Subgroups — #72306739

Order 2048.

At least 6569 conjugacy classes of subgroups; only the 766 normal subgroups are shown.

The subgroup lattice has 306 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72306738	C₂	1	2	yes	maximal, complemented
2	#72306736	C₂	1	2	yes	maximal, complemented
3	#72306734	C₂	1	2	yes	maximal, complemented
4	#72306733	C₂	1	2	yes	maximal, complemented
5	#72306726	C₂	1	1	yes	maximal, characteristic, complemented
6	#72306624	C₂	1	8	yes	maximal, complemented
7	#72306581	C₂	1	1	yes	maximal, characteristic, complemented
8	#72286504	C₂	1	1	yes	maximal, characteristic, complemented
9	#72283294	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#152119	C₂	1	1	yes	maximal, characteristic, complemented
11	#72279417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#7487867	C₂	1	1	yes	maximal, characteristic, complemented
13	#72231252	C₂	1	8	yes	maximal, complemented
14	#10586163	C₂²	1	1	yes	characteristic, complemented
15	C₂×#61110	C₂²	1	1	yes	characteristic, complemented
16	C₂×#60801	C₂²	1	1	yes	characteristic
17	#7622903	C₂²	1	2	yes	—
18	#7622874	C₂²	1	2	yes	—
19	C₂×#30497	C₂²	1	2	yes	complemented
20	#7487867	C₂²	1	4	yes	complemented
21	#7487867	C₂²	1	4	yes	complemented
22	C₂×#23994	C₂²	1	2	yes	—
23	C₂×#23361	C₂²	1	1	yes	characteristic, complemented
24	#6349118	C₂²	1	4	yes	complemented
25	#6348994	C₂²	1	4	yes	complemented
26	#6348453	C₂²	1	1	yes	characteristic, complemented
27	#6348453	C₂²	1	1	yes	characteristic, complemented
28	#6348452	C₂²	1	1	yes	characteristic
29	#6348452	C₂²	1	1	yes	characteristic
30	#6344056	C₂²	1	1	yes	characteristic, complemented
31	#512900	C₂²	1	8	yes	complemented
32	#512864	C₂²	1	8	yes	complemented
33	C₂×#13591	C₂²	1	1	yes	characteristic, complemented
34	C₂×#13585	C₂²	1	1	yes	characteristic, complemented
35	C₂×#13580	C₂²	1	2	yes	complemented
36	#512585	C₂²	1	2	yes	complemented
37	#512564	C₂²	1	2	yes	complemented
38	#511842	C₂²	1	8	yes	complemented
39	#511839	C₂²	1	8	yes	complemented
40	#493634	C₂²	1	2	yes	complemented
41	#493605	C₂²	1	2	yes	complemented
42	C₂×#12719	C₂²	1	2	yes	complemented
43	#153150	C₂²	1	16	yes	complemented
44	#153148	C₂²	1	16	yes	complemented
45	#152689	C₂²	1	8	yes	complemented
46	#152688	C₂²	1	8	yes	complemented
47	#152606	C₂²	1	8	yes	complemented
48	#152604	C₂²	1	8	yes	complemented
49	#152119	C₂²	1	4	yes	complemented
50	#152119	C₂²	1	4	yes	complemented
51	#151820	C₂²	1	2	yes	complemented
52	#151784	C₂²	1	2	yes	complemented
53	#62762	C₂³	1	2	yes	complemented
54	#62759	C₂³	1	1	yes	characteristic
55	#62759	C₂³	1	1	yes	characteristic
56	#41²	C₂³	1	2	yes	complemented
57	C₂×#3444	C₂³	1	1	yes	characteristic
58	C₂²×#532	D₄	1	1	yes	characteristic, complemented
59	#61110	C₂³	1	4	yes	—
