Subgroups — B(4,𝔽₂)×C₂₇₇ #31799591

Order 17728.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#120×C₂₇₇	C₂	1	3	yes	maximal, complemented
3	#99×C₂₇₇	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₂₇₇	C₂²	1	1	yes	characteristic, complemented
5	#41×C₂₇₇	—	2	3	no	—
6	C₂×D₄×C₂₇₇	—	2	3	no	—
7	C₂×D₄×C₂₇₇	—	2	3	no	—
8	C₂×D₄×C₂₇₇	C₂²	1	3	yes	—
9	C₂²⋊C₄×C₂₇₇	—	2	3	no	—
10	C₂²⋊C₄×C₂₇₇	C₂²	1	3	yes	complemented
11	C₂³×C₂₇₇	—	2	1	no	—
12	C₂³×C₂₇₇	—	2	3	no	—
13	C₂³×C₂₇₇	—	4	2	no	—
14	C₂³×C₂₇₇	D₄	1	3	yes	complemented
15	C₂³×C₂₇₇	C₂³	1	1	yes	characteristic
16	Q₈×C₂₇₇	—	2	1	no	—
17	D₄×C₂₇₇	—	2	3	no	—
18	D₄×C₂₇₇	—	2	6	no	—
19	D₄×C₂₇₇	—	4	6	no	—
20	C₂×C₄×C₂₇₇	—	2	3	no	—
21	C₂×C₄×C₂₇₇	—	2	3	no	—
22	C₂×C₄×C₂₇₇	D₄	1	3	yes	complemented
23	C₂²×C₂₇₇	—	2	6	no	—
24	C₂²×C₂₇₇	—	2	3	no	—
25	C₂²×C₂₇₇	—	4	1	no	—
26	C₂²×C₂₇₇	—	4	3	no	—
27	C₂²×C₂₇₇	—	4	3	no	—
28	C₂²×C₂₇₇	—	4	1	no	—
29	C₂²×C₂₇₇	—	8	1	no	—
30	C₂²×C₂₇₇	C₂×D₄	1	3	yes	—
31	C₄×C₂₇₇	—	2	3	no	—
32	C₄×C₂₇₇	—	4	3	no	—
33	C₂×C₂₇₇	—	2	3	no	—
34	C₂×C₂₇₇	—	4	2	no	—
35	C₂×C₂₇₇	—	4	3	no	—
36	C₂×C₂₇₇	#120	1	1	yes	characteristic
37	C₂₇₇	B(4,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
38	B(4,𝔽₂)	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	H(5,𝔽₂)	C₂×C₂₇₇	1	1	yes	characteristic, complemented
40	#120	C₂×C₂₇₇	1	3	yes	complemented
41	#99	C₂×C₂₇₇	1	3	yes	complemented
42	C₂⁴	C₂²×C₂₇₇	1	1	yes	characteristic, complemented
43	#41	—	2	3	no	—
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	—	2	3	no	—
46	C₂×D₄	C₂²×C₂₇₇	1	3	yes	—
47	C₂²⋊C₄	—	2	3	no	—
48	C₂²⋊C₄	C₂²×C₂₇₇	1	3	yes	complemented
49	C₂³	—	2	1	no	—
50	C₂³	—	2	3	no	—
51	C₂³	—	4	2	no	—
52	C₂³	D₄×C₂₇₇	1	3	yes	complemented
53	C₂³	C₂³×C₂₇₇	1	1	yes	characteristic
54	Q₈	—	2	1	no	—
55	D₄	—	2	3	no	—
56	D₄	—	2	6	no	—
57	D₄	—	4	6	no	—
58	C₂×C₄	—	2	3	no	—
59	C₂×C₄	—	2	3	no	—
60	C₂×C₄	D₄×C₂₇₇	1	3	yes	complemented
61	C₂²	—	2	6	no	—
62	C₂²	—	2	3	no	—
63	C₂²	—	4	1	no	—
64	C₂²	—	4	3	no	—
65	C₂²	—	4	3	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	8	1	no	—
68	C₂²	C₂×D₄×C₂₇₇	1	3	yes	—
69	C₄	—	2	3	no	—
70	C₄	—	4	3	no	—
71	C₂	—	2	3	no	—
72	C₂	—	4	2	no	—
73	C₂	—	4	3	no	—
74	C₂	#120×C₂₇₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₇₇	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₂₇₇
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₇₇ ◁ C₂³×C₂₇₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×C₂₇₇	C₂	semidirect
2. #120×C₂₇₇	C₂	semidirect
3. #99×C₂₇₇	C₂	semidirect
4. C₂⁴×C₂₇₇	C₂²	semidirect
10. C₂²⋊C₄×C₂₇₇	C₂²	semidirect
14. C₂³×C₂₇₇	D₄	semidirect
22. C₂×C₄×C₂₇₇	D₄	semidirect
37. C₂₇₇	B(4,𝔽₂)	direct (N × H)
38. B(4,𝔽₂)	C₂₇₇	direct (N × H)
39. H(5,𝔽₂)	C₂×C₂₇₇	semidirect
40. #120	C₂×C₂₇₇	semidirect
41. #99	C₂×C₂₇₇	semidirect
42. C₂⁴	C₂²×C₂₇₇	semidirect
48. C₂²⋊C₄	C₂²×C₂₇₇	semidirect
52. C₂³	D₄×C₂₇₇	semidirect
60. C₂×C₄	D₄×C₂₇₇	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×C₂₇₇	2
#120×C₂₇₇	2
#99×C₂₇₇	2
B(4,𝔽₂)	277

Subgroups — B(4,𝔽2)×C277 #31799591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — B(4,𝔽₂)×C₂₇₇ #31799591