Subgroups — #30624270

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 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	2	yes	maximal, complemented
2	#21462211	C₂	1	1	yes	maximal, characteristic, complemented
3	#21462210	C₂	1	2	yes	maximal, complemented
4	#21462209	C₂	1	1	yes	maximal, characteristic, complemented
5	#21462208	C₂	1	1	yes	maximal, characteristic, complemented
6	#21462207	C₂	1	2	yes	maximal, complemented
7	#21462206	C₂	1	1	yes	maximal, characteristic, complemented
8	#21462161	C₂	1	1	yes	maximal, characteristic, complemented
9	#21462062	C₂	1	2	yes	maximal, complemented
10	#21462029	C₂	1	1	yes	maximal, characteristic, complemented
11	#21454719	C₂	1	1	yes	maximal, characteristic
12	#16840821	—	2	2	no	—
13	#16840821	—	2	2	no	—
14	#16840821	C₂²	1	2	yes	complemented
15	#16840821	C₂²	1	2	yes	complemented
16	#16840820	C₂²	1	2	yes	complemented
17	#16840820	C₂²	1	2	yes	complemented
18	C₂×#14519852	C₂²	1	1	yes	characteristic
19	D₆₁⋊M₃₂	—	2	2	no	—
20	D₆₁⋊M₃₂	C₂²	1	2	yes	—
21	C₂×D₆₁⋊C₁₆	—	2	2	no	—
22	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
23	#16840685	C₂²	1	2	yes	complemented
24	#16840684	C₂²	1	1	yes	characteristic, complemented
25	#16840684	C₂²	1	1	yes	characteristic, complemented
26	#16840661	C₂²	1	2	yes	complemented
27	#16840661	C₂²	1	2	yes	complemented
28	#16840660	C₂²	1	2	yes	complemented
29	#16840660	C₂²	1	2	yes	complemented
30	#16840651	C₂²	1	1	yes	characteristic, complemented
31	#16840650	C₂²	1	1	yes	characteristic, complemented
32	#16840648	C₂²	1	1	yes	characteristic, complemented
33	C₄×#11758187	C₂²	1	1	yes	characteristic, complemented
34	#16840559	C₄	1	1	yes	characteristic
35	#131×D₆₁	C₂²	1	2	yes	—
36	#16839885	C₄	1	1	yes	characteristic
37	#16839882	C₄	1	2	yes	—
38	#16839879	C₂²	1	1	yes	characteristic
39	#16839878	C₂²	1	1	yes	characteristic
40	#16839876	C₂²	1	2	yes	—
41	#16839809	C₄	1	1	yes	characteristic
42	#16839777	C₄	1	2	yes	—
43	#16839719	C₂²	1	1	yes	characteristic
44	C₆₁×#444	C₄	1	1	yes	characteristic
45	#14519852	—	2	2	no	—
46	#14519852	—	2	2	no	—
47	#14519852	—	2	2	no	—
48	#14519852	—	2	2	no	—
49	#14519851	—	2	2	no	—
50	#14519851	—	2	2	no	—
51	#14519851	C₂³	1	2	yes	—
52	#14519851	C₂³	1	2	yes	—
53	D₆₁⋊C₁₆	—	2	2	no	—
54	D₆₁⋊C₁₆	—	2	2	no	—
55	D₆₁⋊C₁₆	—	2	2	no	—
56	D₆₁⋊C₁₆	D₄	1	2	yes	—
57	D₆₁⋊C₁₆	D₄	1	2	yes	—
58	C₂×#11758187	—	2	2	no	—
59	C₂×#11758187	—	2	2	no	—
60	C₂×#11758187	C₂³	1	1	yes	characteristic
61	C₂×#11758187	C₂³	1	1	yes	characteristic
62	C₂×#11758187	C₂³	1	1	yes	characteristic
