Subgroups — #72251880

Order 2048.

At least 6336 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

The subgroup lattice has 294 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251879	C₂	1	2	yes	maximal, complemented
2	#72251877	C₂	1	4	yes	maximal, complemented
3	#72251876	C₂	1	4	yes	maximal, complemented
4	#72251875	C₂	1	4	yes	maximal, complemented
5	#72237547	C₂	1	1	yes	maximal, characteristic, complemented
6	#72235088	C₂	1	1	yes	maximal, characteristic, complemented
7	#72234144	C₂	1	4	yes	maximal, complemented
8	#72232342	C₂	1	1	yes	maximal, characteristic, complemented
9	#72231750	C₂	1	4	yes	maximal, complemented
10	#72215568	C₂	1	2	yes	maximal, complemented
11	#72213496	C₂	1	2	yes	maximal, complemented
12	#72210515	C₂	1	2	yes	maximal, complemented
13	#10586169	C₂²	1	1	yes	characteristic, complemented
14	C₂×#52004	C₂²	1	1	yes	characteristic, complemented
15	C₂×#35222	C₂²	1	1	yes	characteristic
16	#6499837	C₂²	1	4	yes	—
17	#6499837	C₂²	1	4	yes	complemented
18	#6470228	C₂²	1	4	yes	complemented
19	#6437449	C₂²	1	2	yes	—
20	#6437449	C₂²	1	2	yes	complemented
21	#6374136	C₂²	1	1	yes	characteristic, complemented
22	#4941683	C₂²	1	2	yes	complemented
23	#4607160	C₂²	1	2	yes	complemented
24	#4312017	C₂²	1	2	yes	complemented
25	#2922862	C₂²	1	2	yes	complemented
26	#1418385	C₂²	1	2	yes	complemented
27	#619512	C₂²	1	2	yes	—
28	#581605	C₂²	1	1	yes	characteristic, complemented
29	#512119	C₂²	1	4	yes	complemented
30	#511771	C₂²	1	4	yes	complemented
31	#508086	C₂²	1	8	yes	complemented
32	#496228	C₂²	1	4	yes	complemented
33	#493372	C₂²	1	2	yes	complemented
34	#493372	C₂²	1	2	yes	complemented
35	#347055	C₂²	1	4	yes	complemented
36	#347052	C₂²	1	4	yes	complemented
37	#347043	C₂²	1	4	yes	complemented
38	#347043	C₂²	1	4	yes	complemented
39	#188029	C₂²	1	2	yes	complemented
40	#187931	C₂²	1	4	yes	complemented
41	#144414	C₂²	1	8	yes	complemented
42	#144414	C₂²	1	8	yes	complemented
43	#144386	C₂²	1	8	yes	complemented
44	#144386	C₂²	1	8	yes	complemented
45	#139827	C₂²	1	4	yes	complemented
46	#139826	C₂²	1	4	yes	complemented
47	#139778	C₂²	1	4	yes	complemented
48	#139776	C₂²	1	4	yes	complemented
49	#139713	C₂²	1	4	yes	complemented
50	#139713	C₂²	1	4	yes	complemented
51	#139706	C₂²	1	8	yes	complemented
52	#133828	C₂²	1	4	yes	complemented
53	#133828	C₂²	1	4	yes	complemented
54	#128074	C₂²	1	4	yes	complemented
55	C₂⁵×D₄	D₄	1	1	yes	characteristic, complemented
56	#62818	C₂³	1	4	yes	—
57	#62811	C₂³	1	2	yes	—
58	C₂²×D₄²	C₂³	1	1	yes	characteristic, complemented
59	C₂²×#535	C₂³	1	1	yes	characteristic
60	C₂²×#535	C₂³	1	1	yes	characteristic
61	C₂²×#525	C₂³	1	1	yes	characteristic
62	C₂×#3021	D₄	1	4	yes	complemented
63	C₂×#2669	C₂³	1	2	yes	complemented
64	C₂×#2534	D₄	1	2	yes	complemented
65	C₂×#2404	C₂³	1	2	yes	—
66	C₂×#2435	D₄	1	2	yes	complemented
