Subgroups — #72343854

Order 6144.

At least 17889 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

The subgroup lattice has 121 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72343791	C₂	1	1	yes	maximal, characteristic, complemented
2	#72343784	C₂	1	4	yes	maximal, complemented
3	#72343782	C₂	1	4	yes	maximal, complemented
4	#72343720	C₂	1	1	yes	maximal, characteristic, complemented
5	#72343269	C₂	1	1	yes	maximal, characteristic, complemented
6	#72343231	C₂²	1	4	yes	complemented
7	C₂×#72215741	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11715710	C₂²	1	2	yes	complemented
9	C₂²×#11714174	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#87522	C₂²	1	1	yes	characteristic
11	#72341994	C₂²	1	4	yes	complemented
12	#72341757	C₂	1	1	yes	maximal, characteristic, complemented
13	#72341710	C₂²	1	4	yes	complemented
14	#72341681	C₂²	1	2	yes	complemented
15	C₂×#72236755	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#3027⋊C₃	C₂²	1	1	yes	characteristic
17	C₂×#32894⋊C₃	C₂²	1	2	yes	complemented
18	C₂×#32895⋊C₃	C₂²	1	2	yes	complemented
19	C₂×#20612⋊C₃	C₂²	1	1	yes	characteristic, complemented
20	#72341492	C₂²	1	1	yes	characteristic, complemented
21	C₂×#11715708	C₂²	1	2	yes	complemented
22	C₂×#11715667	C₂²	1	1	yes	characteristic, complemented
23	C₂×#11710937	C₂²	1	1	yes	characteristic, complemented
24	C₂×#6628885	D₃	1	1	yes	characteristic, complemented
25	C₂²×#87519	C₂²	1	1	yes	characteristic
26	C₂×#11714174	C₂²	1	4	yes	complemented
27	#72227692	C₂²	1	2	yes	complemented
28	C₂×#87519	D₄	1	2	yes	complemented
29	C₂×#87519	C₂³	1	2	yes	—
30	C₂²×#5932	D₄	1	1	yes	characteristic, complemented
31	C₂²×#5934	D₄	1	1	yes	characteristic, complemented
32	C₂×#87522	D₄	1	4	yes	complemented
33	C₂×#87522	C₂³	1	2	yes	—
34	C₂×#85567	C₂³	1	2	yes	complemented
35	#11715659	C₂³	1	2	yes	complemented
36	C₂×#87548	C₂³	1	1	yes	characteristic, complemented
37	C₂×#87547	C₂³	1	1	yes	characteristic, complemented
38	C₂²×#5950	C₂³	1	1	yes	characteristic
39	C₂×#3027⋊C₃	D₄	1	4	yes	complemented
40	C₂×#3027⋊C₃	C₂³	1	2	yes	—
41	C₂×#1791⋊C₃	C₂³	1	2	yes	complemented
42	C₂×#63103	C₂×D₃	1	1	yes	characteristic, complemented
43	C₂²×#3027	C₂×D₃	1	1	yes	characteristic
44	C₂×#32895	C₂×D₃	1	2	yes	complemented
45	C₂×#32894	C₂×D₃	1	2	yes	complemented
46	C₂×#20612	C₂×D₃	1	1	yes	characteristic, complemented
47	C₂×#5950	C₂×D₄	1	2	yes	—
48	C₂×#5950	C₂×D₄	1	1	yes	characteristic
49	C₂×#5950	C₂×D₄	1	2	yes	—
50	C₂×#5950	C₂⁴	1	1	yes	characteristic
51	C₂²×#997	C₂×D₄	1	1	yes	characteristic
52	#87522	C₂×D₄	1	4	yes	complemented
53	C₂×#5934	C₂×D₄	1	2	yes	complemented
54	C₂×#5932	C₂×D₄	1	2	yes	complemented
55	#3027⋊C₃	C₂×D₄	1	4	yes	complemented
56	C₂×2₋¹⁺⁶	C₂²×D₃	1	1	yes	characteristic, complemented
57	C₂×H(7,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
58	C₂²×#585	C₂²×D₃	1	1	yes	characteristic
59	C₂³×AGL(1,ℤ/8)	S₄	1	1	yes	characteristic, complemented
60	C₂×#3027	#67	1	4	yes	complemented
61	C₂×#3027	C₂²×D₃	1	2	yes	—
62	C₂×#1791	C₂²×D₃	1	2	yes	complemented
63	#5950	#120	1	2	yes	—
64	#5950	C₂²×D₄	1	1	yes	characteristic
65	#5950	C₂²×D₄	1	1	yes	characteristic
66	C₂×#997	#120	1	2	yes	—
67	C₂×#997	C₂²×D₄	1	1	yes	characteristic
68	C₂×#585	D₃×D₄	1	2	yes	—
69	C₂×#585	D₃×D₄	1	1	yes	characteristic
70	C₂×#585	C₂×#67	1	2	yes	—
71	C₂×#585	C₂³×D₃	1	1	yes	characteristic
72	C₂²×H(5,𝔽₂)	D₃×D₄	1	1	yes	characteristic
73	C₂³×#41	C₂×S₄	1	1	yes	characteristic
74	C₂⁴×D₄	C₂×S₄	1	1	yes	characteristic
75	C₂³×SD₁₆	C₂×S₄	1	2	yes	complemented
76	C₂³×D₈	C₂×S₄	1	2	yes	complemented
77	C₂³×M₁₆	C₂×S₄	1	1	yes	characteristic
78	#3027	C₂×#67	1	4	yes	—
79	#997	C₂×#120	1	1	yes	characteristic
