Subgroups — #30606473

Order 15616.

418 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊#504	C₂	1	1	yes	maximal, characteristic
2	#21462031	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16840649	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461593	C₂	1	1	yes	maximal, characteristic
5	#21460872	C₂	1	1	yes	maximal, characteristic, complemented
6	#21460862	C₂	1	1	yes	maximal, characteristic, complemented
7	#350×D₆₁	C₂	1	1	yes	maximal, characteristic
8	#16840821	—	2	2	no	—
9	#16840821	C₄	1	2	yes	complemented
10	#16840820	C₄	1	2	yes	complemented
11	D₆₁⋊M₃₂	—	2	1	no	—
12	D₆₁⋊M₃₂	C₂²	1	1	yes	characteristic
13	C₂×D₆₁⋊C₁₆	—	2	1	no	—
14	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	D₆₁⋊#104	C₄	1	1	yes	characteristic
16	#16840752	C₄	1	1	yes	characteristic
17	#16840725	C₂²	1	1	yes	characteristic
18	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
19	#16840649	—	2	2	no	—
20	#16840649	—	2	1	no	—
21	C₁₆×#6923	—	2	1	no	—
22	#16840637	C₄	1	1	yes	characteristic
23	#16840634	C₄	1	1	yes	characteristic
24	#131×D₆₁	C₂²	1	1	yes	characteristic
25	M₃₂×D₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
27	#16839547	C₄	1	1	yes	characteristic, complemented
28	#16839546	C₄	1	1	yes	characteristic, complemented
29	#16839545	C₄	1	1	yes	characteristic, complemented
30	C₆₁×#350	C₄	1	1	yes	characteristic, complemented
31	D₆₁⋊M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
33	#14519856	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
35	#14519852	—	2	1	no	—
36	#14519852	—	2	2	no	—
37	#14519851	—	2	2	no	—
38	#14519851	C₂×C₄	1	2	yes	—
39	D₆₁⋊C₁₆	—	2	1	no	—
40	D₆₁⋊C₁₆	—	2	2	no	—
41	D₆₁⋊C₁₆	D₄	1	1	yes	characteristic
42	D₆₁⋊C₁₆	D₄	1	1	yes	characteristic
43	D₆₁⋊C₁₆	D₄	1	2	yes	—
44	C₂×#11758187	—	2	2	no	—
45	C₂×#11758187	C₂×C₄	1	2	yes	—
46	#14519842	C₈	1	2	yes	—
47	#14519841	C₈	1	2	yes	—
48	#14519840	C₈	1	2	yes	—
49	#14519839	C₈	1	2	yes	—
50	#14519831	—	2	1	no	—
51	C₈×#6923	—	2	1	no	—
52	C₈×#6923	—	2	1	no	—
53	C₈×#6923	—	2	1	no	—
54	#41×D₆₁	C₂×C₄	1	1	yes	characteristic
55	#14519785	—	2	1	no	—
56	#14519785	C₂×C₄	1	1	yes	characteristic
57	#14519784	C₂×C₄	1	1	yes	characteristic
58	#14519783	—	2	1	no	—
59	#14519783	C₂×C₄	1	1	yes	characteristic
60	M₁₆×D₆₁	—	2	1	no	—
61	M₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₆₁	—	2	1	no	—
63	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
64	#14519717	C₂×C₄	1	1	yes	characteristic
65	#14519716	—	2	1	no	—
66	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
67	C₁₆×D₆₁	—	2	1	no	—
68	C₁₆×D₆₁	—	2	1	no	—
69	C₁₆×D₆₁	—	2	1	no	—
70	#131×C₆₁	C₂×C₄	1	1	yes	characteristic
71	M₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₂²×#6923	—	2	1	no	—
74	#41×C₆₁	C₄²	1	1	yes	characteristic
75	#11758226	C₂×C₈	1	1	yes	characteristic
