Subgroups — C₂²×#16840324 #30586732

Order 15616.

15230 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₆₁	C₂	1	2	yes	maximal, complemented
2	C₂×#16840324	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#14519761	C₂	1	4	yes	maximal, complemented
4	C₂²×#14519751	C₂	1	4	yes	maximal, complemented
5	C₂²×#14519747	C₂	1	4	yes	maximal, complemented
6	C₂³×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₆₁	C₂²	1	2	yes	complemented
8	C₂³×D₂₄₄	C₂²	1	4	yes	complemented
9	C₂³×C₂²⋊D₆₁	—	2	32	no	—
10	C₂³×C₂²⋊D₆₁	C₂²	1	4	yes	—
11	C₂³×C₂²⋊D₆₁	C₂²	1	4	yes	complemented
12	C₂³×C₂²⋊D₆₁	C₂²	1	48	yes	complemented
13	C₂⁴×Dic₆₁	C₂²	1	2	yes	complemented
14	#16840324	C₂²	1	256	yes	direct product factor
15	C₂×#14519761	C₂²	1	96	yes	complemented
16	C₂×#14519751	C₂²	1	96	yes	complemented
17	C₂×#14519747	C₂²	1	96	yes	complemented
18	C₂²×C₄⋊Dic₆₁	C₂²	1	4	yes	complemented
19	C₂²×#11758198	C₂²	1	8	yes	complemented
20	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₆₁	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	complemented
23	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	24	yes	complemented
24	C₂⁴×D₆₁	—	2	16	no	—
25	C₂⁴×D₆₁	—	4	16	no	—
26	C₂⁴×D₆₁	C₂³	1	4	yes	—
27	C₂⁴×D₆₁	C₂³	1	24	yes	complemented
28	C₂²×D₂₄₄	—	4	16	no	—
29	C₂²×D₂₄₄	C₂³	1	48	yes	complemented
30	C₂²×C₂²⋊D₆₁	—	2	32	no	—
31	C₂²×C₂²⋊D₆₁	—	2	32	no	—
32	C₂²×C₂²⋊D₆₁	—	2	384	no	—
33	C₂²×C₂²⋊D₆₁	C₂³	1	48	yes	—
34	C₂²×C₂²⋊D₆₁	C₂³	1	48	yes	complemented
35	C₂²×C₂²⋊D₆₁	C₂³	1	128	yes	complemented
36	C₂³×Dic₆₁	—	2	16	no	—
37	C₂³×Dic₆₁	C₂³	1	4	yes	—
38	C₂³×Dic₆₁	C₂³	1	24	yes	complemented
39	#14519761	C₂³	1	256	yes	complemented
40	#14519751	C₂³	1	256	yes	complemented
41	#14519747	C₂³	1	256	yes	complemented
42	C₂×C₄⋊Dic₆₁	C₂³	1	48	yes	complemented
43	C₂×#11758198	C₂³	1	96	yes	complemented
44	C₂⁵×C₆₁	—	4	8	no	—
45	C₂⁵×C₆₁	D₄	1	16	yes	complemented
46	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
47	C₂⁵×C₆₁	C₂³	1	2	yes	—
48	C₂⁵×C₆₁	C₂³	1	12	yes	—
49	C₂³×C₄×C₆₁	—	4	8	no	—
50	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₁	C₂³	1	24	yes	complemented
52	C₂×C₂²⋊C₄×C₆₁	C₂³	1	48	yes	complemented
53	C₂×C₂²⋊C₄×C₆₁	C₂³	1	64	yes	complemented
54	C₂⁴×C₆₁	—	2	16	no	—
55	C₂⁴×C₆₁	—	2	8	no	—
56	C₂⁴×C₆₁	—	4	64	no	—
57	C₂⁴×C₆₁	—	4	48	no	—
