Subgroups — C₂×#72220578 #72341352

Order 3072.

At least 6004 conjugacy classes of subgroups; only the 264 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×#11711332	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14694⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11715572	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11711331	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#1020⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×#1020	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#1020⋊C₂	C₂	1	1	yes	maximal, characteristic
8	#72220578	C₂	1	8	yes	maximal, direct product factor
9	C₂³×#1020	C₂²	1	1	yes	characteristic
10	C₂³×#1020	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1020	C₂²	1	4	yes	complemented
12	C₂²×#5936	C₂²	1	1	yes	characteristic, complemented
13	C₂²×#5936	C₂²	1	1	yes	characteristic, complemented
14	#11715572	C₂²	1	4	yes	complemented
15	C₂²×#1020⋊C₂	C₄	1	8	yes	complemented
16	C₂²×#1020⋊C₂	C₂²	1	4	yes	—
17	C₂×#1020⋊C₄	C₂²	1	4	yes	complemented
18	#11711332	C₂²	1	4	yes	complemented
19	#11711331	C₂²	1	4	yes	complemented
20	C₂²×C₄×C₂²⋊A₄	C₂²	1	1	yes	characteristic, complemented
21	C₂³×#120⋊C₃	C₂²	1	1	yes	characteristic
22	C₂²×#379⋊C₃	C₂²	1	1	yes	characteristic, complemented
23	#14694⋊C₃	C₂²	1	4	yes	complemented
24	C₂×#14694	D₃	1	1	yes	characteristic, complemented
25	C₂³×C₂²⋊A₄	C₂³	1	1	yes	characteristic
26	C₂²×#1020	C₂×C₄	1	4	yes	—
27	C₂²×#1020	C₂×C₄	1	4	yes	complemented
28	C₂²×#1020	D₄	1	4	yes	—
29	C₂²×#1020	D₄	1	4	yes	—
30	C₂²×#1020	C₂³	1	2	yes	—
31	C₂²×#1020	C₂³	1	2	yes	—
32	C₂×C₄×C₂²⋊A₄	C₂³	1	2	yes	complemented
33	C₂×#5936	C₂³	1	2	yes	complemented
34	C₂×#5936	C₂³	1	2	yes	complemented
35	C₂²×#120⋊C₃	C₂×C₄	1	4	yes	complemented
36	C₂²×#120⋊C₃	C₂³	1	2	yes	—
37	C₂×#1020⋊C₂	C₂×C₄	1	16	yes	complemented
38	C₂×#379⋊C₃	C₂³	1	2	yes	complemented
39	C₂⁶×C₄	C₂×D₃	1	1	yes	characteristic, complemented
40	C₂³×#120	C₂×D₃	1	1	yes	characteristic
41	C₂²×#379	C₂×D₃	1	1	yes	characteristic, complemented
42	#14694	C₂×D₃	1	4	yes	complemented
43	C₂²×C₂²⋊A₄	C₂²×C₄	1	2	yes	—
44	C₂²×C₂²⋊A₄	C₂×D₄	1	2	yes	—
45	C₂²×C₂²⋊A₄	C₂×D₄	1	2	yes	—
46	C₂²×C₂²⋊A₄	C₂⁴	1	1	yes	characteristic
47	C₂×#1020	C₂²⋊C₄	1	16	yes	complemented
48	C₂×#1020	C₂²×C₄	1	4	yes	—
49	C₂×#1020	C₂²×C₄	1	4	yes	complemented
50	C₂×#1020	C₂×D₄	1	4	yes	—
51	C₂×#1020	C₂×D₄	1	4	yes	—
52	C₂×#120⋊C₃	C₂²×C₄	1	4	yes	complemented
53	C₂⁷	C₂²×D₃	1	1	yes	characteristic
54	C₂⁵×C₄	C₂²×D₃	1	2	yes	complemented
55	C₂²×#120	C₄×D₃	1	4	yes	complemented
56	C₂²×#120	C₂²×D₃	1	2	yes	—
57	C₂³×C₂²⋊C₄	S₄	1	1	yes	characteristic, complemented
58	C₂×#379	C₂²×D₃	1	2	yes	complemented
59	C₂×C₂²⋊A₄	C₂×C₂²⋊C₄	1	4	yes	complemented
60	C₂×C₂²⋊A₄	C₂³×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊A₄	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊A₄	C₂²×D₄	1	1	yes	characteristic
63	#1020	C₂×C₂²⋊C₄	1	8	yes	complemented
64	C₂⁶	C₂×C₄×D₃	1	2	yes	—
65	C₂⁶	D₃×D₄	1	2	yes	—
66	C₂⁶	D₃×D₄	1	2	yes	—
67	C₂⁶	C₂×S₄	1	1	yes	characteristic
68	C₂⁶	C₂³×D₃	1	1	yes	characteristic
69	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
70	C₂⁴×C₄	C₂×S₄	1	1	yes	characteristic
71	C₂×#120	C₂×C₄×D₃	1	4	yes	complemented
72	C₂²×C₂²⋊C₄	C₂×S₄	1	4	yes	complemented
73	C₂²⋊A₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented
74	C₂⁵	D₃×C₂²⋊C₄	1	4	yes	complemented
75	C₂⁵	C₄×S₄	1	4	yes	complemented
76	C₂⁵	C₂²×C₄×D₃	1	1	yes	characteristic
77	C₂⁵	C₂×D₃×D₄	1	1	yes	characteristic
78	C₂⁵	C₂×D₃×D₄	1	1	yes	characteristic
79	C₂⁵	C₂²×S₄	1	2	yes	—
80	C₂⁵	C₂²×S₄	1	1	yes	characteristic
81	C₂³×C₄	C₂²×S₄	1	2	yes	—
82	C₂³×C₄	C₂²×S₄	1	2	yes	—
83	C₂⁴	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
84	C₂⁴	C₂×C₄×S₄	1	2	yes	—
85	C₂⁴	C₂×C₄×S₄	1	4	yes	complemented
86	C₂⁴	D₄×S₄	1	2	yes	—
87	C₂⁴	D₄×S₄	1	2	yes	—
88	C₂⁴	C₂³×S₄	1	1	yes	characteristic
89	C₂²×C₄	#1020⋊C₂	1	1	yes	characteristic, complemented
90	C₂³	C₂²⋊C₄×S₄	1	4	yes	—
91	C₂³	C₂×#1020⋊C₂	1	1	yes	characteristic
92	C₂³	C₂²×C₄×S₄	1	1	yes	characteristic
93	C₂³	C₂×D₄×S₄	1	1	yes	characteristic
94	C₂³	C₂×D₄×S₄	1	1	yes	characteristic
95	C₂×C₄	C₂×#1020⋊C₂	1	2	yes	complemented
96	C₂²	C₄×#1020⋊C₂	1	2	yes	—
97	C₂²	#11713200	1	2	yes	—
98	C₂²	#11713271	1	2	yes	—
99	C₂²	C₂×C₂²⋊C₄×S₄	1	1	yes	characteristic
100	C₂²	C₂²×#1020⋊C₂	1	1	yes	characteristic
101	C₂	C₂×#11713200	1	1	yes	characteristic
102	C₂	#72220578	1	4	yes	direct product factor
103	C₂	C₂×C₄×#1020⋊C₂	1	1	yes	characteristic
104	C₂	C₂×#11713271	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1020⋊C₂
commutator	C₂×C₂²⋊A₄	C₂³×C₄
Frattini	C₂⁴	C₂³×S₄
Fitting	C₂×#14694	D₃
socle	C₂⁵	C₂²×S₄
prime radical	C₂³×#1020⋊C₂	C₂
cosocle	C₂²×C₂²⋊A₄	C₂⁴
hypercenter	C₂²×C₄	#1020⋊C₂
hypocenter	C₂²⋊A₄	C₂²×C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11711332	C₂	semidirect
2. C₂×#14694⋊C₃	C₂	semidirect
3. C₂×#11715572	C₂	semidirect
4. C₂×#11711331	C₂	semidirect
5. C₂²×#1020⋊C₄	C₂	semidirect
6. C₂²×C₄×#1020	C₂	semidirect
8. #72220578	C₂	direct (N × H)
11. C₂×C₄×#1020	C₂²	semidirect
12. C₂²×#5936	C₂²	semidirect
13. C₂²×#5936	C₂²	semidirect
14. #11715572	C₂²	semidirect
15. C₂²×#1020⋊C₂	C₄	semidirect
17. C₂×#1020⋊C₄	C₂²	semidirect
18. #11711332	C₂²	semidirect
19. #11711331	C₂²	semidirect
20. C₂²×C₄×C₂²⋊A₄	C₂²	semidirect
22. C₂²×#379⋊C₃	C₂²	semidirect
23. #14694⋊C₃	C₂²	semidirect
24. C₂×#14694	D₃	semidirect
27. C₂²×#1020	C₂×C₄	semidirect
32. C₂×C₄×C₂²⋊A₄	C₂³	semidirect
33. C₂×#5936	C₂³	semidirect
34. C₂×#5936	C₂³	semidirect
35. C₂²×#120⋊C₃	C₂×C₄	semidirect
37. C₂×#1020⋊C₂	C₂×C₄	semidirect
38. C₂×#379⋊C₃	C₂³	semidirect
39. C₂⁶×C₄	C₂×D₃	semidirect
41. C₂²×#379	C₂×D₃	semidirect
42. #14694	C₂×D₃	semidirect
47. C₂×#1020	C₂²⋊C₄	semidirect
49. C₂×#1020	C₂²×C₄	semidirect
52. C₂×#120⋊C₃	C₂²×C₄	semidirect
54. C₂⁵×C₄	C₂²×D₃	semidirect
55. C₂²×#120	C₄×D₃	semidirect
57. C₂³×C₂²⋊C₄	S₄	semidirect
58. C₂×#379	C₂²×D₃	semidirect
59. C₂×C₂²⋊A₄	C₂×C₂²⋊C₄	semidirect
63. #1020	C₂×C₂²⋊C₄	semidirect
71. C₂×#120	C₂×C₄×D₃	semidirect
72. C₂²×C₂²⋊C₄	C₂×S₄	semidirect
73. C₂²⋊A₄	C₂²×C₂²⋊C₄	semidirect
74. C₂⁵	D₃×C₂²⋊C₄	semidirect
75. C₂⁵	C₄×S₄	semidirect
83. C₂⁴	C₂×D₃×C₂²⋊C₄	semidirect
85. C₂⁴	C₂×C₄×S₄	semidirect
89. C₂²×C₄	#1020⋊C₂	semidirect
95. C₂×C₄	C₂×#1020⋊C₂	semidirect
102. C₂	#72220578	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×#11711332	2
C₂×#14694⋊C₃	2
C₂×#11715572	2
C₂×#11711331	2
C₂×#364513	3
C₂²×#1020⋊C₄	2
C₂²×C₄×#1020	2
C₂³×#1020⋊C₂	2
#72220578	2
C₂×C₂²⋊C₄×S₄	4

Subgroups — C2×#72220578 #72341352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72220578 #72341352