60	#60801	D₄	1	2	yes	complemented
61	#60801	D₄	1	2	yes	complemented
62	#60801	C₂³	1	2	yes	—
63	#60801	C₂³	1	2	yes	—
64	C₂×#3152	C₂³	1	1	yes	characteristic
65	C₂×#3151	C₂³	1	1	yes	characteristic
66	C₂×#3139	D₄	1	1	yes	characteristic, complemented
67	C₂×#3138	D₄	1	1	yes	characteristic, complemented
68	C₂×#3025	D₄	1	2	yes	complemented
69	C₂×#3018	C₂³	1	2	yes	—
70	C₂×#2920	D₄	1	1	yes	characteristic, complemented
71	C₂×#2901	D₄	1	2	yes	complemented
72	#33567	C₂³	1	8	yes	—
73	#33546	C₂³	1	8	yes	—
74	C₂×#2132	C₂³	1	2	yes	—
75	C₂×#2126	C₂³	1	2	yes	—
76	#30497	C₂³	1	8	yes	—
77	#23994	D₄	1	4	yes	complemented
78	#23994	D₄	1	4	yes	complemented
79	#23994	C₂³	1	4	yes	—
80	#23994	C₂³	1	4	yes	—
81	#23361	C₂³	1	4	yes	—
82	C₂×#1891	D₄	1	2	yes	complemented
83	C₂×#1884	C₂³	1	2	yes	—
84	C₂×#1827	D₄	1	2	yes	complemented
85	#18718	D₄	1	2	yes	complemented
86	#18718	D₄	1	2	yes	complemented
87	#18714	D₄	1	2	yes	complemented
88	#18714	D₄	1	2	yes	complemented
89	#18684	D₄	1	4	yes	complemented
90	#18682	D₄	1	4	yes	complemented
91	#18549	C₂³	1	4	yes	complemented
92	#18548	C₂³	1	4	yes	—
93	#18528	C₂³	1	4	yes	complemented
94	#18527	C₂³	1	4	yes	—
95	#17757	C₂³	1	2	yes	complemented
96	#17752	C₂³	1	2	yes	complemented
97	#17738	C₂³	1	1	yes	characteristic
98	#17738	C₂³	1	1	yes	characteristic
99	#13591	C₂³	1	4	yes	—
100	#13585	C₂³	1	4	yes	—
101	#13580	C₂³	1	8	yes	complemented
102	#13574	C₂³	1	8	yes	—
103	#13562	C₂³	1	8	yes	—
104	#12773	C₂³	1	8	yes	complemented
105	#12765	C₂³	1	8	yes	complemented
106	#12719	C₂³	1	4	yes	complemented
107	#12719	C₂³	1	4	yes	complemented
108	#7496	C₂³	1	8	yes	complemented
109	#7482	C₂³	1	8	yes	complemented
110	C₂×#585	C₂×D₄	1	2	yes	—
111	C₂×#568	C₂×D₄	1	2	yes	—
112	#3550	C₂×D₄	1	1	yes	characteristic
113	#3550	C₂×D₄	1	1	yes	characteristic
114	#3549	C₂×D₄	1	1	yes	characteristic
115	#3549	C₂×D₄	1	1	yes	characteristic
116	#3544	C₂×D₄	1	2	yes	—
117	#3543	C₂×D₄	1	2	yes	—
118	#3483	C₂⁴	1	2	yes	—
119	#3481	C₂⁴	1	2	yes	—
120	#3444	C₂×D₄	1	1	yes	characteristic
121	#3444	C₂×D₄	1	1	yes	characteristic
122	#3444	C₂×D₄	1	2	yes	—
123	#3444	C₂×D₄	1	2	yes	—
124	#3444	C₂⁴	1	1	yes	characteristic
125	#3444	C₂⁴	1	1	yes	characteristic
126	C₂×#533	C₂×D₄	1	1	yes	characteristic
127	C₂×#532	C₂×D₄	1	2	yes	—
128	C₂×#532	C₂×D₄	1	2	yes	—
129	C₂×#532	C₂⁴	1	1	yes	characteristic
130	C₂²×C₄⋊Q₈	C₂×D₄	1	1	yes	characteristic
131	C₂²×#127	C₂×D₄	1	1	yes	characteristic
132	C₂×#525	C₂×D₄	1	2	yes	—
133	C₂×C₄×#41	C₂×D₄	1	2	yes	—
134	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
135	#3152	C₂×D₄	1	2	yes	—
136	#3152	C₂⁴	1	2	yes	—
137	#3151	C₂×D₄	1	2	yes	—
138	#3151	C₂×D₄	1	2	yes	—
139	#3151	C₂×D₄	1	2	yes	—
140	#3151	C₂⁴	1	2	yes	—
141	C₂×#493	C₂×D₄	1	2	yes	—
142	C₂×#492	C₂×D₄	1	2	yes	—
143	#3139	C₂×D₄	1	2	yes	—
