Subgroups — #72341808

Order 6144.

At least 6098 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

The subgroup lattice has 124 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72341790	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341783	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72236850	C₂	1	1	yes	maximal, characteristic
4	#72341764	C₂	1	1	yes	maximal, characteristic, complemented
5	#72341755	C₂	1	1	yes	maximal, characteristic, complemented
6	#72341747	C₂	1	2	yes	maximal, complemented
7	#72341742	C₂	1	2	yes	maximal, complemented
8	C₂×#11711638	C₂²	1	2	yes	—
9	#72341731	C₂²	1	2	yes	complemented
10	#72341722	C₂	1	2	yes	maximal, complemented
11	#72341716	C₂²	1	2	yes	complemented
12	#72341710	C₂²	1	2	yes	complemented
13	#72341702	C₂²	1	2	yes	complemented
14	#72341690	C₂	1	1	yes	maximal, characteristic, complemented
15	#72341685	C₂²	1	2	yes	complemented
16	C₂×#11715736	C₂²	1	1	yes	characteristic
17	C₂×#32948⋊C₃	C₂²	1	1	yes	characteristic
18	#72341665	C₂²	1	1	yes	characteristic, complemented
19	#72341657	C₂²	1	1	yes	characteristic, complemented
20	C₄×#11714184	C₂	1	2	yes	maximal, complemented
21	#72341357	C₂²	1	2	yes	complemented
22	C₄×#3027⋊C₃	C₂²	1	2	yes	complemented
23	#72341338	C₂²	1	2	yes	—
24	#72341273	C₂²	1	2	yes	complemented
25	#72281182	D₃	1	1	yes	characteristic, complemented
26	C₄×#87521	C₂²	1	2	yes	complemented
27	#72236850	C₄	1	8	yes	complemented
28	C₂×#11714184	C₂²	1	2	yes	—
29	#72227323	C₂	1	1	yes	maximal, characteristic, complemented
30	#72227321	C₂²	1	1	yes	characteristic, complemented
31	C₄×#87520	C₂²	1	1	yes	characteristic
32	#72227309	C₂²	1	1	yes	characteristic, complemented
33	#72227309	C₂²	1	1	yes	characteristic, complemented
34	#72227308	C₂²	1	1	yes	characteristic, complemented
35	C₂×#11715681	C₂²	1	1	yes	characteristic
36	C₂×#11715681	C₂²	1	1	yes	characteristic
37	C₂×#87521	C₂³	1	2	yes	—
38	C₂×#87520	C₂³	1	1	yes	characteristic
39	#11715736	C₂×C₄	1	4	yes	complemented
40	#11715681	C₂×C₄	1	4	yes	complemented
41	C₄×#5933	D₄	1	2	yes	—
42	C₄×#5935	D₄	1	2	yes	—
43	C₂×#85567	C₂³	1	1	yes	characteristic
44	C₂×#85567	C₂³	1	1	yes	characteristic
45	#11715663	C₂³	1	1	yes	characteristic
46	#11715659	C₂³	1	2	yes	—
47	#11714184	C₂×C₄	1	8	yes	complemented
48	#11711638	C₂×C₄	1	8	yes	complemented
49	C₂×#85666	C₂³	1	1	yes	characteristic
50	C₂×#85666	C₂³	1	1	yes	characteristic
51	C₄×#5950	C₂³	1	1	yes	characteristic
52	C₂×#3027⋊C₃	C₂³	1	2	yes	—
53	#32948⋊C₃	C₂×C₄	1	4	yes	complemented
54	#20610⋊C₃	C₂³	1	2	yes	—
55	#10574037	C₂×D₃	1	1	yes	characteristic, complemented
56	C₂×#32948	C₂×D₃	1	1	yes	characteristic
57	C₄×#3027	C₂×D₃	1	2	yes	complemented
58	#392717	C₂×D₃	1	2	yes	complemented
59	#392195	C₂×D₃	1	1	yes	characteristic, complemented
60	C₂×#5950	C₂⁴	1	1	yes	characteristic
61	#87521	C₂²×C₄	1	4	yes	complemented
62	#87520	C₂²×C₄	1	2	yes	—
63	#87520	#41	1	2	yes	—
64	C₂×#5935	C₂×D₄	1	1	yes	characteristic
65	C₂×#5933	C₂×D₄	1	1	yes	characteristic
66	#85666	C₂²×C₄	1	2	yes	complemented
67	C₄×#997	C₂×D₄	1	2	yes	—
68	#85567	C₂²×C₄	1	2	yes	complemented
69	#85565	C₂×D₄	1	2	yes	—
70	#3027⋊C₃	C₂²×C₄	1	4	yes	complemented
71	C₂×#3573	C₂²×D₃	1	1	yes	characteristic
72	C₂×#3573	C₂²×D₃	1	1	yes	characteristic
73	C₄×#585	C₂²×D₃	1	1	yes	characteristic
74	C₂×#3027	C₂²×D₃	1	2	yes	—
75	C₂²×#442	S₄	1	1	yes	characteristic, complemented
76	#32948	C₄×D₃	1	4	yes	complemented
77	#20610	C₂²×D₃	1	2	yes	—
78	#5950	C₂³×C₄	1	1	yes	characteristic
79	#5950	C₂×#41	1	1	yes	characteristic
80	C₂×#997	C₂²×D₄	1	1	yes	characteristic
