Subgroups — #72255221

Order 3072.

At least 6480 conjugacy classes of subgroups; only the 343 normal subgroups are shown.

The subgroup lattice has 287 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72255217	C₂	1	2	yes	maximal, complemented
2	#72255216	C₂	1	2	yes	maximal, complemented
3	#72255215	C₂	1	2	yes	maximal, complemented
4	#72255213	C₂	1	2	yes	maximal, complemented
5	#72255069	C₂	1	1	yes	maximal, characteristic, complemented
6	#72254969	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#493370	C₂	1	1	yes	maximal, characteristic, complemented
8	#72249311	C₂	1	1	yes	maximal, characteristic, complemented
9	#72249307	C₂	1	1	yes	maximal, characteristic, complemented
10	#72248732	C₂	1	1	yes	maximal, characteristic, complemented
11	#72248668	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#87239	C₂²	1	1	yes	characteristic
13	C₂×#84052	C₂²	1	1	yes	characteristic
14	#10828423	C₂²	1	2	yes	—
15	#10826826	C₂²	1	2	yes	—
16	#10826818	C₂²	1	2	yes	—
17	#10826769	C₂²	1	2	yes	—
18	C₂×#72129	C₂²	1	1	yes	characteristic
19	#10740386	C₂²	1	1	yes	characteristic, complemented
20	#10733626	C₂²	1	1	yes	characteristic, complemented
21	#10732887	C₂²	1	1	yes	characteristic, complemented
22	#10732882	C₂²	1	1	yes	characteristic, complemented
23	#10717057	C₂²	1	1	yes	characteristic
24	#10711423	C₂²	1	2	yes	complemented
25	#10710646	C₂²	1	2	yes	complemented
26	#10687759	C₂²	1	2	yes	—
27	#10687759	C₂²	1	2	yes	—
28	#10682458	C₂²	1	2	yes	complemented
29	#10682458	C₂²	1	2	yes	complemented
30	C₂²×C₃×#535	C₂²	1	1	yes	characteristic
31	C₂×C₃×#1781	C₂²	1	1	yes	characteristic
32	C₂×C₃×#1781	C₂²	1	1	yes	characteristic
33	C₃×#12694	C₂²	1	2	yes	complemented
34	C₃×#12693	C₂²	1	2	yes	complemented
35	#493370	D₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×#41	D₄	1	1	yes	characteristic, complemented
37	C₂⁴×C₃×D₄	C₂³	1	1	yes	characteristic
38	C₂⁴×#67	D₄	1	1	yes	characteristic
39	C₂²×#1004	D₄	1	1	yes	characteristic, complemented
40	C₂²×C₃×#127	D₄	1	1	yes	characteristic, complemented
41	C₂²×C₃×#127	C₂³	1	1	yes	characteristic
42	C₂²×C₃×#124	D₄	1	1	yes	characteristic, complemented
43	C₂²×C₃×#124	C₂³	1	1	yes	characteristic
44	C₂²×C₃×#117	D₄	1	1	yes	characteristic, complemented
45	C₂³×#217	D₄	1	1	yes	characteristic, complemented
46	#87239	D₄	1	2	yes	complemented
47	#87239	C₂³	1	2	yes	—
48	C₂²×#937	D₄	1	1	yes	characteristic
49	C₂²×#937	C₂³	1	1	yes	characteristic
50	C₂×#5798	D₄	1	1	yes	characteristic, complemented
51	C₂²×C₃×#99	D₄	1	2	yes	complemented
52	C₂×#5247	D₄	1	1	yes	characteristic
53	C₂×#5247	D₄	1	1	yes	characteristic, complemented
54	C₂×#5220	D₄	1	1	yes	characteristic
55	C₂×#5220	C₂³	1	1	yes	characteristic
56	C₂²×#833	D₄	1	2	yes	complemented
57	C₃×#1781	D₄	1	2	yes	complemented
58	C₃×#1781	C₂³	1	2	yes	—
59	C₃×#1781	C₂³	1	2	yes	—
60	#72131	D₄	1	1	yes	characteristic, complemented
61	#72131	D₄	1	1	yes	characteristic, complemented
62	#72131	C₂³	1	1	yes	characteristic
63	#72131	C₂³	1	1	yes	characteristic