76	Q₈×D₆₁	C₂×C₈	1	1	yes	characteristic
77	#11758224	—	2	1	no	—
78	#11758224	C₂×C₈	1	1	yes	characteristic
79	D₄×D₆₁	—	2	1	no	—
80	D₄×D₆₁	C₂×C₈	1	1	yes	characteristic
81	#11758222	—	2	1	no	—
82	#11758222	C₄²	1	1	yes	characteristic
83	C₂×C₄×D₆₁	—	2	1	no	—
84	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊#6923	—	2	1	no	—
86	#11758217	—	2	1	no	—
87	C₂×#92420	—	2	1	no	—
88	C₄⋊#6923	C₂×C₈	1	2	yes	—
89	C₄×#6923	—	2	1	no	—
90	C₄×#6923	—	2	1	no	—
91	C₄×#6923	C₂×C₈	1	2	yes	—
92	#11758213	C₂×C₈	1	2	yes	—
93	D₆₁⋊C₈	—	2	1	no	—
94	D₆₁⋊C₈	—	2	1	no	—
95	D₆₁⋊C₈	C₂×C₈	1	2	yes	—
96	M₁₆×C₆₁	—	2	1	no	—
97	M₁₆×C₆₁	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₆₁	—	2	1	no	—
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	#11758194	—	2	1	no	—
101	#11758194	C₄²	1	1	yes	characteristic
102	C₂×#92418	—	2	1	no	—
103	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
104	#11758189	—	2	1	no	—
105	#11758189	—	2	1	no	—
106	#11758189	—	2	1	no	—
107	C₈×D₆₁	—	2	1	no	—
108	C₈×D₆₁	—	2	1	no	—
109	C₈×D₆₁	—	2	1	no	—
110	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
111	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
112	C₈×D₆₁	#41	1	1	yes	characteristic
113	C₈×D₆₁	#41	1	1	yes	characteristic
114	#11758187	—	2	2	no	—
115	#11758187	C₂²⋊C₄	1	2	yes	—
116	#11758187	C₂²⋊C₄	1	2	yes	—
117	C₁₆×C₆₁	—	2	1	no	—
118	C₁₆×C₆₁	—	2	1	no	—
119	#11758185	—	2	1	no	—
120	#11758185	—	2	1	no	—
121	C₂²×D₆₁	—	2	1	no	—
122	C₂²×D₆₁	#131	1	1	yes	characteristic
123	C₂×#6923	—	2	2	no	—
124	C₂×#6923	—	2	2	no	—
125	C₂×#6923	—	4	2	no	—
126	Q₈×C₆₁	C₄×C₈	1	1	yes	characteristic
127	D₄×C₆₁	—	2	1	no	—
128	D₄×C₆₁	C₄×C₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
131	C₂²⋊D₆₁	—	2	1	no	—
132	C₂²⋊D₆₁	—	2	1	no	—
133	C₂²⋊D₆₁	—	2	1	no	—
134	C₂×Dic₆₁	—	2	1	no	—
135	C₂×Dic₆₁	#131	1	1	yes	characteristic
136	D₂₄₄	—	2	1	no	—
137	D₂₄₄	C₄×C₈	1	1	yes	characteristic
138	C₄×D₆₁	—	2	1	no	—
139	C₄×D₆₁	—	2	1	no	—
140	C₄×D₆₁	—	2	1	no	—
141	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
142	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
143	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
144	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
145	Dic₁₂₂	—	2	1	no	—
146	Dic₁₂₂	C₄×C₈	1	1	yes	characteristic
147	#92420	—	2	2	no	—
148	#92420	—	2	2	no	—
149	C₈×C₆₁	—	2	1	no	—
150	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
151	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
152	#92418	—	2	1	no	—
153	#92418	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	#92418	#117	1	1	yes	characteristic
155	#12615	—	61	1	no	maximal, Sylow
156	C₂²×C₆₁	—	2	1	no	—
157	C₂²×C₆₁	#405	1	1	yes	characteristic
158	C₂×D₆₁	—	2	2	no	—
159	C₂×D₆₁	—	2	1	no	—
160	C₂×D₆₁	—	4	2	no	—
161	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
