Subgroups — C₂×#24915858 #35610498

Order 21248.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14581789	C₂	1	14	yes	maximal, complemented
2	#24915858	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄³×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₃₂	C₂²	1	28	yes	—
5	C₂×#14581789	—	2	112	no	—
6	C₂×#14581789	C₂²	1	112	yes	complemented
7	C₂²×C₄²×C₈₃	C₂²	1	7	yes	—
8	C₄³×C₈₃	C₂²	1	8	yes	complemented
9	C₂⁴×D₈₃	C₂³	1	8	yes	—
10	C₂²×D₃₃₂	—	2	336	no	—
11	C₂²×D₃₃₂	C₂³	1	112	yes	—
12	#14581789	—	2	448	no	—
13	C₂³×C₄×C₈₃	C₂³	1	7	yes	—
14	C₂×C₄²×C₈₃	D₄	1	56	yes	complemented
15	C₂×C₄²×C₈₃	C₂³	1	28	yes	—
16	C₂³×D₈₃	—	2	112	no	—
17	C₂³×D₈₃	C₂⁴	1	16	yes	—
18	C₂×D₃₃₂	—	2	672	no	—
19	C₂×D₃₃₂	—	4	448	no	—
20	C₂⁴×C₈₃	C₂⁴	1	1	yes	characteristic
21	C₂²×C₄×C₈₃	C₂×D₄	1	84	yes	—
22	C₂²×C₄×C₈₃	C₂⁴	1	14	yes	—
23	C₄²×C₈₃	C₂×D₄	1	112	yes	complemented
24	C₂³×C₈₃	C₂²×D₄	1	14	yes	—
25	C₂³×C₈₃	C₂⁵	1	1	yes	characteristic
26	C₂²×D₈₃	—	2	112	no	—
27	C₂²×D₈₃	—	4	224	no	—
28	C₂×C₄×C₈₃	#127	1	112	yes	complemented
29	C₂×C₄×C₈₃	C₂²×D₄	1	84	yes	—
30	D₃₃₂	—	4	448	no	—
31	C₂²×C₈₃	C₂×#127	1	28	yes	—
32	C₂²×C₈₃	C₂³×D₄	1	7	yes	—
33	C₂×D₈₃	—	4	112	no	—
34	C₂×D₈₃	—	8	64	no	—
35	C₄×C₈₃	C₂×#127	1	56	yes	complemented
36	C₂×#2867	—	83	1	no	maximal, Sylow
37	C₂×C₈₃	#2867	1	8	yes	complemented
38	C₂×C₈₃	C₂²×#127	1	7	yes	—
39	D₈₃	—	8	16	no	—
40	C₂²×#127	—	83	14	no	—
41	#2867	—	83	16	no	—
42	C₂×C₄³	D₈₃	1	1	yes	characteristic, complemented
43	C₈₃	C₂×#2867	1	1	yes	characteristic, complemented, Sylow
44	C₂³×D₄	—	83	28	no	—
45	C₂×#127	—	83	112	no	—
46	C₂×#127	—	166	112	no	—
47	C₂²×C₄²	C₂×D₈₃	1	7	yes	—
48	C₄³	C₂×D₈₃	1	8	yes	complemented
49	C₂⁵	—	83	8	no	—
50	C₂²×D₄	—	83	112	no	—
51	C₂²×D₄	—	166	336	no	—
52	C₂³×C₄	C₂²×D₈₃	1	7	yes	—
53	#127	—	166	448	no	—
54	C₂×C₄²	D₃₃₂	1	56	yes	complemented
55	C₂×C₄²	C₂²×D₈₃	1	28	yes	—
56	C₂⁴	—	83	16	no	—
57	C₂⁴	—	166	112	no	—
58	C₂⁴	C₂³×D₈₃	1	1	yes	characteristic
59	C₂×D₄	—	332	448	no	—
60	C₂×D₄	—	166	672	no	—
61	C₂²×C₄	C₂×D₃₃₂	1	84	yes	—
62	C₂²×C₄	C₂³×D₈₃	1	14	yes	—
63	C₄²	C₂×D₃₃₂	1	112	yes	complemented
64	C₂³	—	332	224	no	—
65	C₂³	—	166	112	no	—
66	C₂³	C₂²×D₃₃₂	1	14	yes	—
67	C₂³	C₂⁴×D₈₃	1	1	yes	characteristic
68	D₄	—	332	448	no	—
69	C₂×C₄	#14581789	1	112	yes	complemented
70	C₂×C₄	C₂²×D₃₃₂	1	84	yes	—
71	C₂²	—	332	112	no	—
72	C₂²	—	664	64	no	—
73	C₂²	C₂×#14581789	1	28	yes	—
74	C₂²	C₂³×D₃₃₂	1	7	yes	—
75	C₄	C₂×#14581789	1	56	yes	complemented
76	C₂	—	664	16	no	—
77	C₂	#24915858	1	8	yes	direct product factor
78	C₂	C₂²×#14581789	1	7	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₃
commutator, cosocle	C₂³×C₈₃	C₂⁵
Frattini	C₂³	C₂⁴×D₈₃
Fitting	C₂×C₄³×C₈₃	C₂
socle	C₂⁴×C₈₃	C₂⁴
hypercenter	C₂×C₄³	D₈₃
hypocenter	C₈₃	C₂×#2867
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂²×#14581789	C₂	semidirect
2. #24915858	C₂	direct (N × H)
3. C₂×C₄³×C₈₃	C₂	semidirect
6. C₂×#14581789	C₂²	semidirect
8. C₄³×C₈₃	C₂²	semidirect
14. C₂×C₄²×C₈₃	D₄	semidirect
23. C₄²×C₈₃	C₂×D₄	semidirect
28. C₂×C₄×C₈₃	#127	semidirect
35. C₄×C₈₃	C₂×#127	semidirect
37. C₂×C₈₃	#2867	semidirect
42. C₂×C₄³	D₈₃	semidirect
43. C₈₃	C₂×#2867	semidirect
48. C₄³	C₂×D₈₃	semidirect
54. C₂×C₄²	D₃₃₂	semidirect
63. C₄²	C₂×D₃₃₂	semidirect
69. C₂×C₄	#14581789	semidirect
75. C₄	C₂×#14581789	semidirect
77. C₂	#24915858	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×#14581789	2
#24915858	2
C₂×C₄³×C₈₃	2
C₂×#2867	83

Subgroups — C2×#24915858 #35610498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24915858 #35610498