Subgroups — #36470828

Order 22784.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018330	C₂	1	6	yes	maximal
2	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13083549	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₈₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
6	C₄³×C₈₉	C₄	1	1	yes	characteristic
7	C₂×#13083549	—	2	12	no	—
8	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₈₉	C₈	1	3	yes	—
11	C₂×C₄²×C₈₉	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₈₉	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₈₉	—	2	6	no	—
14	C₄×C₈×C₈₉	—	2	6	no	—
15	#13083549	—	4	4	no	—
16	C₂²×C₄×C₈₉	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₈₉	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₈×C₈₉	—	2	3	no	—
20	C₂×C₈×C₈₉	—	2	3	no	—
21	C₂×C₈×C₈₉	M₁₆	1	6	yes	—
22	C₂×C₈×C₈₉	#41	1	6	yes	—
23	C₄²×C₈₉	—	2	6	no	—
24	C₄²×C₈₉	—	2	3	no	—
25	C₄²×C₈₉	—	2	3	no	—
26	C₄²×C₈₉	C₁₆	1	4	yes	complemented
27	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₈₉	—	2	3	no	—
29	C₂×C₄×C₈₉	—	2	3	no	—
30	C₂×C₄×C₈₉	—	2	3	no	—
31	C₂×C₄×C₈₉	—	2	3	no	—
32	C₂×C₄×C₈₉	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₈₉	M₃₂	1	6	yes	—
34	C₂×C₄×C₈₉	#117	1	3	yes	—
35	C₂×C₄×C₈₉	C₂×M₁₆	1	3	yes	—
36	C₈×C₈₉	—	2	6	no	—
37	C₈×C₈₉	#126	1	4	yes	—
38	C₂²×C₈₉	#431	1	3	yes	—
39	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₈₉	C₂×M₃₂	1	3	yes	—
41	C₄×C₈₉	—	2	6	no	—
42	C₄×C₈₉	—	2	6	no	—
43	C₄×C₈₉	#383	1	1	yes	characteristic
44	C₄×C₈₉	#432	1	3	yes	—
45	#12894	—	89	1	no	maximal, Sylow
46	C₂×C₈₉	#1839	1	1	yes	characteristic
47	C₂×C₈₉	#2162	1	3	yes	—
48	C₂×C₈₉	#2163	1	3	yes	—
49	C₄²×C₈	D₈₉	1	1	yes	characteristic
50	#1374	—	89	6	no	—
51	C₈₉	#12894	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	89	4	no	—
53	C₂×C₄×C₈	Dic₈₉	1	3	yes	—
54	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
55	C₄³	Dic₈₉	1	1	yes	characteristic
56	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
57	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
58	C₂×C₄²	#10623494	1	3	yes	—
59	C₂×C₄²	#10623494	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₈₉	1	3	yes	—
61	C₂×C₁₆	—	178	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×#10623494	1	3	yes	—
65	C₂²×C₄	C₂×#10623494	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#13083558	1	6	yes	—
70	C₂×C₈	#13083574	1	6	yes	—
71	C₄²	—	2	3	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	6	no	—
74	C₄²	#13083549	1	4	yes	complemented
75	C₁₆	—	356	4	no	—
76	C₂³	C₂²×#10623494	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#13083549	1	6	yes	—
82	C₂×C₄	#15667952	1	6	yes	—
83	C₂×C₄	#15667971	1	3	yes	—
84	C₂×C₄	C₂×#13083558	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#15668006	1	4	yes	—
87	C₂²	#19151651	1	3	yes	—
88	C₂²	C₂²×#13083549	1	1	yes	characteristic
89	C₂²	C₂×#15667952	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#19151466	1	1	yes	characteristic
93	C₄	#19151655	1	3	yes	—
94	C₂	#26020395	1	1	yes	characteristic
95	C₂	#26022051	1	3	yes	—
96	C₂	#26022055	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₁₆
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#12894
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
26. C₄²×C₈₉	C₁₆	semidirect
51. C₈₉	#12894	semidirect
74. C₄²	#13083549	semidirect

Maximal subgroups

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

subgroup	index
#26018330	2
C₄²×C₈×C₈₉	2
#12894	89