Subgroups — #72343985

Order 6144.

At least 15456 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

The subgroup lattice has 152 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72343979	C₂	1	1	yes	maximal, characteristic, complemented
2	#72343975	C₂	1	1	yes	maximal, characteristic, complemented
3	#72343929	C₂	1	1	yes	maximal, characteristic, complemented
4	#72343908	C₂	1	1	yes	maximal, characteristic, complemented
5	#72343899	C₂²	1	1	yes	characteristic, complemented
6	#72343896	C₂	1	1	yes	maximal, characteristic, complemented
7	#72343893	C₂²	1	1	yes	characteristic, complemented
8	#72343865	C₂	1	1	yes	maximal, characteristic, complemented
9	#72343782	C₂	1	1	yes	maximal, characteristic, complemented
10	#72343706	C₂	1	1	yes	maximal, characteristic, complemented
11	#72343391	C₂	1	1	yes	maximal, characteristic, complemented
12	#72343239	C₂	1	1	yes	maximal, characteristic, complemented
13	#72343231	C₂²	1	1	yes	characteristic, complemented
14	C₂×#72215704	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#11715682	C₂²	1	1	yes	characteristic, complemented
16	C₂×#11715710	C₂²	1	1	yes	characteristic, complemented
17	#72342866	C₂	1	1	yes	maximal, characteristic, complemented
18	#72341994	C₂²	1	1	yes	characteristic, complemented
19	#72341955	C₂²	1	1	yes	characteristic, complemented
20	#72341845	C₂²	1	1	yes	characteristic, complemented
21	#72341843	C₂²	1	1	yes	characteristic, complemented
22	#72341800	C₂	1	1	yes	maximal, characteristic, complemented
23	#72341797	C₂²	1	1	yes	characteristic, complemented
24	#72341793	C₂²	1	1	yes	characteristic, complemented
25	#72341716	C₂²	1	1	yes	characteristic, complemented
26	#72341710	C₂²	1	1	yes	characteristic, complemented
27	#72341702	C₂²	1	1	yes	characteristic, complemented
28	#72341685	C₂²	1	1	yes	characteristic, complemented
29	#72341681	C₂²	1	1	yes	characteristic, complemented
30	#72341610	C₂²	1	1	yes	characteristic, complemented
31	#72341580	C₂²	1	1	yes	characteristic, complemented
32	C₂×#72236770	C₂	1	1	yes	maximal, characteristic, complemented
33	C₂×#11711643	C₂²	1	1	yes	characteristic, complemented
34	C₂×#11711642	C₂²	1	1	yes	characteristic, complemented
35	C₂×#11715711	C₂²	1	1	yes	characteristic, complemented
36	C₂×#32895⋊C₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×#11711597	C₂²	1	1	yes	characteristic, complemented
38	C₂×#11715830	C₂²	1	1	yes	characteristic, complemented
39	#72341480	C₂²	1	1	yes	characteristic, complemented
40	C₂×#11715708	C₂²	1	1	yes	characteristic, complemented
41	C₂×#11710899	C₂²	1	1	yes	characteristic, complemented
42	#72312259	D₃	1	1	yes	characteristic, complemented
