Subgroups — #72266717

Order 2048.

At least 6509 conjugacy classes of subgroups; only the 672 normal subgroups are shown.

The subgroup lattice has 311 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266716	C₂	1	2	yes	maximal, complemented
2	#72266710	C₂	1	4	yes	maximal, complemented
3	#72266708	C₂	1	4	yes	maximal, complemented
4	#72266307	C₂	1	1	yes	maximal, characteristic, complemented
5	#72263553	C₂	1	2	yes	maximal, complemented
6	#72263134	C₂	1	4	yes	maximal, complemented
7	#72262701	C₂	1	1	yes	maximal, characteristic, complemented
8	#72258912	C₂	1	1	yes	maximal, characteristic, complemented
9	#72240132	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237037	C₂	1	2	yes	maximal, complemented
11	#72233426	C₂	1	4	yes	maximal, complemented
12	#72231950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#139716	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218215	C₂	1	2	yes	maximal, complemented
15	C₂×#347042	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	C₂×#23900	C₂²	1	2	yes	—
18	#6437445	C₂²	1	2	yes	—
19	C₂×#21746	C₂²	1	1	yes	characteristic
20	#5063443	C₂²	1	2	yes	complemented
21	#5063442	C₂²	1	2	yes	complemented
22	#4464171	C₂²	1	1	yes	characteristic, complemented
23	#2356274	C₂²	1	2	yes	complemented
24	#2108839	C₂²	1	2	yes	complemented
25	#616961	C₂²	1	1	yes	characteristic
26	#616961	C₂²	1	1	yes	characteristic, complemented
27	C₂×#15947	C₂²	1	1	yes	characteristic
28	#556752	C₂²	1	1	yes	characteristic
29	#556752	C₂²	1	1	yes	characteristic, complemented
30	#552671	C₂²	1	1	yes	characteristic, complemented
31	#493370	C₂²	1	2	yes	complemented
32	C₂×#12687	C₂²	1	2	yes	complemented
33	#347042	C₂²	1	4	yes	—
34	#347042	C₂²	1	4	yes	—
35	#346456	C₂²	1	4	yes	complemented
36	#346446	C₂²	1	4	yes	complemented
37	C₂×#8547	C₂²	1	1	yes	characteristic
38	C₂×#8541	C₂²	1	1	yes	characteristic
39	#187944	C₂²	1	2	yes	—
40	#187944	C₂²	1	2	yes	—
41	#187934	C₂²	1	4	yes	complemented
42	#187934	C₂²	1	4	yes	complemented
43	C₂×#8534	C₂²	1	2	yes	—
44	#186835	C₂²	1	2	yes	—
45	#144628	C₂²	1	8	yes	complemented
46	#144627	C₂²	1	8	yes	complemented
47	#144388	C₂²	1	4	yes	complemented
48	#144388	C₂²	1	4	yes	complemented
49	#144386	C₂²	1	4	yes	complemented
50	#144386	C₂²	1	4	yes	complemented
51	#139720	C₂²	1	4	yes	complemented
52	#139718	C₂²	1	4	yes	complemented
53	#139716	C₂²	1	4	yes	complemented
54	#139716	C₂²	1	4	yes	complemented
55	#139707	C₂²	1	8	yes	complemented
56	#139706	C₂²	1	8	yes	complemented
57	#139702	C₂²	1	8	yes	complemented
58	#139700	C₂²	1	8	yes	complemented
59	#135507	C₂²	1	4	yes	complemented
60	#135507	C₂²	1	4	yes	complemented
61	#128033	C₂²	1	2	yes	complemented
62	#128033	C₂²	1	2	yes	complemented
63	C₂×#7304	C₂²	1	2	yes	complemented
64	#128009	C₂²	1	2	yes	complemented
65	#62817	C₂³	1	4	yes	—
66	C₂×#3484	C₂³	1	2	yes	—
67	C₂²×#535	C₂³	1	1	yes	characteristic
68	C₂³×#120	D₄	1	2	yes	—
69	C₂³×#120	C₂³	1	1	yes	characteristic
70	C₂³×#120	C₂³	1	1	yes	characteristic
71	C₂×#2404	D₄	1	2	yes	complemented
72	C₂²×#379	D₄	1	2	yes	complemented
