Group #20251096 — subgroups

Subgroups — #20251096

Order 6016.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#15689040	C₂	1	3	yes	maximal
2	#15689034	C₂	1	2	yes	maximal, complemented
3	#15689015	C₂	1	6	yes	maximal, complemented
4	#15688988	C₂	1	3	yes	maximal, complemented
5	C₄₇×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
8	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
9	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
10	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
11	C₄₇⋊#95	C₂²	1	3	yes	—
12	C₄₇⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
15	#95×C₄₇	C₂²	1	3	yes	—
16	C₂³×D₄₇	C₂³	1	2	yes	—
17	C₂×Q₈×C₄₇	—	4	3	no	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
21	C₂²×Dic₄₇	C₂³	1	6	yes	—
22	C₂×D₁₈₈	—	4	3	no	—
23	C₄⋊C₄×C₄₇	—	4	3	no	—
24	(C₂×D₄₇)⋊C₄	—	2	12	no	—
25	(C₂×D₄₇)⋊C₄	—	4	6	no	—
26	C₄⋊Dic₄₇	—	4	3	no	—
27	C₄×Dic₄₇	—	2	6	no	—
28	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
29	C₂²×D₄₇	—	2	6	no	—
30	C₂²×D₄₇	—	2	6	no	—
31	C₂²×D₄₇	—	2	2	no	—
32	Q₈×C₄₇	—	4	4	no	—
33	C₂×C₄×C₄₇	—	2	3	no	—
34	C₂×C₄×C₄₇	—	2	3	no	—
35	C₂×C₄×C₄₇	—	2	6	no	—
36	C₂×C₄×C₄₇	—	2	3	no	—
37	C₂×C₄×C₄₇	—	2	3	no	—
38	C₂×C₄×C₄₇	#41	1	6	yes	—
39	C₂×Dic₄₇	—	2	6	no	—
40	C₂×Dic₄₇	—	2	6	no	—
41	C₂×Dic₄₇	—	2	6	no	—
42	D₁₈₈	—	4	4	no	—
43	C₂²×C₄₇	C₂×#41	1	3	yes	—
44	C₂²×C₄₇	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
46	C₂×D₄₇	—	4	6	no	—
47	C₂×D₄₇	—	4	6	no	—
48	C₂×D₄₇	—	4	2	no	—
49	C₄×C₄₇	—	2	6	no	—
50	C₄×C₄₇	—	4	1	no	—
51	C₄×C₄₇	—	4	3	no	—
52	Dic₄₇	—	4	6	no	—
53	#2785	—	47	1	no	maximal, Sylow
54	C₂×C₄₇	#539	1	3	yes	—
55	C₂×C₄₇	#555	1	3	yes	—
56	C₂×C₄₇	H(3,𝔽₄)	1	1	yes	characteristic
57	D₄₇	—	8	2	no	—
58	#395	D₄₇	1	1	yes	characteristic, complemented
59	#394	—	47	3	no	—
60	#393	—	47	2	no	—
61	#388	—	47	6	no	—
62	#382	—	47	3	no	—
63	C₄₇	#2785	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
65	C₂²×D₄	—	47	1	no	—
66	C₂×C₄⋊C₄	—	47	3	no	—
67	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
68	C₂×C₂²⋊C₄	—	47	6	no	—
69	C₂×C₂²⋊C₄	—	47	6	no	—
70	C₂×C₄²	—	47	3	no	—
71	#95	—	47	3	no	—
72	#95	—	47	6	no	—
73	#95	C₂×D₄₇	1	3	yes	—
74	C₂⁴	—	47	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	188	3	no	—
77	C₂²×C₄	—	47	6	no	—
78	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
80	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	188	3	no	—
83	C₂²⋊C₄	—	94	12	no	—
84	C₂²⋊C₄	—	188	6	no	—
85	C₄²	—	94	6	no	—
86	C₂³	—	94	6	no	—
87	C₂³	—	94	6	no	—
88	C₂³	—	94	2	no	—
89	C₂³	C₂³×D₄₇	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	188	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	94	6	no	—
98	C₂×C₄	—	94	6	no	—
99	C₂×C₄	—	94	6	no	—
100	C₂×C₄	#10625354	1	6	yes	—
101	C₂²	—	188	6	no	—
102	C₂²	—	188	2	no	—
103	C₂²	—	188	6	no	—
104	C₂²	C₂×#10625354	1	3	yes	—
105	C₂²	#13094069	1	3	yes	—
106	C₂²	#13094071	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	188	6	no	—
111	C₂	—	376	2	no	—
112	C₂	#15689722	1	3	yes	—
113	C₂	#15689839	1	3	yes	—
114	C₂	#15689866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689040

C₂

yes

maximal

#15689034

C₂

yes

maximal, complemented

#15689015

C₂

yes

maximal, complemented

#15688988

C₂

yes

maximal, complemented

C₄₇×#395

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2785

—

maximal, Sylow

C₂×C₄₇

#539

yes

—

C₂×C₄₇

#555

yes

—

C₂×C₄₇

H(3,𝔽₄)

yes

characteristic

D₄₇

—

#395

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#2785

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₄₇

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

188

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₂²⋊C₄

—

C₂²⋊C₄

—

188

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₄₇

yes

characteristic

Q₈

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#10625354

yes

—

101

C₂²

—

188

—

102

C₂²

—

188

—

103

C₂²

—

188

—

104

C₂²

C₂×#10625354

yes

—

105

C₂²

#13094069

yes

—

106

C₂²

#13094071

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

188

—

111

C₂

—

376

—

112

C₂

#15689722

yes

—

113

C₂

#15689839

yes

—

114

C₂

#15689866

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#395	C₂
hypercenter	#395	D₄₇
hypocenter	C₄₇	#2785
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#395

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #15689034	C₂	semidirect
3. #15689015	C₂	semidirect
4. #15688988	C₂	semidirect
5. C₄₇×#395	C₂	semidirect
58. #395	D₄₇	semidirect
63. C₄₇	#2785	semidirect

H ≅ G/N

type

2. #15689034

C₂

semidirect

3. #15689015

C₂

semidirect

4. #15688988

C₂

semidirect

5. C₄₇×#395

C₂

semidirect

58. #395

D₄₇

semidirect

63. C₄₇

#2785

semidirect

Subgroups — #20251096

Conjugacy classes of subgroups

Complements