58	C₂⁴×C₆₁	C₂×D₄	1	16	yes	—
59	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
60	C₂⁴×C₆₁	C₂×D₄	1	96	yes	complemented
61	C₂⁴×C₆₁	C₂⁴	1	12	yes	—
62	C₂⁴×C₆₁	C₂⁴	1	6	yes	—
63	C₂⁴×C₆₁	C₂⁴	1	16	yes	—
64	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
65	C₂³×D₆₁	—	2	16	no	—
66	C₂³×D₆₁	—	2	96	no	—
67	C₂³×D₆₁	—	2	32	no	—
68	C₂³×D₆₁	—	4	96	no	—
69	C₂³×D₆₁	—	4	128	no	—
70	C₂³×D₆₁	C₂⁴	1	24	yes	—
71	C₂³×D₆₁	C₂⁴	1	32	yes	complemented
72	C₂²×C₄×C₆₁	—	2	16	no	—
73	C₂²×C₄×C₆₁	—	4	48	no	—
74	C₂²×C₄×C₆₁	C₂⁴	1	24	yes	—
75	C₂²×C₄×C₆₁	C₂⁴	1	32	yes	complemented
76	C₂×C₂²⋊D₆₁	—	2	192	no	—
77	C₂×C₂²⋊D₆₁	—	2	192	no	—
78	C₂×C₂²⋊D₆₁	—	2	512	no	—
79	C₂×C₂²⋊D₆₁	C₂⁴	1	64	yes	—
80	C₂×C₂²⋊D₆₁	C₂⁴	1	64	yes	complemented
81	C₂²×Dic₆₁	—	2	16	no	—
82	C₂²×Dic₆₁	—	2	96	no	—
83	C₂²×Dic₆₁	C₂⁴	1	24	yes	—
84	C₂²×Dic₆₁	C₂⁴	1	32	yes	complemented
85	C₂×D₂₄₄	—	4	96	no	—
86	C₂×D₂₄₄	C₂⁴	1	64	yes	complemented
87	C₂²⋊C₄×C₆₁	C₂⁴	1	64	yes	complemented
88	#11758198	C₂⁴	1	128	yes	complemented
89	C₄⋊Dic₆₁	C₂⁴	1	64	yes	complemented
90	C₂³×C₆₁	—	2	64	no	—
91	C₂³×C₆₁	—	2	32	no	—
92	C₂³×C₆₁	—	2	24	no	—
93	C₂³×C₆₁	—	2	48	no	—
94	C₂³×C₆₁	—	4	192	no	—
95	C₂³×C₆₁	—	4	32	no	—
96	C₂³×C₆₁	C₂²×D₄	1	48	yes	—
97	C₂³×C₆₁	C₂²×D₄	1	24	yes	—
98	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
99	C₂³×C₆₁	C₂²×D₄	1	64	yes	complemented
100	C₂³×C₆₁	H(5,𝔽₂)	1	8	yes	—
101	C₂³×C₆₁	C₂⁵	1	8	yes	—
102	C₂³×C₆₁	C₂⁵	1	4	yes	—
103	C₂³×C₆₁	C₂⁵	1	3	yes	—
104	C₂²×D₆₁	—	2	48	no	—
105	C₂²×D₆₁	—	2	64	no	—
106	C₂²×D₆₁	—	2	96	no	—
107	C₂²×D₆₁	—	4	64	no	—
108	C₂²×D₆₁	—	4	384	no	—
109	C₂²×D₆₁	—	4	64	no	—
110	C₂²×D₆₁	C₂⁵	1	16	yes	—
111	C₂×C₄×C₆₁	—	2	48	no	—
112	C₂×C₄×C₆₁	—	4	32	no	—
113	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
114	C₂²⋊D₆₁	—	2	128	no	—
115	C₂²⋊D₆₁	—	2	128	no	—
116	C₂×Dic₆₁	—	2	48	no	—
117	C₂×Dic₆₁	—	2	64	no	—
118	C₂×Dic₆₁	C₂⁵	1	16	yes	—
119	D₂₄₄	—	4	64	no	—
120	C₂²×#534	—	61	1	no	maximal, Sylow
121	C₂²×C₆₁	—	2	8	no	—
122	C₂²×C₆₁	—	2	16	no	—
123	C₂²×C₆₁	—	2	96	no	—
124	C₂²×C₆₁	—	2	48	no	—
125	C₂²×C₆₁	—	4	64	no	—
126	C₂²×C₆₁	#534	1	16	yes	complemented
127	C₂²×C₆₁	C₂³×D₄	1	16	yes	—
128	C₂²×C₆₁	C₂³×D₄	1	8	yes	—
129	C₂²×C₆₁	C₂³×D₄	1	6	yes	—
130	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	12	yes	—
