Subgroups — #29677297

Order 15616.

542 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458624	C₂	1	2	yes	maximal
2	#21458620	C₂	1	2	yes	maximal
3	#21445429	C₂	1	1	yes	maximal, characteristic
4	#21445374	C₂	1	2	yes	maximal
5	#21445373	C₂	1	2	yes	maximal
6	#21445064	C₂	1	2	yes	maximal
7	#21445063	C₂	1	2	yes	maximal
8	#21444717	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#1591	C₂	1	1	yes	maximal, characteristic
10	#16840491	C₂²	1	2	yes	—
11	#16840490	C₂²	1	2	yes	—
12	#16840167	C₂²	1	1	yes	characteristic
13	#16840167	C₂²	1	1	yes	characteristic
14	#16840131	—	2	2	no	—
15	#16840129	—	2	2	no	—
16	#16840113	C₂²	1	2	yes	—
17	#16840113	C₂²	1	2	yes	—
18	#16840070	—	2	2	no	—
19	#16840068	—	2	2	no	—
20	#16840048	C₂²	1	2	yes	—
21	#16840047	C₂²	1	2	yes	—
22	#16839973	—	2	2	no	—
23	#16839973	C₂²	1	2	yes	—
24	#16839972	C₂²	1	2	yes	—
25	#16839970	—	2	2	no	—
26	#16839892	C₂²	1	2	yes	—
27	#16839870	—	2	2	no	—
28	#16839866	—	2	2	no	—
29	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
30	#16839515	C₂²	1	1	yes	characteristic
31	#16839509	C₂²	1	2	yes	—
32	#16839507	C₂²	1	2	yes	—
33	#16839494	C₂²	1	2	yes	—
34	C₆₁×#446	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#335	C₂²	1	1	yes	characteristic
37	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₄₄	—	2	2	no	—
40	C₂×Dic₂₄₄	D₄	1	1	yes	characteristic
41	C₂×#11758202	—	2	2	no	—
42	#14519777	—	2	2	no	—
43	#14519777	C₂³	1	2	yes	—
44	#14519777	C₂³	1	2	yes	—
45	#14519775	—	2	2	no	—
46	#14519769	—	2	2	no	—
47	#14519768	—	2	2	no	—
48	#14519768	—	4	2	no	—
49	#14519767	—	4	2	no	—
50	Q₈×Dic₆₁	—	2	2	no	—
51	C₄×Dic₁₂₂	—	2	2	no	—
52	C₄×Dic₁₂₂	—	2	2	no	—
53	C₄×Dic₁₂₂	C₂³	1	2	yes	—
54	#14519742	—	2	2	no	—
55	#14519711	—	4	1	no	—
56	#14519711	—	4	1	no	—
57	#14519711	D₄	1	1	yes	characteristic
58	#14519709	—	2	2	no	—
59	#14519708	—	2	2	no	—
60	C₄⋊#92418	C₂³	1	2	yes	—
61	#14519706	—	2	2	no	—
62	#14519702	—	2	2	no	—
63	#14519702	—	2	2	no	—
64	#14519702	—	2	2	no	—
65	#14519702	—	2	2	no	—
66	#14519702	—	2	2	no	—
67	#14519702	—	2	2	no	—
68	#14519702	D₄	1	2	yes	—
69	#14519701	—	2	2	no	—
70	#14519700	—	2	2	no	—
71	#14519687	—	2	2	no	—
72	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
74	#107×C₆₁	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₁	—	2	2	no	—
76	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
77	#103×C₆₁	D₄	1	2	yes	—
78	#97×C₆₁	—	2	2	no	—
79	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
83	C₂×Dic₁₂₂	—	2	2	no	—
84	C₂×Dic₁₂₂	—	2	2	no	—
85	C₂×Dic₁₂₂	C₂×D₄	1	2	yes	—
86	Q₁₆×C₆₁	—	4	1	no	—
87	C₂×C₈×C₆₁	—	2	1	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
97	#11758202	—	4	4	no	—
98	C₄⋊Dic₆₁	—	2	2	no	—
99	C₄⋊Dic₆₁	—	2	2	no	—
100	C₄⋊Dic₆₁	—	2	2	no	—
101	C₄⋊Dic₆₁	—	2	2	no	—
102	C₄⋊Dic₆₁	—	4	2	no	—
103	C₄⋊Dic₆₁	—	4	2	no	—
104	C₄⋊Dic₆₁	C₂×D₄	1	2	yes	—