67	C₂×#2435	D₄	1	2	yes	complemented
68	C₂×#2433	C₂³	1	2	yes	—
69	C₂×#2433	C₂³	1	2	yes	—
70	C₂×#2380	C₂³	1	1	yes	characteristic, complemented
71	C₂×#2321	D₄	1	2	yes	complemented
72	C₂×#2309	D₄	1	1	yes	characteristic, complemented
73	C₂×#2309	D₄	1	1	yes	characteristic, complemented
74	C₂×#2280	D₄	1	1	yes	characteristic, complemented
75	#24056	C₂³	1	8	yes	complemented
76	#23913	D₄	1	8	yes	complemented
77	#23906	C₂³	1	8	yes	—
78	#23906	C₂³	1	4	yes	—
79	#23906	C₂³	1	4	yes	—
80	#23900	C₂³	1	8	yes	complemented
81	#21752	C₂³	1	4	yes	—
82	#20956	D₄	1	4	yes	complemented
83	#20591	C₂³	1	2	yes	—
84	#20584	C₂³	1	4	yes	—
85	#15984	C₂³	1	2	yes	—
86	#15982	D₄	1	2	yes	complemented
87	#15948	D₄	1	2	yes	complemented
88	#15948	D₄	1	2	yes	complemented
89	#15947	D₄	1	2	yes	complemented
90	#15313	C₂³	1	2	yes	complemented
91	#15313	C₂³	1	2	yes	complemented
92	#15260	C₂³	1	2	yes	—
93	#15258	C₂³	1	2	yes	complemented
94	#15258	C₂³	1	2	yes	complemented
95	#15257	C₂³	1	2	yes	—
96	#15257	C₂³	1	2	yes	—
97	#14661	C₂³	1	1	yes	characteristic, complemented
98	#14419	C₂³	1	2	yes	complemented
99	#14344	D₄	1	2	yes	complemented
100	#14154	C₂³	1	1	yes	characteristic, complemented
101	#13940	D₄	1	2	yes	complemented
102	#13937	C₂³	1	2	yes	—
103	#12852	C₂³	1	8	yes	complemented
104	#12693	C₂³	1	4	yes	complemented
105	#12693	C₂³	1	4	yes	complemented
106	#12693	C₂³	1	8	yes	complemented
107	#12687	C₂³	1	8	yes	complemented
108	#8542	C₂³	1	4	yes	—
109	#8542	C₂³	1	4	yes	—
110	#8534	C₂³	1	8	yes	complemented
111	#8534	C₂³	1	8	yes	complemented
112	#7304	C₂³	1	8	yes	complemented
113	#7304	C₂³	1	8	yes	complemented
114	C₂×#585	C₂×D₄	1	2	yes	complemented
115	C₂³×#41	C₂⁴	1	1	yes	characteristic
116	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
117	C₂⁴×D₄	C₂×D₄	1	2	yes	—
118	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
119	C₂⁴×D₄	C₂×D₄	1	2	yes	—
120	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
121	#3485	C₂×D₄	1	4	yes	complemented
122	#3484	C₂⁴	1	4	yes	—
123	C₂×#535	C₂×D₄	1	4	yes	—
124	C₂×#535	C₂×D₄	1	4	yes	—
125	C₂²×#127	C₂⁴	1	2	yes	—
126	C₂²×#127	C₂⁴	1	1	yes	characteristic
127	C₂²×#127	C₂⁴	1	1	yes	characteristic
128	C₂²×#124	C₂×D₄	1	2	yes	—
129	C₂²×#124	C₂×D₄	1	1	yes	characteristic
130	C₂²×#124	C₂×D₄	1	1	yes	characteristic
131	C₂²×#123	C₂×D₄	1	2	yes	—
132	C₂²×#123	C₂×D₄	1	2	yes	—
133	C₂²×C₂²⋊Q₈	C₂×D₄	1	2	yes	—
134	C₂²×#121	C₂⁴	1	2	yes	—
135	C₂²×#120	C₂×D₄	1	2	yes	—
136	C₂²×#120	C₂×D₄	1	2	yes	—
137	C₂²×#120	C₂×D₄	1	2	yes	—
138	C₂²×#120	C₂×D₄	1	2	yes	—
139	C₂²×#120	C₂⁴	1	2	yes	—
140	#3428	C₂×D₄	1	2	yes	complemented
141	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
142	C₂²×C₄×D₄	C₂×D₄	1	2	yes	—
143	C₂²×#117	C₂×D₄	1	1	yes	characteristic
144	C₂²×#117	C₂×D₄	1	1	yes	characteristic
145	C₂²×#117	C₂×D₄	1	1	yes	characteristic
