Subgroups — C₄×D₁₀₉₆ #22577487

Order 8768.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₀₉₆	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₄₈	C₂	1	2	yes	maximal, complemented
3	#17984022	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984010	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₄₈	C₂²	1	2	yes	—
7	C₂×C₄×D₁₃₇	—	2	2	no	—
8	D₁₀₉₆	C₄	1	4	yes	direct product factor
9	C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
10	#14555487	—	2	2	no	—
11	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×D₁₃₇	—	2	2	no	—
15	D₅₄₈	—	2	2	no	—
16	D₅₄₈	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₁₃₇	—	2	2	no	—
18	C₄×D₁₃₇	—	4	4	no	—
19	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₃₇	—	2	1	no	—
23	C₈×C₁₃₇	C₂×C₄	1	2	yes	complemented
24	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
25	C₂×D₁₃₇	—	2	4	no	—
26	C₂×D₁₃₇	—	4	2	no	—
27	C₂×D₁₃₇	—	4	2	no	—
28	C₄×C₁₃₇	—	2	1	no	—
29	C₄×C₁₃₇	D₈	1	2	yes	complemented
30	C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₁₃₇	#41	1	1	yes	characteristic
32	Dic₁₃₇	—	4	2	no	—
33	C₂×C₁₃₇	C₄×D₄	1	1	yes	characteristic
34	C₂×C₁₃₇	C₂×D₈	1	1	yes	characteristic
35	C₂×C₁₃₇	#135	1	1	yes	characteristic
36	D₁₃₇	—	4	4	no	—
37	C₁₃₇	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	137	1	no	maximal, Sylow
39	C₂×D₈	—	137	1	no	—
40	C₄×D₄	—	137	2	no	—
41	#107	—	137	1	no	—
42	#102	—	137	2	no	—
43	C₄×C₈	D₁₃₇	1	1	yes	characteristic, complemented
44	C₂×D₄	—	137	2	no	—
45	C₂²×C₄	—	274	2	no	—
46	D₈	—	137	4	no	—
47	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
48	C₂×C₈	C₂×D₁₃₇	1	1	yes	characteristic
49	C₄⋊C₄	—	137	2	no	—
50	C₂²⋊C₄	—	274	2	no	—
51	C₄²	C₂×D₁₃₇	1	1	yes	characteristic
52	C₂³	—	274	2	no	—
53	D₄	—	274	2	no	—
54	D₄	—	137	4	no	—
55	C₂×C₄	—	274	2	no	—
56	C₂×C₄	—	548	4	no	—
57	C₂×C₄	D₅₄₈	1	1	yes	characteristic
58	C₂×C₄	D₅₄₈	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₄×D₁₃₇	1	2	yes	complemented
62	C₂²	—	274	4	no	—
63	C₂²	—	548	2	no	—
64	C₂²	—	548	2	no	—
65	C₂²	C₂×D₅₄₈	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	548	2	no	—
68	C₄	D₁₀₉₆	1	2	yes	direct product factor
69	C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
70	C₄	#14555510	1	1	yes	characteristic
71	C₂	—	548	4	no	—
72	C₂	C₄×D₅₄₈	1	1	yes	characteristic
73	C₂	C₂×D₁₀₉₆	1	1	yes	characteristic
74	C₂	#17984110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅₄₈
commutator	C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₇
Fitting	C₄×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₄
hypercenter	C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	C₄×D₈
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₅₄₈	C₂	semidirect
3. #17984022	C₂	semidirect
4. #17984010	C₂	semidirect
5. C₄×C₈×C₁₃₇	C₂	semidirect
8. D₁₀₉₆	C₄	direct (N × H)
16. D₅₄₈	C₂×C₄	semidirect
23. C₈×C₁₃₇	C₂×C₄	semidirect
29. C₄×C₁₃₇	D₈	semidirect
37. C₁₃₇	C₄×D₈	semidirect
43. C₄×C₈	D₁₃₇	semidirect
61. C₈	C₄×D₁₃₇	semidirect
68. C₄	D₁₀₉₆	direct (N × H)

Subgroups — C4×D1096 #22577487

Conjugacy classes of subgroups

Complements

Subgroups — C₄×D₁₀₉₆ #22577487