Subgroups — #26021654

Order 11392.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151646	C₂	1	3	yes	maximal
2	C₂×#15667930	C₂	1	3	yes	maximal
3	#338×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
5	C₂×#13083558	C₂²	1	3	yes	—
6	#15667947	—	2	6	no	—
7	#15667930	—	2	6	no	—
8	C₂×M₁₆×C₈₉	C₂²	1	3	yes	—
9	C₂×Dic₁₇₈	—	2	3	no	—
10	C₂×Dic₁₇₈	—	2	3	no	—
11	C₂²×Dic₈₉	Q₈	1	2	yes	—
12	#13083558	—	2	3	no	—
13	#13083558	D₄	1	6	yes	—
14	C₂×#10623494	—	2	3	no	—
15	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
16	M₁₆×C₈₉	—	2	3	no	—
17	M₁₆×C₈₉	—	2	3	no	—
18	C₂×C₈×C₈₉	—	2	3	no	—
19	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
21	C₂×C₄×C₈₉	#41	1	3	yes	—
22	C₂×Dic₈₉	—	2	6	no	—
23	Dic₁₇₈	—	4	3	no	—
24	Dic₁₇₈	—	4	1	no	—
25	C₈×C₈₉	—	4	3	no	—
26	#10623494	—	2	6	no	—
27	C₂²×C₈₉	—	4	1	no	—
28	C₂²×C₈₉	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₈₉	#121	1	3	yes	—
30	C₄×C₈₉	#126	1	1	yes	characteristic
31	Dic₈₉	—	4	2	no	—
32	C₂×C₈₉	—	2	3	no	—
33	C₂×C₈₉	#396	1	1	yes	characteristic
34	#2070	—	89	1	no	maximal, Sylow
35	C₈₉	#2070	1	1	yes	characteristic, complemented, Sylow
36	#430	—	89	3	no	—
37	C₂×#101	—	89	3	no	—
38	#338	D₈₉	1	1	yes	characteristic
39	C₂²×Q₈	—	89	1	no	—
40	C₂×M₁₆	—	89	3	no	—
41	C₂×M₁₆	C₂×D₈₉	1	3	yes	—
42	#108	—	178	6	no	—
43	#101	—	178	6	no	—
44	C₂×Q₈	—	178	3	no	—
45	C₂×Q₈	—	178	3	no	—
46	C₂²×C₄	—	89	2	no	—
47	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	89	6	no	—
51	M₁₆	—	178	3	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	178	3	no	—
54	C₂³	Q₈×D₈₉	1	1	yes	characteristic
55	Q₈	—	356	3	no	—
56	Q₈	—	356	1	no	—
57	C₂×C₄	—	178	6	no	—
58	C₂×C₄	D₄×D₈₉	1	3	yes	—
59	C₂×C₄	#13083574	1	3	yes	—
60	C₈	—	4	3	no	—
61	C₈	—	178	6	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#15667990	1	3	yes	—
64	C₄	—	356	2	no	—
65	C₄	#15667985	1	3	yes	—
66	C₄	#15668006	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#19151522	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19151522
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#338×C₈₉	C₂
socle	C₂×C₈₉	#396
prime radical	C₂³×C₈₉	C₂×Q₈
hypercenter	#338	D₈₉
hypocenter	C₈₉	#2070
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
35. C₈₉	#2070	semidirect

Maximal subgroups

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

subgroup	index
#19151646	2
C₂×#15667930	2
#338×C₈₉	2
#2070	89