146	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
147	C₂×#453	C₂×D₄	1	8	yes	—
148	C₂×#453	C₂×D₄	1	8	yes	complemented
149	C₂×#413	C₂×D₄	1	4	yes	complemented
150	C₂×#390	C₂×D₄	1	2	yes	—
151	C₂×#390	C₂×D₄	1	2	yes	—
152	C₂×#390	C₂×D₄	1	2	yes	—
153	C₂×#390	C₂×D₄	1	2	yes	—
154	C₂×#386	C₂×D₄	1	2	yes	—
155	C₂×#386	C₂×D₄	1	2	yes	—
156	C₂×#386	C₂⁴	1	2	yes	—
157	C₂×#385	C₂×D₄	1	2	yes	—
158	C₂×#381	C₂×D₄	1	2	yes	—
159	C₂×#381	C₂×D₄	1	2	yes	—
160	C₂×#380	C₂×D₄	1	2	yes	—
161	C₂×#379	C₂×D₄	1	2	yes	—
162	C₂×#379	C₂×D₄	1	2	yes	—
163	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	2	yes	—
164	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	2	yes	—
165	C₂×C₄×C₂²⋊C₄	C₂×D₄	1	2	yes	—
166	#1887	C₂×D₄	1	8	yes	—
167	#1887	C₂×D₄	1	8	yes	complemented
168	#1785	C₂×D₄	1	4	yes	complemented
169	#1783	C₂⁴	1	4	yes	—
170	#1781	C₂⁴	1	8	yes	—
171	#1440	C₂×D₄	1	2	yes	—
172	#1440	C₂×D₄	1	2	yes	—
173	#1440	C₂×D₄	1	2	yes	—
174	#1440	C₂×D₄	1	2	yes	—
175	#1438	C₂×D₄	1	2	yes	—
176	#1438	C₂×D₄	1	2	yes	—
177	#1438	C₂×D₄	1	2	yes	—
178	#1438	C₂×D₄	1	2	yes	—
179	#1438	C₂×D₄	1	2	yes	—
180	#1438	C₂×D₄	1	2	yes	—
181	#1438	C₂⁴	1	2	yes	—
182	#1438	C₂⁴	1	2	yes	—
183	C₂⁶	#120	1	1	yes	characteristic
184	C₂⁶	C₂²×D₄	1	2	yes	—
185	C₂×H(5,𝔽₂)	C₂²×D₄	1	4	yes	—
186	C₂²×#41	C₂²×D₄	1	2	yes	—
187	C₂²×#41	C₂²×D₄	1	2	yes	—
188	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
189	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
190	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
191	C₂³×D₄	#120	1	2	yes	—
192	C₂³×D₄	#120	1	2	yes	—
193	C₂³×D₄	C₂²×D₄	1	2	yes	—
194	C₂³×D₄	C₂²×D₄	1	2	yes	—
195	C₂³×D₄	C₂²×D₄	1	2	yes	—
196	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
197	C₂³×D₄	C₂²×D₄	1	2	yes	—
198	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
199	C₂³×D₄	C₂⁵	1	1	yes	characteristic
200	C₂⁴×C₄	#120	1	1	yes	characteristic
201	C₂⁴×C₄	#120	1	2	yes	—
202	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
203	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
204	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
205	C₂⁴×C₄	C₂²×D₄	1	2	yes	—
206	C₂×#127	C₂²×D₄	1	4	yes	—
207	C₂×#127	C₂²×D₄	1	4	yes	—
208	C₂×#124	#120	1	4	yes	—
209	C₂×#124	#120	1	4	yes	—
210	#518	C₂²×D₄	1	4	yes	—
211	C₂×#117	#120	1	2	yes	—
212	C₂×#117	#120	1	2	yes	—
213	C₂²×C₄⋊C₄	#120	1	2	yes	—
214	C₂²×C₄⋊C₄	#120	1	2	yes	—
215	C₂²×C₄⋊C₄	C₂²×D₄	1	2	yes	—
216	C₂²×C₂²⋊C₄	#120	1	2	yes	—
217	C₂²×C₂²⋊C₄	#120	1	2	yes	—
218	C₂²×C₂²⋊C₄	#120	1	2	yes	—
219	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
220	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
221	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
222	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
223	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
224	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