144	#3139	C₂×D₄	1	2	yes	—
145	#3138	C₂×D₄	1	2	yes	—
146	#3138	C₂×D₄	1	2	yes	—
147	#3025	C₂×D₄	1	4	yes	—
148	#3025	C₂×D₄	1	4	yes	—
149	#3018	C₂×D₄	1	4	yes	—
150	#3018	C₂⁴	1	4	yes	—
151	#2920	C₂×D₄	1	2	yes	—
152	#2920	C₂×D₄	1	2	yes	—
153	#2901	C₂×D₄	1	4	yes	—
154	#2901	C₂×D₄	1	4	yes	—
155	C₂×#413	C₂×D₄	1	2	yes	—
156	C₂×#410	C₂×D₄	1	2	yes	—
157	C₂×C₄×M₁₆	C₂×D₄	1	1	yes	characteristic
158	C₂×C₄×M₁₆	C₂×D₄	1	1	yes	characteristic
159	#2132	C₂⁴	1	4	yes	—
160	#2126	C₂⁴	1	4	yes	—
161	#1891	C₂×D₄	1	4	yes	—
162	#1891	C₂×D₄	1	4	yes	—
163	#1884	C₂×D₄	1	4	yes	—
164	#1884	C₂⁴	1	4	yes	—
165	#1827	C₂×D₄	1	4	yes	—
166	#1827	C₂×D₄	1	4	yes	—
167	#1624	C₂×D₄	1	8	yes	complemented
168	#1623	C₂×D₄	1	8	yes	complemented
169	#1620	C₂×D₄	1	8	yes	complemented
170	#1619	C₂×D₄	1	8	yes	complemented
171	#1560	C₂⁴	1	2	yes	—
172	#1560	C₂⁴	1	2	yes	—
173	#1558	C₂×D₄	1	2	yes	—
174	#1555	C₂×D₄	1	2	yes	—
175	#1543	C₂×D₄	1	2	yes	—
176	#1543	C₂×D₄	1	2	yes	—
177	#585	C₂²×D₄	1	4	yes	—
178	C₂²×#41	C₂²×D₄	1	2	yes	—
179	C₂⁴×C₄	#120	1	1	yes	characteristic
180	#568	C₂²×D₄	1	4	yes	—
181	#559	C₂²×D₄	1	2	yes	—
182	Q₈²	C₂²×D₄	1	2	yes	—
183	#555	C₂²×D₄	1	2	yes	—
184	D₄²	C₂²×D₄	1	2	yes	—
185	#533	#120	1	2	yes	—
186	#533	C₂²×D₄	1	1	yes	characteristic
187	#533	C₂²×D₄	1	1	yes	characteristic
188	#532	C₂²×D₄	1	1	yes	characteristic
189	#532	C₂²×D₄	1	2	yes	—
190	#532	C₂²×D₄	1	1	yes	characteristic
191	#532	C₂²×D₄	1	1	yes	characteristic
192	#532	H(5,𝔽₂)	1	2	yes	—
193	#532	C₂⁵	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	C₂²×D₄	1	2	yes	—
196	C₂×#127	#120	1	2	yes	—
197	C₂×#127	#120	1	2	yes	—
198	C₂×#127	C₂²×D₄	1	1	yes	characteristic
199	C₂×#127	C₂²×D₄	1	2	yes	—
200	C₂×#124	H(5,𝔽₂)	1	2	yes	—
201	#525	#120	1	2	yes	—
202	#525	#120	1	2	yes	—
203	#525	C₂²×D₄	1	2	yes	—
204	#525	C₂²×D₄	1	2	yes	—
205	C₄×#41	#120	1	2	yes	—
206	C₄×#41	#120	1	2	yes	—
207	C₄×#41	C₂²×D₄	1	2	yes	—
208	C₄×#41	C₂²×D₄	1	2	yes	—
209	C₂×#117	#120	1	2	yes	—
210	C₂²×C₄²	#120	1	1	yes	characteristic
211	C₂²×C₄²	#120	1	1	yes	characteristic
212	C₂²×C₄²	#120	1	1	yes	characteristic
213	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
214	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
215	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
216	#493	#120	1	4	yes	—
217	#493	C₂²×D₄	1	4	yes	—
218	#492	#120	1	4	yes	—
219	#492	C₂²×D₄	1	4	yes	—
220	#413	C₂²×D₄	1	4	yes	—
221	#410	C₂²×D₄	1	4	yes	—
222	C₄×M₁₆	#120	1	2	yes	—
223	C₄×M₁₆	C₂²×D₄	1	2	yes	—
224	C₄×M₁₆	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₈	#120	1	2	yes	—
226	#333	C₂²×D₄	1	4	yes	—
227	#331	C₂²×D₄	1	4	yes	—
228	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
229	C₂×#41	C₂×#120	1	2	yes	—
230	C₂×#41	D₄²	1	2	yes	—
231	C₂×#41	D₄²	1	2	yes	—
232	C₂×#41	C₂³×D₄	1	2	yes	—