131	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
132	C₂×D₆₁	—	2	16	no	—
133	C₂×D₆₁	—	2	32	no	—
134	C₂×D₆₁	—	4	128	no	—
135	C₂×D₆₁	—	4	96	no	—
136	C₄×C₆₁	—	2	16	no	—
137	Dic₆₁	—	2	16	no	—
138	C₂⁴×D₄	—	61	2	no	—
139	C₂×#534	—	61	48	no	—
140	C₂²×#123	—	61	4	no	—
141	C₂²×#121	—	61	4	no	—
142	C₂²×#120	—	61	4	no	—
143	C₂³×C₂²⋊C₄	D₆₁	1	1	yes	characteristic, complemented
144	C₂×C₆₁	—	2	16	no	—
145	C₂×C₆₁	—	2	8	no	—
146	C₂×C₆₁	C₂×#534	1	12	yes	complemented
147	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
148	C₂×C₆₁	C₂²×H(5,𝔽₂)	1	2	yes	—
149	D₆₁	—	4	16	no	—
150	C₂⁶	—	61	2	no	—
151	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
152	C₂³×D₄	—	61	4	no	—
153	C₂³×D₄	—	61	4	no	—
154	C₂³×D₄	—	61	48	no	—
155	C₂³×D₄	—	61	4	no	—
156	C₂³×D₄	—	122	32	no	—
157	C₂⁴×C₄	—	61	2	no	—
158	C₂⁴×C₄	C₂×D₆₁	1	2	yes	complemented
159	#534	—	61	256	no	—
160	C₂×#123	—	61	96	no	—
161	C₂×#121	—	61	96	no	—
162	C₂×#120	—	61	96	no	—
163	C₂²×C₄⋊C₄	—	61	4	no	—
164	C₂²×C₂²⋊C₄	—	61	8	no	—
165	C₂²×C₂²⋊C₄	C₂×D₆₁	1	4	yes	complemented
166	C₂²×C₂²⋊C₄	C₂×D₆₁	1	24	yes	complemented
167	C₆₁	C₂²×#534	1	1	yes	characteristic, complemented, Sylow
168	C₂⁵	—	4	8	no	—
169	C₂⁵	—	61	4	no	—
170	C₂⁵	—	61	24	no	—
171	C₂⁵	—	122	16	no	—
172	C₂⁵	—	244	16	no	—
173	C₂⁵	D₂₄₄	1	16	yes	complemented
174	C₂⁵	C₂²×D₆₁	1	2	yes	—
175	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
176	C₂⁵	C₂²×D₆₁	1	12	yes	—
177	C₂²×D₄	—	61	48	no	—
178	C₂²×D₄	—	61	48	no	—
179	C₂²×D₄	—	61	128	no	—
180	C₂²×D₄	—	61	48	no	—
181	C₂²×D₄	—	122	32	no	—
182	C₂²×D₄	—	122	32	no	—
183	C₂²×D₄	—	122	384	no	—
184	C₂²×D₄	—	244	16	no	—
185	C₂³×C₄	—	4	8	no	—
186	C₂³×C₄	—	61	24	no	—
187	C₂³×C₄	—	61	4	no	—
188	C₂³×C₄	—	122	16	no	—
189	C₂³×C₄	C₂²×D₆₁	1	4	yes	—
190	C₂³×C₄	C₂²×D₆₁	1	24	yes	complemented
191	#123	—	61	256	no	—
192	#121	—	61	256	no	—
193	#120	—	61	256	no	—
194	C₂×C₄⋊C₄	—	61	48	no	—
195	C₂×C₂²⋊C₄	—	61	96	no	—
196	C₂×C₂²⋊C₄	C₂²×D₆₁	1	48	yes	complemented
197	C₂×C₂²⋊C₄	C₂²×D₆₁	1	64	yes	complemented
198	C₂⁴	—	2	8	no	—
199	C₂⁴	—	2	16	no	—
200	C₂⁴	—	4	64	no	—
201	C₂⁴	—	4	48	no	—
202	C₂⁴	—	61	24	no	—
203	C₂⁴	—	61	32	no	—
204	C₂⁴	—	122	96	no	—
205	C₂⁴	—	122	16	no	—
206	C₂⁴	—	122	32	no	—
207	C₂⁴	—	244	96	no	—
208	C₂⁴	—	244	128	no	—
209	C₂⁴	C₂×D₂₄₄	1	16	yes	—
210	C₂⁴	C₂×D₂₄₄	1	8	yes	—