43	C₂×#72217027	C₂	1	1	yes	maximal, characteristic, complemented
44	C₂×#60399⋊C₃	C₂²	1	1	yes	characteristic, complemented
45	C₂×#11714174	C₂²	1	1	yes	characteristic, complemented
46	C₂×#11714184	C₂²	1	1	yes	characteristic
47	C₂×#11714162	C₂²	1	1	yes	characteristic
48	C₂×#11714175	C₂²	1	1	yes	characteristic, complemented
49	C₂×#11715739	C₂²	1	1	yes	characteristic, complemented
50	C₂×#11715839	C₂²	1	1	yes	characteristic, complemented
51	C₂×#11715828	C₂²	1	1	yes	characteristic
52	C₂×#87521	C₂³	1	1	yes	characteristic
53	C₂×#87519	C₂³	1	1	yes	characteristic
54	#11715828	D₄	1	4	yes	complemented
55	C₂×#87522	C₂³	1	1	yes	characteristic
56	C₂×#87522	C₂³	1	1	yes	characteristic, complemented
57	C₂×#87520	C₂³	1	1	yes	characteristic
58	C₂×#85567	C₂³	1	1	yes	characteristic, complemented
59	C₂×#85566	C₂³	1	1	yes	characteristic, complemented
60	#11715659	C₂³	1	1	yes	characteristic, complemented
61	#11714184	D₄	1	4	yes	complemented
62	C₂×#73031	D₄	1	1	yes	characteristic, complemented
63	C₂×#73025	D₄	1	1	yes	characteristic, complemented
64	C₂×#73030	D₄	1	1	yes	characteristic, complemented
65	C₂×#73026	D₄	1	1	yes	characteristic, complemented
66	C₂×#87548	C₂³	1	1	yes	characteristic, complemented
67	C₂×#87547	C₂³	1	1	yes	characteristic
68	C₂×#85666	C₂³	1	1	yes	characteristic, complemented
69	C₂×#3027⋊C₃	C₂³	1	1	yes	characteristic
70	C₂×#3025⋊C₃	C₂³	1	1	yes	characteristic
71	C₂×#1791⋊C₃	C₂³	1	1	yes	characteristic, complemented
72	#20610⋊C₃	C₂³	1	1	yes	characteristic, complemented
73	C₂×#63092	C₂×D₃	1	1	yes	characteristic, complemented
74	C₂×#60399	C₂×D₃	1	1	yes	characteristic
75	#6628899	C₂×D₃	1	1	yes	characteristic, complemented
76	C₂×#32895	C₂×D₃	1	1	yes	characteristic, complemented
77	#392717	C₂×D₃	1	1	yes	characteristic, complemented
78	#392665	C₂×D₃	1	1	yes	characteristic, complemented
79	#392141	C₂×D₃	1	1	yes	characteristic, complemented
80	#87547	C₂×D₄	1	2	yes	complemented
81	C₂×#5950	C₂⁴	1	1	yes	characteristic
82	#87522	C₂×D₄	1	2	yes	complemented
83	#87521	C₂×D₄	1	2	yes	complemented
84	#87520	C₂×D₄	1	2	yes	—
85	#87520	C₂×D₄	1	2	yes	—
86	C₂×#5935	C₂×D₄	1	1	yes	characteristic
87	C₂×#5934	C₂×D₄	1	1	yes	characteristic
88	C₂×#5933	C₂×D₄	1	1	yes	characteristic
89	C₂×#5932	C₂×D₄	1	1	yes	characteristic
90	#3027⋊C₃	C₂×D₄	1	2	yes	complemented
91	C₂×#458⋊C₃	C₂×D₄	1	1	yes	characteristic
92	C₂×B(4,𝔽₂)⋊C₃	C₂×D₄	1	1	yes	characteristic
93	C₂×2₋¹⁺⁶	C₂²×D₃	1	1	yes	characteristic, complemented
94	C₂×H(7,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
95	C₂×#3573	C₂²×D₃	1	1	yes	characteristic, complemented
96	C₂×#3027	C₂²×D₃	1	1	yes	characteristic