64	#72129	D₄	1	2	yes	complemented
65	#72129	C₂³	1	2	yes	—
66	C₂²×#535	C₂×D₃	1	1	yes	characteristic
67	C₂×#1781	C₂×D₃	1	1	yes	characteristic
68	C₂×#1781	C₂×D₃	1	1	yes	characteristic
69	#12694	C₂×D₃	1	2	yes	complemented
70	#12693	C₂×D₃	1	2	yes	complemented
71	C₂⁶×C₃	C₂×D₄	1	1	yes	characteristic
72	C₂⁶×C₃	C₂×D₄	1	1	yes	characteristic
73	C₂⁵×D₃	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₃×D₄	#41	1	1	yes	characteristic
86	C₂³×C₃×D₄	#41	1	1	yes	characteristic
87	C₂³×C₃×D₄	C₂⁴	1	1	yes	characteristic
88	C₂⁴×C₃×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂⁴×Dic₃	C₂×D₄	1	1	yes	characteristic
90	C₂×#1004	C₂×D₄	1	2	yes	—
91	C₂²×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂²×#217	C₂×D₄	1	1	yes	characteristic
94	C₂²×#217	C₂×D₄	1	1	yes	characteristic
95	C₂²×#217	C₂×D₄	1	1	yes	characteristic
96	C₂²×#217	C₂×D₄	1	1	yes	characteristic
97	C₂²×#217	C₂×D₄	1	1	yes	characteristic
98	C₂²×#217	C₂×D₄	1	1	yes	characteristic
99	#5798	C₂×D₄	1	2	yes	—
100	C₂×C₃×#99	C₂×D₄	1	4	yes	—
101	C₂×#833	C₂×D₄	1	4	yes	—
102	C₂³×#41	D₁₂	1	1	yes	characteristic, complemented
103	C₂⁴×D₄	C₂²×D₃	1	1	yes	characteristic
104	C₂²×#127	D₁₂	1	1	yes	characteristic, complemented
105	C₂²×#127	C₂²×D₃	1	1	yes	characteristic
106	C₂²×#124	D₁₂	1	1	yes	characteristic, complemented
107	C₂²×#124	C₂²×D₃	1	1	yes	characteristic
108	C₂²×#117	D₁₂	1	1	yes	characteristic, complemented
109	C₂²×#99	#67	1	2	yes	complemented
110	#1781	#67	1	2	yes	complemented
111	#1781	C₂²×D₃	1	2	yes	—
112	#1781	C₂²×D₃	1	2	yes	—
113	C₂⁵×C₃	#120	1	1	yes	characteristic
114	C₂⁵×C₃	#120	1	1	yes	characteristic
115	C₂⁵×C₃	#120	1	1	yes	characteristic
116	C₂⁵×C₃	#120	1	1	yes	characteristic
117	C₂⁵×C₃	#120	1	1	yes	characteristic
118	C₂⁵×C₃	#121	1	1	yes	characteristic
119	C₂⁵×C₃	#121	1	1	yes	characteristic
120	C₂⁵×C₃	#121	1	1	yes	characteristic
121	C₂⁵×C₃	#121	1	1	yes	characteristic
122	C₂⁵×C₃	#127	1	1	yes	characteristic
123	C₂⁵×C₃	#127	1	1	yes	characteristic
124	C₂⁵×C₃	C₂²×D₄	1	1	yes	characteristic
125	C₂⁵×C₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₃×D₄	#120	1	1	yes	characteristic
127	C₂²×C₃×D₄	#120	1	1	yes	characteristic
128	C₂²×C₃×D₄	#120	1	1	yes	characteristic
129	C₂²×C₃×D₄	#120	1	1	yes	characteristic
130	C₂²×C₃×D₄	#121	1	1	yes	characteristic
131	C₂²×C₃×D₄	#121	1	1	yes	characteristic
132	C₂²×C₃×D₄	#121	1	1	yes	characteristic
133	C₂²×C₃×D₄	#121	1	1	yes	characteristic
134	C₂²×C₃×D₄	#121	1	1	yes	characteristic
135	C₂²×C₃×D₄	#121	1	1	yes	characteristic
136	C₂²×C₃×D₄	#127	1	1	yes	characteristic
137	C₂²×C₃×D₄	#127	1	1	yes	characteristic
138	C₂²×C₃×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₃×D₄	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₃×D₄	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₃×D₄	C₂×#41	1	1	yes	characteristic
142	C₂³×C₃×C₄	#120	1	1	yes	characteristic
143	C₂³×C₃×C₄	#120	1	1	yes	characteristic
144	C₂³×C₃×C₄	#120	1	1	yes	characteristic
145	C₂³×C₃×C₄	#121	1	1	yes	characteristic
