Subgroups — #27164494

Order 12736.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₉₉	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×Dic₃₉₈	C₂	1	3	yes	maximal, complemented
4	#20263920	C₂	1	3	yes	maximal, complemented
5	#20263913	C₂	1	3	yes	maximal
6	#20263912	C₂	1	3	yes	maximal
7	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₉₈	C₂²	1	3	yes	—
9	Q₈×D₁₉₉	C₄	1	8	yes	complemented
10	C₂×C₄×D₁₉₉	C₂²	1	3	yes	—
11	C₄⋊Dic₁₉₉	C₂²	1	3	yes	—
12	Dic₁₉₉⋊C₄	C₂²	1	3	yes	—
13	Dic₁₉₉⋊C₄	C₂²	1	3	yes	—
14	(C₂×D₁₉₉)⋊C₄	C₂²	1	1	yes	characteristic
15	(C₂×D₁₉₉)⋊C₄	C₂²	1	3	yes	—
16	C₄×Dic₁₉₉	C₂²	1	3	yes	—
17	C₄×Dic₁₉₉	C₂²	1	6	yes	—
18	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₉₉	C₂²	1	3	yes	—
20	C₄²×C₁₉₉	C₂²	1	3	yes	—
21	C₂²×D₁₉₉	C₂³	1	1	yes	characteristic
22	Dic₃₉₈	C₂×C₄	1	12	yes	—
23	C₂×Dic₁₉₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₁₉₉	C₂³	1	3	yes	—
25	C₂×Dic₁₉₉	C₂³	1	3	yes	—
26	C₄×D₁₉₉	C₂×C₄	1	12	yes	—
27	Q₈×C₁₉₉	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
30	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
31	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
32	C₂×D₁₉₉	—	2	1	no	—
33	C₂×D₁₉₉	—	2	1	no	—
34	C₂×D₁₉₉	C₂²×C₄	1	2	yes	—
35	C₄×C₁₉₉	—	2	3	no	—
36	C₄×C₁₉₉	—	2	1	no	—
37	C₄×C₁₉₉	C₂²×C₄	1	6	yes	—
38	Dic₁₉₉	—	2	1	no	—
39	Dic₁₉₉	—	2	3	no	—
40	Dic₁₉₉	C₂²×C₄	1	6	yes	—
41	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
43	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
44	D₁₉₉	—	2	2	no	—
45	C₁₉₉	#519	1	1	yes	characteristic, complemented, Sylow
46	#519	—	199	1	no	maximal, Sylow
47	C₂²×Q₈	—	199	1	no	—
48	C₄×Q₈	—	199	1	no	—
49	C₄×Q₈	—	199	3	no	—
50	C₄×Q₈	—	199	3	no	—
51	C₄×Q₈	D₁₉₉	1	1	yes	characteristic, complemented
52	#117	—	199	3	no	—
53	#117	—	199	3	no	—
54	C₂×Q₈	—	199	8	no	—
55	C₂×Q₈	—	199	3	no	—
56	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
57	C₂²×C₄	—	199	3	no	—
58	C₄⋊C₄	—	199	3	no	—
59	C₄⋊C₄	—	199	3	no	—
60	C₄⋊C₄	—	199	3	no	—
61	C₄⋊C₄	C₂×D₁₉₉	1	3	yes	—
62	C₂²⋊C₄	—	199	3	no	—
63	C₂²⋊C₄	—	199	1	no	—
64	C₄²	—	199	3	no	—
65	C₄²	—	199	6	no	—
66	C₄²	C₂×D₁₉₉	1	3	yes	—
67	C₂³	—	199	1	no	—
68	Q₈	—	199	12	no	—
69	Q₈	C₄×D₁₉₉	1	4	yes	—
70	C₂×C₄	—	199	1	no	—
71	C₂×C₄	—	199	3	no	—
72	C₂×C₄	—	199	3	no	—
73	C₂×C₄	—	199	12	no	—
74	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
75	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
77	C₂²	—	398	1	no	—
78	C₂²	—	398	1	no	—
79	C₂²	—	199	2	no	—
80	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	1	no	—
83	C₄	—	398	1	no	—
84	C₄	—	398	3	no	—
85	C₄	—	199	6	no	—
86	C₄	C₂×C₄×D₁₉₉	1	6	yes	—
87	C₂	—	398	2	no	—
88	C₂	C₂²×C₄×D₁₉₉	1	1	yes	characteristic
89	C₂	#20264002	1	1	yes	characteristic
90	C₂	#20264003	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator	C₂×C₁₉₉	C₂³×C₄
Fitting	C₄×Q₈×C₁₉₉	C₂
socle, cosocle	C₂²×C₁₉₉	C₂⁴
prime radical	C₂²×D₁₉₉	C₂³
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	#519
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₉₉	C₂	semidirect
3. C₄×Dic₃₉₈	C₂	semidirect
4. #20263920	C₂	semidirect
7. C₄×Q₈×C₁₉₉	C₂	semidirect
9. Q₈×D₁₉₉	C₄	semidirect
45. C₁₉₉	#519	semidirect
51. 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
C₂×Q₈×D₁₉₉	2
Q₈×Dic₁₉₉	2
C₄×Dic₃₉₈	2
#20263920	2
#20263913	2
#20263912	2
C₄×Q₈×C₁₉₉	2
#519	199