Subgroups — #36441165

Order 22784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018316	C₂	1	1	yes	maximal, characteristic
2	#26018169	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1367	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13083549	—	2	1	no	—
5	C₄×#13083549	C₄	1	1	yes	characteristic
6	C₄×#13083549	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₈₉	—	2	1	no	—
8	C₈²×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083549	—	2	1	no	—
10	C₂×#13083549	—	4	1	no	—
11	C₂×#13083549	C₈	1	1	yes	characteristic
12	C₂×#13083549	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	—	4	1	no	—
14	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
16	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
17	#13083549	—	2	2	no	—
18	#13083549	—	2	1	no	—
19	#13083549	—	8	1	no	—
20	#13083549	C₁₆	1	2	yes	complemented
21	C₂×C₈×C₈₉	—	2	1	no	—
22	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
23	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
24	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
25	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
26	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
27	C₁₆×C₈₉	—	8	1	no	—
28	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
30	C₂×C₄×C₈₉	#108	1	1	yes	characteristic
31	C₈×C₈₉	—	2	1	no	—
32	C₈×C₈₉	—	2	1	no	—
33	C₈×C₈₉	—	2	1	no	—
34	C₈×C₈₉	—	4	1	no	—
35	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₈₉	M₃₂	1	1	yes	characteristic
37	C₂²×C₈₉	#334	1	1	yes	characteristic
38	C₄×C₈₉	—	2	1	no	—
39	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
40	C₄×C₈₉	#364	1	1	yes	characteristic
41	C₄×C₈₉	#365	1	1	yes	characteristic
42	C₄×C₈₉	#365	1	1	yes	characteristic
43	#7099	—	89	1	no	maximal, Sylow
44	C₂×C₈₉	#1367	1	1	yes	characteristic
45	C₂×C₈₉	#1368	1	1	yes	characteristic
46	C₂×C₈₉	#1369	1	1	yes	characteristic
47	#1367	—	89	1	no	—
48	#1367	D₈₉	1	1	yes	characteristic
49	#1311	—	89	1	no	—
50	C₈₉	#7099	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊C₁₆	—	2	1	no	—
52	C₄⋊C₁₆	—	178	1	no	—
53	C₄×C₁₆	—	89	1	no	—
54	C₄×C₁₆	—	89	1	no	—
55	C₈²	C₂×D₈₉	1	1	yes	characteristic
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	—	89	1	no	—
58	C₂×C₁₆	—	89	1	no	—
59	C₂×C₁₆	—	356	1	no	—
60	C₂×C₁₆	—	178	1	no	—
61	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
62	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
63	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
66	C₂×C₈	#13083557	1	1	yes	characteristic
67	C₂×C₈	#13083561	1	1	yes	characteristic
68	C₂×C₈	#13083562	1	1	yes	characteristic
69	C₄²	#13083553	1	1	yes	characteristic
70	C₁₆	—	8	1	no	—
71	C₁₆	—	89	2	no	—
72	C₁₆	—	178	2	no	—
73	C₁₆	—	178	1	no	—
74	C₁₆	—	712	1	no	—
75	C₂×C₄	#15667941	1	1	yes	characteristic
76	C₂×C₄	#15667943	1	1	yes	characteristic
77	C₂×C₄	#15667947	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	2	1	no	—
80	C₈	—	2	1	no	—
81	C₈	—	4	1	no	—
82	C₈	C₁₆×D₈₉	1	1	yes	characteristic
83	C₈	#15667951	1	1	yes	characteristic
84	C₂²	#19151348	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	#19151416	1	1	yes	characteristic
87	C₄	#19151418	1	1	yes	characteristic
88	C₄	#19151420	1	1	yes	characteristic
89	C₄	#19151422	1	1	yes	characteristic
90	C₂	#26018315	1	1	yes	characteristic
91	C₂	#26018317	1	1	yes	characteristic
92	C₂	#26018319	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667943
commutator	C₈×C₈₉	C₂×C₁₆
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1367	C₂
socle, prime radical	C₂²×C₈₉	#334
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1367	D₈₉
hypocenter	C₈₉	#7099
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

Complements

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

N	H ≅ G/N	type
20. #13083549	C₁₆	semidirect
50. C₈₉	#7099	semidirect

Maximal subgroups

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

subgroup	index
#26018316	2
#26018169	2
C₈₉×#1367	2
#7099	89