Subgroups — #36442082

Order 22784.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15667918	C₂	1	1	yes	maximal, characteristic
2	#26018324	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667918	—	2	1	no	—
5	#19151351	—	2	1	no	—
6	#500×C₈₉	—	2	1	no	—
7	#366×C₈₉	—	2	1	no	—
8	#366×C₈₉	C₄	1	2	yes	—
9	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₄⋊#10623494	—	4	1	no	—
11	#15667918	—	4	1	no	—
12	C₄⋊Q₈×C₈₉	—	4	1	no	—
13	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
15	#107×C₈₉	—	2	2	no	—
16	#107×C₈₉	—	4	1	no	—
17	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
18	C₂×#10623494	—	8	1	no	—
19	C₂×Q₈×C₈₉	—	8	1	no	—
20	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
22	C₄⋊C₄×C₈₉	—	4	2	no	—
23	C₄⋊C₄×C₈₉	—	8	1	no	—
24	C₄²×C₈₉	D₈	1	1	yes	characteristic
25	C₁₆×C₈₉	—	2	1	no	—
26	C₁₆×C₈₉	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₈₉	Q₁₆	1	1	yes	characteristic
28	Q₈×C₈₉	—	16	2	no	—
29	C₂×C₄×C₈₉	—	8	2	no	—
30	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
31	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
32	C₂×C₄×C₈₉	SD₃₂	1	1	yes	characteristic
33	C₈×C₈₉	—	2	1	no	—
34	C₈×C₈₉	#101	1	1	yes	characteristic
35	C₈×C₈₉	#103	1	1	yes	characteristic
36	#10623494	—	16	1	no	—
37	C₂²×C₈₉	#357	1	1	yes	characteristic
38	C₄×C₈₉	—	2	1	no	—
39	C₄×C₈₉	—	16	2	no	—
40	C₄×C₈₉	#332	1	1	yes	characteristic
41	C₄×C₈₉	#362	1	1	yes	characteristic
42	C₄×C₈₉	SD₆₄	1	1	yes	characteristic
43	C₄×C₈₉	Q₆₄	1	1	yes	characteristic
44	#7451	—	89	1	no	maximal, Sylow
45	C₂×C₈₉	#1364	1	1	yes	characteristic
46	C₂×C₈₉	#1416	1	1	yes	characteristic
47	C₂×C₈₉	#1419	1	1	yes	characteristic
48	#2252	D₈₉	1	1	yes	characteristic
49	C₄⋊C₃₂	—	89	1	no	—
50	#1371	—	89	1	no	—
51	C₈₉	#7451	1	1	yes	characteristic, complemented, Sylow
52	#500	—	2	1	no	—
53	C₂×C₃₂	—	178	1	no	—
54	#366	—	2	1	no	—
55	#366	Dic₈₉	1	2	yes	—
56	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
57	#335	—	178	1	no	—
58	C₄⋊Q₈	—	4	1	no	—
59	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
60	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	#107	—	4	1	no	—
63	C₄⋊C₈	—	356	1	no	—
64	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
65	C₃₂	—	356	1	no	—
66	C₂×Q₈	—	8	1	no	—
67	C₂×C₈	—	712	1	no	—
68	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
69	C₂×C₈	#13083562	1	1	yes	characteristic
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	8	1	no	—
72	C₄²	#13083559	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	#13083561	1	1	yes	characteristic
75	C₁₆	#13083564	1	1	yes	characteristic
76	Q₈	—	16	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	#15667933	1	1	yes	characteristic
79	C₂×C₄	#15667953	1	1	yes	characteristic
80	C₂×C₄	#15667956	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	1424	1	no	—
83	C₈	#15667931	1	1	yes	characteristic
84	C₈	#15667936	1	1	yes	characteristic
85	C₂²	#19151399	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	16	2	no	—
88	C₄	#19151344	1	1	yes	characteristic
89	C₄	#19151414	1	1	yes	characteristic
90	C₄	#19151435	1	1	yes	characteristic
91	C₄	#19151438	1	1	yes	characteristic
92	C₂	#26018307	1	1	yes	characteristic
93	C₂	#26018433	1	1	yes	characteristic
94	C₂	#26018442	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151399
commutator	C₂×C₁₆×C₈₉	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×#2252	C₂
socle, prime radical	C₂²×C₈₉	#357
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#7451
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

Complements

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

N	H ≅ G/N	type
51. C₈₉	#7451	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15667918	2
#26018324	2
C₈₉×#2252	2
#7451	89