Subgroups — C₂×#72215741 #72343089

Order 3072.

At least 12513 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

The subgroup lattice has 106 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11715710	C₂	1	2	yes	maximal, complemented
2	C₂²×#87522	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11715708	C₂	1	2	yes	maximal, complemented
4	C₂×#11715667	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11710937	C₂	1	1	yes	maximal, characteristic, complemented
6	#72215741	C₂	1	8	yes	maximal, direct product factor
7	C₂×#87522	C₂²	1	2	yes	complemented
8	C₂×#87522	C₂²	1	4	yes	complemented
9	#11715710	C₂²	1	8	yes	complemented
10	#11715708	C₂²	1	8	yes	complemented
11	#11715667	C₂²	1	4	yes	complemented
12	C₂×#85567	C₂²	1	2	yes	complemented
13	#11710937	C₂²	1	4	yes	complemented
14	C₂×#87548	C₂²	1	1	yes	characteristic, complemented
15	C₂×#87547	C₂²	1	1	yes	characteristic, complemented
16	C₂²×#5950	C₂²	1	1	yes	characteristic
17	C₂×#63103	D₃	1	1	yes	characteristic, complemented
18	#87548	C₂³	1	2	yes	complemented
19	#87547	C₂³	1	2	yes	complemented
20	C₂×#5950	D₄	1	1	yes	characteristic, complemented
21	C₂×#5950	D₄	1	2	yes	complemented
22	C₂×#5950	C₂³	1	1	yes	characteristic
23	C₂×#5950	C₂³	1	2	yes	—
24	C₂²×#997	D₄	1	1	yes	characteristic, complemented
25	#87522	C₂³	1	4	yes	complemented
26	#85567	C₂³	1	4	yes	complemented
27	#63103	C₂×D₃	1	4	yes	complemented
28	C₂×2₋¹⁺⁶	C₂×D₃	1	1	yes	characteristic, complemented
29	C₂×H(7,𝔽₂)	C₂×D₃	1	1	yes	characteristic, complemented
30	C₂²×#585	C₂×D₃	1	1	yes	characteristic
31	#5950	C₂×D₄	1	2	yes	complemented
32	#5950	C₂×D₄	1	1	yes	characteristic, complemented
33	#5950	C₂⁴	1	1	yes	characteristic
34	C₂×#997	C₂×D₄	1	1	yes	characteristic, complemented
35	C₂×#997	C₂×D₄	1	2	yes	complemented
36	2₋¹⁺⁶	C₂²×D₃	1	2	yes	complemented
37	H(7,𝔽₂)	C₂²×D₃	1	2	yes	complemented
38	C₂×#585	#67	1	1	yes	characteristic
39	C₂×#585	#67	1	2	yes	—
40	C₂×#585	S₄	1	2	yes	complemented
41	C₂×#585	C₂²×D₃	1	1	yes	characteristic
42	C₂×#585	C₂²×D₃	1	2	yes	—
43	C₂²×H(5,𝔽₂)	#67	1	1	yes	characteristic
44	C₂³×#41	S₄	1	1	yes	characteristic
45	#997	C₂²×D₄	1	1	yes	characteristic, complemented
46	#585	C₂×#67	1	2	yes	—
47	#585	C₂×#67	1	1	yes	characteristic
48	#585	C₂×S₄	1	8	yes	complemented
49	#585	C₂³×D₃	1	1	yes	characteristic
50	C₂×2₋¹⁺⁴	C₂×S₄	1	2	yes	complemented
51	C₂×H(5,𝔽₂)	C₂×#67	1	1	yes	characteristic
52	C₂×H(5,𝔽₂)	C₂×#67	1	2	yes	—
53	C₂×H(5,𝔽₂)	C₂×S₄	1	2	yes	complemented
54	C₂²×#41	C₂×S₄	1	4	yes	—
55	C₂²×#41	C₂×S₄	1	2	yes	—
56	C₂³×Q₈	C₂×S₄	1	1	yes	characteristic
57	C₂³×D₄	C₂×S₄	1	1	yes	characteristic
58	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
59	C₂⁵	GL(2,ℤ/4)	1	1	yes	characteristic
60	2₋¹⁺⁴	C₂²×S₄	1	4	yes	complemented
61	H(5,𝔽₂)	C₂²×#67	1	1	yes	characteristic
62	H(5,𝔽₂)	C₂²×S₄	1	4	yes	complemented
63	C₂×#41	GL(2,ℤ/4)	1	2	yes	—
64	C₂×#41	GL(2,ℤ/4)	1	4	yes	—
65	C₂×#41	C₂²×S₄	1	2	yes	—
66	C₂×#41	C₂²×S₄	1	4	yes	—
67	C₂×#41	#1020	1	1	yes	characteristic
