Subgroups — #36442131

Order 22784.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151428	C₂	1	2	yes	maximal
2	C₂²×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667948	—	2	2	no	—
4	#19151428	—	2	2	no	—
5	#19151428	C₄	1	4	yes	—
6	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×C₃₂×C₈₉	—	2	2	no	—
8	C₂×C₃₂×C₈₉	C₄	1	2	yes	—
9	C₂×#13083558	—	4	2	no	—
10	#15667948	—	2	4	no	—
11	#15667948	—	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	#13083558	—	4	4	no	—
19	#13083558	—	8	2	no	—
20	C₂×#10623494	—	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	Q₃₂	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	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
34	C₈×C₈₉	#95	1	1	yes	characteristic
35	C₈×C₈₉	#103	1	2	yes	—
36	C₈×C₈₉	#106	1	1	yes	characteristic
37	#10623494	—	8	4	no	—
38	C₂²×C₈₉	—	2	2	no	—
39	C₂²×C₈₉	#358	1	2	yes	—
40	C₂²×C₈₉	#366	1	1	yes	characteristic
41	C₄×C₈₉	#340	1	1	yes	characteristic
42	C₄×C₈₉	#357	1	2	yes	—
43	C₄×C₈₉	#367	1	1	yes	characteristic
44	#7472	—	89	1	no	maximal, Sylow
45	C₂×C₈₉	—	2	2	no	—
46	C₂×C₈₉	#1380	1	1	yes	characteristic
47	C₂×C₈₉	#1425	1	2	yes	—
48	C₂²×C₃₂	D₈₉	1	1	yes	characteristic
49	C₂×#368	—	89	2	no	—
50	C₈₉	#7472	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
52	C₂×#108	—	178	2	no	—
53	C₂×C₃₂	—	2	2	no	—
54	C₂×C₃₂	Dic₈₉	1	2	yes	—
55	#368	—	89	4	no	—
56	#368	—	178	2	no	—
57	C₂×M₁₆	—	356	2	no	—
58	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
59	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
60	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
61	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
62	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
63	#108	—	356	2	no	—
64	#108	—	178	4	no	—
65	C₃₂	—	2	2	no	—
66	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
67	M₁₆	—	356	4	no	—
68	M₁₆	—	712	2	no	—
69	C₂×C₈	—	712	2	no	—
70	C₂×C₈	#13083551	1	2	yes	—
71	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
72	C₂×C₈	#13083563	1	2	yes	—
73	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
74	C₁₆	C₄×Dic₈₉	1	1	yes	characteristic
75	C₁₆	C₂²⋊Dic₈₉	1	1	yes	characteristic
76	C₁₆	#13083553	1	2	yes	—
77	C₂³	Dic₇₁₂	1	1	yes	characteristic
78	C₂×C₄	#15667934	1	2	yes	—
79	C₂×C₄	#15667946	1	1	yes	characteristic
80	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
81	C₂×C₄	#15667957	1	2	yes	—
82	C₈	—	712	4	no	—
83	C₈	#15667919	1	1	yes	characteristic
84	C₈	#15667937	1	2	yes	—
85	C₈	#15667944	1	1	yes	characteristic
86	C₂²	—	2	2	no	—
87	C₂²	#19151403	1	2	yes	—
88	C₂²	#19151424	1	1	yes	characteristic
89	C₄	#19151361	1	1	yes	characteristic
90	C₄	#19151400	1	2	yes	—
91	C₄	#19151426	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#26018344	1	1	yes	characteristic
94	C₂	#26018456	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₄₂₄
commutator	C₁₆×C₈₉	C₄²
Frattini	C₂²×C₁₆	C₂×D₈₉
Fitting	C₂²×C₃₂×C₈₉	C₂
socle	C₂²×C₈₉	#366
prime radical	C₂³×C₈₉	Q₃₂
cosocle	C₂²×C₁₆×C₈₉	C₂²
hypercenter	C₂²×C₃₂	D₈₉
hypocenter	C₈₉	#7472
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
50. C₈₉	#7472	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151428	2
C₂²×C₃₂×C₈₉	2
#7472	89