73	#23907	C₂³	1	4	yes	—
74	#23906	C₂³	1	4	yes	—
75	#23900	C₂³	1	4	yes	—
76	#23900	C₂³	1	4	yes	—
77	#21746	C₂³	1	4	yes	—
78	C₂×#1783	C₂³	1	2	yes	—
79	C₂×#1781	C₂³	1	2	yes	—
80	C₂×#1781	C₂³	1	2	yes	—
81	#17271	C₂³	1	2	yes	complemented
82	#17137	C₂³	1	2	yes	complemented
83	#15947	D₄	1	1	yes	characteristic
84	#15947	D₄	1	1	yes	characteristic
85	#15947	D₄	1	1	yes	characteristic
86	#15947	D₄	1	1	yes	characteristic
87	#15947	C₂³	1	2	yes	—
88	#15947	C₂³	1	2	yes	—
89	#15259	D₄	1	1	yes	characteristic, complemented
90	#15259	D₄	1	1	yes	characteristic, complemented
91	#15259	C₂³	1	2	yes	—
92	#15259	C₂³	1	1	yes	characteristic, complemented
93	#15259	C₂³	1	1	yes	characteristic, complemented
94	#15258	D₄	1	1	yes	characteristic, complemented
95	#15258	D₄	1	1	yes	characteristic, complemented
96	#15258	C₂³	1	2	yes	—
97	#15258	C₂³	1	1	yes	characteristic, complemented
98	#15258	C₂³	1	1	yes	characteristic, complemented
99	#14657	C₂³	1	2	yes	complemented
100	#14492	C₂³	1	2	yes	complemented
101	#14419	C₂³	1	2	yes	complemented
102	#14341	D₄	1	2	yes	complemented
103	#14335	C₂³	1	1	yes	characteristic
104	#14335	C₂³	1	1	yes	characteristic
105	#14335	C₂³	1	1	yes	characteristic
106	#14255	D₄	1	2	yes	complemented
107	#14255	D₄	1	2	yes	complemented
108	#14255	C₂³	1	2	yes	—
109	#14255	C₂³	1	2	yes	—
110	#14134	C₂³	1	2	yes	complemented
111	#13868	D₄	1	2	yes	complemented
112	C₂×#1438	D₄	1	2	yes	—
113	C₂×#1438	C₂³	1	1	yes	characteristic
114	#12694	C₂³	1	4	yes	complemented
115	#12693	C₂³	1	4	yes	complemented
116	#12687	C₂³	1	4	yes	complemented
117	#12687	C₂³	1	4	yes	complemented
118	#8547	C₂³	1	4	yes	—
119	#8541	C₂³	1	4	yes	—
120	#8534	C₂³	1	4	yes	—
121	#8534	C₂³	1	4	yes	—
122	#8534	C₂³	1	8	yes	—
123	#8534	C₂³	1	8	yes	—
124	#8453	C₂³	1	8	yes	—
125	#7307	C₂³	1	8	yes	complemented
126	#7304	C₂³	1	4	yes	complemented
127	#7304	C₂³	1	4	yes	complemented
128	#7304	C₂³	1	8	yes	complemented
129	#7304	C₂³	1	8	yes	complemented
130	C₂⁷	C₂×D₄	1	1	yes	characteristic
131	C₂⁷	C₂×D₄	1	1	yes	characteristic
132	C₂²×H(5,𝔽₂)	#41	1	2	yes	complemented
133	C₂³×#41	C₂×D₄	1	1	yes	characteristic
134	C₂⁴×D₄	C₂×D₄	1	2	yes	—
135	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
136	#3484	#41	1	4	yes	complemented
137	#3484	C₂⁴	1	4	yes	—
138	C₂²×#127	C₂×D₄	1	1	yes	characteristic
139	C₂²×#127	C₂⁴	1	1	yes	characteristic
140	C₂²×#124	C₂×D₄	1	1	yes	characteristic
141	C₂²×#124	C₂⁴	1	1	yes	characteristic
142	C₂²×#120	C₂×D₄	1	1	yes	characteristic
143	C₂²×#120	C₂×D₄	1	1	yes	characteristic
144	C₂²×#120	C₂×D₄	1	2	yes	—
145	C₂²×#120	C₂×D₄	1	2	yes	—
146	C₂²×#120	C₂×D₄	1	2	yes	—
147	C₂²×#120	C₂×D₄	1	2	yes	—
148	C₂²×#120	C₂×D₄	1	1	yes	characteristic
149	C₂²×#120	C₂×D₄	1	1	yes	characteristic
150	C₂²×#120	C₂×D₄	1	2	yes	—
151	C₂²×#120	C₂×D₄	1	1	yes	characteristic
152	C₂²×#120	C₂×D₄	1	1	yes	characteristic
153	C₂²×#120	C₂⁴	1	1	yes	characteristic
154	C₂²×#120	C₂⁴	1	1	yes	characteristic
155	C₂²×#120	C₂⁴	1	2	yes	—
156	C₂×#518	#41	1	2	yes	complemented
157	C₂²×#117	C₂×D₄	1	1	yes	characteristic
