Subgroups — #29568306

Order 15616.

402 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445962	C₂	1	1	yes	maximal, characteristic
2	#21445936	C₂	1	1	yes	maximal, characteristic
3	#21443645	C₂	1	2	yes	maximal
4	#21443642	C₂	1	2	yes	maximal
5	C₆₁×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	—	2	1	no	—
7	#16839517	—	2	1	no	—
8	#16839517	—	2	1	no	—
9	#16839517	—	2	1	no	—
10	#16839517	C₂²	1	2	yes	—
11	#16839517	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
13	C₆₁×#391	C₄	1	1	yes	characteristic
14	C₆₁×#391	C₂²	1	1	yes	characteristic
15	C₆₁×#389	C₄	1	1	yes	characteristic
16	C₆₁×#389	C₂²	1	1	yes	characteristic
17	C₆₁×#387	—	2	1	no	—
18	C₆₁×#387	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
21	C₆₁×#376	—	2	1	no	—
22	C₄³×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×#92418	—	2	2	no	—
24	C₂²×#92418	—	2	2	no	—
25	C₄⋊#92418	—	4	2	no	—
26	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Q₈×C₆₁	—	2	2	no	—
28	C₄×Q₈×C₆₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	—	2	1	no	—
36	C₂×C₄²×C₆₁	—	2	1	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	#95×C₆₁	—	2	1	no	—
45	#95×C₆₁	—	2	1	no	—
46	#95×C₆₁	—	2	1	no	—
47	#95×C₆₁	—	2	1	no	—
48	#95×C₆₁	—	2	1	no	—
49	#95×C₆₁	—	2	1	no	—
50	#95×C₆₁	—	2	1	no	—
51	C₂×Q₈×C₆₁	—	2	1	no	—
52	C₂×Q₈×C₆₁	—	4	1	no	—
53	C₂×Q₈×C₆₁	—	4	1	no	—
54	C₂×Q₈×C₆₁	M₁₆	1	2	yes	—
55	C₂²×C₄×C₆₁	—	2	1	no	—
56	C₂²×C₄×C₆₁	—	2	1	no	—
57	C₂²×C₄×C₆₁	—	2	1	no	—
58	C₂²×C₄×C₆₁	—	2	1	no	—
59	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₆₁	—	2	4	no	—
67	C₄⋊C₄×C₆₁	—	2	1	no	—
68	C₄⋊C₄×C₆₁	—	4	2	no	—
69	C₄⋊C₄×C₆₁	—	4	1	no	—
70	C₄⋊C₄×C₆₁	—	4	1	no	—
71	C₄⋊C₄×C₆₁	—	4	2	no	—
72	C₄⋊C₄×C₆₁	M₁₆	1	2	yes	—
73	C₄²×C₆₁	—	2	4	no	—
74	C₄²×C₆₁	—	2	1	no	—
75	C₄²×C₆₁	—	2	1	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	4	1	no	—
78	C₄²×C₆₁	—	4	1	no	—
79	C₄²×C₆₁	—	4	1	no	—
80	C₄²×C₆₁	—	4	1	no	—
81	C₄²×C₆₁	—	4	2	no	—
82	C₄²×C₆₁	SD₁₆	1	2	yes	—
83	C₄²×C₆₁	Q₁₆	1	2	yes	—
84	C₂×#92418	—	4	2	no	—
85	C₂×#92418	—	4	2	no	—
86	C₂×#92418	—	4	2	no	—
87	C₂×#92418	—	4	2	no	—
88	C₂×#92418	—	4	2	no	—
89	C₂×#92418	—	4	2	no	—
90	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₆₁	—	4	2	no	—
92	Q₈×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	2	1	no	—
94	C₂×C₄×C₆₁	—	2	1	no	—
95	C₂×C₄×C₆₁	—	2	1	no	—
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	—	2	1	no	—
99	C₂×C₄×C₆₁	—	2	1	no	—
100	C₂×C₄×C₆₁	—	2	1	no	—
101	C₂×C₄×C₆₁	—	2	1	no	—
102	C₂×C₄×C₆₁	—	2	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	—	4	1	no	—
113	C₂×C₄×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	4	1	no	—
116	C₂×C₄×C₆₁	—	4	1	no	—
117	C₂×C₄×C₆₁	—	4	1	no	—
118	C₂×C₄×C₆₁	—	4	1	no	—
119	C₂×C₄×C₆₁	—	4	1	no	—
120	C₂×C₄×C₆₁	—	4	1	no	—
121	C₂×C₄×C₆₁	#103	1	2	yes	—
122	C₂×C₄×C₆₁	#103	1	2	yes	—
123	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₁	#135	1	2	yes	—
128	#92418	—	8	2	no	—
129	#92418	—	8	2	no	—
130	#7644	—	61	1	no	maximal, Sylow
131	C₂²×C₆₁	#407	1	1	yes	characteristic
132	C₂²×C₆₁	#410	1	1	yes	characteristic
133	C₂²×C₆₁	#413	1	1	yes	characteristic
134	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
135	C₂²×C₆₁	#416	1	1	yes	characteristic
136	C₂²×C₆₁	#421	1	1	yes	characteristic
137	C₂²×C₆₁	#421	1	1	yes	characteristic