63	C₂×#11758187	C₂³	1	1	yes	characteristic
64	#41×D₆₁	C₂×C₄	1	2	yes	—
65	#14519809	C₈	1	2	yes	—
66	#14519806	C₈	1	2	yes	—
67	#14519804	C₈	1	2	yes	—
68	#14519801	C₈	1	2	yes	—
69	#14519799	C₂×C₄	1	1	yes	characteristic
70	#14519797	C₂×C₄	1	2	yes	—
71	#14519785	—	2	2	no	—
72	#14519785	C₂³	1	2	yes	—
73	#14519784	C₂³	1	2	yes	—
74	#14519783	—	2	2	no	—
75	#14519783	C₂×C₄	1	2	yes	—
76	C₂×#11758189	C₂×C₄	1	1	yes	characteristic
77	M₁₆×D₆₁	—	2	2	no	—
78	M₁₆×D₆₁	C₂×C₄	1	2	yes	—
79	C₂×C₈×D₆₁	—	2	2	no	—
80	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
81	#14519734	C₂×C₄	1	1	yes	characteristic
82	#14519713	C₂×C₄	1	1	yes	characteristic
83	#14519712	C₂×C₄	1	2	yes	—
84	#14519705	C₂×C₄	1	2	yes	—
85	#14519704	C₂×C₄	1	2	yes	—
86	#14519703	C₂×C₄	1	2	yes	—
87	#14519687	C₂×C₄	1	1	yes	characteristic
88	C₄×#92418	C₂³	1	1	yes	characteristic
89	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
90	#135×C₆₁	C₂×C₄	1	1	yes	characteristic
91	#131×C₆₁	C₂×C₄	1	2	yes	—
92	#108×C₆₁	C₂×C₄	1	1	yes	characteristic
93	#104×C₆₁	C₂×C₄	1	2	yes	—
94	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
95	#41×C₆₁	C₂²×C₄	1	2	yes	—
96	#11758226	C₂×C₈	1	2	yes	—
97	Q₈×D₆₁	C₂×C₈	1	2	yes	—
98	#11758224	—	2	2	no	—
99	#11758224	C₂×C₈	1	2	yes	—
100	D₄×D₆₁	—	2	2	no	—
101	D₄×D₆₁	C₂×C₈	1	2	yes	—
102	#11758222	—	2	2	no	—
103	#11758222	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₆₁	—	2	2	no	—
105	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
106	Q₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
107	SD₁₆×C₆₁	C₂×C₈	1	2	yes	—
108	D₈×C₆₁	C₂×C₈	1	1	yes	characteristic
109	M₁₆×C₆₁	—	2	2	no	—
110	M₁₆×C₆₁	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₆₁	—	2	2	no	—
112	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	#11758202	C₂×C₈	1	2	yes	—
116	#11758201	C₂×C₈	1	2	yes	—
117	#11758200	C₂×C₈	1	2	yes	—
118	#11758199	C₂×C₈	1	2	yes	—
119	#11758198	—	2	1	no	—
120	#11758196	—	2	1	no	—
121	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
122	#11758194	—	2	2	no	—
123	#11758194	C₂²×C₄	1	2	yes	—
124	C₂×#92418	—	2	2	no	—
125	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
126	C₂×#92418	C₂⁴	1	1	yes	characteristic
127	Dic₂₄₄	C₂×C₈	1	1	yes	characteristic
128	D₄₈₈	C₂×C₈	1	1	yes	characteristic
129	#11758190	C₂×C₈	1	2	yes	—
130	#11758189	—	2	2	no	—
131	#11758189	—	2	2	no	—
132	#11758189	—	2	2	no	—
133	#11758189	C₂×C₈	1	2	yes	—
134	#11758189	C₂×C₈	1	2	yes	—
135	C₈×D₆₁	—	2	2	no	—
136	C₈×D₆₁	—	2	2	no	—
137	C₈×D₆₁	—	2	2	no	—