211	C₂⁴	C₂×D₂₄₄	1	96	yes	complemented
212	C₂⁴	C₂³×D₆₁	1	12	yes	—
213	C₂⁴	C₂³×D₆₁	1	6	yes	—
214	C₂⁴	C₂³×D₆₁	1	16	yes	—
215	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
216	C₂×D₄	—	61	64	no	—
217	C₂×D₄	—	61	64	no	—
218	C₂×D₄	—	61	64	no	—
219	C₂×D₄	—	122	192	no	—
220	C₂×D₄	—	122	192	no	—
221	C₂×D₄	—	122	512	no	—
222	C₂×D₄	—	244	96	no	—
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	4	48	no	—
225	C₂²×C₄	—	61	24	no	—
226	C₂²×C₄	—	61	32	no	—
227	C₂²×C₄	—	122	96	no	—
228	C₂²×C₄	—	122	16	no	—
229	C₂²×C₄	C₂³×D₆₁	1	24	yes	—
230	C₂²×C₄	C₂³×D₆₁	1	32	yes	complemented
231	C₄⋊C₄	—	61	64	no	—
232	C₂²⋊C₄	—	61	128	no	—
233	C₂²⋊C₄	C₂³×D₆₁	1	64	yes	complemented
234	C₂³	—	2	64	no	—
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	2	48	no	—
238	C₂³	—	4	192	no	—
239	C₂³	—	4	32	no	—
240	C₂³	—	61	16	no	—
241	C₂³	—	122	64	no	—
242	C₂³	—	122	48	no	—
243	C₂³	—	122	96	no	—
244	C₂³	—	244	64	no	—
245	C₂³	—	244	384	no	—
246	C₂³	—	244	64	no	—
247	C₂³	C₂²×D₂₄₄	1	48	yes	—
248	C₂³	C₂²×D₂₄₄	1	24	yes	—
249	C₂³	C₂²×D₂₄₄	1	4	yes	—
250	C₂³	C₂²×D₂₄₄	1	64	yes	complemented
251	C₂³	#14519822	1	8	yes	—
252	C₂³	C₂⁴×D₆₁	1	8	yes	—
253	C₂³	C₂⁴×D₆₁	1	4	yes	—
254	C₂³	C₂⁴×D₆₁	1	3	yes	—
255	D₄	—	122	128	no	—
256	D₄	—	122	128	no	—
257	D₄	—	244	64	no	—
258	C₂×C₄	—	2	48	no	—
259	C₂×C₄	—	4	32	no	—
260	C₂×C₄	—	61	16	no	—
261	C₂×C₄	—	122	64	no	—
262	C₂×C₄	—	122	48	no	—
263	C₂×C₄	C₂⁴×D₆₁	1	16	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	16	no	—
266	C₂²	—	2	96	no	—
267	C₂²	—	2	48	no	—
268	C₂²	—	4	64	no	—
269	C₂²	—	122	16	no	—
270	C₂²	—	122	32	no	—
271	C₂²	—	244	128	no	—
272	C₂²	—	244	96	no	—
273	C₂²	#16840324	1	16	yes	direct product factor
274	C₂²	C₂³×D₂₄₄	1	16	yes	—
275	C₂²	C₂³×D₂₄₄	1	8	yes	—
276	C₂²	C₂³×D₂₄₄	1	6	yes	—
277	C₂²	C₂×#14519822	1	12	yes	—
278	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
279	C₄	—	2	16	no	—
280	C₄	—	122	16	no	—
281	C₂	—	2	16	no	—
282	C₂	—	2	8	no	—
283	C₂	—	244	16	no	—
284	C₂	C₂×#16840324	1	12	yes	direct product factor
285	C₂	C₂⁴×D₂₄₄	1	1	yes	characteristic
286	C₂	C₂²×#14519822	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator, cosocle	C₂²×C₆₁	C₂⁶
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₂²⋊C₄×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂³×C₂²⋊C₄	D₆₁
