Subgroups — #350×#6923 #30606407

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×#6923	C₂	1	1	yes	maximal, characteristic
2	D₆₁⋊#350	C₂	1	1	yes	maximal, characteristic
3	M₃₂×#6923	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×#6923	C₂	1	1	yes	maximal, characteristic, complemented
5	#21460875	C₂	1	1	yes	maximal, characteristic, complemented
6	#21460872	C₂	1	1	yes	maximal, characteristic, complemented
7	#350×D₆₁	C₂	1	1	yes	maximal, characteristic
8	D₆₁⋊#131	C₄	1	1	yes	characteristic
9	#41×#6923	C₄	1	1	yes	characteristic
10	D₆₁⋊M₃₂	C₂²	1	1	yes	characteristic
11	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16840728	—	2	1	no	—
13	#16840728	C₄	1	1	yes	characteristic
14	#16840725	—	2	1	no	—
15	#16840725	C₂²	1	1	yes	characteristic
16	M₁₆×#6923	—	2	1	no	—
17	M₁₆×#6923	C₄	1	1	yes	characteristic
18	C₂×C₈×#6923	—	2	1	no	—
19	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
20	#16840661	C₄	1	2	yes	complemented
21	#16840660	C₄	1	2	yes	complemented
22	#16840649	—	2	1	no	—
23	C₁₆×#6923	—	2	1	no	—
24	C₁₆×#6923	—	2	1	no	—
25	C₁₆×#6923	—	2	1	no	—
26	#131×D₆₁	C₂²	1	1	yes	characteristic
27	M₃₂×D₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
29	#16839547	C₄	1	1	yes	characteristic, complemented
30	#16839546	C₄	1	1	yes	characteristic, complemented
31	#16839545	C₄	1	1	yes	characteristic, complemented
32	C₆₁×#350	C₄	1	1	yes	characteristic, complemented
33	#14519865	C₈	1	2	yes	—
34	#14519864	—	2	2	no	—
35	#14519864	C₈	1	2	yes	—
36	D₆₁⋊M₁₆	—	2	1	no	—
37	D₆₁⋊M₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×D₆₁⋊C₈	—	2	1	no	—
39	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
40	Q₈×#6923	C₈	1	2	yes	—
41	D₄×#6923	—	2	2	no	—
42	D₄×#6923	C₈	1	2	yes	—
43	#14519856	—	2	1	no	—
44	#14519856	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×#6923	—	2	1	no	—
46	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
47	#14519852	—	2	1	no	—
48	#14519851	C₂×C₄	1	2	yes	complemented
49	D₆₁⋊C₁₆	—	2	1	no	—
50	D₆₁⋊C₁₆	—	2	1	no	—
51	D₆₁⋊C₁₆	—	2	1	no	—
52	C₂×#11758187	C₂×C₄	1	2	yes	complemented
53	#14519831	—	2	1	no	—
54	#14519831	—	2	1	no	—
55	#14519831	—	2	1	no	—
56	C₈×#6923	—	2	1	no	—
57	C₈×#6923	—	2	1	no	—
58	C₈×#6923	—	2	1	no	—
59	C₈×#6923	D₄	1	1	yes	characteristic
60	C₈×#6923	D₄	1	1	yes	characteristic
61	C₈×#6923	D₄	1	1	yes	characteristic
62	C₈×#6923	D₄	1	1	yes	characteristic
63	#41×D₆₁	C₂×C₄	1	1	yes	characteristic
64	#14519785	—	2	1	no	—
65	#14519785	C₂×C₄	1	1	yes	characteristic
66	#14519784	C₂×C₄	1	1	yes	characteristic
67	#14519783	—	2	1	no	—
68	#14519783	C₂×C₄	1	1	yes	characteristic
69	M₁₆×D₆₁	—	2	1	no	—
70	M₁₆×D₆₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₆₁	—	2	1	no	—
72	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
73	#14519717	C₂×C₄	1	1	yes	characteristic, complemented
74	#14519716	—	2	1	no	—