68	C₂²×Q₈	GL(2,ℤ/4)	1	2	yes	—
69	C₂²×Q₈	C₂²×S₄	1	2	yes	—
70	C₂²×D₄	C₂²×S₄	1	2	yes	—
71	C₂³×C₄	GL(2,ℤ/4)	1	1	yes	characteristic
72	C₂³×C₄	GL(2,ℤ/4)	1	2	yes	—
73	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
74	C₂³×C₄	C₂²×S₄	1	2	yes	—
75	C₂⁴	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
76	C₂⁴	C₂×GL(2,ℤ/4)	1	2	yes	—
77	#41	C₂×GL(2,ℤ/4)	1	4	yes	—
78	#41	C₂×GL(2,ℤ/4)	1	2	yes	—
79	#41	C₂³×S₄	1	2	yes	—
80	#41	C₂×#1020	1	4	yes	—
81	C₂×Q₈	C₂×GL(2,ℤ/4)	1	2	yes	—
82	C₂×Q₈	C₂×GL(2,ℤ/4)	1	4	yes	—
83	C₂×Q₈	C₂×#1020	1	1	yes	characteristic
84	C₂×D₄	C₂×#1020	1	1	yes	characteristic
85	C₂²×C₄	C₂×GL(2,ℤ/4)	1	2	yes	—
86	C₂²×C₄	C₂×GL(2,ℤ/4)	1	1	yes	characteristic
87	C₂²×C₄	C₂³×S₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×#1020	1	1	yes	characteristic
89	C₂³	C₂²×GL(2,ℤ/4)	1	1	yes	characteristic
90	C₂³	#87524	1	1	yes	characteristic
91	Q₈	C₂²×GL(2,ℤ/4)	1	2	yes	—
92	Q₈	C₂²×#1020	1	2	yes	—
93	D₄	C₂²×#1020	1	2	yes	—
94	C₂×C₄	#87524	1	2	yes	—
95	C₂×C₄	#87524	1	1	yes	characteristic
96	C₂×C₄	C₂²×#1020	1	2	yes	—
97	C₂×C₄	C₂²×#1020	1	1	yes	characteristic
98	C₂²	C₂×#87524	1	2	yes	—
99	C₂²	C₂×#87524	1	1	yes	characteristic
100	C₄	C₂×#87524	1	2	yes	—
101	C₄	C₂×#87524	1	1	yes	characteristic
102	C₄	C₂³×#1020	1	1	yes	characteristic
103	C₂	#72215741	1	2	yes	direct product factor
104	C₂	C₂²×#87524	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#87524
commutator, cosocle	#5950	C₂⁴
Frattini	C₄	C₂³×#1020
Fitting	C₂×#63103	D₃
hypercenter	C₂×#41	#1020
hypocenter	#997	C₂²×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#11715710	C₂	semidirect
2. C₂²×#87522	C₂	semidirect
3. C₂×#11715708	C₂	semidirect
4. C₂×#11715667	C₂	semidirect
5. C₂×#11710937	C₂	semidirect
6. #72215741	C₂	direct (N × H)
7. C₂×#87522	C₂²	semidirect
8. C₂×#87522	C₂²	semidirect
9. #11715710	C₂²	semidirect
10. #11715708	C₂²	semidirect
11. #11715667	C₂²	semidirect
12. C₂×#85567	C₂²	semidirect
13. #11710937	C₂²	semidirect
14. C₂×#87548	C₂²	semidirect
15. C₂×#87547	C₂²	semidirect
17. C₂×#63103	D₃	semidirect
18. #87548	C₂³	semidirect
19. #87547	C₂³	semidirect
20. C₂×#5950	D₄	semidirect
21. C₂×#5950	D₄	semidirect
24. C₂²×#997	D₄	semidirect
25. #87522	C₂³	semidirect
26. #85567	C₂³	semidirect
27. #63103	C₂×D₃	semidirect
28. C₂×2₋¹⁺⁶	C₂×D₃	semidirect
29. C₂×H(7,𝔽₂)	C₂×D₃	semidirect
31. #5950	C₂×D₄	semidirect
32. #5950	C₂×D₄	semidirect
34. C₂×#997	C₂×D₄	semidirect
35. C₂×#997	C₂×D₄	semidirect
36. 2₋¹⁺⁶	C₂²×D₃	semidirect
37. H(7,𝔽₂)	C₂²×D₃	semidirect
40. C₂×#585	S₄	semidirect
45. #997	C₂²×D₄	semidirect
48. #585	C₂×S₄	semidirect
50. C₂×2₋¹⁺⁴	C₂×S₄	semidirect
53. C₂×H(5,𝔽₂)	C₂×S₄	semidirect
60. 2₋¹⁺⁴	C₂²×S₄	semidirect
62. H(5,𝔽₂)	C₂²×S₄	semidirect
103. C₂	#72215741	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×#11715710	2
C₂²×#87522	2
C₂×#11715708	2
C₂×#11715667	2
C₂×#11710937	2
C₂×#6628885	3
#72215741	2
C₂×#85543	4
C₂×#85513	4

Subgroups — C2×#72215741 #72343089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72215741 #72343089