Subgroups — C₂×C₄×Dic₃₉₈ #27164483

Order 12736.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₉₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₉₉⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₉₉	C₂	1	2	yes	maximal
6	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₃₉₈	C₄	1	8	yes	direct product factor
8	C₂×Dic₃₉₈	C₂²	1	4	yes	—
9	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
10	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
11	C₄⋊Dic₁₉₉	C₂²	1	4	yes	—
12	Dic₁₉₉⋊C₄	C₂²	1	8	yes	—
13	C₄×Dic₁₉₉	C₂²	1	8	yes	—
14	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₄²×C₁₉₉	C₂²	1	4	yes	—
18	Dic₃₉₈	C₂×C₄	1	16	yes	direct product factor
19	C₂×Dic₁₉₉	—	2	4	no	—
20	C₂×Dic₁₉₉	C₂×C₄	1	8	yes	—
21	C₂×Dic₁₉₉	C₂³	1	4	yes	—
22	C₂×Dic₁₉₉	C₂³	1	4	yes	—
23	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	—	2	2	no	—
25	C₂×C₄×C₁₉₉	C₂×C₄	1	4	yes	—
26	C₂×C₄×C₁₉₉	Q₈	1	4	yes	complemented
27	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
29	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
30	C₂²×C₁₉₉	C₂²×C₄	1	2	yes	—
31	C₂²×C₁₉₉	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₉₉	#41	1	2	yes	—
33	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
34	C₄×C₁₉₉	—	2	2	no	—
35	C₄×C₁₉₉	C₂²×C₄	1	4	yes	—
36	C₄×C₁₉₉	C₂×Q₈	1	4	yes	complemented
37	Dic₁₉₉	—	2	4	no	—
38	Dic₁₉₉	C₂²×C₄	1	8	yes	—
39	C₂×C₁₉₉	C₄×Q₈	1	4	yes	complemented
40	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₁₉₉	C₂×#41	1	1	yes	characteristic
43	C₁₉₉	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄×Q₈	—	199	1	no	maximal, Sylow
45	C₂²×Q₈	—	199	1	no	—
46	C₄×Q₈	—	199	8	no	—
47	C₂×C₄⋊C₄	—	199	2	no	—
48	C₂×C₄⋊C₄	—	199	1	no	—
49	C₂×C₄²	—	199	2	no	—
50	C₂×C₄²	D₁₉₉	1	1	yes	characteristic
51	C₂×Q₈	—	199	8	no	—
52	C₂×Q₈	—	199	4	no	—
53	C₂²×C₄	—	199	2	no	—
54	C₂²×C₄	—	199	2	no	—
55	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
58	C₄⋊C₄	—	199	8	no	—
59	C₄⋊C₄	—	199	4	no	—
60	C₄²	—	199	8	no	—
61	C₄²	C₂×D₁₉₉	1	4	yes	—
62	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
63	Q₈	—	199	16	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	398	4	no	—
66	C₂×C₄	—	199	4	no	—
67	C₂×C₄	—	199	8	no	—
68	C₂×C₄	—	199	4	no	—
69	C₂×C₄	C₄×D₁₉₉	1	4	yes	—
70	C₂×C₄	Dic₃₉₈	1	4	yes	direct product factor
71	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
72	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
73	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
74	C₂²	C₂×C₄×D₁₉₉	1	2	yes	—
75	C₂²	C₂×Dic₃₉₈	1	2	yes	—
76	C₂²	#15691886	1	2	yes	—
77	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	398	4	no	—
80	C₄	—	199	8	no	—
81	C₄	C₂×C₄×D₁₉₉	1	4	yes	—
82	C₄	C₂×Dic₃₉₈	1	4	yes	direct product factor
83	C₂	C₄×Dic₃₉₈	1	4	yes	direct product factor
84	C₂	C₂²×C₄×D₁₉₉	1	1	yes	characteristic
85	C₂	C₂²×Dic₃₉₈	1	1	yes	characteristic
86	C₂	C₂×#15691886	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₃₉₈	C₂	direct (N × H)
7. C₂×Dic₃₉₈	C₄	direct (N × H)
18. Dic₃₉₈	C₂×C₄	direct (N × H)
26. C₂×C₄×C₁₉₉	Q₈	semidirect
36. C₄×C₁₉₉	C₂×Q₈	semidirect
39. C₂×C₁₉₉	C₄×Q₈	semidirect
43. C₁₉₉	C₂×C₄×Q₈	semidirect
70. C₂×C₄	Dic₃₉₈	direct (N × H)
82. C₄	C₂×Dic₃₉₈	direct (N × H)
83. C₂	C₄×Dic₃₉₈	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
C₄×Dic₃₉₈	2
C₂×C₄⋊Dic₁₉₉	2
C₂×Dic₁₉₉⋊C₄	2
C₂×C₄×Dic₁₉₉	2
C₂×C₄²×C₁₉₉	2
C₂×C₄×Q₈	199

Subgroups — C2×C4×Dic398 #27164483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₃₉₈ #27164483