162	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
163	#6923	—	4	2	no	—
164	#6923	—	4	2	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
168	Dic₆₁	—	2	1	no	—
169	Dic₆₁	C₈×D₄	1	1	yes	characteristic
170	Dic₆₁	C₈×D₄	1	1	yes	characteristic
171	C₂×#504	—	61	1	no	—
172	C₂×#350	—	61	1	no	—
173	#2108	—	61	1	no	—
174	C₂×#346	—	61	1	no	—
175	#1764	—	61	1	no	—
176	#1379	—	61	1	no	—
177	#1376	—	61	1	no	—
178	C₂×C₆₁	—	2	1	no	—
179	C₂×C₆₁	—	4	1	no	—
180	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
181	D₆₁	—	2	1	no	—
182	D₆₁	—	4	1	no	—
183	D₆₁	#2171	1	2	yes	—
184	C₂×#131	—	61	1	no	—
185	#504	—	61	2	no	—
186	#504	—	61	2	no	—
187	#504	—	122	2	no	—
188	C₂×M₃₂	—	61	1	no	—
189	C₂×M₃₂	—	61	1	no	—
190	C₂×M₃₂	—	122	1	no	—
191	C₂²×C₁₆	—	61	1	no	—
192	C₂²×C₁₆	—	61	1	no	—
193	C₂²×C₁₆	—	122	1	no	—
194	C₂×#104	—	61	1	no	—
195	#416	—	61	1	no	—
196	#405	—	61	1	no	—
197	C₂×C₄×C₈	—	61	1	no	—
198	#350	—	61	1	no	—
199	#350	—	61	1	no	—
200	#350	—	61	1	no	—
201	#350	#6923	1	1	yes	characteristic, complemented
202	#346	—	122	2	no	—
203	#346	—	122	1	no	—
204	C₄×C₁₆	—	122	1	no	—
205	#341	—	61	1	no	—
206	#339	—	61	1	no	—
207	C₆₁	#12615	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	61	1	no	—
209	#131	—	61	1	no	—
210	#131	—	61	1	no	—
211	#131	—	61	1	no	—
212	#131	—	122	1	no	—
213	#131	—	122	1	no	—
214	#131	C₂×#6923	1	1	yes	characteristic
215	C₂×M₁₆	—	61	1	no	—
216	C₂×M₁₆	—	61	1	no	—
217	C₂×M₁₆	—	122	1	no	—
218	C₂²×C₈	—	61	1	no	—
219	C₂²×C₈	—	61	1	no	—
220	C₂²×C₈	—	122	1	no	—
221	#117	—	61	1	no	—
222	C₂×C₄²	—	61	1	no	—
223	M₃₂	—	61	2	no	—
224	M₃₂	—	61	1	no	—
225	M₃₂	—	122	1	no	—
226	M₃₂	—	122	2	no	—
227	M₃₂	—	122	2	no	—
228	M₃₂	—	122	1	no	—
229	M₃₂	C₂×#6923	1	1	yes	characteristic
230	C₂×C₁₆	—	61	2	no	—
231	C₂×C₁₆	—	61	1	no	—
232	C₂×C₁₆	—	61	1	no	—
233	C₂×C₁₆	—	61	2	no	—
234	C₂×C₁₆	—	61	1	no	—
235	C₂×C₁₆	—	122	1	no	—
236	C₂×C₁₆	—	122	1	no	—
237	C₂×C₁₆	—	122	1	no	—
238	C₂×C₁₆	—	122	2	no	—
239	C₂×C₁₆	—	122	2	no	—
240	C₂×C₁₆	—	122	1	no	—
241	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
242	#104	—	61	2	no	—
243	#104	—	61	2	no	—
244	#103	—	61	2	no	—
245	#102	—	61	2	no	—
246	#97	—	122	1	no	—
247	C₄×C₈	—	122	1	no	—
248	C₄×C₈	—	122	1	no	—
249	C₄×C₈	—	122	1	no	—
250	#41	—	61	1	no	—
251	#41	—	61	1	no	—
252	#41	—	61	1	no	—
253	#41	—	122	1	no	—
254	#41	—	122	1	no	—
255	#41	C₄×#6923	1	1	yes	characteristic
256	C₂×Q₈	—	61	1	no	—
257	C₂×D₄	—	61	1	no	—
258	C₂×D₄	—	122	1	no	—
259	C₂²×C₄	—	61	1	no	—
260	C₂²×C₄	—	122	1	no	—
261	C₂²×C₄	—	122	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	61	2	no	—
264	M₁₆	—	61	1	no	—
265	M₁₆	—	122	1	no	—
266	M₁₆	—	122	1	no	—
267	M₁₆	—	122	1	no	—
268	M₁₆	—	122	1	no	—
269	M₁₆	—	122	1	no	—
270	M₁₆	C₄×#6923	1	1	yes	characteristic