97	C₂×#3025	C₂²×D₃	1	1	yes	characteristic
98	C₂²×#450	S₄	1	1	yes	characteristic, complemented
99	C₂×#1791	C₂²×D₃	1	1	yes	characteristic, complemented
100	#20610	C₂²×D₃	1	1	yes	characteristic, complemented
101	#5950	C₂²×D₄	1	1	yes	characteristic
102	#5950	C₂²×D₄	1	1	yes	characteristic
103	C₂×#997	C₂²×D₄	1	1	yes	characteristic
104	#5935	#120	1	2	yes	—
105	#5933	#120	1	2	yes	—
106	H(7,𝔽₂)	D₃×D₄	1	2	yes	—
107	C₂×#585	C₂³×D₃	1	1	yes	characteristic
108	C₂⁴×D₄	C₂×S₄	1	1	yes	characteristic
109	C₂³×SD₁₆	C₂×S₄	1	1	yes	characteristic
110	C₂³×SD₁₆	C₂×S₄	1	1	yes	characteristic, complemented
111	C₂²×C₂²⋊Q₈	C₂×S₄	1	1	yes	characteristic, complemented
112	#3027	D₃×D₄	1	2	yes	—
113	C₂×#458	D₃×D₄	1	1	yes	characteristic
114	C₂×B(4,𝔽₂)	D₃×D₄	1	1	yes	characteristic
115	C₂²×#102	C₂×S₄	1	1	yes	characteristic
116	C₂²×#102	C₂×S₄	1	1	yes	characteristic, complemented
117	C₂²×C₂²⋊C₈	C₂×S₄	1	1	yes	characteristic, complemented
118	#997	C₂×#120	1	1	yes	characteristic
119	#585	C₂×D₃×D₄	1	1	yes	characteristic
120	#585	C₂×D₃×D₄	1	1	yes	characteristic
121	C₂×H(5,𝔽₂)	C₂×D₃×D₄	1	1	yes	characteristic
122	C₂³×Q₈	C₂²×S₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
124	C₂³×D₄	C₂²×S₄	1	1	yes	characteristic
125	C₂⁴×C₄	C₂²×S₄	1	1	yes	characteristic
126	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
127	C₂³×C₈	C₂²×S₄	1	1	yes	characteristic
128	C₂²×C₄⋊C₄	C₂²×S₄	1	1	yes	characteristic
129	C₂⁵	D₄×S₄	1	1	yes	characteristic
130	H(5,𝔽₂)	D₃×#120	1	1	yes	characteristic
131	C₂²×D₄	D₄×S₄	1	2	yes	—
132	C₂²×D₄	D₄×S₄	1	2	yes	—
133	C₂³×C₄	D₄×S₄	1	1	yes	characteristic
134	C₂³×C₄	C₂³×S₄	1	1	yes	characteristic
135	C₂×SD₁₆	#1020⋊C₂	1	1	yes	characteristic
136	C₂⁴	C₂×D₄×S₄	1	1	yes	characteristic
137	C₂×Q₈	C₂×#1020⋊C₂	1	1	yes	characteristic
138	C₂×D₄	C₂×#1020⋊C₂	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×S₄	1	1	yes	characteristic
141	C₂×C₈	C₂×#1020⋊C₂	1	1	yes	characteristic
142	C₂³	S₄×#120	1	1	yes	characteristic
143	D₄	#11713200	1	2	yes	—
144	C₂×C₄	C₂²×#1020⋊C₂	1	1	yes	characteristic
145	C₂²	D₄×#1020⋊C₂	1	1	yes	characteristic
146	C₄	C₂×#11713200	1	1	yes	characteristic
147	C₄	D₄×#1020⋊C₂	1	1	yes	characteristic
148	C₂	#72222918	1	1	yes	characteristic
149	C₂	#72240982	1	1	yes	characteristic
150	C₂	#72240996	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₄×#1020⋊C₂
commutator, cosocle	C₂×#5950	C₂⁴
Frattini	C₂³×C₄	C₂³×S₄
Fitting	#72312259	D₃
hypercenter	C₂×SD₁₆	#1020⋊C₂
hypocenter	#997	C₂×#120
LCS	G ▷ C₂×#5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #72343979	C₂	semidirect
2. #72343975	C₂	semidirect
3. #72343929	C₂	semidirect
4. #72343908	C₂	semidirect
