Subgroups — C₂×#20269996 #27213925

Order 13504.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809961	C₂	1	2	yes	maximal, complemented
2	#20269996	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16809950	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#16809961	—	2	4	no	—
8	#16809961	C₂²	1	8	yes	complemented
9	C₂×Dic₄₂₂	C₂²	1	2	yes	—
10	C₂×C₂²⋊D₂₁₁	—	2	2	no	—
11	C₂²×Dic₂₁₁	—	2	2	no	—
12	C₂×C₄×D₂₁₁	C₂²	1	1	yes	characteristic
13	Dic₈₄₄	C₂²	1	4	yes	complemented
14	#16809950	C₂²	1	8	yes	complemented
15	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₈×D₂₁₁	C₂²	1	4	yes	complemented
17	C₂×D₄×C₂₁₁	C₂²	1	2	yes	—
18	D₈×C₂₁₁	C₂²	1	4	yes	complemented
19	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
20	C₂²×D₂₁₁	D₄	1	1	yes	characteristic
21	Dic₄₂₂	—	2	2	no	—
22	Dic₄₂₂	C₂³	1	4	yes	—
23	C₂²⋊D₂₁₁	—	2	8	no	—
24	C₂×Dic₂₁₁	—	2	2	no	—
25	C₂×Dic₂₁₁	—	2	8	no	—
26	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
27	C₄×D₂₁₁	D₄	1	2	yes	—
28	C₄×D₂₁₁	C₂³	1	2	yes	—
29	C₂₁₁⋊C₈	C₂³	1	2	yes	—
30	C₂³×C₂₁₁	—	2	2	no	—
31	D₄×C₂₁₁	—	2	2	no	—
32	D₄×C₂₁₁	C₂³	1	4	yes	—
33	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
34	C₈×C₂₁₁	C₂³	1	2	yes	—
35	C₂²×C₂₁₁	—	2	4	no	—
36	C₂²×C₂₁₁	—	4	4	no	—
37	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×D₂₁₁	—	2	2	no	—
39	C₂×D₂₁₁	C₂×D₄	1	2	yes	—
40	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
41	C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
42	Dic₂₁₁	—	2	4	no	—
43	Dic₂₁₁	C₂×D₄	1	2	yes	—
44	C₂×C₂₁₁	—	4	4	no	—
45	C₂×C₂₁₁	#135	1	2	yes	complemented
46	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
47	D₂₁₁	—	2	2	no	—
48	C₂₁₁	C₂×#135	1	1	yes	characteristic, complemented, Sylow
49	C₂×#135	—	211	1	no	maximal, Sylow
50	C₂×#41	—	211	2	no	—
51	#135	—	211	8	no	—
52	C₂×Q₁₆	—	211	1	no	—
53	C₂×SD₁₆	—	211	2	no	—
54	C₂×D₈	D₂₁₁	1	1	yes	characteristic, complemented
55	C₂²×C₈	—	211	1	no	—
56	#41	—	422	4	no	—
57	#41	—	211	8	no	—
58	C₂×Q₈	—	211	2	no	—
59	C₂×D₄	—	422	2	no	—
60	C₂×D₄	C₂×D₂₁₁	1	2	yes	—
61	C₂²×C₄	—	422	2	no	—
62	C₂²×C₄	—	211	1	no	—
63	Q₁₆	—	211	4	no	—
64	SD₁₆	—	211	8	no	—
65	D₈	C₂×D₂₁₁	1	4	yes	complemented
66	C₂×C₈	—	211	1	no	—
67	C₂×C₈	—	211	4	no	—
68	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
69	C₂³	—	2	2	no	—
70	C₂³	—	211	1	no	—
71	Q₈	—	422	2	no	—
72	Q₈	—	211	4	no	—
73	D₄	—	2	2	no	—
74	D₄	—	422	8	no	—
75	D₄	C₂²×D₂₁₁	1	4	yes	—
76	C₂×C₄	—	422	2	no	—
77	C₂×C₄	—	422	8	no	—
78	C₂×C₄	—	211	1	no	—
79	C₂×C₄	—	211	2	no	—
80	C₂×C₄	—	211	2	no	—
81	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
82	C₈	—	211	2	no	—
83	C₈	C₂²×D₂₁₁	1	2	yes	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	4	no	—
86	C₂²	—	422	2	no	—
87	C₂²	—	211	2	no	—
88	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
89	C₄	—	422	4	no	—
90	C₄	—	211	2	no	—
91	C₄	D₄×D₂₁₁	1	1	yes	characteristic
92	C₄	C₂³×D₂₁₁	1	1	yes	characteristic
93	C₂	—	4	4	no	—
94	C₂	—	422	2	no	—
95	C₂	#20269996	1	2	yes	direct product factor
96	C₂	C₂×D₄×D₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₄×C₂₁₁	C₂⁴
Frattini	C₄	C₂³×D₂₁₁
Fitting	C₂×D₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
hypercenter	C₂×D₈	D₂₁₁
hypocenter	C₂₁₁	C₂×#135
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16809961	C₂	semidirect
2. #20269996	C₂	direct (N × H)
3. C₂×Dic₈₄₄	C₂	semidirect
4. C₂×#16809950	C₂	semidirect
5. C₂×C₈×D₂₁₁	C₂	semidirect
6. C₂×D₈×C₂₁₁	C₂	semidirect
8. #16809961	C₂²	semidirect
13. Dic₈₄₄	C₂²	semidirect
14. #16809950	C₂²	semidirect
16. C₈×D₂₁₁	C₂²	semidirect
18. D₈×C₂₁₁	C₂²	semidirect
45. C₂×C₂₁₁	#135	semidirect
48. C₂₁₁	C₂×#135	semidirect
54. C₂×D₈	D₂₁₁	semidirect
65. D₈	C₂×D₂₁₁	semidirect
95. C₂	#20269996	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×#16809961	2
#20269996	2
C₂×Dic₈₄₄	2
C₂×#16809950	2
C₂×C₈×D₂₁₁	2
C₂×D₈×C₂₁₁	2
C₂×#135	211

Subgroups — C2×#20269996 #27213925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20269996 #27213925