Subgroups — C₂×C₄₇×#2867 #26076340

Order 12032.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₄₇	C₂	1	14	yes	maximal, complemented
2	C₄₇×#2867	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄×C₄₇	C₂²	1	28	yes	—
5	C₂×#127×C₄₇	—	2	112	no	—
6	C₂×#127×C₄₇	C₂²	1	112	yes	complemented
7	C₂²×C₄²×C₄₇	C₂²	1	7	yes	—
8	C₄³×C₄₇	C₂²	1	8	yes	complemented
9	C₂⁵×C₄₇	C₂³	1	8	yes	—
10	C₂²×D₄×C₄₇	—	2	336	no	—
11	C₂²×D₄×C₄₇	C₂³	1	112	yes	—
12	C₂³×C₄×C₄₇	C₂³	1	7	yes	—
13	#127×C₄₇	—	2	448	no	—
14	C₂×C₄²×C₄₇	D₄	1	56	yes	complemented
15	C₂×C₄²×C₄₇	C₂³	1	28	yes	—
16	C₂⁴×C₄₇	—	2	112	no	—
17	C₂⁴×C₄₇	C₂⁴	1	16	yes	—
18	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
19	C₂×D₄×C₄₇	—	2	672	no	—
20	C₂×D₄×C₄₇	—	4	448	no	—
21	C₂²×C₄×C₄₇	C₂×D₄	1	84	yes	—
22	C₂²×C₄×C₄₇	C₂⁴	1	14	yes	—
23	C₄²×C₄₇	C₂×D₄	1	112	yes	complemented
24	C₂³×C₄₇	—	2	112	no	—
25	C₂³×C₄₇	—	4	224	no	—
26	C₂³×C₄₇	C₂²×D₄	1	14	yes	—
27	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
28	D₄×C₄₇	—	4	448	no	—
29	C₂×C₄×C₄₇	#127	1	112	yes	complemented
30	C₂×C₄×C₄₇	C₂²×D₄	1	84	yes	—
31	C₂×#2867	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₄₇	—	4	112	no	—
33	C₂²×C₄₇	—	8	64	no	—
34	C₂²×C₄₇	C₂×#127	1	28	yes	—
35	C₂²×C₄₇	C₂³×D₄	1	7	yes	—
36	C₄×C₄₇	C₂×#127	1	56	yes	complemented
37	C₂²×#127	C₂×C₄₇	1	14	yes	complemented
38	#2867	C₂×C₄₇	1	16	yes	direct product factor
39	C₂×C₄³	C₂×C₄₇	1	1	yes	characteristic, complemented
40	C₂×C₄₇	—	8	16	no	—
41	C₂×C₄₇	#2867	1	8	yes	direct product factor
42	C₂×C₄₇	C₂²×#127	1	7	yes	—
43	C₂³×D₄	C₂²×C₄₇	1	28	yes	—
44	C₂×#127	—	2	112	no	—
45	C₂×#127	C₂²×C₄₇	1	112	yes	complemented
46	C₂²×C₄²	C₂²×C₄₇	1	7	yes	—
47	C₄³	C₂²×C₄₇	1	8	yes	complemented
48	C₄₇	C₂×#2867	1	1	yes	characteristic, direct product factor, Sylow
49	C₂⁵	C₂³×C₄₇	1	8	yes	—
50	C₂²×D₄	—	2	336	no	—
51	C₂²×D₄	C₂³×C₄₇	1	112	yes	—
52	C₂³×C₄	C₂³×C₄₇	1	7	yes	—
53	#127	—	2	448	no	—
54	C₂×C₄²	D₄×C₄₇	1	56	yes	complemented
55	C₂×C₄²	C₂³×C₄₇	1	28	yes	—
56	C₂⁴	—	2	112	no	—
57	C₂⁴	C₂⁴×C₄₇	1	16	yes	—
58	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
59	C₂×D₄	—	2	672	no	—
60	C₂×D₄	—	4	448	no	—
61	C₂²×C₄	C₂×D₄×C₄₇	1	84	yes	—
62	C₂²×C₄	C₂⁴×C₄₇	1	14	yes	—
63	C₄²	C₂×D₄×C₄₇	1	112	yes	complemented
64	C₂³	—	2	112	no	—
65	C₂³	—	4	224	no	—
66	C₂³	C₂²×D₄×C₄₇	1	14	yes	—
67	C₂³	C₂⁵×C₄₇	1	1	yes	characteristic
68	D₄	—	4	448	no	—
69	C₂×C₄	#127×C₄₇	1	112	yes	complemented
70	C₂×C₄	C₂²×D₄×C₄₇	1	84	yes	—
71	C₂²	—	4	112	no	—
72	C₂²	—	8	64	no	—
73	C₂²	C₂×#127×C₄₇	1	28	yes	—
74	C₂²	C₂³×D₄×C₄₇	1	7	yes	—
75	C₄	C₂×#127×C₄₇	1	56	yes	complemented
76	C₂	—	8	16	no	—
77	C₂	C₄₇×#2867	1	8	yes	direct product factor
78	C₂	C₂²×#127×C₄₇	1	7	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#127×C₄₇	C₂	semidirect
2. C₄₇×#2867	C₂	direct (N × H)
3. C₂×C₄³×C₄₇	C₂	semidirect
6. C₂×#127×C₄₇	C₂²	semidirect
8. C₄³×C₄₇	C₂²	semidirect
14. C₂×C₄²×C₄₇	D₄	semidirect
23. C₄²×C₄₇	C₂×D₄	semidirect
29. C₂×C₄×C₄₇	#127	semidirect
31. C₂×#2867	C₄₇	direct (N × H)
36. C₄×C₄₇	C₂×#127	semidirect
37. C₂²×#127	C₂×C₄₇	semidirect
38. #2867	C₂×C₄₇	direct (N × H)
39. C₂×C₄³	C₂×C₄₇	semidirect
41. C₂×C₄₇	#2867	direct (N × H)
45. C₂×#127	C₂²×C₄₇	semidirect
47. C₄³	C₂²×C₄₇	semidirect
48. C₄₇	C₂×#2867	direct (N × H)
54. C₂×C₄²	D₄×C₄₇	semidirect
63. C₄²	C₂×D₄×C₄₇	semidirect
69. C₂×C₄	#127×C₄₇	semidirect
75. C₄	C₂×#127×C₄₇	semidirect
77. C₂	C₄₇×#2867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#127×C₄₇	2
C₄₇×#2867	2
C₂×C₄³×C₄₇	2
C₂×#2867	47

Subgroups — C2×C47×#2867 #26076340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2867 #26076340