146	C₂³×C₃×C₄	#121	1	1	yes	characteristic
147	C₂³×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
148	C₂⁶	D₃×D₄	1	1	yes	characteristic
149	C₂⁶	D₃×D₄	1	1	yes	characteristic
150	C₂³×Q₈	C₂×D₁₂	1	1	yes	characteristic
151	C₂³×D₄	C₂×D₁₂	1	1	yes	characteristic
152	C₂³×D₄	C₂×D₁₂	1	1	yes	characteristic
153	C₂³×D₄	#235	1	1	yes	characteristic
154	C₂³×D₄	#235	1	1	yes	characteristic
155	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
156	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
157	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
158	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
159	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
160	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
161	C₂³×D₄	C₂×#67	1	1	yes	characteristic
162	C₂³×D₄	C₂×#67	1	1	yes	characteristic
163	C₂³×D₄	C₂³×D₃	1	1	yes	characteristic
164	C₂⁴×C₄	C₂×D₁₂	1	1	yes	characteristic
165	C₂²×C₄²	C₂×D₁₂	1	1	yes	characteristic
166	C₂²×C₄²	C₂×D₁₂	1	1	yes	characteristic
167	C₂×#99	C₂×#67	1	4	yes	—
168	C₂⁴×C₃	#392	1	1	yes	characteristic
169	C₂⁴×C₃	#392	1	1	yes	characteristic
170	C₂⁴×C₃	#392	1	1	yes	characteristic
171	C₂⁴×C₃	#392	1	1	yes	characteristic
172	C₂⁴×C₃	#392	1	1	yes	characteristic
173	C₂⁴×C₃	#392	1	1	yes	characteristic
174	C₂⁴×C₃	B(4,𝔽₂)	1	1	yes	characteristic
175	C₂⁴×C₃	B(4,𝔽₂)	1	1	yes	characteristic
176	C₂⁴×C₃	B(4,𝔽₂)	1	1	yes	characteristic
177	C₂⁴×C₃	B(4,𝔽₂)	1	1	yes	characteristic
178	C₂⁴×C₃	C₂×#120	1	1	yes	characteristic
179	C₂⁴×C₃	C₂×#120	1	1	yes	characteristic
180	C₂⁴×C₃	C₂×#121	1	1	yes	characteristic
181	C₂⁴×C₃	C₂×#121	1	1	yes	characteristic
182	C₂⁴×C₃	C₂×#127	1	1	yes	characteristic
183	C₂×C₃×D₄	B(4,𝔽₂)	1	4	yes	—
184	C₂²×C₃×C₄	#392	1	1	yes	characteristic
185	C₂²×C₃×C₄	#392	1	1	yes	characteristic
186	C₂²×C₃×C₄	C₂×#120	1	1	yes	characteristic
187	C₂²×C₃×C₄	C₂×#121	1	1	yes	characteristic
188	C₂⁵	#882	1	1	yes	characteristic
189	C₂⁵	#882	1	1	yes	characteristic
190	C₂⁵	#882	1	1	yes	characteristic
191	C₂⁵	#884	1	1	yes	characteristic
192	C₂⁵	#884	1	1	yes	characteristic
193	C₂⁵	#884	1	1	yes	characteristic
194	C₂⁵	#884	1	1	yes	characteristic
195	C₂⁵	#937	1	1	yes	characteristic
196	C₂⁵	#937	1	1	yes	characteristic
197	C₂⁵	#940	1	1	yes	characteristic
198	C₂⁵	#940	1	1	yes	characteristic
199	C₂⁵	C₂×D₃×D₄	1	1	yes	characteristic
200	C₂⁵	C₂×D₃×D₄	1	1	yes	characteristic
201	C₂²×D₄	#882	1	1	yes	characteristic
202	C₂²×D₄	#882	1	1	yes	characteristic
203	C₂²×D₄	#884	1	1	yes	characteristic
204	C₂²×D₄	#884	1	1	yes	characteristic
205	C₂²×D₄	#884	1	1	yes	characteristic
206	C₂²×D₄	#884	1	1	yes	characteristic
207	C₂²×D₄	#930	1	1	yes	characteristic
208	C₂²×D₄	#930	1	1	yes	characteristic
209	C₂²×D₄	#937	1	1	yes	characteristic
210	C₂²×D₄	#937	1	1	yes	characteristic
211	C₂²×D₄	#940	1	1	yes	characteristic
212	C₂²×D₄	#940	1	1	yes	characteristic
213	C₂²×D₄	C₂×#235	1	1	yes	characteristic
214	C₂²×D₄	C₂×D₃×D₄	1	1	yes	characteristic
215	C₂²×D₄	C₂×D₃×D₄	1	1	yes	characteristic
216	C₂²×D₄	C₂²×#67	1	1	yes	characteristic