105	#11758196	—	2	2	no	—
106	#11758196	—	4	2	no	—
107	#11758196	—	4	2	no	—
108	#11758196	—	4	2	no	—
109	#11758196	—	4	2	no	—
110	C₄×Dic₆₁	—	2	2	no	—
111	C₄×Dic₆₁	—	4	2	no	—
112	C₄×Dic₆₁	—	4	2	no	—
113	C₂×#92418	—	2	2	no	—
114	Dic₂₄₄	—	4	1	no	—
115	Dic₂₄₄	—	4	4	no	—
116	Q₈×C₆₁	—	2	2	no	—
117	Q₈×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×Dic₆₁	—	2	2	no	—
124	C₂×Dic₆₁	—	2	2	no	—
125	C₂×Dic₆₁	—	4	2	no	—
126	C₂×Dic₆₁	—	4	2	no	—
127	Dic₁₂₂	—	2	2	no	—
128	Dic₁₂₂	—	2	2	no	—
129	Dic₁₂₂	—	4	2	no	—
130	Dic₁₂₂	—	4	2	no	—
131	Dic₁₂₂	—	4	2	no	—
132	Dic₁₂₂	—	4	2	no	—
133	C₈×C₆₁	—	2	2	no	—
134	C₈×C₆₁	—	2	1	no	—
135	C₈×C₆₁	—	2	1	no	—
136	C₈×C₆₁	—	2	1	no	—
137	C₈×C₆₁	—	2	1	no	—
138	C₈×C₆₁	—	4	2	no	—
139	#92418	—	4	2	no	—
140	#20321	—	61	1	no	maximal, Sylow
141	C₂²×C₆₁	D₄²	1	1	yes	characteristic
142	C₄×C₆₁	—	2	1	no	—
143	C₄×C₆₁	—	4	2	no	—
144	C₄×C₆₁	—	4	2	no	—
145	C₄×C₆₁	#575	1	1	yes	characteristic
146	C₄×C₆₁	#575	1	1	yes	characteristic
147	C₄×C₆₁	#578	1	1	yes	characteristic
148	C₄×C₆₁	#578	1	1	yes	characteristic
149	Dic₆₁	—	4	2	no	—
150	Dic₆₁	—	4	2	no	—
151	Dic₆₁	—	8	2	no	—
152	Dic₆₁	—	8	2	no	—
153	#3374	—	61	2	no	—
154	#3374	—	61	2	no	—
155	#1710	—	61	1	no	—
156	#1700	—	61	2	no	—
157	#1700	—	61	2	no	—
158	#1648	—	61	2	no	—
159	#1648	—	61	2	no	—
160	#1591	—	61	1	no	—
161	#1591	D₆₁	1	1	yes	characteristic
162	C₂×C₆₁	#3277	1	1	yes	characteristic
163	C₂×C₆₁	#3277	1	1	yes	characteristic
164	C₂×C₆₁	#3293	1	1	yes	characteristic
165	#557	—	61	2	no	—
166	#557	—	61	2	no	—
167	#500	—	61	1	no	—
168	#500	—	61	1	no	—
169	#491	—	122	2	no	—
170	#491	—	122	2	no	—
171	#487	—	61	2	no	—
172	#487	—	61	2	no	—
173	#479	—	122	2	no	—
174	#479	—	122	2	no	—
175	#475	—	61	2	no	—
176	#475	—	61	2	no	—
177	#462	—	61	2	no	—
178	#462	—	61	2	no	—
179	#462	—	122	2	no	—
180	#462	—	122	2	no	—
181	#446	—	61	2	no	—
182	#446	C₂×D₆₁	1	2	yes	—
183	#441	—	122	2	no	—
184	#441	—	122	2	no	—
185	C₄×Q₁₆	—	61	1	no	—
186	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
187	#335	—	61	1	no	—
188	#335	C₂×D₆₁	1	1	yes	characteristic
189	#332	—	61	2	no	—
190	#332	—	61	2	no	—
191	Q₈⋊C₈	—	61	2	no	—
192	Q₈⋊C₈	C₂×D₆₁	1	2	yes	—
193	C₈²	C₂×D₆₁	1	1	yes	characteristic
194	C₆₁	#20321	1	1	yes	characteristic, complemented, Sylow
195	C₂×Q₁₆	—	61	1	no	—
196	C₂×Q₁₆	—	122	2	no	—
197	C₂×Q₁₆	—	122	2	no	—
198	C₂×Q₁₆	D₂₄₄	1	1	yes	characteristic
199	C₄⋊Q₈	—	61	2	no	—
200	C₄⋊Q₈	—	61	2	no	—
201	C₄⋊Q₈	—	122	2	no	—
202	C₄⋊Q₈	—	122	2	no	—
203	#125	—	122	2	no	—
204	#125	—	122	2	no	—
205	#125	—	244	2	no	—
206	#125	—	244	2	no	—
207	C₄×Q₈	—	61	2	no	—
208	C₄×Q₈	—	122	2	no	—
209	C₄×Q₈	—	122	2	no	—
210	C₄×Q₈	—	122	2	no	—
211	C₄×Q₈	—	122	2	no	—
212	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
213	#107	—	61	1	no	—
214	#107	—	244	1	no	—
215	#107	—	244	1	no	—
216	#107	D₂₄₄	1	1	yes	characteristic
217	#106	—	122	2	no	—
