Subgroups — #36475426

Order 22784.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄₂₄	C₂	1	4	yes	maximal
2	#26022505	C₂	1	2	yes	maximal
3	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₁₂	—	2	2	no	—
5	C₂×Dic₇₁₂	C₂²	1	4	yes	—
6	#19151978	—	2	2	no	—
7	Dic₁₄₂₄	—	2	8	no	—
8	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
9	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₅₆	—	2	4	no	—
11	C₂×Dic₃₅₆	—	4	2	no	—
12	#15668012	—	4	2	no	—
13	Dic₇₁₂	—	2	8	no	—
14	Dic₇₁₂	—	4	4	no	—
15	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
18	C₃₂×C₈₉	D₄	1	4	yes	—
19	C₂×Dic₁₇₈	—	4	4	no	—
20	Dic₃₅₆	—	4	4	no	—
21	Dic₃₅₆	—	4	4	no	—
22	Dic₃₅₆	—	8	4	no	—
23	C₄⋊Dic₈₉	—	8	4	no	—
24	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
25	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₈₉	D₈	1	1	yes	characteristic
27	C₁₆×C₈₉	D₈	1	2	yes	—
28	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
30	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
31	C₂×Dic₈₉	—	8	4	no	—
32	Dic₁₇₈	—	8	4	no	—
33	Dic₁₇₈	—	8	4	no	—
34	C₈×C₈₉	D₁₆	1	2	yes	—
35	C₈×C₈₉	#127	1	1	yes	characteristic
36	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
37	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
38	C₄×C₈₉	Q₆₄	1	4	yes	complemented
39	C₄×C₈₉	#493	1	1	yes	characteristic
40	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
41	Dic₈₉	—	16	4	no	—
42	#13726	—	89	1	no	maximal, Sylow
43	C₂×C₈₉	#2246	1	1	yes	characteristic
44	C₂×C₈₉	C₂×Q₆₄	1	2	yes	—
45	C₂×Q₆₄	—	89	4	no	—
46	#2247	—	89	2	no	—
47	C₄×C₃₂	D₈₉	1	1	yes	characteristic
48	C₈₉	#13726	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₃₂	—	89	4	no	—
50	C₂×Q₃₂	—	178	2	no	—
51	#494	—	178	2	no	—
52	Q₆₄	—	178	8	no	—
53	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
54	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
55	C₂×Q₁₆	—	356	2	no	—
56	C₂×Q₁₆	—	178	4	no	—
57	C₄⋊Q₈	—	356	2	no	—
58	Q₃₂	—	356	4	no	—
59	Q₃₂	—	178	8	no	—
60	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
62	C₄×C₈	D₃₅₆	1	1	yes	characteristic
63	C₃₂	D₃₅₆	1	4	yes	—
64	C₂×Q₈	—	356	4	no	—
65	Q₁₆	—	356	4	no	—
66	Q₁₆	—	356	4	no	—
67	Q₁₆	—	712	4	no	—
68	C₂×C₈	D₇₁₂	1	1	yes	characteristic
69	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
70	C₄⋊C₄	—	712	4	no	—
71	C₄²	D₇₁₂	1	1	yes	characteristic
72	C₁₆	D₇₁₂	1	2	yes	—
73	C₁₆	C₂×D₃₅₆	1	2	yes	—
74	Q₈	—	712	4	no	—
75	Q₈	—	712	4	no	—
76	C₂×C₄	—	712	4	no	—
77	C₂×C₄	D₁₄₂₄	1	2	yes	—
78	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
79	C₈	D₁₄₂₄	1	2	yes	—
80	C₈	#15668009	1	1	yes	characteristic
81	C₈	C₂×D₇₁₂	1	1	yes	characteristic
82	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
83	C₄	—	1424	4	no	—
84	C₄	Dic₁₄₂₄	1	4	yes	complemented
85	C₄	#19151975	1	1	yes	characteristic
86	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
87	C₂	#26022502	1	1	yes	characteristic
88	C₂	C₂×Dic₁₄₂₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₄₂₄
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₄×C₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₈₉
hypocenter	C₈₉	#13726
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
38. C₄×C₈₉	Q₆₄	semidirect
48. C₈₉	#13726	semidirect
84. C₄	Dic₁₄₂₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₁₄₂₄	2
#26022505	2
C₄×C₃₂×C₈₉	2
#13726	89