225	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
226	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
227	C₂²×C₄²	#120	1	1	yes	characteristic
228	C₂²×C₄²	#120	1	1	yes	characteristic
229	C₂²×C₄²	C₂²×D₄	1	2	yes	—
230	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
231	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
232	C₂×#100	C₂²×D₄	1	8	yes	—
233	C₂⁵	C₂×#120	1	1	yes	characteristic
234	C₂⁵	C₂×#120	1	2	yes	—
235	C₂⁵	C₂³×D₄	1	2	yes	—
236	C₂²×Q₈	C₂×#120	1	2	yes	—
237	C₂²×Q₈	#534	1	2	yes	—
238	C₂²×D₄	C₂×#120	1	2	yes	—
239	C₂²×D₄	C₂×#120	1	2	yes	—
240	C₂²×D₄	D₄²	1	4	yes	—
241	C₂²×D₄	D₄²	1	4	yes	—
242	C₂²×D₄	C₂³×D₄	1	2	yes	—
243	C₂²×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
244	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
245	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
246	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
247	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
248	C₂³×C₄	C₂×#120	1	1	yes	characteristic
249	C₂³×C₄	C₂×#120	1	1	yes	characteristic
250	C₂³×C₄	C₂×#120	1	2	yes	—
251	C₂³×C₄	C₂×#120	1	2	yes	—
252	C₂³×C₄	C₂×#120	1	2	yes	—
253	C₂³×C₄	C₂×#120	1	2	yes	—
254	C₂³×C₄	C₂×#120	1	1	yes	characteristic
255	C₂³×C₄	#534	1	1	yes	characteristic
256	C₂³×C₄	#534	1	1	yes	characteristic
257	C₂³×C₄	D₄²	1	4	yes	—
258	C₂³×C₄	D₄²	1	4	yes	—
259	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
260	C₂×C₄²	C₂×#120	1	2	yes	—
261	C₂×C₄²	C₂×#120	1	2	yes	—
262	C₂⁴	#2403	1	1	yes	characteristic
263	C₂⁴	#2403	1	1	yes	characteristic
264	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
265	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
266	C₂⁴	C₂²×#120	1	1	yes	characteristic
267	C₂⁴	C₂×D₄²	1	4	yes	—
268	C₂×D₄	#3026	1	4	yes	—
269	C₂²×C₄	#2403	1	4	yes	—
270	C₂²×C₄	#2403	1	1	yes	characteristic
271	C₂²×C₄	#2403	1	1	yes	characteristic
272	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
273	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
274	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
275	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
276	C₂²×C₄	C₂²×#120	1	1	yes	characteristic
277	C₂²×C₄	C₂×#534	1	1	yes	characteristic
278	C₂³	#23895	1	2	yes	—
279	C₂³	#33553	1	4	yes	—
280	C₂³	C₂×#2403	1	1	yes	characteristic
281	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
282	C₂×C₄	#23900	1	1	yes	characteristic
283	C₂×C₄	#23900	1	2	yes	—
284	C₂×C₄	#23900	1	1	yes	characteristic
285	C₂×C₄	C₂×#3026	1	2	yes	—
286	C₂²	C₂×#23900	1	1	yes	characteristic
287	C₂²	#6437621	1	1	yes	characteristic
288	C₂²	C₂×#23895	1	1	yes	characteristic
289	C₂²	C₂×#33553	1	2	yes	—
290	C₂²	#7622878	1	2	yes	—
291	C₂	#72233065	1	1	yes	characteristic
292	C₂	#72251874	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#23900
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#525 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72251879	C₂	semidirect
