Subgroups — #72286190

Order 4096.

At least 9132 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

The subgroup lattice has 201 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72286180	C₂	1	1	yes	maximal, characteristic, complemented
2	#72286121	C₂	1	1	yes	maximal, characteristic, complemented
3	#72286086	C₂	1	1	yes	maximal, characteristic, complemented
4	#72286044	C₂	1	1	yes	maximal, characteristic, complemented
5	#72285917	C₂²	1	1	yes	characteristic
6	#72285770	C₂	1	1	yes	maximal, characteristic, complemented
7	#72285722	C₂	1	1	yes	maximal, characteristic, complemented
8	#72285694	C₂	1	1	yes	maximal, characteristic, complemented
9	#72285637	C₂	1	1	yes	maximal, characteristic, complemented
10	#72285629	C₂²	1	1	yes	characteristic, complemented
11	#72285553	C₂	1	1	yes	maximal, characteristic, complemented
12	#72285544	C₂²	1	1	yes	characteristic, complemented
13	#72285311	C₂	1	1	yes	maximal, characteristic, complemented
14	#72285183	C₂²	1	1	yes	characteristic
15	#72285127	C₂²	1	1	yes	characteristic, complemented
16	#72284918	C₂	1	1	yes	maximal, characteristic, complemented
17	#72284401	C₂²	1	1	yes	characteristic
18	#72284051	C₂	1	1	yes	maximal, characteristic, complemented
19	#72283811	C₂	1	1	yes	maximal, characteristic, complemented
20	#72283784	C₂	1	1	yes	maximal, characteristic, complemented
21	#72283777	C₂²	1	1	yes	characteristic, complemented
22	#72283773	C₂²	1	1	yes	characteristic, complemented
23	#72283758	C₂²	1	1	yes	characteristic
24	#72283743	C₂²	1	1	yes	characteristic, complemented
25	#72283735	C₂²	1	1	yes	characteristic, complemented
26	#72283717	C₂²	1	1	yes	characteristic
27	#72283686	C₂²	1	1	yes	characteristic, complemented
28	#72283656	C₂²	1	1	yes	characteristic
29	#72283656	C₂²	1	1	yes	characteristic, complemented
30	#72283656	C₂²	1	1	yes	characteristic, complemented
31	#72283562	C₂²	1	1	yes	characteristic, complemented
32	#72283505	C₂²	1	1	yes	characteristic, complemented
33	#72241387	C₂²	1	1	yes	characteristic
34	#72241361	C₂	1	1	yes	maximal, characteristic, complemented
35	#72241347	C₂²	1	1	yes	characteristic
36	#72237008	C₂²	1	1	yes	characteristic, complemented
37	#72236998	C₂²	1	1	yes	characteristic, complemented
38	#72236934	C₂²	1	1	yes	characteristic
39	#72236934	C₂²	1	1	yes	characteristic, complemented
40	#72236934	C₂²	1	1	yes	characteristic, complemented
41	#72236933	C₂²	1	1	yes	characteristic
42	#72236933	C₂²	1	1	yes	characteristic, complemented
43	#72236933	C₂²	1	1	yes	characteristic, complemented
44	#72235225	C₂²	1	1	yes	characteristic, complemented
45	#72235140	C₂²	1	1	yes	characteristic, complemented
46	#72213804	C₂²	1	1	yes	characteristic
47	#72213804	C₂²	1	1	yes	characteristic, complemented
48	#72213804	C₂²	1	1	yes	characteristic, complemented
49	#72213574	C₂²	1	1	yes	characteristic, complemented
50	#72213523	C₂²	1	1	yes	characteristic, complemented
51	#8932760	D₄	1	1	yes	characteristic, complemented
52	#8930050	D₄	1	1	yes	characteristic, complemented
53	#8924809	D₄	1	1	yes	characteristic
54	#6470244	D₄	1	1	yes	characteristic
55	#6470231	D₄	1	1	yes	characteristic
56	#6437630	C₂³	1	1	yes	characteristic
57	#6432959	C₂³	1	1	yes	characteristic
58	#357286	C₂³	1	1	yes	characteristic
59	#346284	D₄	1	1	yes	characteristic
60	#346284	D₄	1	1	yes	characteristic
61	#346283	D₄	1	1	yes	characteristic, complemented
62	#346283	D₄	1	1	yes	characteristic, complemented
63	#346274	D₄	1	1	yes	characteristic, complemented
64	#346269	D₄	1	1	yes	characteristic
65	#346267	D₄	1	1	yes	characteristic
66	#346246	D₄	1	1	yes	characteristic, complemented
67	#346239	D₄	1	1	yes	characteristic
68	#346239	D₄	1	1	yes	characteristic
