Subgroups — #72343675

Order 6144.

Only the 298 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 130 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72343642	C₂	1	1	yes	maximal, characteristic, complemented
2	#72343425	C₂	1	1	yes	maximal, characteristic, complemented
3	#72343406	C₂	1	4	yes	maximal, complemented
4	#72343391	C₂	1	4	yes	maximal, complemented
5	#72341845	C₂²	1	4	yes	complemented
6	#72341757	C₂	1	1	yes	maximal, characteristic, complemented
7	#72341716	C₂²	1	4	yes	complemented
8	#72341681	C₂²	1	2	yes	complemented
9	#72341614	C₂	1	1	yes	maximal, characteristic, complemented
10	#72341610	C₂²	1	4	yes	complemented
11	C₂×#72236781	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11715709	C₂²	1	2	yes	complemented
13	C₂×#11715711	C₂²	1	2	yes	complemented
14	C₂×#11715661	C₂²	1	1	yes	characteristic, complemented
15	#72341492	C₂²	1	1	yes	characteristic, complemented
16	C₂×#72215685	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×#11710873	C₂²	1	1	yes	characteristic, complemented
18	C₂×#11710899	C₂²	1	2	yes	complemented
19	C₂²×#11715839	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂²×#87521	C₂²	1	1	yes	characteristic
21	C₂×#11710937	C₂²	1	1	yes	characteristic, complemented
22	C₂²×#87547	C₂²	1	1	yes	characteristic
23	C₂×#10586999	D₃	1	1	yes	characteristic, complemented
24	C₂²×#87519	C₂²	1	1	yes	characteristic
25	#72227692	C₂²	1	2	yes	complemented
26	C₂×#11710898	C₂²	1	2	yes	complemented
27	C₂×#11715839	C₂²	1	4	yes	complemented
28	C₂×#87521	D₄	1	4	yes	complemented
29	C₂×#87521	C₂³	1	2	yes	—
30	C₂×#87519	D₄	1	2	yes	—
31	C₂×#87519	C₂³	1	2	yes	—
32	C₂²×#5932	D₄	1	1	yes	characteristic
33	C₂²×#5934	D₄	1	1	yes	characteristic
34	C₂×#85566	C₂³	1	2	yes	complemented
35	#11715659	C₂³	1	2	yes	complemented
36	C₂×#87548	C₂³	1	1	yes	characteristic, complemented
37	C₂×#87547	D₄	1	4	yes	complemented
38	C₂×#87547	C₂³	1	2	yes	—
39	C₂×#87547	C₂³	1	1	yes	characteristic, complemented
40	C₂²×#5950	C₂³	1	1	yes	characteristic
41	C₂×#85666	C₂³	1	2	yes	complemented
42	C₂×#63103	C₂×D₃	1	1	yes	characteristic, complemented
43	C₂²×H(7,𝔽₂)	C₂×D₃	1	1	yes	characteristic
44	C₂×#63092	C₂×D₃	1	2	yes	complemented
45	C₂×#63091	C₂×D₃	1	2	yes	complemented
46	C₂×#63076	C₂×D₃	1	1	yes	characteristic, complemented
47	#87547	C₂×D₄	1	4	yes	complemented
48	C₂×#5950	C₂×D₄	1	2	yes	—
49	C₂×#5950	C₂×D₄	1	1	yes	characteristic
50	C₂×#5950	C₂×D₄	1	2	yes	—
51	C₂×#5950	C₂⁴	1	1	yes	characteristic
52	C₂²×#997	C₂×D₄	1	1	yes	characteristic
53	#87521	C₂×D₄	1	4	yes	complemented
54	C₂×#5934	C₂×D₄	1	2	yes	—
55	C₂×#5932	C₂×D₄	1	2	yes	—
56	C₂×2₋¹⁺⁶	C₂²×D₃	1	1	yes	characteristic, complemented
57	C₂×H(7,𝔽₂)	#67	1	4	yes	complemented
58	C₂×H(7,𝔽₂)	C₂²×D₃	1	2	yes	—
59	C₂×H(7,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
60	C₂²×#585	C₂²×D₃	1	1	yes	characteristic
61	C₂³×AGL(1,ℤ/8)	S₄	1	3	yes	complemented
62	C₂×#3573	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	H(7,𝔽₂)	C₂×#67	1	4	yes	—
69	C₂×#585	D₃×D₄	1	2	yes	—
70	C₂×#585	D₃×D₄	1	1	yes	characteristic
71	C₂×#585	C₂×#67	1	2	yes	—
72	C₂×#585	C₂³×D₃	1	1	yes	characteristic
73	C₂²×H(5,𝔽₂)	D₃×D₄	1	1	yes	characteristic
74	C₂³×#41	C₂×S₄	1	3	yes	complemented
75	C₂⁴×D₄	C₂×S₄	1	3	yes	—
76	C₂³×SD₁₆	C₂×S₄	1	6	yes	complemented
77	C₂³×D₈	C₂×S₄	1	6	yes	complemented
78	C₂³×M₁₆	C₂×S₄	1	3	yes	complemented
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	3	yes	complemented
86	C₂³×D₄	GL(2,ℤ/4)	1	12	yes	complemented