2. #72251877	C₂	semidirect
3. #72251876	C₂	semidirect
4. #72251875	C₂	semidirect
5. #72237547	C₂	semidirect
6. #72235088	C₂	semidirect
7. #72234144	C₂	semidirect
8. #72232342	C₂	semidirect
9. #72231750	C₂	semidirect
10. #72215568	C₂	semidirect
11. #72213496	C₂	semidirect
12. #72210515	C₂	semidirect
13. #10586169	C₂²	semidirect
14. C₂×#52004	C₂²	semidirect
17. #6499837	C₂²	semidirect
18. #6470228	C₂²	semidirect
20. #6437449	C₂²	semidirect
21. #6374136	C₂²	semidirect
22. #4941683	C₂²	semidirect
23. #4607160	C₂²	semidirect
24. #4312017	C₂²	semidirect
25. #2922862	C₂²	semidirect
26. #1418385	C₂²	semidirect
28. #581605	C₂²	semidirect
29. #512119	C₂²	semidirect
30. #511771	C₂²	semidirect
31. #508086	C₂²	semidirect
32. #496228	C₂²	semidirect
33. #493372	C₂²	semidirect
34. #493372	C₂²	semidirect
35. #347055	C₂²	semidirect
36. #347052	C₂²	semidirect
37. #347043	C₂²	semidirect
38. #347043	C₂²	semidirect
39. #188029	C₂²	semidirect
40. #187931	C₂²	semidirect
41. #144414	C₂²	semidirect
42. #144414	C₂²	semidirect
43. #144386	C₂²	semidirect
44. #144386	C₂²	semidirect
45. #139827	C₂²	semidirect
46. #139826	C₂²	semidirect
47. #139778	C₂²	semidirect
48. #139776	C₂²	semidirect
49. #139713	C₂²	semidirect
50. #139713	C₂²	semidirect
51. #139706	C₂²	semidirect
52. #133828	C₂²	semidirect
53. #133828	C₂²	semidirect
54. #128074	C₂²	semidirect
55. C₂⁵×D₄	D₄	semidirect
58. C₂²×D₄²	C₂³	semidirect
62. C₂×#3021	D₄	semidirect
63. C₂×#2669	C₂³	semidirect
64. C₂×#2534	D₄	semidirect
66. C₂×#2435	D₄	semidirect
67. C₂×#2435	D₄	semidirect
70. C₂×#2380	C₂³	semidirect
71. C₂×#2321	D₄	semidirect
72. C₂×#2309	D₄	semidirect
73. C₂×#2309	D₄	semidirect
74. C₂×#2280	D₄	semidirect
75. #24056	C₂³	semidirect
76. #23913	D₄	semidirect
80. #23900	C₂³	semidirect
82. #20956	D₄	semidirect
86. #15982	D₄	semidirect
87. #15948	D₄	semidirect
88. #15948	D₄	semidirect
89. #15947	D₄	semidirect
90. #15313	C₂³	semidirect
91. #15313	C₂³	semidirect
93. #15258	C₂³	semidirect
94. #15258	C₂³	semidirect
97. #14661	C₂³	semidirect
98. #14419	C₂³	semidirect
99. #14344	D₄	semidirect
100. #14154	C₂³	semidirect
101. #13940	D₄	semidirect
103. #12852	C₂³	semidirect
104. #12693	C₂³	semidirect
105. #12693	C₂³	semidirect
106. #12693	C₂³	semidirect
107. #12687	C₂³	semidirect
110. #8534	C₂³	semidirect
111. #8534	C₂³	semidirect
112. #7304	C₂³	semidirect
113. #7304	C₂³	semidirect
114. C₂×#585	C₂×D₄	semidirect
121. #3485	C₂×D₄	semidirect
140. #3428	C₂×D₄	semidirect
148. C₂×#453	C₂×D₄	semidirect
149. C₂×#413	C₂×D₄	semidirect
167. #1887	C₂×D₄	semidirect
168. #1785	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72251879	2
#72251877	2
#72251876	2
#72251875	2
#72237547	2
#72235088	2
#72234144	2
#72232342	2
#72231750	2
#72215568	2
#72213496	2
#72210515	2