Group #20251784 — subgroups

Subgroups — #20251784

Order 6016.

648 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 245 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	C₂×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂×#13094022	C₂	1	4	yes	maximal
3	#15688992	C₂	1	8	yes	maximal
4	C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	2	yes	—
7	#13094024	—	2	4	no	—
8	#13094022	—	2	16	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂×Dic₄₇⋊C₄	C₂²	1	16	yes	—
11	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
13	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₄₇	—	2	4	no	—
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₇	—	2	4	no	—
18	C₂×Q₈×C₄₇	—	2	8	no	—
19	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
22	C₂²×Dic₄₇	C₂³	1	4	yes	—
23	C₂²×Dic₄₇	C₂³	1	4	yes	—
24	C₂×Dic₉₄	—	2	8	no	—
25	C₂×Dic₉₄	—	2	4	no	—
26	C₄⋊C₄×C₄₇	—	2	8	no	—
27	C₄²×C₄₇	D₄	1	4	yes	—
28	C₄⋊Dic₄₇	—	2	8	no	—
29	Dic₄₇⋊C₄	—	2	32	no	—
30	C₄×Dic₄₇	D₄	1	8	yes	—
31	C₄×Dic₄₇	Q₈	1	16	yes	complemented
32	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
33	Q₈×C₄₇	—	4	8	no	—
34	C₂×C₄×C₄₇	—	2	8	no	—
35	C₂×C₄×C₄₇	—	2	4	no	—
36	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₄₇	C₂×Q₈	1	8	yes	—
40	C₂×Dic₄₇	—	2	8	no	—
41	C₂×Dic₄₇	—	2	4	no	—
42	C₂×Dic₄₇	C₂×D₄	1	8	yes	—
43	C₂×Dic₄₇	C₂×Q₈	1	16	yes	—
44	Dic₉₄	—	4	8	no	—
45	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
46	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₄₇	C₂²×Q₈	1	4	yes	—
48	C₄×C₄₇	—	4	4	no	—
49	C₄×C₄₇	#127	1	4	yes	—
50	C₄×C₄₇	C₄⋊Q₈	1	8	yes	complemented
51	Dic₄₇	—	4	4	no	—
52	Dic₄₇	C₄⋊Q₈	1	16	yes	complemented
53	#2868	—	47	1	no	maximal, Sylow
54	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
55	C₂×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
56	C₂×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
57	C₂×C₄⋊Q₈	—	47	4	no	—
58	C₂×C₄⋊Q₈	—	47	1	no	—
59	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
60	#384	—	47	8	no	—
61	C₄³	—	47	1	no	—
62	C₄₇	#2868	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	47	2	no	—
64	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
65	C₄⋊Q₈	—	2	4	no	—
66	C₄⋊Q₈	—	94	16	no	—
67	C₄⋊Q₈	—	94	4	no	—
68	C₂×C₄⋊C₄	—	47	16	no	—
69	C₂×C₄⋊C₄	—	47	4	no	—
70	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
71	C₂×C₄²	—	47	2	no	—
72	C₂×C₄²	—	47	4	no	—
73	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
74	C₂×Q₈	—	2	8	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	94	8	no	—
77	C₂×Q₈	—	94	4	no	—
78	C₂²×C₄	—	47	4	no	—
79	C₂²×C₄	—	47	4	no	—
80	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
81	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	94	32	no	—
85	C₄⋊C₄	—	94	8	no	—
86	C₄²	—	47	8	no	—
87	C₄²	—	47	16	no	—
88	C₄²	C₂²⋊D₄₇	1	4	yes	—
89	C₂³	C₂³×D₄₇	1	1	yes	characteristic
90	Q₈	—	4	8	no	—
91	Q₈	—	188	8	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	47	8	no	—
95	C₂×C₄	—	47	16	no	—
96	C₂×C₄	—	94	8	no	—
97	C₂×C₄	—	94	4	no	—
98	C₂×C₄	D₄×D₄₇	1	4	yes	—
99	C₂×C₄	Q₈×D₄₇	1	8	yes	—
100	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
102	C₂²	C₂×D₄×D₄₇	1	2	yes	—
103	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
104	C₂²	C₂×Q₈×D₄₇	1	4	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	47	16	no	—
107	C₄	—	188	4	no	—
108	C₄	#13094020	1	4	yes	—
109	C₄	#13094022	1	8	yes	complemented
110	C₂	C₂×#13094020	1	1	yes	characteristic
111	C₂	C₄⋊Q₈×D₄₇	1	4	yes	—
112	C₂	C₂×#13094022	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal, characteristic

C₂×#13094022

C₂

yes

maximal

#15688992

C₂

yes

maximal

C₄²×Dic₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

#13094024

—

#13094022

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

D₄

yes

—

C₄×Dic₄₇

Q₈

yes

complemented

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×Q₈

yes

—

Dic₉₄

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

#127

yes

—

C₄×C₄₇

C₄⋊Q₈

yes

complemented

Dic₄₇

—

Dic₄₇

C₄⋊Q₈

yes

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊Q₈

yes

—

C₂×C₄₇

C₂×C₄⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₄₇

yes

characteristic

#384

—

C₄³

—

C₄₇

#2868

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₄₇

yes

—

C₂³

C₂³×D₄₇

yes

characteristic

Q₈

—

Q₈

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₄₇

yes

—

C₂×C₄

Q₈×D₄₇

yes

—

100

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

102

C₂²

C₂×D₄×D₄₇

yes

—

103

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

104

C₂²

C₂×Q₈×D₄₇

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

188

—

108

C₄

#13094020

yes

—

109

C₄

#13094022

yes

complemented

110

C₂

C₂×#13094020

yes

characteristic

111

C₂

C₄⋊Q₈×D₄₇

yes

—

112

C₂

C₂×#13094022

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#2868
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₄₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
31. C₄×Dic₄₇	Q₈	semidirect
50. C₄×C₄₇	C₄⋊Q₈	semidirect
52. Dic₄₇	C₄⋊Q₈	semidirect
62. C₄₇	#2868	semidirect
109. C₄	#13094022	semidirect

H ≅ G/N

type

31. C₄×Dic₄₇

Q₈

semidirect

50. C₄×C₄₇

C₄⋊Q₈

semidirect

52. Dic₄₇

C₄⋊Q₈

semidirect

62. C₄₇

#2868

semidirect

109. C₄

#13094022

semidirect

Subgroups — #20251784

Conjugacy classes of subgroups

Complements