Subgroups — #72341785

Order 3072.

At least 6220 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

The subgroup lattice has 118 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72341777	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341771	C₂	1	1	yes	maximal, characteristic, complemented
3	#72341767	C₂	1	1	yes	maximal, characteristic, complemented
4	#72341762	C₂	1	1	yes	maximal, characteristic, complemented
5	#72341646	C₂	1	1	yes	maximal, characteristic, complemented
6	#72341599	C₂	1	1	yes	maximal, characteristic, complemented
7	#72341595	C₂	1	1	yes	maximal, characteristic, complemented
8	#72341578	C₂	1	1	yes	maximal, characteristic, complemented
9	#72341558	C₂	1	1	yes	maximal, characteristic, complemented
10	#72341550	C₂	1	1	yes	maximal, characteristic, complemented
11	#72341462	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11715827	C₂	1	1	yes	maximal, characteristic, complemented
13	#72245678	C₂	1	1	yes	maximal, characteristic, complemented
14	#72241395	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#11714162	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#87519	C₂²	1	1	yes	characteristic, complemented
17	C₂×#87518	C₂²	1	1	yes	characteristic, complemented
18	C₂×#87522	C₂²	1	1	yes	characteristic
19	C₂×#87516	C₂²	1	1	yes	characteristic
20	C₂×#87516	C₂²	1	1	yes	characteristic, complemented
21	C₂×#87520	C₂²	1	1	yes	characteristic
22	C₂×#87520	C₂²	1	1	yes	characteristic, complemented
23	#11715719	C₂²	1	1	yes	characteristic, complemented
24	#11715718	C₂²	1	1	yes	characteristic, complemented
25	#11715717	C₂²	1	1	yes	characteristic
26	#11715716	C₂²	1	1	yes	characteristic, complemented
27	#11715657	C₂²	1	1	yes	characteristic, complemented
28	C₂×#73031	C₂²	1	1	yes	characteristic, complemented
29	C₂×#73025	C₂²	1	1	yes	characteristic, complemented
30	C₂×#73030	C₂²	1	1	yes	characteristic, complemented
31	C₂×#73026	C₂²	1	1	yes	characteristic, complemented
32	#11711637	C₂²	1	1	yes	characteristic, complemented
33	#11711636	C₂²	1	1	yes	characteristic, complemented
34	#11711635	C₂²	1	1	yes	characteristic, complemented
35	#11711634	C₂²	1	1	yes	characteristic, complemented
36	#11711633	C₂²	1	1	yes	characteristic, complemented
37	#11711632	C₂²	1	1	yes	characteristic, complemented
38	#11711631	C₂²	1	1	yes	characteristic, complemented
39	#11711630	C₂²	1	1	yes	characteristic, complemented
40	#11711595	C₂²	1	1	yes	characteristic, complemented
41	#11711595	C₂²	1	1	yes	characteristic, complemented
42	#11711594	C₂²	1	1	yes	characteristic, complemented
43	#11711594	C₂²	1	1	yes	characteristic, complemented
44	C₂²×#5950	C₂²	1	1	yes	characteristic, complemented
45	C₂×#3025⋊C₃	C₂²	1	1	yes	characteristic, complemented
46	#32901⋊C₃	C₂²	1	1	yes	characteristic, complemented
47	#32900⋊C₃	C₂²	1	1	yes	characteristic, complemented
48	#32899⋊C₃	C₂²	1	1	yes	characteristic
49	#32898⋊C₃	C₂²	1	1	yes	characteristic, complemented
50	#20608⋊C₃	C₂²	1	1	yes	characteristic, complemented
51	#6628891	D₃	1	1	yes	characteristic, complemented
52	C₂×#5950	C₂³	1	1	yes	characteristic
53	C₂×#5950	C₂³	1	1	yes	characteristic
54	C₂²×#997	C₂³	1	1	yes	characteristic
55	#87520	D₄	1	4	yes	complemented
56	C₂×#5935	C₂³	1	1	yes	characteristic
57	C₂×#5935	C₂³	1	1	yes	characteristic
58	C₂×#5934	C₂³	1	1	yes	characteristic
59	C₂×#5933	C₂³	1	1	yes	characteristic
60	C₂×#5933	C₂³	1	1	yes	characteristic
61	C₂×#5932	C₂³	1	1	yes	characteristic
62	#85564	C₂³	1	1	yes	characteristic
63	#85564	C₂³	1	1	yes	characteristic
64	C₂×#458⋊C₃	C₂³	1	1	yes	characteristic
65	C₂×B(4,𝔽₂)⋊C₃	C₂³	1	1	yes	characteristic
66	#1790⋊C₃	C₂³	1	1	yes	characteristic
67	#1790⋊C₃	C₂³	1	1	yes	characteristic
68	C₂²×#585	C₂×D₃	1	1	yes	characteristic, complemented
69	C₂×#3025	C₂×D₃	1	1	yes	characteristic
70	#32901	C₂×D₃	1	1	yes	characteristic, complemented
71	#32900	C₂×D₃	1	1	yes	characteristic, complemented
72	#32899	C₂×D₃	1	1	yes	characteristic
73	#32898	C₂×D₃	1	1	yes	characteristic
74	#20608	C₂×D₃	1	1	yes	characteristic, complemented
