₁₈₈

yes

—

Q₈

C₄⋊Dic₄₇

yes

—

D₄

C₄⋊Dic₄₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

376

—

C₂×C₄

C₂×Dic₉₄

yes

—

C₂×C₄

C₂×D₁₈₈

yes

—

100

C₂×C₄

C₂²×Dic₄₇

yes

—

101

C₂×C₄

C₂³×D₄₇

yes

characteristic

102

C₈

C₂²×Dic₄₇

yes

—

103

C₈

C₂²×Dic₄₇

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×C₄⋊Dic₄₇

yes

—

107

C₂²

C₂²×D₁₈₈

yes

characteristic

108

C₄

—

188

—

109

C₄

—

188

—

110

C₄

C₂×C₄⋊Dic₄₇

yes

—

111

C₄

C₂³×Dic₄₇

yes

characteristic

112

C₄

C₂²×Dic₉₄

yes

characteristic

113

C₂

—

114

C₂

C₂²×C₄⋊Dic₄₇

yes

characteristic

115

C₂

#15689928

yes

characteristic

116

C₂

#15689940

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₄×C₄₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×#131	D₄₇
hypocenter	C₄₇	#2913
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₄×C₄₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#131×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

N	H ≅ G/N	type
2. #15689165	C₂	semidirect
3. #15689161	C₂	semidirect
4. C₂×#13093958	C₂	semidirect
15. #131×C₄₇	C₄	semidirect
24. M₁₆×C₄₇	C₂×C₄	semidirect
25. C₂×C₈×C₄₇	C₂×C₄	semidirect
64. C₄₇	#2913	semidirect
66. #131	Dic₄₇	semidirect
83. M₁₆	C₂×Dic₄₇	semidirect
84. C₂×C₈	C₂×Dic₄₇	semidirect

H ≅ G/N

type

2. #15689165

C₂

semidirect

3. #15689161

C₂

semidirect

4. C₂×#13093958

C₂

semidirect

15. #131×C₄₇

C₄

semidirect

24. M₁₆×C₄₇

C₂×C₄

semidirect

25. C₂×C₈×C₄₇

C₂×C₄

semidirect

64. C₄₇

#2913

semidirect

66. #131

Dic₄₇

semidirect

83. M₁₆

C₂×Dic₄₇

semidirect

84. C₂×C₈

C₂×Dic₄₇

semidirect

Subgroups — #20251988

Conjugacy classes of subgroups

Complements