233	C₂²×Q₈	D₄²	1	2	yes	—
234	C₂²×D₄	D₄²	1	2	yes	—
235	C₂³×C₄	B(4,𝔽₂)	1	1	yes	characteristic
236	C₂³×C₄	B(4,𝔽₂)	1	1	yes	characteristic
237	C₂³×C₄	B(4,𝔽₂)	1	1	yes	characteristic
238	C₂³×C₄	C₂×#120	1	1	yes	characteristic
239	C₂³×C₄	C₂×#120	1	1	yes	characteristic
240	C₂³×C₄	C₂×#120	1	1	yes	characteristic
241	C₄⋊Q₈	C₂×#120	1	1	yes	characteristic
242	C₄⋊Q₈	D₄²	1	2	yes	—
243	C₄⋊Q₈	C₂³×D₄	1	1	yes	characteristic
244	#127	C₂×#120	1	4	yes	—
245	#127	C₂×#120	1	1	yes	characteristic
246	#127	D₄²	1	2	yes	—
247	#127	C₂³×D₄	1	1	yes	characteristic
248	#124	#534	1	2	yes	—
249	#124	D₄²	1	2	yes	—
250	#124	D₄²	1	2	yes	—
251	#124	C₂×H(5,𝔽₂)	1	2	yes	—
252	#117	C₂×#120	1	2	yes	—
253	#117	C₂×#120	1	2	yes	—
254	C₂×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
255	C₂×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
257	C₂×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄²	C₂×#120	1	1	yes	characteristic
259	C₂×C₄²	C₂×#120	1	1	yes	characteristic
260	C₂×C₄²	C₂×#120	1	1	yes	characteristic
261	C₂×C₄²	C₂×#120	1	1	yes	characteristic
262	C₂×C₄²	C₂×#120	1	1	yes	characteristic
263	C₂×C₄²	#534	1	1	yes	characteristic
264	C₂×C₄²	#534	1	1	yes	characteristic
265	C₂×C₄²	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₈	C₂×#120	1	4	yes	—
267	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
268	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
269	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
270	C₂×Q₈	#2403	1	2	yes	—
271	C₂×Q₈	C₂×D₄²	1	2	yes	—
272	C₂×D₄	#2403	1	2	yes	—
273	C₂×D₄	C₂×D₄²	1	2	yes	—
274	C₂²×C₄	#2403	1	1	yes	characteristic
275	C₂²×C₄	#2403	1	1	yes	characteristic
276	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
277	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
278	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
279	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
280	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
281	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
282	C₄²	#2403	1	1	yes	characteristic
283	C₄²	#2403	1	1	yes	characteristic
284	C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
285	C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
286	C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
287	C₄²	C₂×B(4,𝔽₂)	1	1	yes	characteristic
288	C₄²	C₂²×#120	1	1	yes	characteristic
289	C₄²	C₂×#534	1	1	yes	characteristic
290	C₂³	#23900	1	1	yes	characteristic
291	C₂³	#23900	1	1	yes	characteristic
292	C₂³	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
293	C₂×C₄	#23900	1	1	yes	characteristic
294	C₂×C₄	#23900	1	1	yes	characteristic
295	C₂×C₄	#33553	1	2	yes	—
296	C₂×C₄	#33553	1	2	yes	—
297	C₂×C₄	C₂×#2403	1	1	yes	characteristic
298	C₂×C₄	C₂²×B(4,𝔽₂)	1	1	yes	characteristic
299	C₂²	#499831	1	1	yes	characteristic
