Subgroups — #520×C₁₉₉ #27163670

Order 12736.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₁₉₉	C₂	1	2	yes	maximal, complemented
3	C₄×D₄×C₁₉₉	C₂	1	6	yes	maximal, complemented
4	#117×C₁₉₉	C₂	1	3	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₁₉₉	C₂	1	3	yes	maximal, complemented
6	#41×C₁₉₉	C₄	1	8	yes	complemented
7	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₁₉₉	C₂²	1	3	yes	—
9	C₂²×C₄×C₁₉₉	C₂²	1	6	yes	—
10	C₂²×C₄×C₁₉₉	C₂²	1	3	yes	—
11	C₄⋊C₄×C₁₉₉	C₂²	1	3	yes	—
12	C₄⋊C₄×C₁₉₉	C₂²	1	6	yes	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₉₉	C₂²	1	6	yes	—
15	C₄²×C₁₉₉	C₂²	1	6	yes	—
16	C₂³×C₁₉₉	C₂³	1	3	yes	—
17	Q₈×C₁₉₉	C₂×C₄	1	4	yes	complemented
18	D₄×C₁₉₉	C₂×C₄	1	12	yes	complemented
19	C₂×C₄×C₁₉₉	—	4	6	no	—
20	C₂×C₄×C₁₉₉	C₂×C₄	1	12	yes	complemented
21	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
22	C₂×C₄×C₁₉₉	C₂³	1	6	yes	—
23	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
24	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	—	2	3	no	—
26	C₂²×C₁₉₉	—	2	3	no	—
27	C₂²×C₁₉₉	C₂²×C₄	1	6	yes	—
28	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
29	C₄×C₁₉₉	—	2	6	no	—
30	C₄×C₁₉₉	—	2	2	no	—
31	C₄×C₁₉₉	C₂²×C₄	1	2	yes	—
32	C₄×C₁₉₉	C₂²×C₄	1	6	yes	—
33	C₂×C₁₉₉	—	2	6	no	—
34	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
35	C₂×C₁₉₉	H(5,𝔽₂)	1	1	yes	characteristic
36	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
37	C₁₉₉	#520	1	1	yes	characteristic, direct product factor, Sylow
38	#520	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×#41	C₂×C₁₉₉	1	1	yes	characteristic
40	C₄×Q₈	C₂×C₁₉₉	1	2	yes	complemented
41	C₄×D₄	C₂×C₁₉₉	1	6	yes	complemented
42	#117	C₂×C₁₉₉	1	3	yes	complemented
43	C₂×C₄⋊C₄	C₂×C₁₉₉	1	3	yes	complemented
44	#41	C₄×C₁₉₉	1	8	yes	complemented
45	C₂×Q₈	C₂²×C₁₉₉	1	1	yes	characteristic
46	C₂×D₄	C₂²×C₁₉₉	1	3	yes	—
47	C₂²×C₄	C₂²×C₁₉₉	1	6	yes	—
48	C₂²×C₄	C₂²×C₁₉₉	1	3	yes	—
49	C₄⋊C₄	C₂²×C₁₉₉	1	3	yes	—
50	C₄⋊C₄	C₂²×C₁₉₉	1	6	yes	—
51	C₄⋊C₄	C₂²×C₁₉₉	1	1	yes	characteristic
52	C₂²⋊C₄	C₂²×C₁₉₉	1	6	yes	—
53	C₄²	C₂²×C₁₉₉	1	6	yes	—
54	C₂³	C₂³×C₁₉₉	1	3	yes	—
55	Q₈	C₂×C₄×C₁₉₉	1	4	yes	complemented
56	D₄	C₂×C₄×C₁₉₉	1	12	yes	complemented
57	C₂×C₄	—	4	6	no	—
58	C₂×C₄	C₂×C₄×C₁₉₉	1	12	yes	complemented
59	C₂×C₄	C₂³×C₁₉₉	1	2	yes	—
60	C₂×C₄	C₂³×C₁₉₉	1	6	yes	—
61	C₂×C₄	C₂³×C₁₉₉	1	3	yes	—
62	C₂×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
63	C₂²	—	2	3	no	—
64	C₂²	—	2	3	no	—
65	C₂²	C₂²×C₄×C₁₉₉	1	6	yes	—
66	C₂²	C₂⁴×C₁₉₉	1	1	yes	characteristic
67	C₄	—	2	6	no	—
68	C₄	—	2	2	no	—
69	C₄	C₂²×C₄×C₁₉₉	1	2	yes	—
70	C₄	C₂²×C₄×C₁₉₉	1	6	yes	—
71	C₂	—	2	6	no	—
72	C₂	C₂³×C₄×C₁₉₉	1	1	yes	characteristic
73	C₂	H(5,𝔽₂)×C₁₉₉	1	1	yes	characteristic
74	C₂	2₋¹⁺⁴×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉₉	C₂⁴
commutator	C₂	C₂³×C₄×C₁₉₉
Frattini, cosocle	C₂²	C₂⁴×C₁₉₉
prime radical	C₂×#41×C₁₉₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×C₁₉₉	C₂	semidirect
3. C₄×D₄×C₁₉₉	C₂	semidirect
4. #117×C₁₉₉	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₉₉	C₂	semidirect
6. #41×C₁₉₉	C₄	semidirect
17. Q₈×C₁₉₉	C₂×C₄	semidirect
18. D₄×C₁₉₉	C₂×C₄	semidirect
20. C₂×C₄×C₁₉₉	C₂×C₄	semidirect
37. C₁₉₉	#520	direct (N × H)
38. #520	C₁₉₉	direct (N × H)
40. C₄×Q₈	C₂×C₁₉₉	semidirect
41. C₄×D₄	C₂×C₁₉₉	semidirect
42. #117	C₂×C₁₉₉	semidirect
43. C₂×C₄⋊C₄	C₂×C₁₉₉	semidirect
44. #41	C₄×C₁₉₉	semidirect
55. Q₈	C₂×C₄×C₁₉₉	semidirect
56. D₄	C₂×C₄×C₁₉₉	semidirect
58. C₂×C₄	C₂×C₄×C₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₁₉₉	2
C₄×Q₈×C₁₉₉	2
C₄×D₄×C₁₉₉	2
#117×C₁₉₉	2
C₂×C₄⋊C₄×C₁₉₉	2
#520	199

Subgroups — #520×C199 #27163670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #520×C₁₉₉ #27163670