Subgroups — #36442139

Order 22784.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022036	C₂	1	2	yes	maximal
2	C₂×M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151636	—	2	2	no	—
4	#19151426	C₄	1	4	yes	complemented
5	#19151424	—	2	2	no	—
6	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
7	M₆₄×C₈₉	—	2	2	no	—
8	C₂×C₃₂×C₈₉	C₄	1	2	yes	complemented
9	#15667971	—	4	2	no	—
10	#15667946	—	2	4	no	—
11	#15667944	—	4	2	no	—
12	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
16	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
17	C₃₂×C₈₉	—	2	2	no	—
18	C₄⋊Dic₈₉	—	4	4	no	—
19	C₂²⋊Dic₈₉	—	8	2	no	—
20	C₄×Dic₈₉	—	8	2	no	—
21	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
22	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
23	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
25	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
26	C₁₆×C₈₉	C₄²	1	1	yes	characteristic
27	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
29	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₈₉	#102	1	2	yes	—
31	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
32	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
33	C₂×C₄×C₈₉	Q₃₂	1	1	yes	characteristic
34	C₂×Dic₈₉	—	8	4	no	—
35	C₈×C₈₉	#95	1	1	yes	characteristic
36	C₈×C₈₉	#103	1	2	yes	—
37	C₈×C₈₉	#106	1	1	yes	characteristic
38	C₂²×C₈₉	—	4	1	no	—
39	C₂²×C₈₉	#357	1	2	yes	—
40	C₂²×C₈₉	#366	1	1	yes	characteristic
41	C₄×C₈₉	#340	1	1	yes	characteristic
42	C₄×C₈₉	#358	1	2	yes	—
43	C₄×C₈₉	#367	1	1	yes	characteristic
44	Dic₈₉	—	16	4	no	—
45	#7475	—	89	1	no	maximal, Sylow
46	C₂×C₈₉	—	2	2	no	—
47	C₂×C₈₉	—	2	1	no	—
48	C₂×C₈₉	#1380	1	1	yes	characteristic
49	C₂×M₆₄	D₈₉	1	1	yes	characteristic
50	#2158	—	89	2	no	—
51	C₈₉	#7475	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
53	#427	—	178	2	no	—
54	M₆₄	—	2	2	no	—
55	C₂×C₃₂	Dic₈₉	1	2	yes	complemented
56	#367	—	89	4	no	—
57	#366	—	178	2	no	—
58	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
59	#117	—	356	2	no	—
60	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
61	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
62	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
63	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
64	#107	—	178	4	no	—
65	#106	—	356	2	no	—
66	C₃₂	—	2	2	no	—
67	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
68	C₂×C₈	#13083551	1	2	yes	—
69	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
70	C₂×C₈	#13083563	1	2	yes	—
71	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
72	C₄⋊C₄	—	356	4	no	—
73	C₂²⋊C₄	—	712	2	no	—
74	C₄²	—	712	2	no	—
75	C₁₆	C₄×Dic₈₉	1	1	yes	characteristic
76	C₁₆	C₂²⋊Dic₈₉	1	1	yes	characteristic
77	C₁₆	#13083553	1	2	yes	—
78	C₂³	D₁₄₂₄	1	1	yes	characteristic
79	C₂×C₄	—	712	4	no	—
80	C₂×C₄	#15667934	1	2	yes	—
81	C₂×C₄	#15667946	1	1	yes	characteristic
82	C₂×C₄	#15667957	1	2	yes	—
83	C₂×C₄	Dic₇₁₂	1	1	yes	characteristic
84	C₈	#15667919	1	1	yes	characteristic
85	C₈	#15667937	1	2	yes	—
86	C₈	#15667944	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	#19151400	1	2	yes	—
89	C₂²	#19151424	1	1	yes	characteristic
90	C₄	—	1424	4	no	—
91	C₄	#19151361	1	1	yes	characteristic
92	C₄	#19151403	1	2	yes	—
93	C₄	#19151426	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	#26018344	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151426
commutator	C₁₆×C₈₉	C₄²
Frattini	C₂²×C₁₆	C₂×D₈₉
Fitting	C₂×M₆₄×C₈₉	C₂
socle	C₂×C₈₉	#1380
prime radical	C₂³×C₈₉	D₁₆
cosocle	C₂²×C₁₆×C₈₉	C₂²
hypercenter	C₂×M₆₄	D₈₉
hypocenter	C₈₉	#7475
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
4. #19151426	C₄	semidirect
8. C₂×C₃₂×C₈₉	C₄	semidirect
51. C₈₉	#7475	semidirect
55. C₂×C₃₂	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022036	2
C₂×M₆₄×C₈₉	2
#7475	89