300	C₂²	#499831	1	1	yes	characteristic
301	C₂²	C₂×#23900	1	1	yes	characteristic
302	C₄	#499831	1	2	yes	—
303	C₄	C₂×#33553	1	2	yes	—
304	C₂	C₂×#499831	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#499831
commutator, Frattini, cosocle	#532	C₂⁵
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72306738	C₂	semidirect
2. #72306736	C₂	semidirect
3. #72306734	C₂	semidirect
4. #72306733	C₂	semidirect
5. #72306726	C₂	semidirect
6. #72306624	C₂	semidirect
7. #72306581	C₂	semidirect
8. #72286504	C₂	semidirect
9. #72283294	C₂	semidirect
10. C₂×#152119	C₂	semidirect
11. #72279417	C₂	semidirect
12. C₂×#7487867	C₂	semidirect
13. #72231252	C₂	semidirect
14. #10586163	C₂²	semidirect
15. C₂×#61110	C₂²	semidirect
19. C₂×#30497	C₂²	semidirect
20. #7487867	C₂²	semidirect
21. #7487867	C₂²	semidirect
23. C₂×#23361	C₂²	semidirect
24. #6349118	C₂²	semidirect
25. #6348994	C₂²	semidirect
26. #6348453	C₂²	semidirect
27. #6348453	C₂²	semidirect
30. #6344056	C₂²	semidirect
31. #512900	C₂²	semidirect
32. #512864	C₂²	semidirect
33. C₂×#13591	C₂²	semidirect
34. C₂×#13585	C₂²	semidirect
35. C₂×#13580	C₂²	semidirect
36. #512585	C₂²	semidirect
37. #512564	C₂²	semidirect
38. #511842	C₂²	semidirect
39. #511839	C₂²	semidirect
40. #493634	C₂²	semidirect
41. #493605	C₂²	semidirect
42. C₂×#12719	C₂²	semidirect
43. #153150	C₂²	semidirect
44. #153148	C₂²	semidirect
45. #152689	C₂²	semidirect
46. #152688	C₂²	semidirect
47. #152606	C₂²	semidirect
48. #152604	C₂²	semidirect
49. #152119	C₂²	semidirect
50. #152119	C₂²	semidirect
51. #151820	C₂²	semidirect
52. #151784	C₂²	semidirect
53. #62762	C₂³	semidirect
56. #41²	C₂³	semidirect
58. C₂²×#532	D₄	semidirect
60. #60801	D₄	semidirect
61. #60801	D₄	semidirect
66. C₂×#3139	D₄	semidirect
67. C₂×#3138	D₄	semidirect
68. C₂×#3025	D₄	semidirect
70. C₂×#2920	D₄	semidirect
71. C₂×#2901	D₄	semidirect
77. #23994	D₄	semidirect
78. #23994	D₄	semidirect
82. C₂×#1891	D₄	semidirect
84. C₂×#1827	D₄	semidirect
85. #18718	D₄	semidirect
86. #18718	D₄	semidirect
87. #18714	D₄	semidirect
88. #18714	D₄	semidirect
89. #18684	D₄	semidirect
90. #18682	D₄	semidirect
91. #18549	C₂³	semidirect
93. #18528	C₂³	semidirect
95. #17757	C₂³	semidirect
96. #17752	C₂³	semidirect
101. #13580	C₂³	semidirect
104. #12773	C₂³	semidirect
105. #12765	C₂³	semidirect
106. #12719	C₂³	semidirect
107. #12719	C₂³	semidirect
108. #7496	C₂³	semidirect
109. #7482	C₂³	semidirect
167. #1624	C₂×D₄	semidirect
168. #1623	C₂×D₄	semidirect
169. #1620	C₂×D₄	semidirect
170. #1619	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72306738	2
#72306736	2
#72306734	2
#72306733	2
#72306726	2
#72306624	2
#72306581	2
#72286504	2
#72283294	2
C₂×#152119	2
#72279417	2
C₂×#7487867	2
#72231252	2