Subgroups — C₂²×C₂²⋊C₄×C₂₇₇ #31799646

Order 17728.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×C₂₇₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₄×C₂₇₇	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₂₇₇	—	2	8	no	—
5	C₂⁴×C₂₇₇	C₄	1	8	yes	complemented
6	C₂⁴×C₂₇₇	C₂²	1	6	yes	—
7	C₂⁴×C₂₇₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₂₇₇	—	2	8	no	—
9	C₂²×C₄×C₂₇₇	C₂²	1	12	yes	complemented
10	C₂²⋊C₄×C₂₇₇	C₂²	1	16	yes	direct product factor
11	C₂³×C₂₇₇	—	2	32	no	—
12	C₂³×C₂₇₇	—	2	24	no	—
13	C₂³×C₂₇₇	C₂×C₄	1	4	yes	—
14	C₂³×C₂₇₇	C₂×C₄	1	24	yes	complemented
15	C₂³×C₂₇₇	D₄	1	8	yes	—
16	C₂³×C₂₇₇	C₂³	1	4	yes	—
17	C₂³×C₂₇₇	C₂³	1	3	yes	—
18	C₂×C₄×C₂₇₇	—	2	24	no	—
19	C₂×C₄×C₂₇₇	C₂³	1	8	yes	complemented
20	C₂²×C₂₇₇	—	2	48	no	—
21	C₂²×C₂₇₇	—	2	8	no	—
22	C₂²×C₂₇₇	C₂²⋊C₄	1	16	yes	direct product factor
23	C₂²×C₂₇₇	C₂²×C₄	1	6	yes	—
24	C₂²×C₂₇₇	C₂²×C₄	1	8	yes	complemented
25	C₂²×C₂₇₇	C₂×D₄	1	12	yes	—
26	C₂²×C₂₇₇	C₂⁴	1	1	yes	characteristic
27	C₄×C₂₇₇	—	2	8	no	—
28	C₂×C₂₇₇	—	2	8	no	—
29	C₂×C₂₇₇	C₂×C₂²⋊C₄	1	12	yes	direct product factor
30	C₂×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
31	C₂×C₂₇₇	C₂²×D₄	1	2	yes	—
32	C₂₇₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
33	C₂²×C₂²⋊C₄	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂⁵	C₂×C₂₇₇	1	1	yes	characteristic
35	C₂³×C₄	C₂×C₂₇₇	1	2	yes	complemented
36	C₂×C₂²⋊C₄	C₂×C₂₇₇	1	12	yes	direct product factor
37	C₂⁴	—	2	8	no	—
38	C₂⁴	C₄×C₂₇₇	1	8	yes	complemented
39	C₂⁴	C₂²×C₂₇₇	1	6	yes	—
40	C₂⁴	C₂²×C₂₇₇	1	1	yes	characteristic
41	C₂²×C₄	—	2	8	no	—
42	C₂²×C₄	C₂²×C₂₇₇	1	12	yes	complemented
43	C₂²⋊C₄	C₂²×C₂₇₇	1	16	yes	direct product factor
44	C₂³	—	2	32	no	—
45	C₂³	—	2	24	no	—
46	C₂³	C₂×C₄×C₂₇₇	1	4	yes	—
47	C₂³	C₂×C₄×C₂₇₇	1	24	yes	complemented
48	C₂³	D₄×C₂₇₇	1	8	yes	—
49	C₂³	C₂³×C₂₇₇	1	4	yes	—
50	C₂³	C₂³×C₂₇₇	1	3	yes	—
51	C₂×C₄	—	2	24	no	—
52	C₂×C₄	C₂³×C₂₇₇	1	8	yes	complemented
53	C₂²	—	2	48	no	—
54	C₂²	—	2	8	no	—
55	C₂²	C₂²⋊C₄×C₂₇₇	1	16	yes	direct product factor
56	C₂²	C₂²×C₄×C₂₇₇	1	6	yes	—
57	C₂²	C₂²×C₄×C₂₇₇	1	8	yes	complemented
58	C₂²	C₂×D₄×C₂₇₇	1	12	yes	—
59	C₂²	C₂⁴×C₂₇₇	1	1	yes	characteristic
60	C₄	—	2	8	no	—
61	C₂	—	2	8	no	—
62	C₂	C₂×C₂²⋊C₄×C₂₇₇	1	12	yes	direct product factor
63	C₂	C₂³×C₄×C₂₇₇	1	1	yes	characteristic
64	C₂	C₂²×D₄×C₂₇₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₇₇	C₂²
commutator	C₂	C₂³×C₄×C₂₇₇
Frattini, cosocle	C₂²	C₂⁴×C₂₇₇
prime radical	C₂⁵×C₂₇₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₇₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×C₂₇₇	C₂	semidirect
3. C₂×C₂²⋊C₄×C₂₇₇	C₂	direct (N × H)
5. C₂⁴×C₂₇₇	C₄	semidirect
9. C₂²×C₄×C₂₇₇	C₂²	semidirect
10. C₂²⋊C₄×C₂₇₇	C₂²	direct (N × H)
14. C₂³×C₂₇₇	C₂×C₄	semidirect
19. C₂×C₄×C₂₇₇	C₂³	semidirect
22. C₂²×C₂₇₇	C₂²⋊C₄	direct (N × H)
24. C₂²×C₂₇₇	C₂²×C₄	semidirect
29. C₂×C₂₇₇	C₂×C₂²⋊C₄	direct (N × H)
32. C₂₇₇	C₂²×C₂²⋊C₄	direct (N × H)
33. C₂²×C₂²⋊C₄	C₂₇₇	direct (N × H)
35. C₂³×C₄	C₂×C₂₇₇	semidirect
36. C₂×C₂²⋊C₄	C₂×C₂₇₇	direct (N × H)
38. C₂⁴	C₄×C₂₇₇	semidirect
42. C₂²×C₄	C₂²×C₂₇₇	semidirect
43. C₂²⋊C₄	C₂²×C₂₇₇	direct (N × H)
47. C₂³	C₂×C₄×C₂₇₇	semidirect
52. C₂×C₄	C₂³×C₂₇₇	semidirect
55. C₂²	C₂²⋊C₄×C₂₇₇	direct (N × H)
57. C₂²	C₂²×C₄×C₂₇₇	semidirect
62. C₂	C₂×C₂²⋊C₄×C₂₇₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₂₇₇	2
C₂³×C₄×C₂₇₇	2
C₂×C₂²⋊C₄×C₂₇₇	2
C₂²×C₂²⋊C₄	277

Subgroups — C22×C22⋊C4×C277 #31799646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊C₄×C₂₇₇ #31799646