Subgroups — C₂²×D₁₆₉₆ #27542208

Order 13568.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄₈	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₆₉₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄₂₄	—	2	2	no	—
5	C₂×D₈₄₈	—	2	6	no	—
6	C₂×D₈₄₈	C₂²	1	12	yes	complemented
7	D₁₆₉₆	C₂²	1	16	yes	direct product factor
8	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₅₃	C₂²	1	6	yes	complemented
10	C₂²×D₂₁₂	—	4	2	no	—
11	C₂×D₄₂₄	—	2	12	no	—
12	C₂×D₄₂₄	—	4	6	no	—
13	D₈₄₈	—	2	12	no	—
14	D₈₄₈	C₂³	1	8	yes	complemented
15	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	D₄	1	3	yes	—
17	C₂×C₁₆×C₅₃	C₂³	1	3	yes	—
18	C₃₂×C₅₃	C₂³	1	4	yes	complemented
19	C₂³×D₅₃	—	8	2	no	—
20	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
21	C₂×D₂₁₂	—	4	12	no	—
22	C₂×D₂₁₂	—	8	6	no	—
23	C₂×C₈×C₅₃	D₈	1	3	yes	—
24	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
25	D₄₂₄	—	2	8	no	—
26	D₄₂₄	—	4	12	no	—
27	C₁₆×C₅₃	C₂×D₄	1	3	yes	—
28	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
29	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
30	C₂²×D₅₃	—	8	12	no	—
31	C₂²×D₅₃	—	16	8	no	—
32	C₂×C₄×C₅₃	D₁₆	1	3	yes	—
33	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
34	D₂₁₂	—	4	8	no	—
35	D₂₁₂	—	8	12	no	—
36	C₈×C₅₃	C₂×D₈	1	3	yes	—
37	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
38	C₂²×D₃₂	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	D₃₂	1	4	yes	complemented
40	C₂²×C₅₃	C₂×D₁₆	1	3	yes	—
41	C₂×D₅₃	—	8	8	no	—
42	C₂×D₅₃	—	16	24	no	—
43	C₄×C₅₃	C₂×D₁₆	1	3	yes	—
44	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
45	C₂²×D₁₆	—	53	2	no	—
46	C₂×D₃₂	—	53	12	no	—
47	C₂²×C₃₂	D₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	C₂×D₃₂	1	6	yes	complemented
49	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
50	D₅₃	—	16	8	no	—
51	C₂²×D₈	—	106	2	no	—
52	C₂×D₁₆	—	53	12	no	—
53	C₂×D₁₆	—	106	6	no	—
54	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
55	D₃₂	—	53	16	no	—
56	C₂×C₃₂	C₂×D₅₃	1	6	yes	complemented
57	C₅₃	C₂²×D₃₂	1	1	yes	characteristic, complemented, Sylow
58	C₂²×D₄	—	212	2	no	—
59	C₂×D₈	—	106	12	no	—
60	C₂×D₈	—	212	6	no	—
61	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
62	D₁₆	—	53	8	no	—
63	D₁₆	—	106	12	no	—
64	C₂×C₁₆	D₂₁₂	1	3	yes	—
65	C₂×C₁₆	C₂²×D₅₃	1	3	yes	—
66	C₃₂	C₂²×D₅₃	1	4	yes	complemented
67	C₂⁴	—	424	2	no	—
68	C₂×D₄	—	424	6	no	—
69	C₂×D₄	—	212	12	no	—
70	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
71	D₈	—	106	8	no	—
72	D₈	—	212	12	no	—
73	C₂×C₈	D₄₂₄	1	3	yes	—
74	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
75	C₁₆	C₂×D₂₁₂	1	3	yes	—
76	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
77	C₂³	—	848	8	no	—
78	C₂³	—	424	12	no	—
79	C₂³	D₈₄₈	1	1	yes	characteristic
80	D₄	—	424	12	no	—
81	D₄	—	212	8	no	—
82	C₂×C₄	D₈₄₈	1	3	yes	—
83	C₂×C₄	C₂×D₄₂₄	1	3	yes	—
84	C₈	C₂×D₄₂₄	1	3	yes	—
85	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
86	C₂²	—	848	24	no	—
87	C₂²	—	424	8	no	—
88	C₂²	D₁₆₉₆	1	4	yes	direct product factor
89	C₂²	C₂×D₈₄₈	1	3	yes	—
90	C₄	C₂×D₈₄₈	1	3	yes	—
91	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
92	C₂	—	848	8	no	—
93	C₂	C₂×D₁₆₉₆	1	6	yes	direct product factor
94	C₂	C₂²×D₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈₄₈
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂²×C₃₂×C₅₃	C₂
socle	C₂³×C₅₃	D₁₆
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	C₂²×D₃₂
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈₄₈	C₂	semidirect
2. C₂×D₁₆₉₆	C₂	direct (N × H)
3. C₂²×C₃₂×C₅₃	C₂	semidirect
6. C₂×D₈₄₈	C₂²	semidirect
7. D₁₆₉₆	C₂²	direct (N × H)
9. C₂×C₃₂×C₅₃	C₂²	semidirect
14. D₈₄₈	C₂³	semidirect
18. C₃₂×C₅₃	C₂³	semidirect
39. C₂²×C₅₃	D₃₂	semidirect
47. C₂²×C₃₂	D₅₃	semidirect
48. C₂×C₅₃	C₂×D₃₂	semidirect
56. C₂×C₃₂	C₂×D₅₃	semidirect
57. C₅₃	C₂²×D₃₂	semidirect
66. C₃₂	C₂²×D₅₃	semidirect
88. C₂²	D₁₆₉₆	direct (N × H)
93. C₂	C₂×D₁₆₉₆	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×D₈₄₈	2
C₂×D₁₆₉₆	2
C₂²×C₃₂×C₅₃	2
C₂²×D₃₂	53

Subgroups — C22×D1696 #27542208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₆₉₆ #27542208