158	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
159	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
160	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
161	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
162	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	complemented
163	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	complemented
164	C₂×#393	C₂×D₄	1	2	yes	complemented
165	C₂×#393	C₂×D₄	1	2	yes	complemented
166	C₂×#392	C₂×D₄	1	2	yes	complemented
167	C₂×#392	C₂×D₄	1	2	yes	complemented
168	C₂×#381	C₂×D₄	1	2	yes	—
169	C₂×#379	C₂×D₄	1	2	yes	—
170	C₂×#379	C₂×D₄	1	2	yes	—
171	C₂×#379	C₂×D₄	1	2	yes	complemented
172	C₂×#379	C₂×D₄	1	2	yes	complemented
173	#1783	C₂⁴	1	4	yes	—
174	#1781	C₂⁴	1	4	yes	—
175	#1781	C₂⁴	1	4	yes	—
176	#1440	C₂×D₄	1	2	yes	complemented
177	#1440	C₂×D₄	1	2	yes	complemented
178	#1440	C₂×D₄	1	2	yes	complemented
179	#1440	C₂×D₄	1	2	yes	complemented
180	#1438	C₂×D₄	1	1	yes	characteristic
181	#1438	C₂×D₄	1	1	yes	characteristic
182	#1438	C₂×D₄	1	2	yes	—
183	#1438	C₂×D₄	1	2	yes	—
184	#1438	C₂×D₄	1	2	yes	—
185	#1438	C₂×D₄	1	2	yes	—
186	#1438	C₂×D₄	1	1	yes	characteristic
187	#1438	C₂×D₄	1	1	yes	characteristic
188	#1438	C₂×D₄	1	2	yes	—
189	#1438	C₂×D₄	1	1	yes	characteristic
190	#1438	C₂×D₄	1	1	yes	characteristic
191	#1438	C₂⁴	1	2	yes	—
192	#1438	C₂⁴	1	2	yes	—
193	#1438	C₂⁴	1	2	yes	—
194	#1438	C₂⁴	1	1	yes	characteristic
195	#1438	C₂⁴	1	1	yes	characteristic
196	#1437	C₂×D₄	1	2	yes	complemented
197	#1437	C₂×D₄	1	2	yes	complemented
198	#1437	C₂×D₄	1	2	yes	complemented
199	#1437	C₂×D₄	1	2	yes	complemented
200	C₂⁶	#120	1	1	yes	characteristic
201	C₂⁶	#120	1	1	yes	characteristic
202	C₂⁶	#120	1	2	yes	—
203	C₂⁶	C₂²×D₄	1	2	yes	—
204	C₂⁶	C₂²×D₄	1	1	yes	characteristic
205	C₂⁶	C₂²×D₄	1	1	yes	characteristic
206	C₂⁶	C₂²×D₄	1	1	yes	characteristic
207	C₂⁶	C₂²×D₄	1	1	yes	characteristic
208	C₂×H(5,𝔽₂)	C₂×#41	1	4	yes	—
209	C₂³×Q₈	C₂²×D₄	1	1	yes	characteristic
210	C₂³×D₄	#120	1	2	yes	—
211	C₂³×D₄	#120	1	2	yes	—
212	C₂³×D₄	C₂²×D₄	1	2	yes	—
213	C₂³×D₄	C₂²×D₄	1	2	yes	—
214	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
215	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
216	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
217	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
218	C₂³×D₄	C₂×#41	1	2	yes	—
219	C₂³×D₄	C₂×#41	1	2	yes	—
220	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
221	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
222	C₂³×D₄	C₂⁵	1	1	yes	characteristic
223	C₂⁴×C₄	#120	1	2	yes	—
224	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
225	#518	C₂×#41	1	4	yes	—
226	C₂²×C₂²⋊C₄	#120	1	2	yes	—
227	C₂²×C₂²⋊C₄	#120	1	2	yes	—
228	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
229	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
230	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
231	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
232	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
233	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
