Subgroups — #27164041

Order 12736.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691870	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	Dic₁₉₉⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₉	C₄	1	2	yes	—
6	#15691870	—	2	2	no	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₁₉₉	C₄	1	1	yes	characteristic
9	C₄×Dic₁₉₉	C₄	1	1	yes	characteristic
10	C₄×Dic₁₉₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	M₁₆×C₁₉₉	—	2	2	no	—
13	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
14	C₂×Dic₁₉₉	—	2	4	no	—
15	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
17	C₁₉₉⋊C₈	—	2	2	no	—
18	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
21	C₂×C₄×C₁₉₉	Q₈	1	2	yes	—
22	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₈×C₁₉₉	—	2	2	no	—
24	C₂²×C₁₉₉	—	2	2	no	—
25	C₂²×C₁₉₉	C₄⋊C₄	1	2	yes	—
26	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
28	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
30	Dic₁₉₉	—	2	2	no	—
31	Dic₁₉₉	M₁₆	1	4	yes	complemented
32	C₂×C₁₉₉	—	2	2	no	—
33	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₁₉₉	C₂×M₁₆	1	2	yes	—
35	C₁₉₉	#423	1	1	yes	characteristic, complemented, Sylow
36	#423	—	199	1	no	maximal, Sylow
37	C₂×M₁₆	—	199	1	no	—
38	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
39	C₂×C₄²	—	199	1	no	—
40	C₄⋊C₈	—	199	4	no	—
41	C₂²×C₄	—	199	2	no	—
42	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
43	M₁₆	—	2	2	no	—
44	M₁₆	—	398	2	no	—
45	C₂×C₈	—	199	2	no	—
46	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
47	C₄²	—	199	1	no	—
48	C₄²	—	199	1	no	—
49	C₄²	—	199	2	no	—
50	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
51	C₂×C₄	—	398	4	no	—
52	C₂×C₄	—	199	2	no	—
53	C₂×C₄	—	199	2	no	—
54	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
56	C₂×C₄	Dic₃₉₈	1	2	yes	—
57	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	398	2	no	—
60	C₂²	—	2	2	no	—
61	C₂²	Dic₁₉₉⋊C₄	1	2	yes	—
62	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
63	C₄	—	398	2	no	—
64	C₄	—	199	4	no	—
65	C₄	Dic₁₉₉⋊C₄	1	2	yes	—
66	C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
67	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
70	C₂	M₁₆×D₁₉₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator, socle	C₂²×C₁₉₉	C₂²×C₄
Fitting	C₂×M₁₆×C₁₉₉	C₂
prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	#423
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. Dic₁₉₉⋊C₈	C₂	semidirect
31. Dic₁₉₉	M₁₆	semidirect
35. C₁₉₉	#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691870	2
C₂×C₄×Dic₁₉₉	2
Dic₁₉₉⋊C₈	2
C₂×M₁₆×C₁₉₉	2
#423	199