Subgroups — C₂×#26019867 #36542119

Order 22784.

1342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151975	C₂	1	6	yes	maximal, complemented
2	#26019867	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₇₁₂	—	2	12	no	—
5	C₂×#15668009	C₂²	1	4	yes	—
6	#19151975	—	2	12	no	—
7	#19151975	C₂²	1	24	yes	complemented
8	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
9	C₈²×C₈₉	C₂²	1	4	yes	complemented
10	C₂²×D₃₅₆	—	2	12	no	—
11	C₂×D₇₁₂	—	2	48	no	—
12	C₂×D₇₁₂	—	4	48	no	—
13	#15668009	—	2	12	no	—
14	#15668009	C₂³	1	8	yes	—
15	C₂²×C₈×C₈₉	D₄	1	6	yes	—
16	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₈₉	D₄	1	6	yes	—
18	C₄×C₈×C₈₉	C₂³	1	6	yes	—
19	C₂³×D₈₉	—	4	4	no	—
20	C₂×D₃₅₆	—	2	24	no	—
21	C₂×D₃₅₆	—	4	48	no	—
22	C₂×D₃₅₆	—	4	12	no	—
23	D₇₁₂	—	4	96	no	—
24	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
25	C₂×C₈×C₈₉	D₈	1	24	yes	complemented
26	C₂×C₈×C₈₉	C₂×D₄	1	12	yes	—
27	C₄²×C₈₉	C₂×D₄	1	3	yes	—
28	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
29	C₂³×C₈₉	#127	1	1	yes	characteristic
30	C₂²×D₈₉	—	4	8	no	—
31	C₂²×D₈₉	—	8	24	no	—
32	C₂×C₄×C₈₉	#127	1	3	yes	—
33	C₂×C₄×C₈₉	C₂×D₈	1	12	yes	—
34	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
35	D₃₅₆	—	4	48	no	—
36	D₃₅₆	—	8	24	no	—
37	C₈×C₈₉	C₂×D₈	1	24	yes	complemented
38	C₂²×C₈₉	#493	1	6	yes	—
39	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
40	C₂×D₈₉	—	8	24	no	—
41	C₂×D₈₉	—	16	16	no	—
42	C₄×C₈₉	#493	1	6	yes	—
43	C₄×C₈₉	C₂²×D₈	1	6	yes	—
44	C₂×#1712	—	89	1	no	maximal, Sylow
45	C₂×C₈₉	#1712	1	4	yes	complemented
46	C₂×C₈₉	C₂×#493	1	3	yes	—
47	D₈₉	—	16	8	no	—
48	C₂×#493	—	89	6	no	—
49	#1712	—	89	8	no	—
50	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
51	C₈₉	C₂×#1712	1	1	yes	characteristic, complemented, Sylow
52	C₂²×D₈	—	178	12	no	—
53	C₂×#127	—	89	4	no	—
54	#493	—	89	24	no	—
55	#493	—	178	12	no	—
56	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
57	C₈²	C₂×D₈₉	1	4	yes	complemented
58	C₂²×D₄	—	178	12	no	—
59	C₂×D₈	—	356	48	no	—
60	C₂×D₈	—	178	48	no	—
61	C₂²×C₈	D₃₅₆	1	6	yes	—
62	#127	—	89	8	no	—
63	#127	—	178	12	no	—
64	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
65	C₄×C₈	D₃₅₆	1	6	yes	—
66	C₄×C₈	C₂²×D₈₉	1	6	yes	—
67	C₂⁴	—	356	4	no	—
68	C₂×D₄	—	356	12	no	—
69	C₂×D₄	—	356	48	no	—
70	C₂×D₄	—	178	24	no	—
71	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
72	D₈	—	356	96	no	—
73	C₂×C₈	D₇₁₂	1	24	yes	complemented
74	C₂×C₈	C₂×D₃₅₆	1	12	yes	—
75	C₄²	C₂×D₃₅₆	1	3	yes	—
76	C₄²	C₂³×D₈₉	1	1	yes	characteristic
77	C₂³	—	356	8	no	—
78	C₂³	—	712	24	no	—
79	C₂³	#15668009	1	1	yes	characteristic
80	D₄	—	356	48	no	—
81	D₄	—	712	24	no	—
82	C₂×C₄	#15668009	1	3	yes	—
83	C₂×C₄	C₂×D₇₁₂	1	12	yes	—
84	C₂×C₄	C₂²×D₃₅₆	1	3	yes	—
85	C₈	C₂×D₇₁₂	1	24	yes	complemented
86	C₂²	—	1424	16	no	—
87	C₂²	—	712	24	no	—
88	C₂²	#19151975	1	6	yes	—
89	C₂²	C₂×#15668009	1	1	yes	characteristic
90	C₄	#19151975	1	6	yes	—
91	C₄	C₂²×D₇₁₂	1	6	yes	—
92	C₂	—	1424	8	no	—
93	C₂	#26019867	1	4	yes	direct product factor
94	C₂	C₂×#19151975	1	3	yes	—

	subgroup	G/H
center	C₂³	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂³×C₈₉	#127
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	C₂×#1712
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. C₂×#19151975	C₂	semidirect
2. #26019867	C₂	direct (N × H)
3. C₂×C₈²×C₈₉	C₂	semidirect
7. #19151975	C₂²	semidirect
9. C₈²×C₈₉	C₂²	semidirect
25. C₂×C₈×C₈₉	D₈	semidirect
37. C₈×C₈₉	C₂×D₈	semidirect
45. C₂×C₈₉	#1712	semidirect
50. C₂×C₈²	D₈₉	semidirect
51. C₈₉	C₂×#1712	semidirect
57. C₈²	C₂×D₈₉	semidirect
73. C₂×C₈	D₇₁₂	semidirect
85. C₈	C₂×D₇₁₂	semidirect
93. C₂	#26019867	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19151975	2
#26019867	2
C₂×C₈²×C₈₉	2
C₂×#1712	89

Subgroups — C2×#26019867 #36542119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019867 #36542119