Subgroups — C₂×#20263955 #27164562

Order 12736.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₉₈	C₂	1	2	yes	maximal
2	#20263955	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₉₉	C₂	1	4	yes	maximal
4	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₉₈	—	2	8	no	—
6	C₂×Dic₃₉₈	C₂²	1	8	yes	—
7	C₂²×Dic₁₉₉	C₂²	1	4	yes	—
8	C₄⋊Dic₁₉₉	C₂²	1	16	yes	—
9	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₁₉₉	C₂²	1	2	yes	—
11	C₄²×C₁₉₉	C₂²	1	4	yes	—
12	Dic₃₉₈	—	2	16	no	—
13	C₂×Dic₁₉₉	—	2	8	no	—
14	C₂×Dic₁₉₉	C₂³	1	8	yes	—
15	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₁₉₉	D₄	1	4	yes	—
17	C₂×C₄×C₁₉₉	Q₈	1	8	yes	complemented
18	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
19	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
20	C₂²×C₁₉₉	C₂×D₄	1	2	yes	—
21	C₂²×C₁₉₉	C₂×Q₈	1	4	yes	—
22	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
23	C₄×C₁₉₉	C₂×D₄	1	4	yes	—
24	C₄×C₁₉₉	C₂×Q₈	1	8	yes	complemented
25	Dic₁₉₉	—	2	8	no	—
26	C₂×C₁₉₉	C₄⋊Q₈	1	4	yes	complemented
27	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
28	C₂×C₁₉₉	C₂²×Q₈	1	2	yes	—
29	C₁₉₉	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₄⋊Q₈	—	199	1	no	maximal, Sylow
31	C₂²×Q₈	—	199	2	no	—
32	C₄⋊Q₈	—	199	8	no	—
33	C₂×C₄⋊C₄	—	199	4	no	—
34	C₂×C₄²	D₁₉₉	1	1	yes	characteristic
35	C₂×Q₈	—	398	8	no	—
36	C₂×Q₈	—	199	8	no	—
37	C₂²×C₄	—	199	4	no	—
38	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₁₉₉	1	2	yes	—
40	C₄⋊C₄	—	199	16	no	—
41	C₄²	C₂×D₁₉₉	1	4	yes	—
42	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
43	Q₈	—	398	16	no	—
44	C₂×C₄	—	398	8	no	—
45	C₂×C₄	—	199	8	no	—
46	C₂×C₄	D₇₉₆	1	4	yes	—
47	C₂×C₄	Dic₃₉₈	1	8	yes	complemented
48	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
49	C₂×C₄	C₂²×D₁₉₉	1	4	yes	—
50	C₂²	C₂×D₇₉₆	1	2	yes	—
51	C₂²	C₂×Dic₃₉₈	1	4	yes	—
52	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
53	C₄	—	398	8	no	—
54	C₄	C₂×D₇₉₆	1	4	yes	—
55	C₄	C₂×Dic₃₉₈	1	8	yes	complemented
56	C₂	#20263955	1	4	yes	direct product factor
57	C₂	C₂²×D₇₉₆	1	1	yes	characteristic
58	C₂	C₂²×Dic₃₉₈	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₉₉
commutator, cosocle	C₂²×C₁₉₉	C₂⁴
Frattini	C₂²	C₂³×D₁₉₉
Fitting	C₂×C₄²×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂³
hypercenter	C₂×C₄²	D₁₉₉
hypocenter	C₁₉₉	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #20263955	C₂	direct (N × H)
17. C₂×C₄×C₁₉₉	Q₈	semidirect
24. C₄×C₁₉₉	C₂×Q₈	semidirect
26. C₂×C₁₉₉	C₄⋊Q₈	semidirect
29. C₁₉₉	C₂×C₄⋊Q₈	semidirect
47. C₂×C₄	Dic₃₉₈	semidirect
55. C₄	C₂×Dic₃₉₈	semidirect
56. C₂	#20263955	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₉₈	2
#20263955	2
C₂×C₄⋊Dic₁₉₉	2
C₂×C₄²×C₁₉₉	2
C₂×C₄⋊Q₈	199

Subgroups — C2×#20263955 #27164562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263955 #27164562