Subgroups — #541×C₁₉₉ #27163691

Order 12736.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#126×C₁₉₉	C₂	1	4	yes	maximal
3	#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₁₉₉	C₂	1	4	yes	maximal, complemented
5	C₂²⋊Q₈×C₁₉₉	C₂	1	2	yes	maximal
6	C₄×Q₈×C₁₉₉	C₂	1	2	yes	maximal, complemented
7	#117×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
9	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
11	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
12	C₄⋊C₄×C₁₉₉	C₂²	1	4	yes	—
13	C₄⋊C₄×C₁₉₉	C₂²	1	4	yes	—
14	C₄⋊C₄×C₁₉₉	C₂²	1	4	yes	—
15	C₄⋊C₄×C₁₉₉	C₂²	1	4	yes	—
16	C₂²⋊C₄×C₁₉₉	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₁₉₉	C₂²	1	4	yes	—
18	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
19	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
20	C₄²×C₁₉₉	C₂²	1	4	yes	—
21	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
22	Q₈×C₁₉₉	—	2	4	no	—
23	C₂×C₄×C₁₉₉	—	2	2	no	—
24	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
25	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
26	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
27	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
28	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉₉	—	2	1	no	—
31	C₂²×C₁₉₉	—	2	2	no	—
32	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
33	C₄×C₁₉₉	—	2	1	no	—
34	C₄×C₁₉₉	—	2	4	no	—
35	C₄×C₁₉₉	—	2	2	no	—
36	C₄×C₁₉₉	—	2	2	no	—
37	C₄×C₁₉₉	—	2	4	no	—
38	C₄×C₁₉₉	#41	1	2	yes	—
39	C₂×C₁₉₉	—	4	1	no	—
40	C₂×C₁₉₉	C₂×#41	1	1	yes	characteristic
41	C₂×C₁₉₉	2₋¹⁺⁴	1	2	yes	—
42	C₁₉₉	#541	1	1	yes	characteristic, direct product factor, Sylow
43	#541	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₄⋊Q₈	C₂×C₁₉₉	1	1	yes	characteristic, complemented
45	#126	C₂×C₁₉₉	1	4	yes	—
46	#125	C₂×C₁₉₉	1	1	yes	characteristic, complemented
47	#125	C₂×C₁₉₉	1	4	yes	complemented
48	C₂²⋊Q₈	C₂×C₁₉₉	1	2	yes	—
49	C₄×Q₈	C₂×C₁₉₉	1	2	yes	complemented
50	#117	C₂×C₁₉₉	1	1	yes	characteristic
51	C₂×Q₈	C₂²×C₁₉₉	1	2	yes	—
52	C₂²×C₄	C₂²×C₁₉₉	1	1	yes	characteristic
53	C₄⋊C₄	C₂²×C₁₉₉	1	2	yes	—
54	C₄⋊C₄	C₂²×C₁₉₉	1	2	yes	—
55	C₄⋊C₄	C₂²×C₁₉₉	1	4	yes	—
56	C₄⋊C₄	C₂²×C₁₉₉	1	4	yes	—
57	C₄⋊C₄	C₂²×C₁₉₉	1	4	yes	—
58	C₄⋊C₄	C₂²×C₁₉₉	1	4	yes	—
59	C₂²⋊C₄	C₂²×C₁₉₉	1	2	yes	—
60	C₂²⋊C₄	C₂²×C₁₉₉	1	4	yes	—
61	C₄²	C₂²×C₁₉₉	1	1	yes	characteristic
62	C₄²	C₂²×C₁₉₉	1	1	yes	characteristic
63	C₄²	C₂²×C₁₉₉	1	4	yes	—
64	C₂³	C₂³×C₁₉₉	1	1	yes	characteristic
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₂³×C₁₉₉	1	2	yes	—
68	C₂×C₄	C₂³×C₁₉₉	1	2	yes	—
69	C₂×C₄	C₂³×C₁₉₉	1	4	yes	—
70	C₂×C₄	C₂³×C₁₉₉	1	4	yes	—
71	C₂×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
72	C₂×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂⁴×C₁₉₉	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	4	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	2	4	no	—
81	C₄	#41×C₁₉₉	1	2	yes	—
82	C₂	—	4	1	no	—
83	C₂	C₂×#41×C₁₉₉	1	1	yes	characteristic
84	C₂	2₋¹⁺⁴×C₁₉₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₁₉₉	C₂	semidirect
3. #125×C₁₉₉	C₂	semidirect
4. #125×C₁₉₉	C₂	semidirect
6. C₄×Q₈×C₁₉₉	C₂	semidirect
42. C₁₉₉	#541	direct (N × H)
43. #541	C₁₉₉	direct (N × H)
44. C₄⋊Q₈	C₂×C₁₉₉	semidirect
46. #125	C₂×C₁₉₉	semidirect
47. #125	C₂×C₁₉₉	semidirect
49. C₄×Q₈	C₂×C₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₁₉₉	2
#126×C₁₉₉	2
#125×C₁₉₉	2
C₂²⋊Q₈×C₁₉₉	2
C₄×Q₈×C₁₉₉	2
#117×C₁₉₉	2
#541	199

Subgroups — #541×C199 #27163691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #541×C₁₉₉ #27163691