hypocenter	C₆₁	C₂²×#534
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊D₆₁	C₂	semidirect
2. C₂×#16840324	C₂	direct (N × H)
3. C₂²×#14519761	C₂	semidirect
4. C₂²×#14519751	C₂	semidirect
5. C₂²×#14519747	C₂	semidirect
6. C₂³×C₂²⋊C₄×C₆₁	C₂	semidirect
7. C₂⁵×D₆₁	C₂²	semidirect
8. C₂³×D₂₄₄	C₂²	semidirect
11. C₂³×C₂²⋊D₆₁	C₂²	semidirect
12. C₂³×C₂²⋊D₆₁	C₂²	semidirect
13. C₂⁴×Dic₆₁	C₂²	semidirect
14. #16840324	C₂²	direct (N × H)
15. C₂×#14519761	C₂²	semidirect
16. C₂×#14519751	C₂²	semidirect
17. C₂×#14519747	C₂²	semidirect
18. C₂²×C₄⋊Dic₆₁	C₂²	semidirect
19. C₂²×#11758198	C₂²	semidirect
21. C₂⁴×C₄×C₆₁	C₂²	semidirect
22. C₂²×C₂²⋊C₄×C₆₁	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₆₁	C₂²	semidirect
27. C₂⁴×D₆₁	C₂³	semidirect
29. C₂²×D₂₄₄	C₂³	semidirect
34. C₂²×C₂²⋊D₆₁	C₂³	semidirect
35. C₂²×C₂²⋊D₆₁	C₂³	semidirect
38. C₂³×Dic₆₁	C₂³	semidirect
39. #14519761	C₂³	semidirect
40. #14519751	C₂³	semidirect
41. #14519747	C₂³	semidirect
42. C₂×C₄⋊Dic₆₁	C₂³	semidirect
43. C₂×#11758198	C₂³	semidirect
45. C₂⁵×C₆₁	D₄	semidirect
51. C₂³×C₄×C₆₁	C₂³	semidirect
52. C₂×C₂²⋊C₄×C₆₁	C₂³	semidirect
53. C₂×C₂²⋊C₄×C₆₁	C₂³	semidirect
60. C₂⁴×C₆₁	C₂×D₄	semidirect
71. C₂³×D₆₁	C₂⁴	semidirect
75. C₂²×C₄×C₆₁	C₂⁴	semidirect
80. C₂×C₂²⋊D₆₁	C₂⁴	semidirect
84. C₂²×Dic₆₁	C₂⁴	semidirect
86. C₂×D₂₄₄	C₂⁴	semidirect
87. C₂²⋊C₄×C₆₁	C₂⁴	semidirect
88. #11758198	C₂⁴	semidirect
89. C₄⋊Dic₆₁	C₂⁴	semidirect
99. C₂³×C₆₁	C₂²×D₄	semidirect
126. C₂²×C₆₁	#534	semidirect
143. C₂³×C₂²⋊C₄	D₆₁	semidirect
146. C₂×C₆₁	C₂×#534	semidirect
158. C₂⁴×C₄	C₂×D₆₁	semidirect
165. C₂²×C₂²⋊C₄	C₂×D₆₁	semidirect
166. C₂²×C₂²⋊C₄	C₂×D₆₁	semidirect
167. C₆₁	C₂²×#534	semidirect
173. C₂⁵	D₂₄₄	semidirect
190. C₂³×C₄	C₂²×D₆₁	semidirect
196. C₂×C₂²⋊C₄	C₂²×D₆₁	semidirect
197. C₂×C₂²⋊C₄	C₂²×D₆₁	semidirect
211. C₂⁴	C₂×D₂₄₄	semidirect
230. C₂²×C₄	C₂³×D₆₁	semidirect
233. C₂²⋊C₄	C₂³×D₆₁	semidirect
250. C₂³	C₂²×D₂₄₄	semidirect
273. C₂²	#16840324	direct (N × H)
284. C₂	C₂×#16840324	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂²⋊D₆₁	2
C₂×#16840324	2
C₂²×#14519761	2
C₂²×#14519751	2
C₂²×#14519747	2
C₂³×C₂²⋊C₄×C₆₁	2
C₂²×#534	61

Subgroups — C22×#16840324 #30586732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840324 #30586732