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	—	2	1	no	—
140	C₄×C₆₁	—	2	1	no	—
141	C₄×C₆₁	—	2	2	no	—
142	C₄×C₆₁	—	2	1	no	—
143	C₄×C₆₁	—	2	1	no	—
144	C₄×C₆₁	—	2	2	no	—
145	C₄×C₆₁	—	4	2	no	—
146	C₄×C₆₁	—	4	1	no	—
147	C₄×C₆₁	—	4	2	no	—
148	C₄×C₆₁	—	4	1	no	—
149	C₄×C₆₁	—	8	1	no	—
150	C₄×C₆₁	—	8	1	no	—
151	#2295	D₆₁	1	1	yes	characteristic
152	#1839	—	61	1	no	—
153	#1831	—	61	1	no	—
154	#1272	—	61	2	no	—
155	#1271	—	61	2	no	—
156	C₂×C₆₁	#1464	1	1	yes	characteristic
157	C₂×C₆₁	#1487	1	1	yes	characteristic
158	C₂×C₆₁	#1488	1	1	yes	characteristic
159	C₂×C₆₁	#1518	1	1	yes	characteristic
160	C₂×C₆₁	#1521	1	1	yes	characteristic
161	C₂×C₆₁	#1535	1	1	yes	characteristic
162	C₂×C₆₁	#1542	1	1	yes	characteristic
163	C₂×C₄×Q₈	Dic₆₁	1	1	yes	characteristic
164	C₂×C₄⋊C₈	—	122	1	no	—
165	#391	Dic₆₁	1	1	yes	characteristic
166	#391	C₂×D₆₁	1	1	yes	characteristic
167	#389	Dic₆₁	1	1	yes	characteristic
168	#389	C₂×D₆₁	1	1	yes	characteristic
169	#387	—	2	1	no	—
170	#387	—	2	1	no	—
171	C₄×C₄⋊C₄	—	2	1	no	—
172	C₄×C₄⋊C₄	Dic₆₁	1	1	yes	characteristic
173	#376	—	2	1	no	—
174	C₄³	C₂×D₆₁	1	1	yes	characteristic
175	#336	—	61	2	no	—
176	#336	—	61	2	no	—
177	#336	—	122	1	no	—
178	#336	—	122	1	no	—
179	#336	—	122	1	no	—
180	C₆₁	#7644	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
182	C₂²×C₈	—	122	2	no	—
183	C₂²×C₈	—	122	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
201	C₄⋊C₈	—	244	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	#95	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	#11758194	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
224	C₂×C₈	—	244	2	no	—
225	C₂×C₈	—	244	2	no	—
226	C₂×C₈	—	244	2	no	—
227	C₂×C₈	—	244	2	no	—
228	C₂×C₈	—	244	2	no	—
229	C₂×C₈	—	244	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	#11758194	1	2	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	#11758201	1	2	yes	—
247	C₄²	#11758202	1	2	yes	—
248	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	#14519701	1	2	yes	—
280	C₂×C₄	#14519701	1	2	yes	—
281	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
282	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
283	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
284	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
285	C₂×C₄	#14519797	1	2	yes	—
286	C₈	—	488	2	no	—
287	C₈	—	488	2	no	—
288	C₂²	#16839726	1	1	yes	characteristic
289	C₂²	#16839736	1	1	yes	characteristic
290	C₂²	#16839746	1	1	yes	characteristic
291	C₂²	C₂×#14519701	1	1	yes	characteristic
292	C₂²	#16839758	1	1	yes	characteristic
293	C₂²	#16839779	1	1	yes	characteristic
294	C₂²	#16839779	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	#21444153	1	1	yes	characteristic
309	C₂	#21444248	1	1	yes	characteristic
310	C₂	#21444253	1	1	yes	characteristic
311	C₂	#21444383	1	1	yes	characteristic
312	C₂	#21444396	1	1	yes	characteristic
313	C₂	#21444450	1	1	yes	characteristic
314	C₂	#21444476	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2295	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#2295	D₆₁
hypocenter	C₆₁	#7644
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
180. C₆₁	#7644	semidirect

Maximal subgroups

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

subgroup	index
#21445962	2
#21445936	2
#21443645	2
#21443642	2
C₆₁×#2295	2
#7644	61