234	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
235	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
236	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
237	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
238	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
239	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
240	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
241	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
242	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
243	C₂×#95	C₂²×D₄	1	2	yes	—
244	C₂×#95	C₂²×D₄	1	2	yes	—
245	C₂⁵	B(4,𝔽₂)	1	2	yes	—
246	C₂⁵	B(4,𝔽₂)	1	2	yes	—
247	C₂⁵	B(4,𝔽₂)	1	2	yes	—
248	C₂⁵	B(4,𝔽₂)	1	2	yes	—
249	C₂⁵	C₂×#120	1	2	yes	—
250	C₂⁵	C₂×#120	1	1	yes	characteristic
251	C₂⁵	C₂×#120	1	1	yes	characteristic
252	C₂⁵	C₂×#120	1	2	yes	—
253	C₂⁵	C₂×#120	1	1	yes	characteristic
254	C₂⁵	#525	1	1	yes	characteristic
255	C₂⁵	#525	1	1	yes	characteristic
256	C₂⁵	#535	1	1	yes	characteristic
257	C₂⁵	#535	1	1	yes	characteristic
258	C₂⁵	#546	1	2	yes	—
259	C₂⁵	#546	1	2	yes	—
260	C₂⁵	C₂³×D₄	1	1	yes	characteristic
261	C₂⁵	C₂³×D₄	1	1	yes	characteristic
262	C₂²×D₄	#525	1	2	yes	—
263	C₂²×D₄	#535	1	2	yes	—
264	C₂²×D₄	#546	1	2	yes	—
265	C₂²×D₄	#546	1	2	yes	—
266	C₂²×D₄	#546	1	2	yes	—
267	C₂²×D₄	#546	1	2	yes	—
268	C₂²×D₄	C₂²×#41	1	2	yes	—
269	C₂²×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
270	C₂³×C₄	C₂×#120	1	2	yes	—
271	C₂³×C₄	C₂×#120	1	2	yes	—
272	C₂³×C₄	C₂×#120	1	1	yes	characteristic
273	C₂³×C₄	#546	1	2	yes	—
274	C₂³×C₄	#546	1	2	yes	—
275	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
276	C₂⁴	#2404	1	1	yes	characteristic
277	C₂⁴	#2404	1	1	yes	characteristic
278	C₂⁴	#2404	1	2	yes	—
279	C₂⁴	#2404	1	1	yes	characteristic
280	C₂⁴	#2404	1	1	yes	characteristic
281	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
282	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
283	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
284	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
285	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
286	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
287	C₂⁴	C₂²×#120	1	1	yes	characteristic
288	C₂⁴	C₂×#525	1	1	yes	characteristic
289	C₂⁴	C₂×#535	1	1	yes	characteristic
290	C₂⁴	C₂×#546	1	2	yes	—
291	C₂²×C₄	#2404	1	2	yes	—
292	C₂²×C₄	C₂×#546	1	2	yes	—
293	C₂³	#23984	1	2	yes	—
294	C₂³	#23984	1	1	yes	characteristic
295	C₂³	#23984	1	1	yes	characteristic
296	C₂³	#23996	1	2	yes	—
297	C₂³	#23996	1	2	yes	—
298	C₂³	#33551	1	4	yes	—
299	C₂³	C₂×#2404	1	1	yes	characteristic
300	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
301	C₂²	#251827	1	2	yes	—
302	C₂²	#251827	1	2	yes	—
303	C₂²	C₂×#23996	1	1	yes	characteristic
304	C₂²	C₂×#23996	1	1	yes	characteristic
305	C₂²	C₂×#23984	1	1	yes	characteristic
306	C₂²	C₂×#33551	1	2	yes	—
307	C₂²	#7622875	1	2	yes	—
308	C₂	#72238500	1	2	yes	—
309	C₂	C₂×#251827	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#23984
