Subgroups — #36442100

Order 22784.

98 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	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic
3	C₈₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667918	—	2	1	no	—
5	C₄×#13083549	C₄	1	1	yes	characteristic
6	C₈×#10623494	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₈₉	—	2	1	no	—
8	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083549	—	2	1	no	—
10	C₄×#10623494	C₈	1	1	yes	characteristic
11	C₄×#10623494	C₈	1	1	yes	characteristic
12	#15667918	—	4	1	no	—
13	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₈₉	—	4	1	no	—
17	C₂×#10623494	—	2	1	no	—
18	C₂×#10623494	C₁₆	1	1	yes	characteristic
19	C₂×#10623494	C₁₆	1	1	yes	characteristic
20	#13083549	—	2	2	no	—
21	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
23	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₈₉	—	2	1	no	—
25	C₁₆×C₈₉	D₈	1	1	yes	characteristic
26	C₁₆×C₈₉	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
28	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
29	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
30	C₈×C₈₉	—	2	1	no	—
31	C₈×C₈₉	#107	1	1	yes	characteristic
32	C₈×C₈₉	#108	1	1	yes	characteristic
33	#10623494	—	2	2	no	—
34	#10623494	—	2	1	no	—
35	#10623494	C₃₂	1	2	yes	complemented
36	C₂²×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₈₉	—	2	1	no	—
38	C₄×C₈₉	#335	1	1	yes	characteristic
39	C₄×C₈₉	#364	1	1	yes	characteristic
40	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₈₉	M₆₄	1	1	yes	characteristic
42	#7458	—	89	1	no	maximal, Sylow
43	C₂×C₈₉	#1370	1	1	yes	characteristic
44	C₂×C₈₉	C₄⋊C₃₂	1	1	yes	characteristic
45	C₂×C₈₉	#1422	1	1	yes	characteristic
46	C₄⋊C₃₂	—	89	1	no	—
47	C₄⋊C₃₂	D₈₉	1	1	yes	characteristic
48	C₈×C₁₆	—	89	1	no	—
49	C₈₉	#7458	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃₂	—	2	1	no	—
51	C₂×C₃₂	—	178	1	no	—
52	C₄×C₁₆	—	89	1	no	—
53	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
54	C₈²	—	89	1	no	—
55	C₂×C₁₆	—	178	1	no	—
56	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
58	C₄×C₈	—	89	1	no	—
59	C₄×C₈	—	89	1	no	—
60	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₃₂	—	356	1	no	—
63	C₂×C₈	—	89	1	no	—
64	C₂×C₈	—	89	1	no	—
65	C₂×C₈	—	178	1	no	—
66	C₂×C₈	#13083553	1	1	yes	characteristic
67	C₂×C₈	#13083557	1	1	yes	characteristic
68	C₄²	C₈×D₈₉	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	—	178	2	no	—
71	C₁₆	#13083559	1	1	yes	characteristic
72	C₁₆	#13083564	1	1	yes	characteristic
73	C₂×C₄	#15667941	1	1	yes	characteristic
74	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
75	C₂×C₄	#15667951	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	89	2	no	—
78	C₈	—	178	2	no	—
79	C₈	—	178	1	no	—
80	C₈	#15667945	1	1	yes	characteristic
81	C₈	#15667947	1	1	yes	characteristic
82	C₂²	#19151416	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#19151350	1	1	yes	characteristic
85	C₄	#19151418	1	1	yes	characteristic
86	C₄	C₃₂×D₈₉	1	1	yes	characteristic
87	C₄	#19151431	1	1	yes	characteristic
88	C₂	#26018321	1	1	yes	characteristic
89	C₂	#26018447	1	1	yes	characteristic
90	C₂	#26018449	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₈₉	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	C₄⋊C₃₂	D₈₉
hypocenter	C₈₉	#7458
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
35. #10623494	C₃₂	semidirect
49. C₈₉	#7458	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
C₁₆×#10623494	2
C₈₉×C₄⋊C₃₂	2
#7458	89