69	#346228	D₄	1	1	yes	characteristic, complemented
70	#346228	D₄	1	1	yes	characteristic, complemented
71	#251850	D₄	1	1	yes	characteristic, complemented
72	#251850	C₂³	1	1	yes	characteristic
73	#251830	D₄	1	1	yes	characteristic, complemented
74	#251830	D₄	1	1	yes	characteristic, complemented
75	#251828	D₄	1	1	yes	characteristic, complemented
76	#251828	C₂³	1	1	yes	characteristic
77	#251828	C₂³	1	1	yes	characteristic
78	#251828	C₂³	1	1	yes	characteristic
79	#245290	D₄	1	1	yes	characteristic, complemented
80	#245282	D₄	1	1	yes	characteristic
81	#245233	D₄	1	1	yes	characteristic, complemented
82	#245225	D₄	1	1	yes	characteristic
83	#213862	D₄	1	1	yes	characteristic, complemented
84	#213861	C₂³	1	1	yes	characteristic
85	#213854	C₂³	1	1	yes	characteristic
86	#213843	D₄	1	1	yes	characteristic, complemented
87	#213839	D₄	1	1	yes	characteristic
88	#149742	C₂³	1	1	yes	characteristic
89	#149742	C₂³	1	1	yes	characteristic
90	#149742	C₂³	1	1	yes	characteristic
91	#149742	C₂³	1	1	yes	characteristic
92	#128064	C₂³	1	1	yes	characteristic
93	#128062	C₂³	1	1	yes	characteristic
94	#62842	C₂×D₄	1	1	yes	characteristic
95	#47195	C₂×D₄	1	1	yes	characteristic
96	#46384	C₂×D₄	1	1	yes	characteristic
97	#46384	C₂×D₄	1	1	yes	characteristic
98	#46384	C₂×D₄	1	1	yes	characteristic
99	#35626	C₂×D₄	1	1	yes	characteristic
100	#35616	C₂×D₄	1	1	yes	characteristic
101	#35616	C₂×D₄	1	1	yes	characteristic
102	#35527	C₂×D₄	1	1	yes	characteristic
103	#35346	C₂×D₄	1	1	yes	characteristic
104	#20690	C₂×D₄	1	1	yes	characteristic
105	#20586	C₂×D₄	1	1	yes	characteristic
106	#20586	C₂×D₄	1	1	yes	characteristic
107	#20586	C₂⁴	1	1	yes	characteristic
108	#20585	C₂×D₄	1	1	yes	characteristic
109	#10392	C₂×D₄	1	1	yes	characteristic
110	#10392	C₂×D₄	1	1	yes	characteristic
111	#10392	C₂×D₄	1	1	yes	characteristic
112	#10392	C₂×D₄	1	1	yes	characteristic
113	#10392	C₂×D₄	1	1	yes	characteristic
114	#10392	C₂×D₄	1	1	yes	characteristic
115	#10392	C₂×D₄	1	1	yes	characteristic
116	#10392	C₂×D₄	1	1	yes	characteristic
117	#10338	C₂×D₄	1	1	yes	characteristic
118	#10338	C₂×D₄	1	1	yes	characteristic
119	#10336	C₂×D₄	1	1	yes	characteristic
120	#10336	C₂×D₄	1	1	yes	characteristic
121	#10336	C₂×D₄	1	1	yes	characteristic
122	#10336	C₂×D₄	1	1	yes	characteristic
123	#10336	C₂×D₄	1	1	yes	characteristic
124	#10336	C₂×D₄	1	1	yes	characteristic
125	#8748	C₂×D₄	1	1	yes	characteristic
126	#8748	C₂×D₄	1	1	yes	characteristic
127	#8748	C₂×D₄	1	1	yes	characteristic
128	#8748	C₂×D₄	1	1	yes	characteristic
129	#8748	C₂×D₄	1	1	yes	characteristic
130	#8748	C₂×D₄	1	1	yes	characteristic
131	#8747	C₂×D₄	1	1	yes	characteristic
132	#8747	C₂×D₄	1	1	yes	characteristic
133	#8546	C₂×D₄	1	1	yes	characteristic
134	#8546	C₂×D₄	1	1	yes	characteristic
135	#8541	C₂×D₄	1	1	yes	characteristic
136	#8541	C₂×D₄	1	1	yes	characteristic
137	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
138	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
139	C₂²×#127	#120	1	1	yes	characteristic
140	C₂²×#124	#120	1	1	yes	characteristic
141	C₂²×#120	#120	1	1	yes	characteristic
142	C₂×#518	#120	1	1	yes	characteristic
143	#2842	#120	1	1	yes	characteristic
144	#2842	#120	1	1	yes	characteristic
145	#2842	#120	1	1	yes	characteristic
146	#2842	#120	1	1	yes	characteristic
147	#2483	#120	1	1	yes	characteristic
148	#2403	#120	1	1	yes	characteristic
149	#2403	#120	1	1	yes	characteristic
150	#2403	#120	1	1	yes	characteristic
151	#2309	#120	1	1	yes	characteristic
152	#2309	#120	1	1	yes	characteristic
153	#2309	C₂²×D₄	1	1	yes	characteristic
154	#2309	C₂²×D₄	1	1	yes	characteristic
155	#2308	#120	1	1	yes	characteristic
156	#1781	#120	1	1	yes	characteristic
157	#1781	#120	1	1	yes	characteristic
158	#1781	#120	1	1	yes	characteristic
