Subgroups — #27164629

Order 12736.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263950	C₂	1	4	yes	maximal
2	#20263947	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263945	C₂	1	4	yes	maximal, complemented
4	#20263931	C₂	1	2	yes	maximal
5	Q₈×Dic₁₉₉	C₂	1	2	yes	maximal, complemented
6	#20263912	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₉₉	C₂²	1	4	yes	—
10	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
11	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
12	Dic₁₉₉⋊C₄	C₂²	1	4	yes	—
13	Dic₁₉₉⋊C₄	C₂²	1	4	yes	—
14	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
15	(C₂×D₁₉₉)⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₁₉₉	C₂²	1	4	yes	—
17	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₁₉₉	C₂²	1	4	yes	—
20	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₉₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₉₉	C₂³	1	2	yes	—
23	C₂×Dic₁₉₉	C₂³	1	4	yes	—
24	C₂×Dic₁₉₉	C₂³	1	1	yes	characteristic
25	C₄×D₁₉₉	—	2	2	no	—
26	Q₈×C₁₉₉	—	2	4	no	—
27	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
29	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
31	C₂×D₁₉₉	—	2	1	no	—
32	C₂×D₁₉₉	—	2	2	no	—
33	C₄×C₁₉₉	—	2	4	no	—
34	C₄×C₁₉₉	—	2	2	no	—
35	C₄×C₁₉₉	#41	1	2	yes	—
36	Dic₁₉₉	—	2	1	no	—
37	Dic₁₉₉	—	2	2	no	—
38	Dic₁₉₉	—	2	4	no	—
39	C₂×C₁₉₉	C₂×#41	1	1	yes	characteristic
40	C₂×C₁₉₉	2₋¹⁺⁴	1	2	yes	—
41	D₁₉₉	—	4	1	no	—
42	C₁₉₉	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	199	1	no	maximal, Sylow
44	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic, complemented
45	#126	—	199	4	no	—
46	#125	—	199	4	no	—
47	#125	—	199	1	no	—
48	C₂²⋊Q₈	—	199	2	no	—
49	C₄×Q₈	—	199	2	no	—
50	#117	—	199	1	no	—
51	C₂×Q₈	C₂×D₁₉₉	1	2	yes	—
52	C₂²×C₄	—	199	1	no	—
53	C₄⋊C₄	—	199	2	no	—
54	C₄⋊C₄	—	199	4	no	—
55	C₄⋊C₄	—	199	4	no	—
56	C₄⋊C₄	—	199	2	no	—
57	C₄⋊C₄	—	199	4	no	—
58	C₄⋊C₄	C₂×D₁₉₉	1	4	yes	—
59	C₂²⋊C₄	—	199	2	no	—
60	C₂²⋊C₄	—	199	4	no	—
61	C₄²	—	199	1	no	—
62	C₄²	—	199	4	no	—
63	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
64	C₂³	—	199	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	398	2	no	—
67	C₂×C₄	—	199	2	no	—
68	C₂×C₄	—	199	4	no	—
69	C₂×C₄	—	199	1	no	—
70	C₂×C₄	C₂²×D₁₉₉	1	4	yes	—
71	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
72	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
73	C₂²	—	398	1	no	—
74	C₂²	—	398	2	no	—
75	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	398	1	no	—
79	C₄	—	398	2	no	—
80	C₄	—	398	4	no	—
81	C₄	#15691885	1	2	yes	—
82	C₂	—	796	1	no	—
83	C₂	C₂×#15691885	1	1	yes	characteristic
84	C₂	#20264003	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	C₄⋊Q₈×C₁₉₉	C₂
prime radical	C₂²×D₁₉₉	C₂³
hypercenter	C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#541
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #20263947	C₂	semidirect
3. #20263945	C₂	semidirect
5. Q₈×Dic₁₉₉	C₂	semidirect
7. C₄⋊Q₈×C₁₉₉	C₂	semidirect
42. C₁₉₉	#541	semidirect
44. C₄⋊Q₈	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263950	2
#20263947	2
#20263945	2
#20263931	2
Q₈×Dic₁₉₉	2
#20263912	2
C₄⋊Q₈×C₁₉₉	2
#541	199