75	C₂×#11758185	C₂×C₄	1	1	yes	characteristic, complemented
76	C₁₆×D₆₁	—	2	1	no	—
77	C₁₆×D₆₁	—	2	1	no	—
78	C₁₆×D₆₁	—	2	1	no	—
79	#131×C₆₁	C₂×C₄	1	1	yes	characteristic
80	M₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
81	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
82	C₂²×#6923	—	2	2	no	—
83	C₂²×#6923	M₁₆	1	2	yes	—
84	#41×C₆₁	C₄²	1	1	yes	characteristic
85	#11758226	C₂×C₈	1	1	yes	characteristic
86	Q₈×D₆₁	C₂×C₈	1	1	yes	characteristic
87	#11758224	—	2	1	no	—
88	#11758224	C₂×C₈	1	1	yes	characteristic
89	D₄×D₆₁	—	2	1	no	—
90	D₄×D₆₁	C₂×C₈	1	1	yes	characteristic
91	#11758222	—	2	1	no	—
92	#11758222	C₄²	1	1	yes	characteristic
93	C₂×C₄×D₆₁	—	2	1	no	—
94	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²⋊#6923	—	2	1	no	—
96	C₂²⋊#6923	—	2	2	no	—
97	#11758217	—	2	1	no	—
98	#11758217	—	2	2	no	—
99	C₂×#92420	—	2	2	no	—
100	C₂×#92420	M₁₆	1	2	yes	—
101	C₄⋊#6923	—	2	2	no	—
102	C₄⋊#6923	C₂×C₈	1	2	yes	—
103	C₄×#6923	—	2	2	no	—
104	C₄×#6923	—	2	1	no	—
105	C₄×#6923	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×#6923	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×#6923	C₂×C₈	1	2	yes	—
108	#11758213	—	2	2	no	—
109	#11758213	C₂×C₈	1	2	yes	—
110	D₆₁⋊C₈	—	2	1	no	—
111	D₆₁⋊C₈	—	2	2	no	—
112	D₆₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	D₆₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	D₆₁⋊C₈	C₂×C₈	1	2	yes	—
115	M₁₆×C₆₁	—	2	1	no	—
116	M₁₆×C₆₁	C₄²	1	1	yes	characteristic
117	C₂×C₈×C₆₁	—	2	1	no	—
118	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
119	#11758194	—	2	1	no	—
120	#11758194	C₄²	1	1	yes	characteristic
121	C₂×#92418	—	2	1	no	—
122	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
123	#11758189	—	2	1	no	—
124	#11758189	—	2	1	no	—
125	#11758189	—	2	1	no	—
126	C₈×D₆₁	—	2	1	no	—
127	C₈×D₆₁	—	2	1	no	—
128	C₈×D₆₁	—	2	1	no	—
129	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
130	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
131	C₈×D₆₁	#41	1	1	yes	characteristic
132	C₈×D₆₁	#41	1	1	yes	characteristic
133	#11758187	—	2	2	no	—
134	#11758187	—	2	2	no	—
135	C₁₆×C₆₁	—	2	1	no	—
136	C₁₆×C₆₁	—	2	1	no	—
137	#11758185	—	2	1	no	—
138	#11758185	—	2	1	no	—
139	C₂²×D₆₁	—	2	1	no	—
140	C₂²×D₆₁	C₂×M₁₆	1	1	yes	characteristic
141	C₂×#6923	—	2	4	no	—
142	C₂×#6923	—	2	2	no	—
143	C₂×#6923	—	4	4	no	—
144	C₂×#6923	C₂²⋊C₈	1	2	yes	—
145	C₂×#6923	C₂²⋊C₈	1	2	yes	—
146	Q₈×C₆₁	C₄×C₈	1	1	yes	characteristic
147	D₄×C₆₁	—	2	1	no	—
148	D₄×C₆₁	C₄×C₈	1	1	yes	characteristic
149	C₂×C₄×C₆₁	—	2	1	no	—
150	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
151	C₂²⋊D₆₁	—	2	1	no	—
152	C₂²⋊D₆₁	—	2	1	no	—
153	C₂²⋊D₆₁	—	2	1	no	—
154	C₂×Dic₆₁	—	2	1	no	—
155	C₂×Dic₆₁	C₂×M₁₆	1	1	yes	characteristic
156	D₂₄₄	—	2	1	no	—
157	D₂₄₄	C₄×C₈	1	1	yes	characteristic
158	C₄×D₆₁	—	2	1	no	—
159	C₄×D₆₁	—	2	1	no	—
160	C₄×D₆₁	—	2	1	no	—
