Subgroups — C₄×Dic₇₉₆ #27164114

Order 12736.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₉₈	C₂	1	2	yes	maximal
3	#20263889	C₂	1	1	yes	maximal, characteristic
4	#20263880	C₂	1	2	yes	maximal
5	C₄×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₂²	1	2	yes	—
7	Dic₇₉₆	C₄	1	4	yes	direct product factor
8	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	Dic₁₉₉⋊C₄	—	2	2	no	—
10	C₄×Dic₁₉₉	—	2	2	no	—
11	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
14	Dic₃₉₈	—	2	2	no	—
15	Dic₃₉₈	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₂×Dic₁₉₉	—	2	2	no	—
18	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
21	C₈×C₁₉₉	—	2	1	no	—
22	C₈×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
24	C₄×C₁₉₉	—	2	1	no	—
25	C₄×C₁₉₉	Q₁₆	1	2	yes	complemented
26	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₉₉	#41	1	1	yes	characteristic
28	Dic₁₉₉	—	2	4	no	—
29	Dic₁₉₉	—	4	2	no	—
30	C₂×C₁₉₉	C₄×D₄	1	1	yes	characteristic
31	C₂×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₁₉₉	#135	1	1	yes	characteristic
33	C₁₉₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×Q₁₆	—	199	1	no	maximal, Sylow
35	C₂×Q₁₆	—	199	1	no	—
36	C₄×Q₈	—	199	2	no	—
37	#107	—	199	1	no	—
38	#103	—	199	2	no	—
39	C₄×C₈	D₁₉₉	1	1	yes	characteristic
40	C₂×Q₈	—	199	2	no	—
41	Q₁₆	—	199	4	no	—
42	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
43	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
44	C₄⋊C₄	—	398	2	no	—
45	C₄⋊C₄	—	199	2	no	—
46	C₄²	—	398	2	no	—
47	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
48	Q₈	—	398	2	no	—
49	Q₈	—	199	4	no	—
50	C₂×C₄	—	398	2	no	—
51	C₂×C₄	—	398	2	no	—
52	C₂×C₄	D₇₉₆	1	1	yes	characteristic
53	C₂×C₄	D₇₉₆	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₄×D₁₉₉	1	2	yes	—
57	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	796	2	no	—
60	C₄	—	398	4	no	—
61	C₄	Dic₇₉₆	1	2	yes	direct product factor
62	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
63	C₄	#15691886	1	1	yes	characteristic
64	C₂	C₄×D₇₉₆	1	1	yes	characteristic
65	C₂	C₂×Dic₇₉₆	1	1	yes	characteristic
66	C₂	#20263977	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇₉₆
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₄×C₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₄×C₈	D₁₉₉
hypocenter	C₁₉₉	C₄×Q₁₆
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
7. Dic₇₉₆	C₄	direct (N × H)
25. C₄×C₁₉₉	Q₁₆	semidirect
33. C₁₉₉	C₄×Q₁₆	semidirect
61. C₄	Dic₇₉₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Dic₇₉₆	2
C₄×Dic₃₉₈	2
#20263889	2
#20263880	2
C₄×C₈×C₁₉₉	2
C₄×Q₁₆	199

Subgroups — C4×Dic796 #27164114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Dic₇₉₆ #27164114