Subgroups — C₂×#26028647 #36760522

Order 22784.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15668009	C₂	1	14	yes	maximal, complemented
2	#26028647	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₅₆	C₂²	1	28	yes	—
5	C₂×#15668009	—	2	112	no	—
6	C₂×#15668009	C₂²	1	112	yes	complemented
7	C₂²×C₄²×C₈₉	C₂²	1	7	yes	—
8	C₄³×C₈₉	C₂²	1	8	yes	complemented
9	C₂⁴×D₈₉	C₂³	1	8	yes	—
10	C₂²×D₃₅₆	—	2	336	no	—
11	C₂²×D₃₅₆	C₂³	1	112	yes	—
12	#15668009	—	2	448	no	—
13	C₂³×C₄×C₈₉	C₂³	1	7	yes	—
14	C₂×C₄²×C₈₉	D₄	1	56	yes	complemented
15	C₂×C₄²×C₈₉	C₂³	1	28	yes	—
16	C₂³×D₈₉	—	2	112	no	—
17	C₂³×D₈₉	C₂⁴	1	16	yes	—
18	C₂×D₃₅₆	—	2	672	no	—
19	C₂×D₃₅₆	—	4	448	no	—
20	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	C₂×D₄	1	84	yes	—
22	C₂²×C₄×C₈₉	C₂⁴	1	14	yes	—
23	C₄²×C₈₉	C₂×D₄	1	112	yes	complemented
24	C₂³×C₈₉	C₂²×D₄	1	14	yes	—
25	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
26	C₂²×D₈₉	—	2	112	no	—
27	C₂²×D₈₉	—	4	224	no	—
28	C₂×C₄×C₈₉	#127	1	112	yes	complemented
29	C₂×C₄×C₈₉	C₂²×D₄	1	84	yes	—
30	D₃₅₆	—	4	448	no	—
31	C₂²×C₈₉	C₂×#127	1	28	yes	—
32	C₂²×C₈₉	C₂³×D₄	1	7	yes	—
33	C₂×D₈₉	—	4	112	no	—
34	C₂×D₈₉	—	8	64	no	—
35	C₄×C₈₉	C₂×#127	1	56	yes	complemented
36	C₂×#2867	—	89	1	no	maximal, Sylow
37	C₂×C₈₉	#2867	1	8	yes	complemented
38	C₂×C₈₉	C₂²×#127	1	7	yes	—
39	D₈₉	—	8	16	no	—
40	C₂²×#127	—	89	14	no	—
41	#2867	—	89	16	no	—
42	C₂×C₄³	D₈₉	1	1	yes	characteristic, complemented
43	C₈₉	C₂×#2867	1	1	yes	characteristic, complemented, Sylow
44	C₂³×D₄	—	89	28	no	—
45	C₂×#127	—	89	112	no	—
46	C₂×#127	—	178	112	no	—
47	C₂²×C₄²	C₂×D₈₉	1	7	yes	—
48	C₄³	C₂×D₈₉	1	8	yes	complemented
49	C₂⁵	—	89	8	no	—
50	C₂²×D₄	—	89	112	no	—
51	C₂²×D₄	—	178	336	no	—
52	C₂³×C₄	C₂²×D₈₉	1	7	yes	—
53	#127	—	178	448	no	—
54	C₂×C₄²	D₃₅₆	1	56	yes	complemented
55	C₂×C₄²	C₂²×D₈₉	1	28	yes	—
56	C₂⁴	—	89	16	no	—
57	C₂⁴	—	178	112	no	—
58	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
59	C₂×D₄	—	356	448	no	—
60	C₂×D₄	—	178	672	no	—
61	C₂²×C₄	C₂×D₃₅₆	1	84	yes	—
62	C₂²×C₄	C₂³×D₈₉	1	14	yes	—
63	C₄²	C₂×D₃₅₆	1	112	yes	complemented
64	C₂³	—	356	224	no	—
65	C₂³	—	178	112	no	—
66	C₂³	C₂²×D₃₅₆	1	14	yes	—
67	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
68	D₄	—	356	448	no	—
69	C₂×C₄	#15668009	1	112	yes	complemented
70	C₂×C₄	C₂²×D₃₅₆	1	84	yes	—
71	C₂²	—	356	112	no	—
72	C₂²	—	712	64	no	—
73	C₂²	C₂×#15668009	1	28	yes	—
74	C₂²	C₂³×D₃₅₆	1	7	yes	—
75	C₄	C₂×#15668009	1	56	yes	complemented
76	C₂	—	712	16	no	—
77	C₂	#26028647	1	8	yes	direct product factor
78	C₂	C₂²×#15668009	1	7	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×#2867
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂²×#15668009	C₂	semidirect
2. #26028647	C₂	direct (N × H)
3. C₂×C₄³×C₈₉	C₂	semidirect
6. C₂×#15668009	C₂²	semidirect
8. C₄³×C₈₉	C₂²	semidirect
14. C₂×C₄²×C₈₉	D₄	semidirect
23. C₄²×C₈₉	C₂×D₄	semidirect
28. C₂×C₄×C₈₉	#127	semidirect
35. C₄×C₈₉	C₂×#127	semidirect
37. C₂×C₈₉	#2867	semidirect
42. C₂×C₄³	D₈₉	semidirect
43. C₈₉	C₂×#2867	semidirect
48. C₄³	C₂×D₈₉	semidirect
54. C₂×C₄²	D₃₅₆	semidirect
63. C₄²	C₂×D₃₅₆	semidirect
69. C₂×C₄	#15668009	semidirect
75. C₄	C₂×#15668009	semidirect
77. C₂	#26028647	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15668009	2
#26028647	2
C₂×C₄³×C₈₉	2
C₂×#2867	89

Subgroups — C2×#26028647 #36760522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028647 #36760522