161	C₄×D₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×D₆₁	#117	1	1	yes	characteristic
163	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
164	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
165	Dic₁₂₂	—	2	1	no	—
166	Dic₁₂₂	C₄×C₈	1	1	yes	characteristic
167	#92420	—	2	2	no	—
168	#92420	C₂²⋊C₈	1	2	yes	—
169	#92420	C₂²⋊C₈	1	2	yes	—
170	C₈×C₆₁	—	2	1	no	—
171	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
172	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
173	#92418	—	2	1	no	—
174	#92418	C₄×D₄	1	1	yes	characteristic
175	#92418	C₄×D₄	1	1	yes	characteristic
176	C₄×#350	—	61	1	no	maximal, Sylow
177	C₂²×C₆₁	—	2	1	no	—
178	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
179	C₂×D₆₁	—	2	2	no	—
180	C₂×D₆₁	—	2	1	no	—
181	C₂×D₆₁	—	4	2	no	—
182	C₂×D₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
183	C₂×D₆₁	#431	1	1	yes	characteristic
184	#6923	—	2	2	no	—
185	#6923	—	4	2	no	—
186	#6923	#350	1	4	yes	direct product factor
187	C₄×C₆₁	—	2	1	no	—
188	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
190	Dic₆₁	—	2	1	no	—
191	Dic₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
192	Dic₆₁	#431	1	1	yes	characteristic
193	C₄×#131	—	61	1	no	—
194	C₂×#350	—	61	1	no	—
195	C₂×#350	—	61	1	no	—
196	C₄×M₃₂	—	61	1	no	—
197	C₂×C₄×C₁₆	—	61	1	no	—
198	#1379	—	61	1	no	—
199	#1379	—	61	1	no	—
200	C₂×C₆₁	—	2	1	no	—
201	C₂×C₆₁	—	4	1	no	—
202	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
203	D₆₁	—	2	1	no	—
204	D₆₁	—	4	1	no	—
205	D₆₁	C₂×#350	1	2	yes	—
206	C₂×#131	—	61	1	no	—
207	C₂×#131	—	61	1	no	—
208	C₄×#41	—	61	1	no	—
209	C₂×M₃₂	—	61	1	no	—
210	C₂×M₃₂	—	61	1	no	—
211	C₂²×C₁₆	—	61	1	no	—
212	C₂²×C₁₆	—	61	1	no	—
213	#405	—	61	1	no	—
214	#405	—	61	1	no	—
215	#405	—	122	1	no	—
216	#405	—	122	1	no	—
217	C₄×M₁₆	—	61	1	no	—
218	C₄×M₁₆	—	122	1	no	—
219	C₂×C₄×C₈	—	61	1	no	—
220	C₂×C₄×C₈	—	122	1	no	—
221	#350	—	61	2	no	—
222	#350	—	61	2	no	—
223	#350	—	61	1	no	—
224	#350	—	61	1	no	—
225	#350	—	61	1	no	—
226	#350	#6923	1	1	yes	characteristic, direct product factor
227	#346	—	122	1	no	—
228	C₄×C₁₆	—	122	1	no	—
229	C₄×C₁₆	—	122	1	no	—
230	C₄×C₁₆	—	122	1	no	—
231	C₆₁	C₄×#350	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	61	1	no	—
233	#131	—	61	2	no	—
234	#131	—	61	2	no	—
235	#131	—	61	1	no	—
236	#131	—	61	1	no	—
237	#131	—	61	1	no	—
238	#131	—	122	1	no	—
239	#131	—	122	1	no	—
240	#131	—	122	2	no	—
241	#131	C₂×#6923	1	1	yes	characteristic
242	C₂×M₁₆	—	61	1	no	—
243	C₂×M₁₆	—	61	1	no	—
244	C₂×M₁₆	—	122	1	no	—
245	C₂×M₁₆	—	122	1	no	—
246	C₂²×C₈	—	61	1	no	—
247	C₂²×C₈	—	61	1	no	—
248	C₂²×C₈	—	122	1	no	—
249	C₂²×C₈	—	122	1	no	—
250	C₄×Q₈	—	61	2	no	—
251	C₄×D₄	—	61	2	no	—
252	C₄×D₄	—	122	2	no	—
253	#117	—	61	1	no	—
254	#117	—	122	1	no	—
255	C₂×C₄²	—	61	1	no	—
256	C₂×C₄²	—	122	1	no	—
257	M₃₂	—	61	2	no	—
258	M₃₂	—	61	1	no	—