87	C₂³×D₄	C₂²×S₄	1	6	yes	—
88	C₂³×D₄	C₂²×S₄	1	3	yes	complemented
89	C₂⁴×C₄	C₂²×S₄	1	3	yes	—
90	C₂×AGL(1,ℤ/8)	#1020	1	1	yes	characteristic
91	C₂³×C₈	C₂²×S₄	1	6	yes	complemented
92	C₂⁵	D₄×S₄	1	3	yes	—
93	H(5,𝔽₂)	C₂×#937	1	1	yes	characteristic
94	C₂×#41	C₂×#1020	1	1	yes	characteristic
95	C₂²×D₄	C₂×GL(2,ℤ/4)	1	12	yes	—
96	C₂²×D₄	C₂×#1020	1	1	yes	characteristic
97	C₂³×C₄	D₄×S₄	1	6	yes	—
98	C₂³×C₄	D₄×S₄	1	3	yes	—
99	C₂³×C₄	C₂×GL(2,ℤ/4)	1	6	yes	—
100	C₂³×C₄	C₂³×S₄	1	3	yes	—
101	C₂×SD₁₆	C₂×#1020	1	2	yes	—
102	C₂×D₈	C₂×#1020	1	2	yes	—
103	C₂×M₁₆	C₂×#1020	1	1	yes	characteristic
104	C₂⁴	#85493	1	6	yes	—
105	C₂⁴	C₂×D₄×S₄	1	3	yes	—
106	C₂×Q₈	C₂²×#1020	1	1	yes	characteristic
107	C₂×D₄	#87524	1	4	yes	—
108	C₂×D₄	C₂²×#1020	1	1	yes	characteristic
109	C₂×D₄	C₂²×#1020	1	2	yes	—
110	C₂²×C₄	#85493	1	6	yes	—
111	C₂²×C₄	C₂×D₄×S₄	1	3	yes	—
112	C₂²×C₄	C₂²×GL(2,ℤ/4)	1	3	yes	—
113	C₂²×C₄	C₂²×#1020	1	1	yes	characteristic
114	C₂×C₈	C₂²×#1020	1	2	yes	—
115	C₂³	C₂×#85493	1	3	yes	—
116	C₂³	D₄×#1020	1	1	yes	characteristic
117	D₄	C₂×#87524	1	4	yes	—
118	C₂×C₄	D₄×#1020	1	1	yes	characteristic
119	C₂×C₄	D₄×#1020	1	2	yes	—
120	C₂×C₄	C₂×#87524	1	2	yes	—
121	C₂×C₄	C₂³×#1020	1	1	yes	characteristic
122	C₂²	C₂×D₄×#1020	1	1	yes	characteristic
123	C₂²	#72216669	1	2	yes	—
124	C₄	C₂×D₄×#1020	1	1	yes	characteristic
125	C₄	#72216669	1	2	yes	—
126	C₄	C₂²×#87524	1	1	yes	characteristic
127	C₂	#72240994	1	2	yes	—
128	C₂	C₂×#72216669	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄×#1020
commutator, cosocle	C₂×#5950	C₂⁴
Frattini	C₂×C₄	C₂³×#1020
Fitting	C₂×#10586999	D₃
hypercenter	C₂×AGL(1,ℤ/8)	#1020
hypocenter	#997	C₂×#120
LCS	G ▷ C₂×#5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #72343642	C₂	semidirect
2. #72343425	C₂	semidirect
3. #72343406	C₂	semidirect
4. #72343391	C₂	semidirect
5. #72341845	C₂²	semidirect
6. #72341757	C₂	semidirect
7. #72341716	C₂²	semidirect
8. #72341681	C₂²	semidirect
9. #72341614	C₂	semidirect
10. #72341610	C₂²	semidirect
11. C₂×#72236781	C₂	semidirect
12. C₂×#11715709	C₂²	semidirect
13. C₂×#11715711	C₂²	semidirect
14. C₂×#11715661	C₂²	semidirect
15. #72341492	C₂²	semidirect
16. C₂×#72215685	C₂	semidirect
17. C₂×#11710873	C₂²	semidirect
18. C₂×#11710899	C₂²	semidirect
19. C₂²×#11715839	C₂	semidirect
21. C₂×#11710937	C₂²	semidirect
23. C₂×#10586999	D₃	semidirect
25. #72227692	C₂²	semidirect
26. C₂×#11710898	C₂²	semidirect
27. C₂×#11715839	C₂²	semidirect
28. C₂×#87521	D₄	semidirect
34. C₂×#85566	C₂³	semidirect
35. #11715659	C₂³	semidirect
36. C₂×#87548	C₂³	semidirect
37. C₂×#87547	D₄	semidirect
39. C₂×#87547	C₂³	semidirect
41. C₂×#85666	C₂³	semidirect
42. C₂×#63103	C₂×D₃	semidirect
44. C₂×#63092	C₂×D₃	semidirect
45. C₂×#63091	C₂×D₃	semidirect
46. C₂×#63076	C₂×D₃	semidirect
47. #87547	C₂×D₄	semidirect
53. #87521	C₂×D₄	semidirect
56. C₂×2₋¹⁺⁶	C₂²×D₃	semidirect
57. C₂×H(7,𝔽₂)	#67	semidirect
59. C₂×H(7,𝔽₂)	C₂²×D₃	semidirect
61. C₂³×AGL(1,ℤ/8)	S₄	semidirect
62. C₂×#3573	C₂²×D₃	semidirect
74. C₂³×#41	C₂×S₄	semidirect
76. C₂³×SD₁₆	C₂×S₄	semidirect
77. C₂³×D₈	C₂×S₄	semidirect
78. C₂³×M₁₆	C₂×S₄	semidirect
85. C₂³×Q₈	C₂²×S₄	semidirect
86. C₂³×D₄	GL(2,ℤ/4)	semidirect
88. C₂³×D₄	C₂²×S₄	semidirect
91. C₂³×C₈	C₂²×S₄	semidirect

Maximal subgroups

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

subgroup	index
#72343642	2
#72343425	2
#72343406	2
#72343391	2
#72341757	2
#72341614	2
C₂×#72236781	2
C₂×#72215685	2
#72312410	3
C₂²×#11715839	2
#72258866	4