80	#585	#937	1	2	yes	—
81	#585	C₂×D₃×D₄	1	1	yes	characteristic
82	#585	C₂²×#67	1	1	yes	characteristic
83	C₂×H(5,𝔽₂)	#937	1	2	yes	—
84	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
85	C₂³×Q₈	C₂²×S₄	1	1	yes	characteristic
86	C₂³×D₄	GL(2,ℤ/4)	1	4	yes	complemented
87	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
88	C₂³×D₄	C₂²×S₄	1	2	yes	—
89	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
90	C₂³×C₈	C₂²×S₄	1	2	yes	—
91	C₂⁵	D₄×S₄	1	1	yes	characteristic
92	H(5,𝔽₂)	C₂×#937	1	1	yes	characteristic
93	C₂×#41	#1020⋊C₂	1	1	yes	characteristic
94	C₂²×D₄	C₂×GL(2,ℤ/4)	1	4	yes	—
95	C₂³×C₄	D₄×S₄	1	2	yes	—
96	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
97	C₂³×C₄	C₂×GL(2,ℤ/4)	1	2	yes	—
98	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
99	C₂⁴	#85493	1	2	yes	—
100	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
101	C₂×Q₈	C₂×#1020⋊C₂	1	1	yes	characteristic
102	C₂×D₄	C₂×#1020⋊C₂	1	1	yes	characteristic
103	C₂²×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
104	C₂²×C₄	#85493	1	2	yes	—
105	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
106	C₂²×C₄	C₂²×GL(2,ℤ/4)	1	1	yes	characteristic
107	C₂³	#11713200	1	1	yes	characteristic
108	C₂³	C₂×#85493	1	1	yes	characteristic
109	C₂×C₄	#11713200	1	2	yes	—
110	C₂×C₄	#11713200	1	1	yes	characteristic
111	C₂×C₄	#11713201	1	2	yes	—
112	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
113	C₂²	C₂×#11713200	1	1	yes	characteristic
114	C₂²	#72220367	1	2	yes	—
115	C₄	C₂×#11713200	1	1	yes	characteristic
116	C₄	#72220367	1	2	yes	—
117	C₄	C₂×#11713201	1	1	yes	characteristic
118	C₂	#72240998	1	2	yes	—
119	C₂	C₂×#72220367	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#11713200
commutator, cosocle	C₂×#5950	C₂⁴
Frattini	C₂³×C₄	C₂³×S₄
Fitting	C₂×#6628885	D₃
hypercenter	C₂×#41	#1020⋊C₂
hypocenter	#997	C₂×#120
LCS	G ▷ C₂×#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. #72343791	C₂	semidirect
2. #72343784	C₂	semidirect
3. #72343782	C₂	semidirect
4. #72343720	C₂	semidirect
5. #72343269	C₂	semidirect
6. #72343231	C₂²	semidirect
7. C₂×#72215741	C₂	semidirect
8. C₂×#11715710	C₂²	semidirect
9. C₂²×#11714174	C₂	semidirect
11. #72341994	C₂²	semidirect
12. #72341757	C₂	semidirect
13. #72341710	C₂²	semidirect
14. #72341681	C₂²	semidirect
15. C₂×#72236755	C₂	semidirect
17. C₂×#32894⋊C₃	C₂²	semidirect
18. C₂×#32895⋊C₃	C₂²	semidirect
19. C₂×#20612⋊C₃	C₂²	semidirect
20. #72341492	C₂²	semidirect
21. C₂×#11715708	C₂²	semidirect
22. C₂×#11715667	C₂²	semidirect
23. C₂×#11710937	C₂²	semidirect
24. C₂×#6628885	D₃	semidirect
26. C₂×#11714174	C₂²	semidirect
27. #72227692	C₂²	semidirect
28. C₂×#87519	D₄	semidirect
30. C₂²×#5932	D₄	semidirect
31. C₂²×#5934	D₄	semidirect
32. C₂×#87522	D₄	semidirect
34. C₂×#85567	C₂³	semidirect
35. #11715659	C₂³	semidirect
36. C₂×#87548	C₂³	semidirect
37. C₂×#87547	C₂³	semidirect
39. C₂×#3027⋊C₃	D₄	semidirect
41. C₂×#1791⋊C₃	C₂³	semidirect
42. C₂×#63103	C₂×D₃	semidirect
44. C₂×#32895	C₂×D₃	semidirect
45. C₂×#32894	C₂×D₃	semidirect
46. C₂×#20612	C₂×D₃	semidirect
52. #87522	C₂×D₄	semidirect
53. C₂×#5934	C₂×D₄	semidirect
54. C₂×#5932	C₂×D₄	semidirect
55. #3027⋊C₃	C₂×D₄	semidirect
56. C₂×2₋¹⁺⁶	C₂²×D₃	semidirect
57. C₂×H(7,𝔽₂)	C₂²×D₃	semidirect
59. C₂³×AGL(1,ℤ/8)	S₄	semidirect
60. C₂×#3027	#67	semidirect
62. C₂×#1791	C₂²×D₃	semidirect
75. C₂³×SD₁₆	C₂×S₄	semidirect
76. C₂³×D₈	C₂×S₄	semidirect
86. C₂³×D₄	GL(2,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
#72343791	2
#72343784	2
#72343782	2
#72343720	2
#72343269	2
C₂×#72215741	2
C₂²×#11714174	2
#72341757	2
C₂×#72236755	2
#72337509	3
#72258866	4