Subgroups — C₂²×Dic₆₈₈ #25254720

Order 11008.

462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₄₄	C₂	1	2	yes	maximal
2	C₂×Dic₆₈₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₇₂	—	2	2	no	—
5	C₂×Dic₃₄₄	—	2	6	no	—
6	C₂×Dic₃₄₄	C₂²	1	12	yes	—
7	Dic₆₈₈	C₂²	1	16	yes	direct product factor
8	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₄₃	C₂²	1	6	yes	—
10	C₂²×Dic₈₆	—	4	2	no	—
11	C₂×Dic₁₇₂	—	2	12	no	—
12	C₂×Dic₁₇₂	—	4	6	no	—
13	Dic₃₄₄	—	2	12	no	—
14	Dic₃₄₄	C₂³	1	8	yes	—
15	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₄₃	D₄	1	3	yes	—
17	C₂×C₁₆×C₄₃	C₂³	1	3	yes	—
18	C₃₂×C₄₃	C₂³	1	4	yes	—
19	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
20	C₂²×Dic₄₃	—	8	2	no	—
21	C₂×Dic₈₆	—	4	12	no	—
22	C₂×Dic₈₆	—	8	6	no	—
23	C₂×C₈×C₄₃	D₈	1	3	yes	—
24	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
25	Dic₁₇₂	—	2	8	no	—
26	Dic₁₇₂	—	4	12	no	—
27	C₁₆×C₄₃	C₂×D₄	1	3	yes	—
28	C₁₆×C₄₃	C₂⁴	1	1	yes	characteristic
29	C₂³×C₄₃	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₄₃	D₁₆	1	3	yes	—
31	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
32	C₂×Dic₄₃	—	8	12	no	—
33	Dic₈₆	—	4	8	no	—
34	Dic₈₆	—	8	12	no	—
35	C₈×C₄₃	C₂×D₈	1	3	yes	—
36	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
37	C₂²×Q₆₄	—	43	1	no	maximal, Sylow
38	C₂²×C₄₃	Q₆₄	1	4	yes	complemented
39	C₂²×C₄₃	C₂×D₁₆	1	3	yes	—
40	C₄×C₄₃	C₂×D₁₆	1	3	yes	—
41	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
42	Dic₄₃	—	8	8	no	—
43	C₂²×Q₃₂	—	43	2	no	—
44	C₂×Q₆₄	—	43	12	no	—
45	C₂²×C₃₂	D₄₃	1	1	yes	characteristic
46	C₂×C₄₃	C₂×Q₆₄	1	6	yes	complemented
47	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
48	C₂²×Q₁₆	—	86	2	no	—
49	C₂×Q₃₂	—	43	12	no	—
50	C₂×Q₃₂	—	86	6	no	—
51	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
52	Q₆₄	—	43	16	no	—
53	C₂×C₃₂	C₂×D₄₃	1	6	yes	—
54	C₄₃	C₂²×Q₆₄	1	1	yes	characteristic, complemented, Sylow
55	C₂²×Q₈	—	172	2	no	—
56	C₂×Q₁₆	—	86	12	no	—
57	C₂×Q₁₆	—	172	6	no	—
58	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
59	Q₃₂	—	43	8	no	—
60	Q₃₂	—	86	12	no	—
61	C₂×C₁₆	D₁₇₂	1	3	yes	—
62	C₂×C₁₆	C₂²×D₄₃	1	3	yes	—
63	C₃₂	C₂²×D₄₃	1	4	yes	—
64	C₂×Q₈	—	344	6	no	—
65	C₂×Q₈	—	172	12	no	—
66	C₂²×C₄	—	344	2	no	—
67	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
68	Q₁₆	—	86	8	no	—
69	Q₁₆	—	172	12	no	—
70	C₂×C₈	D₃₄₄	1	3	yes	—
71	C₂×C₈	C₂×D₁₇₂	1	3	yes	—
72	C₁₆	C₂×D₁₇₂	1	3	yes	—
73	C₁₆	C₂³×D₄₃	1	1	yes	characteristic
74	C₂³	D₆₈₈	1	1	yes	characteristic
75	Q₈	—	344	12	no	—
76	Q₈	—	172	8	no	—
77	C₂×C₄	—	344	12	no	—
78	C₂×C₄	D₆₈₈	1	3	yes	—
79	C₂×C₄	C₂×D₃₄₄	1	3	yes	—
80	C₈	C₂×D₃₄₄	1	3	yes	—
81	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
82	C₂²	Dic₆₈₈	1	4	yes	direct product factor
83	C₂²	C₂×D₆₈₈	1	3	yes	—
84	C₄	—	344	8	no	—
85	C₄	C₂×D₆₈₈	1	3	yes	—
86	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
87	C₂	C₂×Dic₆₈₈	1	6	yes	direct product factor
88	C₂	C₂²×D₆₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₆₈₈
commutator, cosocle	C₁₆×C₄₃	C₂⁴
Frattini	C₁₆	C₂³×D₄₃
Fitting	C₂²×C₃₂×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	D₁₆
hypercenter	C₂²×C₃₂	D₄₃
hypocenter	C₄₃	C₂²×Q₆₄
LCS	G ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
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
2. C₂×Dic₆₈₈	C₂	direct (N × H)
7. Dic₆₈₈	C₂²	direct (N × H)
38. C₂²×C₄₃	Q₆₄	semidirect
46. C₂×C₄₃	C₂×Q₆₄	semidirect
54. C₄₃	C₂²×Q₆₄	semidirect
82. C₂²	Dic₆₈₈	direct (N × H)
87. C₂	C₂×Dic₆₈₈	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×Dic₃₄₄	2
C₂×Dic₆₈₈	2
C₂²×C₃₂×C₄₃	2
C₂²×Q₆₄	43

Subgroups — C22×Dic688 #25254720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₆₈₈ #25254720