217	C₂³×C₄	#882	1	1	yes	characteristic
218	C₂³×C₄	#882	1	1	yes	characteristic
219	C₂³×C₄	#882	1	1	yes	characteristic
220	C₂³×C₄	#930	1	1	yes	characteristic
221	C₂³×C₄	#930	1	1	yes	characteristic
222	C₂³×C₄	C₂²×D₁₂	1	1	yes	characteristic
223	C₂³×C₃	#2021	1	2	yes	—
224	C₂³×C₃	#2021	1	1	yes	characteristic
225	C₂³×C₃	#2021	1	1	yes	characteristic
226	C₂³×C₃	#2021	1	1	yes	characteristic
227	C₂³×C₃	#2021	1	1	yes	characteristic
228	C₂³×C₃	C₂×#392	1	1	yes	characteristic
229	C₂³×C₃	C₂×B(4,𝔽₂)	1	1	yes	characteristic
230	C₂³×C₃	C₂×B(4,𝔽₂)	1	1	yes	characteristic
231	C₂³×C₃	C₂×B(4,𝔽₂)	1	2	yes	—
232	C₂×C₃×C₄	#2021	1	2	yes	—
233	C₂×C₃×C₄	#3026	1	2	yes	—
234	C₂⁴	#4741	1	1	yes	characteristic
235	C₂⁴	#4741	1	1	yes	characteristic
236	C₂⁴	#4741	1	1	yes	characteristic
237	C₂⁴	#4741	1	1	yes	characteristic
238	C₂⁴	#4960	1	1	yes	characteristic
239	C₂⁴	#4960	1	1	yes	characteristic
240	C₂⁴	#4960	1	1	yes	characteristic
241	C₂⁴	#4960	1	1	yes	characteristic
242	C₂⁴	#4960	1	1	yes	characteristic
243	C₂⁴	#4960	1	1	yes	characteristic
244	C₂⁴	C₂×#882	1	1	yes	characteristic
245	C₂⁴	C₂×#884	1	1	yes	characteristic
246	C₂⁴	C₂×#884	1	1	yes	characteristic
247	C₂⁴	C₂×#937	1	1	yes	characteristic
248	C₂⁴	C₂×#940	1	1	yes	characteristic
249	C₂×D₄	#5032	1	4	yes	—
250	C₂²×C₄	#4960	1	1	yes	characteristic
251	C₂²×C₄	#4960	1	1	yes	characteristic
252	C₂²×C₄	C₂×#882	1	1	yes	characteristic
253	C₂²×C₄	C₂×#930	1	1	yes	characteristic
254	C₂²×C₃	C₂×#2021	1	1	yes	characteristic
255	C₂²×C₃	C₂×#2021	1	1	yes	characteristic
256	C₂²×C₃	#23444	1	1	yes	characteristic
257	C₂²×C₃	#33551	1	1	yes	characteristic
258	C₂²×C₃	#33551	1	1	yes	characteristic
259	C₂²×C₃	#33551	1	1	yes	characteristic
260	C₂²×C₃	#33552	1	1	yes	characteristic
261	C₂³	#69500	1	2	yes	—
262	C₂³	#70037	1	1	yes	characteristic
263	C₂³	#70037	1	1	yes	characteristic
264	C₂³	#70037	1	1	yes	characteristic
265	C₂³	#70037	1	1	yes	characteristic
266	C₂³	C₂×#4741	1	1	yes	characteristic
267	C₂³	C₂×#4741	1	1	yes	characteristic
268	C₂³	C₂×#4960	1	1	yes	characteristic
269	C₂³	C₂×#5032	1	2	yes	—
270	C₂×C₄	#69500	1	2	yes	—
271	C₂×C₄	#78686	1	2	yes	—
272	C₂×C₃	#233541	1	1	yes	characteristic
273	C₂×C₃	#498927	1	1	yes	characteristic
274	C₂×C₃	#498931	1	1	yes	characteristic
275	C₂²	C₂×#70037	1	1	yes	characteristic
276	C₂²	C₂×#69500	1	1	yes	characteristic
277	C₂²	#10822438	1	1	yes	characteristic
278	C₂²	#10949251	1	1	yes	characteristic
279	C₂²	#10949251	1	1	yes	characteristic
280	C₂²	#10949251	1	1	yes	characteristic
281	C₂²	#10949262	1	1	yes	characteristic
282	C₃	#72213293	1	1	yes	characteristic, complemented, Sylow
283	C₂	#72248395	1	1	yes	characteristic
284	C₂	#72248443	1	1	yes	characteristic
285	C₂	#72248688	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#70037
commutator, cosocle	C₂³×C₃×D₄	C₂⁴
Frattini	C₂³×D₄	C₂³×D₃
Fitting	C₃×#493370	C₂
socle	C₂²×C₃	C₂×#2021
hypercenter	#493370	D₃
hypocenter	C₃	#72213293
LCS	G ▷ C₂³×C₃×D₄ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ #493370

