Subgroups — #27163779

Order 12736.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263914	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₉	—	2	1	no	—
5	C₄⋊Dic₁₉₉	C₄	1	2	yes	complemented
6	C₂²⋊Dic₁₉₉	—	2	1	no	—
7	C₄×Dic₁₉₉	—	2	1	no	—
8	C₄×Dic₁₉₉	—	2	1	no	—
9	C₄×Dic₁₉₉	C₄	1	2	yes	complemented
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₁₉₉	—	2	1	no	—
12	M₁₆×C₁₉₉	C₄	1	2	yes	complemented
13	C₂×C₈×C₁₉₉	—	2	1	no	—
14	C₂×Dic₁₉₉	—	2	4	no	—
15	C₂×Dic₁₉₉	—	2	2	no	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	2	no	—
25	C₂²×C₁₉₉	—	2	2	no	—
26	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₁₉₉	C₄²	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₉₉	—	2	4	no	—
34	Dic₁₉₉	—	4	2	no	—
35	C₂×C₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	—	2	1	no	—
37	C₂×C₁₉₉	#95	1	1	yes	characteristic
38	C₂×C₁₉₉	#104	1	2	yes	—
39	C₁₉₉	#339	1	1	yes	characteristic, complemented, Sylow
40	#339	—	199	1	no	maximal, Sylow
41	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
42	#117	—	199	1	no	—
43	C₂×C₄²	—	199	1	no	—
44	C₂²×C₄	—	398	1	no	—
45	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
46	M₁₆	—	2	1	no	—
47	M₁₆	Dic₁₉₉	1	2	yes	complemented
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	199	2	no	—
50	C₂²⋊C₄	—	398	1	no	—
51	C₄²	—	398	1	no	—
52	C₄²	—	398	1	no	—
53	C₄²	—	199	2	no	—
54	C₂³	D₇₉₆	1	1	yes	characteristic
55	C₂×C₄	—	398	2	no	—
56	C₂×C₄	—	398	4	no	—
57	C₂×C₄	—	398	2	no	—
58	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
59	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
63	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
67	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
68	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
69	C₄	—	796	2	no	—
70	C₄	—	398	4	no	—
71	C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
72	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
73	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
74	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
78	C₂	#20263881	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₉₉
commutator	C₄×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂×M₁₆×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂³×C₁₉₉	D₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	#339
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₁₉₉	C₄	semidirect
9. C₄×Dic₁₉₉	C₄	semidirect
12. M₁₆×C₁₉₉	C₄	semidirect
39. C₁₉₉	#339	semidirect
47. M₁₆	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263914	2
C₂×C₄×Dic₁₉₉	2
C₂×M₁₆×C₁₉₉	2
#339	199