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ 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. #72266716	C₂	semidirect
2. #72266710	C₂	semidirect
3. #72266708	C₂	semidirect
4. #72266307	C₂	semidirect
5. #72263553	C₂	semidirect
6. #72263134	C₂	semidirect
7. #72262701	C₂	semidirect
8. #72258912	C₂	semidirect
9. #72240132	C₂	semidirect
10. #72237037	C₂	semidirect
11. #72233426	C₂	semidirect
12. #72231950	C₂	semidirect
13. C₂×#139716	C₂	semidirect
14. #72218215	C₂	semidirect
15. C₂×#347042	C₂	semidirect
20. #5063443	C₂²	semidirect
21. #5063442	C₂²	semidirect
22. #4464171	C₂²	semidirect
23. #2356274	C₂²	semidirect
24. #2108839	C₂²	semidirect
26. #616961	C₂²	semidirect
29. #556752	C₂²	semidirect
30. #552671	C₂²	semidirect
31. #493370	C₂²	semidirect
32. C₂×#12687	C₂²	semidirect
35. #346456	C₂²	semidirect
36. #346446	C₂²	semidirect
41. #187934	C₂²	semidirect
42. #187934	C₂²	semidirect
45. #144628	C₂²	semidirect
46. #144627	C₂²	semidirect
47. #144388	C₂²	semidirect
48. #144388	C₂²	semidirect
49. #144386	C₂²	semidirect
50. #144386	C₂²	semidirect
51. #139720	C₂²	semidirect
52. #139718	C₂²	semidirect
53. #139716	C₂²	semidirect
54. #139716	C₂²	semidirect
55. #139707	C₂²	semidirect
56. #139706	C₂²	semidirect
57. #139702	C₂²	semidirect
58. #139700	C₂²	semidirect
59. #135507	C₂²	semidirect
60. #135507	C₂²	semidirect
61. #128033	C₂²	semidirect
62. #128033	C₂²	semidirect
63. C₂×#7304	C₂²	semidirect
64. #128009	C₂²	semidirect
71. C₂×#2404	D₄	semidirect
72. C₂²×#379	D₄	semidirect
81. #17271	C₂³	semidirect
82. #17137	C₂³	semidirect
89. #15259	D₄	semidirect
90. #15259	D₄	semidirect
92. #15259	C₂³	semidirect
93. #15259	C₂³	semidirect
94. #15258	D₄	semidirect
95. #15258	D₄	semidirect
97. #15258	C₂³	semidirect
98. #15258	C₂³	semidirect
99. #14657	C₂³	semidirect
100. #14492	C₂³	semidirect
101. #14419	C₂³	semidirect
102. #14341	D₄	semidirect
106. #14255	D₄	semidirect
107. #14255	D₄	semidirect
110. #14134	C₂³	semidirect
111. #13868	D₄	semidirect
114. #12694	C₂³	semidirect
115. #12693	C₂³	semidirect
116. #12687	C₂³	semidirect
117. #12687	C₂³	semidirect
125. #7307	C₂³	semidirect
126. #7304	C₂³	semidirect
127. #7304	C₂³	semidirect
128. #7304	C₂³	semidirect
129. #7304	C₂³	semidirect
132. C₂²×H(5,𝔽₂)	#41	semidirect
136. #3484	#41	semidirect
156. C₂×#518	#41	semidirect
162. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
163. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
164. C₂×#393	C₂×D₄	semidirect
165. C₂×#393	C₂×D₄	semidirect
166. C₂×#392	C₂×D₄	semidirect
167. C₂×#392	C₂×D₄	semidirect
171. C₂×#379	C₂×D₄	semidirect
172. C₂×#379	C₂×D₄	semidirect
176. #1440	C₂×D₄	semidirect
177. #1440	C₂×D₄	semidirect
178. #1440	C₂×D₄	semidirect
179. #1440	C₂×D₄	semidirect
196. #1437	C₂×D₄	semidirect
197. #1437	C₂×D₄	semidirect
198. #1437	C₂×D₄	semidirect
199. #1437	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72266716	2
#72266710	2
#72266708	2
#72266307	2
#72263553	2
#72263134	2
#72262701	2
#72258912	2
#72240132	2
#72237037	2
#72233426	2
#72231950	2
C₂×#139716	2
#72218215	2
C₂×#347042	2