Complements

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

N	H ≅ G/N	type
1. #72255217	C₂	semidirect
2. #72255216	C₂	semidirect
3. #72255215	C₂	semidirect
4. #72255213	C₂	semidirect
5. #72255069	C₂	semidirect
6. #72254969	C₂	semidirect
7. C₃×#493370	C₂	semidirect
8. #72249311	C₂	semidirect
9. #72249307	C₂	semidirect
10. #72248732	C₂	semidirect
11. #72248668	C₂	semidirect
19. #10740386	C₂²	semidirect
20. #10733626	C₂²	semidirect
21. #10732887	C₂²	semidirect
22. #10732882	C₂²	semidirect
24. #10711423	C₂²	semidirect
25. #10710646	C₂²	semidirect
28. #10682458	C₂²	semidirect
29. #10682458	C₂²	semidirect
33. C₃×#12694	C₂²	semidirect
34. C₃×#12693	C₂²	semidirect
35. #493370	D₃	semidirect
36. C₂³×C₃×#41	D₄	semidirect
39. C₂²×#1004	D₄	semidirect
40. C₂²×C₃×#127	D₄	semidirect
42. C₂²×C₃×#124	D₄	semidirect
44. C₂²×C₃×#117	D₄	semidirect
45. C₂³×#217	D₄	semidirect
46. #87239	D₄	semidirect
50. C₂×#5798	D₄	semidirect
51. C₂²×C₃×#99	D₄	semidirect
53. C₂×#5247	D₄	semidirect
56. C₂²×#833	D₄	semidirect
57. C₃×#1781	D₄	semidirect
60. #72131	D₄	semidirect
61. #72131	D₄	semidirect
64. #72129	D₄	semidirect
69. #12694	C₂×D₃	semidirect
70. #12693	C₂×D₃	semidirect
102. C₂³×#41	D₁₂	semidirect
104. C₂²×#127	D₁₂	semidirect
106. C₂²×#124	D₁₂	semidirect
108. C₂²×#117	D₁₂	semidirect
109. C₂²×#99	#67	semidirect
110. #1781	#67	semidirect
282. C₃	#72213293	semidirect

Maximal subgroups

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

subgroup	index
#72255217	2
#72255216	2
#72255215	2
#72255213	2
#72255069	2
#72254969	2
C₃×#493370	2
#72249311	2
#72249307	2
#72248732	2
#72248668	2
#72213293	3