81	#5935	C₄×D₄	1	2	yes	—
82	#5933	C₄×D₄	1	2	yes	—
83	C₂×#585	C₂³×D₃	1	1	yes	characteristic
84	C₂³×D₈	C₂×S₄	1	1	yes	characteristic
85	#3573	C₂×C₄×D₃	1	2	yes	complemented
86	C₄×H(5,𝔽₂)	D₃×D₄	1	2	yes	—
87	C₂²×C₄×D₄	C₂×S₄	1	2	yes	—
88	#3027	C₂×C₄×D₃	1	4	yes	complemented
89	C₂²×#106	C₂×S₄	1	1	yes	characteristic
90	C₂²×#102	C₂×S₄	1	2	yes	—
91	C₂²×#97	C₂×S₄	1	1	yes	characteristic
92	#997	C₂×C₄×D₄	1	1	yes	characteristic
93	#585	C₂²×C₄×D₃	1	1	yes	characteristic
94	#585	D₃×#41	1	1	yes	characteristic
95	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
96	C₂³×D₄	C₂²×S₄	1	2	yes	—
97	C₂²×D₈	C₄×S₄	1	4	yes	complemented
98	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
99	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
100	C₂²×C₄⋊C₄	C₂²×S₄	1	2	yes	—
101	C₂²×C₄²	C₂²×S₄	1	1	yes	characteristic
102	H(5,𝔽₂)	C₄×D₃×D₄	1	1	yes	characteristic
103	C₂²×D₄	C₂×C₄×S₄	1	4	yes	—
104	C₂³×C₄	D₄×S₄	1	2	yes	—
105	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×S₄	1	2	yes	—
107	#97	#1020⋊C₂	1	1	yes	characteristic
108	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
109	C₂²×C₄	C₂²×C₄×S₄	1	1	yes	characteristic
110	C₂²×C₄	#41×S₄	1	1	yes	characteristic
111	C₂×C₈	C₂×#1020⋊C₂	1	1	yes	characteristic
112	C₂×C₈	C₂×#1020⋊C₂	1	1	yes	characteristic
113	C₄²	C₂×#1020⋊C₂	1	1	yes	characteristic
114	C₂³	C₄×D₄×S₄	1	1	yes	characteristic
115	C₂×C₄	#11713271	1	2	yes	—
116	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
117	C₈	C₄×#1020⋊C₂	1	2	yes	—
118	C₂²	C₂×#11713271	1	1	yes	characteristic
119	C₄	#72220637	1	1	yes	characteristic
120	C₄	C₂×C₄×#1020⋊C₂	1	1	yes	characteristic
121	C₂	#72240988	1	2	yes	—
122	C₂	C₄×#11713271	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#11713271
commutator	#5950	C₂³×C₄
Frattini	C₂³×C₄	C₂³×S₄
Fitting	#72281182	D₃
prime radical	C₂×#72236850	C₂
cosocle	C₂×#5950	C₂⁴
hypercenter	#97	#1020⋊C₂
hypocenter	#997	C₂×C₄×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. #72341790	C₂	semidirect
2. #72341783	C₂	semidirect
4. #72341764	C₂	semidirect
5. #72341755	C₂	semidirect
6. #72341747	C₂	semidirect
7. #72341742	C₂	semidirect
9. #72341731	C₂²	semidirect
10. #72341722	C₂	semidirect
11. #72341716	C₂²	semidirect
12. #72341710	C₂²	semidirect
13. #72341702	C₂²	semidirect
14. #72341690	C₂	semidirect
15. #72341685	C₂²	semidirect
18. #72341665	C₂²	semidirect
19. #72341657	C₂²	semidirect
20. C₄×#11714184	C₂	semidirect
21. #72341357	C₂²	semidirect
22. C₄×#3027⋊C₃	C₂²	semidirect
24. #72341273	C₂²	semidirect
25. #72281182	D₃	semidirect
26. C₄×#87521	C₂²	semidirect
27. #72236850	C₄	semidirect
29. #72227323	C₂	semidirect
30. #72227321	C₂²	semidirect
32. #72227309	C₂²	semidirect
33. #72227309	C₂²	semidirect
34. #72227308	C₂²	semidirect
39. #11715736	C₂×C₄	semidirect
40. #11715681	C₂×C₄	semidirect
47. #11714184	C₂×C₄	semidirect
48. #11711638	C₂×C₄	semidirect
53. #32948⋊C₃	C₂×C₄	semidirect
55. #10574037	C₂×D₃	semidirect
57. C₄×#3027	C₂×D₃	semidirect
58. #392717	C₂×D₃	semidirect
59. #392195	C₂×D₃	semidirect
61. #87521	C₂²×C₄	semidirect
66. #85666	C₂²×C₄	semidirect
68. #85567	C₂²×C₄	semidirect
70. #3027⋊C₃	C₂²×C₄	semidirect
75. C₂²×#442	S₄	semidirect
76. #32948	C₄×D₃	semidirect
85. #3573	C₂×C₄×D₃	semidirect
88. #3027	C₂×C₄×D₃	semidirect
97. C₂²×D₈	C₄×S₄	semidirect

Maximal subgroups

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

subgroup	index
#72341790	2
#72341783	2
C₂×#72236850	2
#72341764	2
#72341755	2
#72341747	2
#72341742	2
#72341722	2
#72341690	2
C₄×#11714184	2
#72311898	3
S₄×#442	4
#72227323	2