159	#1781	#120	1	1	yes	characteristic
160	#1781	C₂²×D₄	1	1	yes	characteristic
161	#1781	C₂²×D₄	1	1	yes	characteristic
162	#1781	C₂²×D₄	1	1	yes	characteristic
163	#1781	C₂²×D₄	1	1	yes	characteristic
164	#1304	#120	1	1	yes	characteristic
165	#1304	#120	1	1	yes	characteristic
166	#1304	#120	1	1	yes	characteristic
167	#1304	#120	1	1	yes	characteristic
168	#1304	#120	1	1	yes	characteristic
169	#1304	#120	1	1	yes	characteristic
170	#1304	#120	1	1	yes	characteristic
171	#1304	#120	1	1	yes	characteristic
172	C₂⁶	#453	1	1	yes	characteristic
173	C₂³×Q₈	B(4,𝔽₂)	1	1	yes	characteristic
174	C₂³×D₄	#453	1	1	yes	characteristic
175	C₂³×D₄	C₂×#120	1	1	yes	characteristic
176	C₂³×D₄	C₂×#120	1	1	yes	characteristic
177	C₂³×D₄	C₂×#120	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	#453	1	1	yes	characteristic
179	C₂²×C₄²	#453	1	1	yes	characteristic
180	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
181	#379	C₂×#120	1	1	yes	characteristic
182	#379	C₂×#120	1	1	yes	characteristic
183	#379	C₂×#120	1	1	yes	characteristic
184	#379	C₂×#120	1	1	yes	characteristic
185	C₂⁵	#2846	1	1	yes	characteristic
186	C₂⁵	C₂×#453	1	1	yes	characteristic
187	C₂⁵	#3019	1	1	yes	characteristic
188	C₂⁵	#3026	1	1	yes	characteristic
189	C₂³×C₄	C₂×#453	1	1	yes	characteristic
190	C₂³×C₄	#3019	1	1	yes	characteristic
191	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
192	C₂⁴	#32888	1	1	yes	characteristic
193	C₂⁴	#32888	1	1	yes	characteristic
194	C₂⁴	#32898	1	1	yes	characteristic
195	C₂³	#346446	1	1	yes	characteristic
196	C₂³	#511790	1	1	yes	characteristic
197	C₂³	#511790	1	1	yes	characteristic
198	C₂²	#72216464	1	1	yes	characteristic
199	C₂	#72237817	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72237817
commutator, Frattini, cosocle	#20586	C₂⁴
LCS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6432959 ◁ G
LECS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72286180	C₂	semidirect
2. #72286121	C₂	semidirect
3. #72286086	C₂	semidirect
4. #72286044	C₂	semidirect
6. #72285770	C₂	semidirect
7. #72285722	C₂	semidirect
8. #72285694	C₂	semidirect
9. #72285637	C₂	semidirect
10. #72285629	C₂²	semidirect
11. #72285553	C₂	semidirect
12. #72285544	C₂²	semidirect
13. #72285311	C₂	semidirect
15. #72285127	C₂²	semidirect
16. #72284918	C₂	semidirect
18. #72284051	C₂	semidirect
19. #72283811	C₂	semidirect
20. #72283784	C₂	semidirect
21. #72283777	C₂²	semidirect
22. #72283773	C₂²	semidirect
24. #72283743	C₂²	semidirect
25. #72283735	C₂²	semidirect
27. #72283686	C₂²	semidirect
29. #72283656	C₂²	semidirect
30. #72283656	C₂²	semidirect
31. #72283562	C₂²	semidirect
32. #72283505	C₂²	semidirect
34. #72241361	C₂	semidirect
36. #72237008	C₂²	semidirect
37. #72236998	C₂²	semidirect
39. #72236934	C₂²	semidirect
40. #72236934	C₂²	semidirect
42. #72236933	C₂²	semidirect
43. #72236933	C₂²	semidirect
44. #72235225	C₂²	semidirect
45. #72235140	C₂²	semidirect
47. #72213804	C₂²	semidirect
48. #72213804	C₂²	semidirect
49. #72213574	C₂²	semidirect
50. #72213523	C₂²	semidirect
51. #8932760	D₄	semidirect
52. #8930050	D₄	semidirect
61. #346283	D₄	semidirect
62. #346283	D₄	semidirect
63. #346274	D₄	semidirect
66. #346246	D₄	semidirect
69. #346228	D₄	semidirect
70. #346228	D₄	semidirect
71. #251850	D₄	semidirect
73. #251830	D₄	semidirect
74. #251830	D₄	semidirect
75. #251828	D₄	semidirect
79. #245290	D₄	semidirect
81. #245233	D₄	semidirect
83. #213862	D₄	semidirect
86. #213843	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72286180	2
#72286121	2
#72286086	2
#72286044	2
#72285770	2
#72285722	2
#72285694	2
#72285637	2
#72285553	2
#72285311	2
#72284918	2
#72284051	2
#72283811	2
#72283784	2
#72241361	2