Subgroups — #36470842

Order 22784.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13083549	C₂	1	6	yes	maximal
2	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13083549	C₂²	1	4	yes	—
4	C₄⋊#13083549	—	2	12	no	—
5	C₂×#97×C₈₉	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
7	#376×C₈₉	C₄	1	1	yes	characteristic
8	C₂×#13083549	—	2	12	no	—
9	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₈₉	C₂×C₄	1	3	yes	—
12	#97×C₈₉	—	2	6	no	—
13	C₄×C₈×C₈₉	D₄	1	6	yes	—
14	C₄×C₈×C₈₉	Q₈	1	6	yes	—
15	#95×C₈₉	C₈	1	4	yes	—
16	#13083549	—	4	4	no	—
17	C₂²×C₄×C₈₉	C₂×C₈	1	6	yes	—
18	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₈₉	—	2	6	no	—
20	C₂×C₈×C₈₉	C₄⋊C₄	1	6	yes	—
21	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
22	C₂×C₈×C₈₉	C₂×Q₈	1	3	yes	—
23	C₄²×C₈₉	C₄⋊C₄	1	6	yes	—
24	C₄²×C₈₉	M₁₆	1	6	yes	—
25	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₈₉	—	2	12	no	—
27	C₂×C₄×C₈₉	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₈₉	C₂×M₁₆	1	3	yes	—
30	C₈×C₈₉	—	2	6	no	—
31	C₈×C₈₉	#127	1	1	yes	characteristic
32	C₈×C₈₉	C₄⋊Q₈	1	3	yes	—
33	C₂²×C₈₉	C₂×C₄⋊C₈	1	3	yes	—
34	C₂²×C₈₉	#504	1	4	yes	—
35	C₄×C₈₉	—	2	12	no	—
36	C₄×C₈₉	#384	1	1	yes	characteristic
37	C₄×C₈₉	#423	1	3	yes	—
38	#12899	—	89	1	no	maximal, Sylow
39	C₂×C₈₉	#1842	1	1	yes	characteristic
40	C₂×C₈₉	#2151	1	6	yes	—
41	C₂×C₄⋊C₁₆	—	89	6	no	—
42	#1726	D₈₉	1	1	yes	characteristic
43	C₈₉	#12899	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₁₆	—	89	4	no	—
45	C₂×#97	Dic₈₉	1	3	yes	—
46	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
47	#376	Dic₈₉	1	1	yes	characteristic
48	C₄⋊C₁₆	—	178	12	no	—
49	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
50	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
51	C₂×C₄²	C₂×Dic₈₉	1	3	yes	—
52	C₂×C₁₆	—	178	12	no	—
53	#97	—	2	6	no	—
54	C₄×C₈	Dic₁₇₈	1	6	yes	—
55	C₄×C₈	D₃₅₆	1	6	yes	—
56	#95	#10623494	1	4	yes	—
57	C₂²×C₄	C₂×#10623494	1	6	yes	—
58	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊Dic₈₉	1	6	yes	—
61	C₂×C₈	C₂×D₃₅₆	1	3	yes	—
62	C₂×C₈	C₂×Dic₁₇₈	1	3	yes	—
63	C₄²	C₄⋊Dic₈₉	1	6	yes	—
64	C₄²	#13083558	1	6	yes	—
65	C₁₆	—	356	4	no	—
66	C₂³	C₂²×#10623494	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#10623494	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
70	C₂×C₄	C₂×#13083558	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	#15668009	1	1	yes	characteristic
73	C₈	#15668012	1	3	yes	—
74	C₂²	C₂×C₄⋊#10623494	1	3	yes	—
75	C₂²	#19152015	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#19151468	1	1	yes	characteristic
78	C₄	#19151622	1	3	yes	—
79	C₂	#26020404	1	1	yes	characteristic
80	C₂	#26022011	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂²×C₈
Fitting	C₈₉×#1726	C₂
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#12899
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
43. C₈₉	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#13083549	2
C₈₉×#1726	2
#12899	89