138	C₈×D₆₁	C₂×D₄	1	2	yes	—
139	C₈×D₆₁	#41	1	2	yes	—
140	#11758187	—	2	2	no	—
141	#11758187	—	2	2	no	—
142	#11758187	—	2	1	no	—
143	#11758187	—	2	1	no	—
144	#11758187	C₂×D₄	1	1	yes	characteristic
145	#11758187	C₂×D₄	1	2	yes	—
146	#11758187	C₂×D₄	1	1	yes	characteristic
147	C₂²×D₆₁	—	2	2	no	—
148	C₂²×D₆₁	#131	1	1	yes	characteristic
149	Q₈×C₆₁	C₂²×C₈	1	2	yes	—
150	D₄×C₆₁	—	2	2	no	—
151	D₄×C₆₁	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₆₁	—	2	2	no	—
153	C₂×C₄×C₆₁	—	2	1	no	—
154	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₆₁	—	2	2	no	—
156	C₂²⋊D₆₁	—	2	2	no	—
157	C₂²⋊D₆₁	—	2	2	no	—
158	C₂×Dic₆₁	—	2	2	no	—
159	C₂×Dic₆₁	—	2	1	no	—
160	C₂×Dic₆₁	#131	1	1	yes	characteristic
161	D₂₄₄	—	2	2	no	—
162	D₂₄₄	C₂²×C₈	1	2	yes	—
163	C₄×D₆₁	—	2	2	no	—
164	C₄×D₆₁	—	2	2	no	—
165	C₄×D₆₁	—	2	2	no	—
166	C₄×D₆₁	C₄×D₄	1	2	yes	—
167	C₄×D₆₁	C₂²×C₈	1	2	yes	—
168	Dic₁₂₂	—	2	2	no	—
169	Dic₁₂₂	C₂²×C₈	1	2	yes	—
170	C₈×C₆₁	—	2	2	no	—
171	C₈×C₆₁	C₄×D₄	1	2	yes	—
172	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
174	#92418	—	2	2	no	—
175	#92418	C₂²×C₈	1	2	yes	—
176	#92418	C₂²×D₄	1	1	yes	characteristic
177	#92418	C₂×#41	1	1	yes	characteristic
178	#33441	—	61	1	no	maximal, Sylow
179	C₂²×C₆₁	—	2	2	no	—
180	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
181	C₂×D₆₁	—	2	2	no	—
182	C₂×D₆₁	—	4	1	no	—
183	C₂×D₆₁	—	4	4	no	—
184	C₂×D₆₁	C₈×D₄	1	2	yes	—
185	C₄×C₆₁	—	2	2	no	—
186	C₄×C₆₁	—	4	1	no	—
187	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
189	Dic₆₁	—	2	2	no	—
190	Dic₆₁	—	4	1	no	—
191	Dic₆₁	C₈×D₄	1	2	yes	—
192	C₂×#504	—	61	2	no	—
193	#2955	—	61	1	no	—
194	#2171	—	61	1	no	—
195	#2171	—	61	2	no	—
196	#2171	—	61	1	no	—
197	#2170	—	61	1	no	—
198	#2170	—	61	2	no	—
199	#2170	—	61	1	no	—
200	#2153	—	61	1	no	—
201	#2117	—	61	2	no	—
202	#2108	—	61	1	no	—
203	C₂×C₆₁	—	2	1	no	—
204	C₂×C₆₁	—	4	2	no	—
205	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
206	D₆₁	—	2	2	no	—
207	D₆₁	—	4	2	no	—
208	#575	—	61	1	no	—
209	C₂×#131	—	61	2	no	—
210	#504	—	61	2	no	—
211	#504	—	61	2	no	—
212	#504	—	61	2	no	—
213	#504	—	61	2	no	—
214	#504	—	122	2	no	—
215	#504	—	122	2	no	—
216	C₂×M₃₂	—	61	1	no	—
217	C₂×M₃₂	—	61	2	no	—
218	C₂×M₃₂	—	122	2	no	—
219	C₂²×C₁₆	—	61	1	no	—
220	C₂²×C₁₆	—	122	2	no	—
221	#444	—	61	2	no	—
222	#444	—	61	1	no	—
223	#444	#6923	1	1	yes	characteristic
224	#443	—	61	1	no	—
225	#443	—	61	1	no	—
226	#443	—	61	2	no	—
227	#430	—	61	1	no	—
228	#421	—	61	2	no	—
229	#405	—	61	1	no	—
230	#364	—	61	1	no	—
231	#364	—	61	1	no	—
232	#364	—	61	2	no	—
