Group #20239151 — subgroups

Subgroups — C₂×C₄₇×#333 #20239151

Order 6016.

256 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 136 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×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₄₇	C₂	1	2	yes	maximal
3	C₄₇×#333	C₂	1	4	yes	maximal, direct product factor
4	C₂²×Q₈×C₄₇	C₄	1	2	yes	—
5	C₂²×C₈×C₄₇	—	2	2	no	—
6	C₄⋊Q₈×C₄₇	—	2	2	no	—
7	C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
8	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
10	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₇	—	2	2	no	—
12	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
13	C₂×Q₈×C₄₇	—	2	2	no	—
14	C₂×Q₈×C₄₇	—	4	4	no	—
15	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₄₇	—	2	2	no	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	D₄	1	2	yes	—
19	C₂×C₈×C₄₇	—	2	4	no	—
20	C₂×C₈×C₄₇	—	4	4	no	—
21	C₄⋊C₄×C₄₇	—	2	4	no	—
22	C₄⋊C₄×C₄₇	—	2	4	no	—
23	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₄₇	D₄	1	2	yes	—
25	C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₄₇	—	4	4	no	—
28	Q₈×C₄₇	—	4	4	no	—
29	C₂×C₄×C₄₇	—	2	2	no	—
30	C₂×C₄×C₄₇	—	2	4	no	—
31	C₂×C₄×C₄₇	—	4	4	no	—
32	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₄₇	SD₁₆	1	4	yes	—
35	C₂×C₄×C₄₇	Q₁₆	1	4	yes	—
36	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₈×C₄₇	—	4	4	no	—
39	C₂²×C₄₇	#100	1	2	yes	—
40	C₂²×C₄₇	#103	1	4	yes	—
41	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₄₇	—	2	1	no	—
43	C₄×C₄₇	—	2	1	no	—
44	C₄×C₄₇	—	4	4	no	—
45	C₄×C₄₇	#103	1	4	yes	—
46	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
47	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
48	C₂×#333	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×C₄₇	#333	1	4	yes	direct product factor
50	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
51	C₂×C₄₇	C₂×#103	1	2	yes	—
52	C₂×C₄⋊Q₈	C₂×C₄₇	1	1	yes	characteristic
53	C₂×C₄⋊C₈	C₂×C₄₇	1	2	yes	—
54	#333	C₂×C₄₇	1	4	yes	direct product factor
55	C₄₇	C₂×#333	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×Q₈	C₄×C₄₇	1	2	yes	—
57	C₂²×C₈	—	2	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	C₄×C₄₇	1	2	yes	—
60	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	4	4	no	—
67	C₂×Q₈	C₂×C₄×C₄₇	1	4	yes	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
70	C₂²×C₄	D₄×C₄₇	1	2	yes	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	4	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
76	C₄²	D₄×C₄₇	1	2	yes	—
77	C₄²	C₂³×C₄₇	1	1	yes	characteristic
78	C₂³	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
79	Q₈	—	4	4	no	—
80	Q₈	—	4	4	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
86	C₂×C₄	SD₁₆×C₄₇	1	4	yes	—
87	C₂×C₄	Q₁₆×C₄₇	1	4	yes	—
88	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
89	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
90	C₈	—	4	4	no	—
91	C₂²	#100×C₄₇	1	2	yes	—
92	C₂²	#103×C₄₇	1	4	yes	—
93	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	2	1	no	—
96	C₄	—	4	4	no	—
97	C₄	#103×C₄₇	1	4	yes	—
98	C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
99	C₄	C₂×Q₁₆×C₄₇	1	2	yes	—
100	C₂	C₄₇×#333	1	4	yes	direct product factor
101	C₂	C₂×#100×C₄₇	1	1	yes	characteristic
102	C₂	C₂×#103×C₄₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal

C₄₇×#333

C₂

yes

maximal, direct product factor

C₂²×Q₈×C₄₇

C₄

yes

—

C₂²×C₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₈×C₄₇

—

C₂²×C₄₇

#100

yes

—

C₂²×C₄₇

#103

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#103

yes

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×#333

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

#333

yes

direct product factor

C₂×C₄₇

C₂×#100

yes

characteristic

C₂×C₄₇

C₂×#103

yes

—

C₂×C₄⋊Q₈

C₂×C₄₇

yes

characteristic

C₂×C₄⋊C₈

C₂×C₄₇

yes

—

#333

C₂×C₄₇

yes

direct product factor

C₄₇

C₂×#333

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₄×C₄₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₄₇

yes

—

C₄⋊Q₈

C₂²×C₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×C₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₄₇

yes

characteristic

C₂²×C₄

D₄×C₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₄₇

yes

characteristic

C₄²

D₄×C₄₇

yes

—

C₄²

C₂³×C₄₇

yes

characteristic

C₂³

C₂²⋊C₄×C₄₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₄₇

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₄₇

yes

—

C₂×C₄

SD₁₆×C₄₇

yes

—

C₂×C₄

Q₁₆×C₄₇

yes

—

C₂×C₄

C₂²×C₄×C₄₇

yes

characteristic

C₂×C₄

C₂×D₄×C₄₇

yes

—

C₈

—

C₂²

#100×C₄₇

yes

—

C₂²

#103×C₄₇

yes

—

C₂²

C₂×C₂²⋊C₄×C₄₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#103×C₄₇

yes

—

C₄

C₂×SD₁₆×C₄₇

yes

—

C₄

C₂×Q₁₆×C₄₇

yes

—

100

C₂

C₄₇×#333

yes

direct product factor

101

C₂

C₂×#100×C₄₇

yes

characteristic

102

C₂

C₂×#103×C₄₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₄²	C₂³×C₄₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₄₇

C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₄₇

Frattini, cosocle

C₄²

C₂³×C₄₇

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
3. C₄₇×#333	C₂	direct (N × H)
48. C₂×#333	C₄₇	direct (N × H)
49. C₂×C₄₇	#333	direct (N × H)
54. #333	C₂×C₄₇	direct (N × H)
55. C₄₇	C₂×#333	direct (N × H)
100. C₂	C₄₇×#333	direct (N × H)

H ≅ G/N

type

3. C₄₇×#333

C₂

direct (N × H)

48. C₂×#333

C₄₇

direct (N × H)

49. C₂×C₄₇

#333

direct (N × H)

54. #333

C₂×C₄₇

direct (N × H)

55. C₄₇

C₂×#333

direct (N × H)

100. C₂

C₄₇×#333

direct (N × H)

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

subgroup	index
C₂×C₄⋊Q₈×C₄₇	2
C₂×C₄⋊C₈×C₄₇	2
C₄₇×#333	2
C₂×#333	47

subgroup

index

C₂×C₄⋊Q₈×C₄₇

C₂×C₄⋊C₈×C₄₇

C₄₇×#333

C₂×#333

Subgroups — C₂×C₄₇×#333 #20239151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C47×#333 #20239151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#333 #20239151