Subgroups — #27164392

Order 12736.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691873	C₂	1	4	yes	maximal, complemented
2	#20263952	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₉₆	—	2	2	no	—
6	C₂×D₇₉₆	C₂²	1	2	yes	—
7	#15691873	—	2	8	no	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₉₉	—	2	2	no	—
11	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂²×D₁₉₉	—	2	2	no	—
13	D₇₉₆	—	2	4	no	—
14	D₇₉₆	—	4	4	no	—
15	C₁₉₉⋊C₈	D₄	1	4	yes	complemented
16	Q₈×C₁₉₉	—	2	4	no	—
17	C₂×C₄×C₁₉₉	—	2	2	no	—
18	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
19	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
20	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
21	C₂×D₁₉₉	—	4	2	no	—
22	C₂×D₁₉₉	—	4	4	no	—
23	C₄×C₁₉₉	—	4	2	no	—
24	C₄×C₁₉₉	SD₁₆	1	4	yes	complemented
25	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
26	C₂×C₁₉₉	#127	1	1	yes	characteristic
27	C₂×C₁₉₉	C₂×SD₁₆	1	2	yes	—
28	D₁₉₉	—	8	2	no	—
29	C₁₉₉	#492	1	1	yes	characteristic, complemented, Sylow
30	#492	—	199	1	no	maximal, Sylow
31	C₂×SD₁₆	—	199	4	no	—
32	C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic, complemented
33	#127	—	199	1	no	—
34	C₄×C₈	—	199	1	no	—
35	C₂×Q₈	C₂×D₁₉₉	1	2	yes	—
36	C₂×D₄	—	398	2	no	—
37	C₂×D₄	—	199	2	no	—
38	SD₁₆	—	398	8	no	—
39	C₂×C₈	—	199	2	no	—
40	C₄⋊C₄	—	2	2	no	—
41	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
42	C₂³	—	398	2	no	—
43	Q₈	—	2	4	no	—
44	D₄	—	796	4	no	—
45	D₄	—	398	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
48	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
49	C₈	—	199	4	no	—
50	C₂²	—	796	2	no	—
51	C₂²	—	796	4	no	—
52	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	#15691873	1	4	yes	complemented
55	C₄	D₄×D₁₉₉	1	2	yes	—
56	C₂	—	1592	2	no	—
57	C₂	#20263951	1	1	yes	characteristic
58	C₂	C₂×#15691873	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	C₂²×C₁₉₉	C₂×D₄
prime radical	#20263952	C₂
hypercenter	C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#492
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
1. C₂×#15691873	C₂	semidirect
3. C₄×C₁₉₉⋊C₈	C₂	semidirect
4. C₄⋊Q₈×C₁₉₉	C₂	semidirect
15. C₁₉₉⋊C₈	D₄	semidirect
24. C₄×C₁₉₉	SD₁₆	semidirect
29. C₁₉₉	#492	semidirect
32. C₄⋊Q₈	D₁₉₉	semidirect
54. C₄	#15691873	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691873	2
#20263952	2
C₄×C₁₉₉⋊C₈	2
C₄⋊Q₈×C₁₉₉	2
#492	199