233	#350	—	61	2	no	—
234	#350	—	61	2	no	—
235	#350	—	61	2	no	—
236	#350	—	61	2	no	—
237	#346	—	61	1	no	—
238	#346	—	61	1	no	—
239	C₄×C₁₆	—	61	1	no	—
240	C₄×C₁₆	—	61	1	no	—
241	C₆₁	#33441	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	61	2	no	—
243	#137	—	61	2	no	—
244	#137	—	61	2	no	—
245	AGL(1,ℤ/8)	—	61	2	no	—
246	AGL(1,ℤ/8)	—	61	2	no	—
247	#135	—	61	2	no	—
248	#135	—	61	1	no	—
249	#135	C₂×#6923	1	1	yes	characteristic
250	#131	—	61	2	no	—
251	#131	—	61	2	no	—
252	#131	—	61	2	no	—
253	#131	—	122	2	no	—
254	#131	—	122	2	no	—
255	#131	C₂×#6923	1	2	yes	—
256	C₂×M₁₆	—	61	1	no	—
257	C₂×M₁₆	—	61	2	no	—
258	C₂×M₁₆	—	122	2	no	—
259	C₂²×C₈	—	61	1	no	—
260	C₂²×C₈	—	122	2	no	—
261	#117	—	61	1	no	—
262	M₃₂	—	61	2	no	—
263	M₃₂	—	61	2	no	—
264	M₃₂	—	122	2	no	—
265	M₃₂	—	122	2	no	—
266	M₃₂	—	122	2	no	—
267	M₃₂	—	122	2	no	—
268	M₃₂	—	122	2	no	—
269	M₃₂	—	122	2	no	—
270	C₂×C₁₆	—	61	1	no	—
271	C₂×C₁₆	—	61	1	no	—
272	C₂×C₁₆	—	61	2	no	—
273	C₂×C₁₆	—	61	2	no	—
274	C₂×C₁₆	—	61	1	no	—
275	C₂×C₁₆	—	61	1	no	—
276	C₂×C₁₆	—	122	2	no	—
277	C₂×C₁₆	—	122	2	no	—
278	C₂×C₁₆	—	122	2	no	—
279	C₂×C₁₆	—	122	2	no	—
280	C₂×C₁₆	—	122	2	no	—
281	#108	—	61	2	no	—
282	#108	—	61	1	no	—
283	#108	C₂×#6923	1	1	yes	characteristic
284	#104	—	61	2	no	—
285	#104	—	61	2	no	—
286	#104	—	61	2	no	—
287	#104	C₂×#6923	1	2	yes	—
288	#97	—	61	1	no	—
289	#97	C₂×#6923	1	1	yes	characteristic
290	C₄×C₈	—	61	1	no	—
291	C₄×C₈	—	61	1	no	—
292	#41	—	61	2	no	—
293	#41	—	61	2	no	—
294	#41	—	61	2	no	—
295	#41	—	122	2	no	—
296	#41	—	122	2	no	—
297	#41	C₂²×#6923	1	2	yes	—
298	C₂×Q₈	—	61	2	no	—
299	C₂×D₄	—	61	2	no	—
300	C₂×D₄	—	122	2	no	—
301	C₂²×C₄	—	61	1	no	—
302	C₂²×C₄	—	122	2	no	—
303	Q₁₆	—	61	2	no	—
304	Q₁₆	—	61	1	no	—
305	Q₁₆	D₆₁⋊C₈	1	1	yes	characteristic
306	SD₁₆	—	61	2	no	—
307	SD₁₆	—	61	2	no	—
308	SD₁₆	—	61	2	no	—
309	SD₁₆	D₆₁⋊C₈	1	2	yes	—
310	D₈	—	61	2	no	—
311	D₈	—	61	1	no	—
312	D₈	D₆₁⋊C₈	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	61	2	no	—
315	M₁₆	—	61	2	no	—
316	M₁₆	—	61	2	no	—
317	M₁₆	—	122	2	no	—
318	M₁₆	—	122	2	no	—
319	M₁₆	—	122	2	no	—
320	M₁₆	—	122	2	no	—
321	M₁₆	C₂²×#6923	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	61	1	no	—
324	C₂×C₈	—	61	1	no	—
325	C₂×C₈	—	61	2	no	—
326	C₂×C₈	—	61	2	no	—
327	C₂×C₈	—	122	2	no	—
328	C₂×C₈	—	122	2	no	—
329	C₂×C₈	—	122	2	no	—
330	C₂×C₈	—	122	2	no	—
331	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
332	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
333	C₄⋊C₄	—	122	1	no	—
334	C₂²⋊C₄	—	122	1	no	—