5. #72343899	C₂²	semidirect
6. #72343896	C₂	semidirect
7. #72343893	C₂²	semidirect
8. #72343865	C₂	semidirect
9. #72343782	C₂	semidirect
10. #72343706	C₂	semidirect
11. #72343391	C₂	semidirect
12. #72343239	C₂	semidirect
13. #72343231	C₂²	semidirect
14. C₂×#72215704	C₂	semidirect
15. C₂×#11715682	C₂²	semidirect
16. C₂×#11715710	C₂²	semidirect
17. #72342866	C₂	semidirect
18. #72341994	C₂²	semidirect
19. #72341955	C₂²	semidirect
20. #72341845	C₂²	semidirect
21. #72341843	C₂²	semidirect
22. #72341800	C₂	semidirect
23. #72341797	C₂²	semidirect
24. #72341793	C₂²	semidirect
25. #72341716	C₂²	semidirect
26. #72341710	C₂²	semidirect
27. #72341702	C₂²	semidirect
28. #72341685	C₂²	semidirect
29. #72341681	C₂²	semidirect
30. #72341610	C₂²	semidirect
31. #72341580	C₂²	semidirect
32. C₂×#72236770	C₂	semidirect
33. C₂×#11711643	C₂²	semidirect
34. C₂×#11711642	C₂²	semidirect
35. C₂×#11715711	C₂²	semidirect
36. C₂×#32895⋊C₃	C₂²	semidirect
37. C₂×#11711597	C₂²	semidirect
38. C₂×#11715830	C₂²	semidirect
39. #72341480	C₂²	semidirect
40. C₂×#11715708	C₂²	semidirect
41. C₂×#11710899	C₂²	semidirect
42. #72312259	D₃	semidirect
43. C₂×#72217027	C₂	semidirect
44. C₂×#60399⋊C₃	C₂²	semidirect
45. C₂×#11714174	C₂²	semidirect
48. C₂×#11714175	C₂²	semidirect
49. C₂×#11715739	C₂²	semidirect
50. C₂×#11715839	C₂²	semidirect
54. #11715828	D₄	semidirect
56. C₂×#87522	C₂³	semidirect
58. C₂×#85567	C₂³	semidirect
59. C₂×#85566	C₂³	semidirect
60. #11715659	C₂³	semidirect
61. #11714184	D₄	semidirect
62. C₂×#73031	D₄	semidirect
63. C₂×#73025	D₄	semidirect
64. C₂×#73030	D₄	semidirect
65. C₂×#73026	D₄	semidirect
66. C₂×#87548	C₂³	semidirect
68. C₂×#85666	C₂³	semidirect
71. C₂×#1791⋊C₃	C₂³	semidirect
72. #20610⋊C₃	C₂³	semidirect
73. C₂×#63092	C₂×D₃	semidirect
75. #6628899	C₂×D₃	semidirect
76. C₂×#32895	C₂×D₃	semidirect
77. #392717	C₂×D₃	semidirect
78. #392665	C₂×D₃	semidirect
79. #392141	C₂×D₃	semidirect
80. #87547	C₂×D₄	semidirect
82. #87522	C₂×D₄	semidirect
83. #87521	C₂×D₄	semidirect
90. #3027⋊C₃	C₂×D₄	semidirect
93. C₂×2₋¹⁺⁶	C₂²×D₃	semidirect
95. C₂×#3573	C₂²×D₃	semidirect
98. C₂²×#450	S₄	semidirect
99. C₂×#1791	C₂²×D₃	semidirect
100. #20610	C₂²×D₃	semidirect
110. C₂³×SD₁₆	C₂×S₄	semidirect
111. C₂²×C₂²⋊Q₈	C₂×S₄	semidirect
116. C₂²×#102	C₂×S₄	semidirect
117. C₂²×C₂²⋊C₈	C₂×S₄	semidirect

Maximal subgroups

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

subgroup	index
#72343979	2
#72343975	2
#72343968	4
#72343929	2
#72343908	2
#72343896	2
#72343865	2
#72343782	2
#72343706	2
#72343391	2
#72343239	2
C₂×#72215704	2
#72342866	2
#72341800	2
C₂×#72236770	2
#72337466	3
C₂×#72217027	2