218	#106	—	122	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	61	2	no	—
221	C₄⋊C₈	—	122	2	no	—
222	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
223	#103	—	61	2	no	—
224	#103	—	122	2	no	—
225	#103	—	122	2	no	—
226	#103	—	122	2	no	—
227	#103	—	122	2	no	—
228	#103	—	122	2	no	—
229	#103	—	122	2	no	—
230	#103	—	122	2	no	—
231	#103	—	122	2	no	—
232	#103	D₂₄₄	1	2	yes	—
233	#97	—	2	2	no	—
234	#97	—	122	2	no	—
235	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
238	C₂×Q₈	—	61	2	no	—
239	C₂×Q₈	—	122	2	no	—
240	C₂×Q₈	—	122	2	no	—
241	C₂×Q₈	C₂×D₂₄₄	1	2	yes	—
242	Q₁₆	—	4	1	no	—
243	Q₁₆	—	244	1	no	—
244	Q₁₆	—	244	4	no	—
245	Q₁₆	—	244	4	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	122	2	no	—
249	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
250	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
251	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
252	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	61	2	no	—
255	C₄⋊C₄	—	122	2	no	—
256	C₄⋊C₄	—	122	2	no	—
257	C₄⋊C₄	—	122	2	no	—
258	C₄⋊C₄	—	122	2	no	—
259	C₄⋊C₄	—	122	2	no	—
260	C₄⋊C₄	—	244	2	no	—
261	C₄⋊C₄	—	244	2	no	—
262	C₄⋊C₄	—	244	2	no	—
263	C₄⋊C₄	—	244	2	no	—
264	C₄⋊C₄	—	244	2	no	—
265	C₄⋊C₄	—	244	2	no	—
266	C₄⋊C₄	C₂×D₂₄₄	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	122	2	no	—
269	C₄²	—	244	2	no	—
270	C₄²	—	244	2	no	—
271	C₄²	C₂³×D₆₁	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	122	2	no	—
275	Q₈	—	122	2	no	—
276	Q₈	—	244	2	no	—
277	Q₈	—	244	2	no	—
278	Q₈	—	244	2	no	—
279	Q₈	—	244	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	122	2	no	—
283	C₂×C₄	—	122	2	no	—
284	C₂×C₄	—	244	2	no	—
285	C₂×C₄	—	244	2	no	—
286	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
288	C₂×C₄	#14519823	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	4	2	no	—
295	C₈	—	244	2	no	—
296	C₂²	D₄×D₂₄₄	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	488	2	no	—
301	C₄	—	488	2	no	—
302	C₄	—	244	2	no	—
303	C₄	—	244	2	no	—
304	C₄	#16840559	1	1	yes	characteristic
305	C₄	#16840562	1	1	yes	characteristic
306	C₄	#16840575	1	1	yes	characteristic
307	C₄	#16840579	1	1	yes	characteristic
308	C₂	#21457475	1	1	yes	characteristic
309	C₂	#21457477	1	1	yes	characteristic
310	C₂	#21457624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₄₄
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1591	C₂
socle, prime radical	C₂²×C₆₁	D₄²
hypercenter	#1591	D₆₁
hypocenter	C₆₁	#20321
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1591

Complements

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

N	H ≅ G/N	type
194. C₆₁	#20321	semidirect

Maximal subgroups

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

subgroup	index
#21458624	2
#21458620	2
#21445429	2
#21445374	2
#21445373	2
#21445064	2
#21445063	2
#21444717	2
C₆₁×#1591	2
#20321	61