75	#5950	C₂×D₄	1	2	yes	complemented
76	C₂×#997	C₂⁴	1	1	yes	characteristic
77	#5935	C₂×D₄	1	2	yes	—
78	#5933	C₂×D₄	1	2	yes	complemented
79	C₂×#585	C₂²×D₃	1	1	yes	characteristic
80	C₂×#585	C₂²×D₃	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
82	C₂²×C₄×D₄	S₄	1	1	yes	characteristic, complemented
83	C₂×#458	C₂²×D₃	1	1	yes	characteristic
84	C₂×B(4,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
85	#1790	C₂²×D₃	1	1	yes	characteristic
86	#1790	C₂²×D₃	1	1	yes	characteristic
87	#997	C₂²×D₄	1	1	yes	characteristic
88	#585	D₃×D₄	1	2	yes	—
89	C₂×H(5,𝔽₂)	C₂³×D₃	1	1	yes	characteristic
90	C₂³×D₄	C₂×S₄	1	1	yes	characteristic
91	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
92	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
93	C₂²×C₄⋊C₄	C₂×S₄	1	1	yes	characteristic
94	C₂²×C₂²⋊C₄	C₂×S₄	1	1	yes	characteristic
95	C₂²×C₂²⋊C₄	C₂×S₄	1	1	yes	characteristic
96	C₂²×C₄²	C₂×S₄	1	1	yes	characteristic
97	C₂⁵	C₂²×S₄	1	1	yes	characteristic
98	C₂⁵	C₂²×S₄	1	1	yes	characteristic
99	H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
100	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
101	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
102	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
103	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
104	C₂³×C₄	C₂²×S₄	1	1	yes	characteristic
105	C₂⁴	C₂³×S₄	1	1	yes	characteristic
106	C₂²×C₄	#1020⋊C₂	1	1	yes	characteristic
107	C₂²×C₄	D₄×S₄	1	2	yes	—
108	C₂³	C₂×#1020⋊C₂	1	1	yes	characteristic
109	C₂³	C₂×D₄×S₄	1	1	yes	characteristic
110	C₂×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
111	C₂×C₄	C₂×#1020⋊C₂	1	1	yes	characteristic
112	C₂²	C₂²×#1020⋊C₂	1	1	yes	characteristic
113	C₄	#11713200	1	2	yes	—
114	C₂	C₂×#11713200	1	1	yes	characteristic
115	C₂	#72233471	1	1	yes	characteristic
116	C₂	#72236838	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1020⋊C₂
commutator, cosocle	C₂×#997	C₂⁴
Frattini	C₂⁴	C₂³×S₄
Fitting	#6628891	D₃
socle	C₂²	C₂²×#1020⋊C₂
hypercenter	C₂²×C₄	#1020⋊C₂
hypocenter	#997	C₂²×D₄
LCS	G ▷ C₂×#997 ▷ #997
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #72341777	C₂	semidirect
2. #72341771	C₂	semidirect
3. #72341767	C₂	semidirect
4. #72341762	C₂	semidirect
5. #72341646	C₂	semidirect
6. #72341599	C₂	semidirect
7. #72341595	C₂	semidirect
8. #72341578	C₂	semidirect
9. #72341558	C₂	semidirect
10. #72341550	C₂	semidirect
11. #72341462	C₂	semidirect
12. C₂×#11715827	C₂	semidirect
13. #72245678	C₂	semidirect
14. #72241395	C₂	semidirect
15. C₂×#11714162	C₂	semidirect
16. C₂×#87519	C₂²	semidirect
17. C₂×#87518	C₂²	semidirect
20. C₂×#87516	C₂²	semidirect
22. C₂×#87520	C₂²	semidirect
23. #11715719	C₂²	semidirect
24. #11715718	C₂²	semidirect
26. #11715716	C₂²	semidirect
27. #11715657	C₂²	semidirect
28. C₂×#73031	C₂²	semidirect
29. C₂×#73025	C₂²	semidirect
30. C₂×#73030	C₂²	semidirect
31. C₂×#73026	C₂²	semidirect
32. #11711637	C₂²	semidirect
33. #11711636	C₂²	semidirect
34. #11711635	C₂²	semidirect
35. #11711634	C₂²	semidirect
36. #11711633	C₂²	semidirect
37. #11711632	C₂²	semidirect
38. #11711631	C₂²	semidirect
39. #11711630	C₂²	semidirect
40. #11711595	C₂²	semidirect
41. #11711595	C₂²	semidirect
42. #11711594	C₂²	semidirect
43. #11711594	C₂²	semidirect
44. C₂²×#5950	C₂²	semidirect
45. C₂×#3025⋊C₃	C₂²	semidirect
46. #32901⋊C₃	C₂²	semidirect
47. #32900⋊C₃	C₂²	semidirect
49. #32898⋊C₃	C₂²	semidirect
50. #20608⋊C₃	C₂²	semidirect
51. #6628891	D₃	semidirect
55. #87520	D₄	semidirect
68. C₂²×#585	C₂×D₃	semidirect
70. #32901	C₂×D₃	semidirect
71. #32900	C₂×D₃	semidirect
74. #20608	C₂×D₃	semidirect
75. #5950	C₂×D₄	semidirect
78. #5933	C₂×D₄	semidirect
82. C₂²×C₄×D₄	S₄	semidirect

Maximal subgroups

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

subgroup	index
#72341777	2
#72341771	2
#72341767	2
#72341762	2
#72341646	2
#72341599	2
#72341595	2
#72341578	2
#72341558	2
#72341550	2
#72341462	2
#72303239	3
C₂×#11715827	2
#72245678	2
#72241395	2
C₂×#11714162	2
#11714331	4