Subgroups — #27164397

Order 12736.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691876	C₂	1	4	yes	maximal
2	#20263955	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₉₈	C₂²	1	2	yes	—
6	#15691876	—	2	8	no	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₉	—	2	2	no	—
9	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₉₉	—	2	2	no	—
11	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
12	Dic₃₉₈	—	2	4	no	—
13	C₂×Dic₁₉₉	—	2	2	no	—
14	C₁₉₉⋊C₈	D₄	1	4	yes	—
15	Q₈×C₁₉₉	—	2	4	no	—
16	C₂×C₄×C₁₉₉	—	2	2	no	—
17	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
18	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
19	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₉₉	—	4	2	no	—
21	C₄×C₁₉₉	Q₁₆	1	4	yes	complemented
22	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
23	Dic₁₉₉	—	4	2	no	—
24	C₂×C₁₉₉	#127	1	1	yes	characteristic
25	C₂×C₁₉₉	C₂×Q₁₆	1	2	yes	—
26	C₁₉₉	#494	1	1	yes	characteristic, complemented, Sylow
27	#494	—	199	1	no	maximal, Sylow
28	C₂×Q₁₆	—	199	4	no	—
29	C₄⋊Q₈	—	199	1	no	—
30	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic
31	C₄×C₈	—	199	1	no	—
32	C₂×Q₈	—	199	2	no	—
33	C₂×Q₈	C₂×D₁₉₉	1	2	yes	—
34	Q₁₆	—	398	8	no	—
35	C₂×C₈	—	199	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	398	2	no	—
38	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	398	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	398	2	no	—
43	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
44	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
45	C₈	—	199	4	no	—
46	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	796	2	no	—
49	C₄	#15691876	1	4	yes	complemented
50	C₄	D₄×D₁₉₉	1	2	yes	—
51	C₂	#20263951	1	1	yes	characteristic
52	C₂	C₂×#15691876	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄⋊Q₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#494
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×C₁₉₉	Q₁₆	semidirect
26. C₁₉₉	#494	semidirect
49. C₄	#15691876	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691876	2
#20263955	2
C₄×C₁₉₉⋊C₈	2
C₄⋊Q₈×C₁₉₉	2
#494	199