Subgroups — C₂²×Dic₇₉₆ #27164795

Order 12736.

258 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	C₂×Dic₇₉₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₉₈	—	2	6	no	—
5	C₂×Dic₃₉₈	C₂²	1	12	yes	—
6	C₂²×Dic₁₉₉	—	2	2	no	—
7	Dic₇₉₆	C₂²	1	16	yes	direct product factor
8	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₁₉₉	C₂²	1	6	yes	—
10	Dic₃₉₈	—	2	12	no	—
11	Dic₃₉₈	C₂³	1	8	yes	—
12	C₂×Dic₁₉₉	—	2	12	no	—
13	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₉₉	D₄	1	3	yes	—
15	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
16	C₈×C₁₉₉	C₂³	1	4	yes	—
17	C₂²×C₁₉₉	Q₁₆	1	4	yes	complemented
18	C₂²×C₁₉₉	C₂×D₄	1	3	yes	—
19	C₄×C₁₉₉	C₂×D₄	1	3	yes	—
20	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
21	Dic₁₉₉	—	2	8	no	—
22	C₂×C₁₉₉	C₂×Q₁₆	1	6	yes	complemented
23	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
24	C₁₉₉	C₂²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
25	C₂²×Q₁₆	—	199	1	no	maximal, Sylow
26	C₂²×Q₈	—	199	2	no	—
27	C₂×Q₁₆	—	199	12	no	—
28	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
29	C₂×Q₈	—	398	6	no	—
30	C₂×Q₈	—	199	12	no	—
31	C₂²×C₄	—	398	2	no	—
32	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
33	Q₁₆	—	199	16	no	—
34	C₂×C₈	C₂×D₁₉₉	1	6	yes	—
35	C₂³	D₇₉₆	1	1	yes	characteristic
36	Q₈	—	398	12	no	—
37	Q₈	—	199	8	no	—
38	C₂×C₄	—	398	12	no	—
39	C₂×C₄	D₇₉₆	1	3	yes	—
40	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
41	C₈	C₂²×D₁₉₉	1	4	yes	—
42	C₂²	Dic₇₉₆	1	4	yes	direct product factor
43	C₂²	C₂×D₇₉₆	1	3	yes	—
44	C₄	—	398	8	no	—
45	C₄	C₂×D₇₉₆	1	3	yes	—
46	C₄	C₂³×D₁₉₉	1	1	yes	characteristic
47	C₂	C₂×Dic₇₉₆	1	6	yes	direct product factor
48	C₂	C₂²×D₇₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇₉₆
commutator, cosocle	C₄×C₁₉₉	C₂⁴
Frattini	C₄	C₂³×D₁₉₉
Fitting	C₂²×C₈×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	D₄
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	C₂²×Q₁₆
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉₆	C₂	direct (N × H)
7. Dic₇₉₆	C₂²	direct (N × H)
17. C₂²×C₁₉₉	Q₁₆	semidirect
22. C₂×C₁₉₉	C₂×Q₁₆	semidirect
24. C₁₉₉	C₂²×Q₁₆	semidirect
42. C₂²	Dic₇₉₆	direct (N × H)
47. C₂	C₂×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
C₂²×Dic₃₉₈	2
C₂×Dic₇₉₆	2
C₂²×C₈×C₁₉₉	2
C₂²×Q₁₆	199

Subgroups — C22×Dic796 #27164795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₇₉₆ #27164795