335	C₄²	—	61	1	no	—
336	C₄²	C₂²×#6923	1	1	yes	characteristic
337	C₁₆	—	61	1	no	—
338	C₁₆	—	61	2	no	—
339	C₁₆	—	61	1	no	—
340	C₁₆	—	122	1	no	—
341	C₁₆	—	122	1	no	—
342	C₁₆	—	122	2	no	—
343	C₁₆	—	122	2	no	—
344	C₂³	—	61	1	no	—
345	C₂³	—	122	2	no	—
346	Q₈	—	61	2	no	—
347	Q₈	—	122	2	no	—
348	Q₈	C₂×D₆₁⋊C₈	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	61	2	no	—
351	D₄	—	122	2	no	—
352	D₄	—	122	2	no	—
353	D₄	—	122	2	no	—
354	D₄	—	122	2	no	—
355	D₄	C₂×D₆₁⋊C₈	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	61	1	no	—
359	C₂×C₄	—	61	2	no	—
360	C₂×C₄	—	61	2	no	—
361	C₂×C₄	—	122	2	no	—
362	C₂×C₄	—	122	2	no	—
363	C₂×C₄	—	122	2	no	—
364	C₂×C₄	—	122	2	no	—
365	C₂×C₄	—	122	1	no	—
366	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	61	1	no	—
369	C₈	—	61	1	no	—
370	C₈	—	61	2	no	—
371	C₈	—	122	2	no	—
372	C₈	D₄×#6923	1	2	yes	—
373	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
374	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	61	2	no	—
377	C₂²	—	122	2	no	—
378	C₂²	—	244	1	no	—
379	C₂²	—	244	4	no	—
380	C₂²	D₆₁⋊#131	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	61	2	no	—
384	C₄	—	122	2	no	—
385	C₄	—	244	1	no	—
386	C₄	C₂×D₄×#6923	1	1	yes	characteristic
387	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	122	2	no	—
391	C₂	—	244	2	no	—
392	C₂	D₄×D₆₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#444	C₄
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	#16840559	C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	#444	#6923
hypocenter	C₆₁	#33441
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
1. D₆₁⋊#504	C₂	semidirect
2. #21462211	C₂	semidirect
3. #21462210	C₂	semidirect
4. #21462209	C₂	semidirect
5. #21462208	C₂	semidirect
6. #21462207	C₂	semidirect
7. #21462206	C₂	semidirect
8. #21462161	C₂	semidirect
9. #21462062	C₂	semidirect
10. #21462029	C₂	semidirect
14. #16840821	C₂²	semidirect
15. #16840821	C₂²	semidirect
16. #16840820	C₂²	semidirect
17. #16840820	C₂²	semidirect
23. #16840685	C₂²	semidirect
24. #16840684	C₂²	semidirect
25. #16840684	C₂²	semidirect
26. #16840661	C₂²	semidirect
27. #16840661	C₂²	semidirect
28. #16840660	C₂²	semidirect
29. #16840660	C₂²	semidirect
30. #16840651	C₂²	semidirect
31. #16840650	C₂²	semidirect
32. #16840648	C₂²	semidirect
33. C₄×#11758187	C₂²	semidirect
241. C₆₁	#33441	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
#21462211	2
#21462210	2
#21462209	2
#21462208	2
#21462207	2
#21462206	2
#21462161	2
#21462062	2
#21462029	2
#21454719	2
#33441	61