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	61	1	no	—
273	C₂×C₈	—	61	1	no	—
274	C₂×C₈	—	61	1	no	—
275	C₂×C₈	—	61	1	no	—
276	C₂×C₈	—	61	1	no	—
277	C₂×C₈	—	61	2	no	—
278	C₂×C₈	—	122	1	no	—
279	C₂×C₈	—	122	1	no	—
280	C₂×C₈	—	122	1	no	—
281	C₂×C₈	—	122	1	no	—
282	C₂×C₈	—	122	1	no	—
283	C₂×C₈	—	122	1	no	—
284	C₂×C₈	—	122	1	no	—
285	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
286	C₄⋊C₄	—	61	2	no	—
287	C₂²⋊C₄	—	122	1	no	—
288	C₄²	—	61	2	no	—
289	C₄²	—	122	1	no	—
290	C₄²	—	122	1	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	61	2	no	—
294	C₁₆	—	61	2	no	—
295	C₁₆	—	122	2	no	—
296	C₁₆	—	122	1	no	—
297	C₁₆	—	122	1	no	—
298	C₂³	—	61	1	no	—
299	C₂³	—	122	1	no	—
300	Q₈	—	61	1	no	—
301	Q₈	—	122	1	no	—
302	Q₈	C₈×#6923	1	1	yes	characteristic
303	D₄	—	2	1	no	—
304	D₄	—	61	1	no	—
305	D₄	—	122	1	no	—
306	D₄	—	122	1	no	—
307	D₄	—	122	1	no	—
308	D₄	—	122	1	no	—
309	D₄	C₈×#6923	1	1	yes	characteristic
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	61	1	no	—
312	C₂×C₄	—	61	1	no	—
313	C₂×C₄	—	61	1	no	—
314	C₂×C₄	—	61	1	no	—
315	C₂×C₄	—	61	1	no	—
316	C₂×C₄	—	122	1	no	—
317	C₂×C₄	—	122	1	no	—
318	C₂×C₄	—	122	1	no	—
319	C₂×C₄	—	122	1	no	—
320	C₂×C₄	—	122	2	no	—
321	C₂×C₄	—	122	2	no	—
322	C₂×C₄	—	244	2	no	—
323	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	61	1	no	—
326	C₈	—	61	1	no	—
327	C₈	—	122	2	no	—
328	C₈	—	122	2	no	—
329	C₈	—	122	1	no	—
330	C₈	D₄×#6923	1	1	yes	characteristic
331	C₈	D₄×#6923	1	1	yes	characteristic
332	C₂²	—	2	1	no	—
333	C₂²	—	61	1	no	—
334	C₂²	—	61	1	no	—
335	C₂²	—	122	2	no	—
336	C₂²	—	122	1	no	—
337	C₂²	—	244	2	no	—
338	C₂²	#16840728	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	61	1	no	—
341	C₄	—	61	1	no	—
342	C₄	—	122	1	no	—
343	C₄	—	244	2	no	—
344	C₄	—	244	2	no	—
345	C₄	C₂²⋊C₄×#6923	1	1	yes	characteristic
346	C₄	C₂×C₈×#6923	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	4	1	no	—
349	C₂	—	61	2	no	—
350	C₂	—	122	1	no	—
351	C₂	—	244	1	no	—
352	C₂	#21461554	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#350	C₄
socle	C₂×C₆₁	C₈×C₂²⋊C₄
prime radical	#41×D₆₁	C₂×C₄
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#350	#6923
hypocenter	C₆₁	#12615
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

Complements

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

N	H ≅ G/N	type
2. #21462031	C₂	semidirect
3. C₂×#16840649	C₂	semidirect
5. #21460872	C₂	semidirect
6. #21460862	C₂	semidirect
9. #16840821	C₄	semidirect
10. #16840820	C₄	semidirect
27. #16839547	C₄	semidirect
28. #16839546	C₄	semidirect
29. #16839545	C₄	semidirect
30. C₆₁×#350	C₄	semidirect
201. #350	#6923	semidirect
207. C₆₁	#12615	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊#504	2
#21462031	2
C₂×#16840649	2
#21461593	2
#21460872	2
#21460862	2
#350×D₆₁	2
#12615	61