259	M₃₂	—	122	1	no	—
260	M₃₂	—	122	1	no	—
261	M₃₂	C₂×#6923	1	1	yes	characteristic, complemented
262	C₂×C₁₆	—	61	2	no	—
263	C₂×C₁₆	—	61	1	no	—
264	C₂×C₁₆	—	122	1	no	—
265	C₂×C₁₆	—	122	1	no	—
266	C₂×C₁₆	—	122	1	no	—
267	C₂×C₁₆	—	122	1	no	—
268	C₂×C₁₆	—	122	1	no	—
269	C₂×C₁₆	—	122	1	no	—
270	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic, complemented
271	#97	—	122	1	no	—
272	#97	—	122	1	no	—
273	#97	—	122	1	no	—
274	C₄×C₈	—	61	1	no	—
275	C₄×C₈	—	61	1	no	—
276	C₄×C₈	—	61	1	no	—
277	C₄×C₈	—	61	1	no	—
278	C₄×C₈	—	122	1	no	—
279	C₄×C₈	—	122	1	no	—
280	C₄×C₈	—	122	1	no	—
281	#41	—	61	1	no	—
282	#41	—	61	1	no	—
283	#41	—	61	1	no	—
284	#41	—	122	1	no	—
285	#41	—	122	1	no	—
286	#41	C₄×#6923	1	1	yes	characteristic
287	C₂×Q₈	—	61	1	no	—
288	C₂×D₄	—	61	1	no	—
289	C₂×D₄	—	122	1	no	—
290	C₂²×C₄	—	61	1	no	—
291	C₂²×C₄	—	61	2	no	—
292	C₂²×C₄	—	122	1	no	—
293	C₂²×C₄	—	122	2	no	—
294	M₁₆	—	2	1	no	—
295	M₁₆	—	61	2	no	—
296	M₁₆	—	61	1	no	—
297	M₁₆	—	122	1	no	—
298	M₁₆	—	122	1	no	—
299	M₁₆	—	122	1	no	—
300	M₁₆	—	122	2	no	—
301	M₁₆	—	122	2	no	—
302	M₁₆	—	122	1	no	—
303	M₁₆	—	122	1	no	—
304	M₁₆	C₄×#6923	1	1	yes	characteristic
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	61	1	no	—
307	C₂×C₈	—	61	1	no	—
308	C₂×C₈	—	61	2	no	—
309	C₂×C₈	—	61	2	no	—
310	C₂×C₈	—	61	1	no	—
311	C₂×C₈	—	61	1	no	—
312	C₂×C₈	—	61	1	no	—
313	C₂×C₈	—	61	1	no	—
314	C₂×C₈	—	61	1	no	—
315	C₂×C₈	—	122	1	no	—
316	C₂×C₈	—	122	1	no	—
317	C₂×C₈	—	122	1	no	—
318	C₂×C₈	—	122	2	no	—
319	C₂×C₈	—	122	2	no	—
320	C₂×C₈	—	122	1	no	—
321	C₂×C₈	—	122	1	no	—
322	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
323	C₄⋊C₄	—	61	2	no	—
324	C₄⋊C₄	—	122	2	no	—
325	C₂²⋊C₄	—	122	2	no	—
326	C₂²⋊C₄	—	122	1	no	—
327	C₄²	—	61	1	no	—
328	C₄²	—	61	1	no	—
329	C₄²	—	61	2	no	—
330	C₄²	—	122	1	no	—
331	C₄²	—	122	2	no	—
332	C₁₆	—	2	1	no	—
333	C₁₆	—	2	1	no	—
334	C₁₆	—	122	2	no	—
335	C₁₆	—	122	2	no	—
336	C₁₆	—	122	1	no	—
337	C₁₆	—	122	1	no	—
338	C₂³	—	61	1	no	—
339	C₂³	—	122	1	no	—
340	Q₈	—	61	1	no	—
341	Q₈	—	122	1	no	—
342	Q₈	C₈×#6923	1	1	yes	characteristic
343	D₄	—	2	1	no	—
344	D₄	—	61	1	no	—
345	D₄	—	122	1	no	—
346	D₄	—	122	1	no	—
347	D₄	—	122	1	no	—
348	D₄	—	122	1	no	—
349	D₄	C₈×#6923	1	1	yes	characteristic
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	61	2	no	—
352	C₂×C₄	—	61	1	no	—
353	C₂×C₄	—	61	1	no	—
354	C₂×C₄	—	61	1	no	—
355	C₂×C₄	—	61	1	no	—
356	C₂×C₄	—	61	1	no	—
357	C₂×C₄	—	61	2	no	—
358	C₂×C₄	—	122	1	no	—
359	C₂×C₄	—	122	1	no	—
360	C₂×C₄	—	122	1	no	—
361	C₂×C₄	—	122	4	no	—
362	C₂×C₄	—	122	1	no	—
363	C₂×C₄	—	122	2	no	—
364	C₂×C₄	—	244	4	no	—
365	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	61	2	no	—
368	C₈	—	61	2	no	—
369	C₈	—	61	1	no	—
370	C₈	—	61	1	no	—
371	C₈	—	122	2	no	—
372	C₈	—	122	1	no	—
373	C₈	D₄×#6923	1	1	yes	characteristic
374	C₈	D₄×#6923	1	1	yes	characteristic
375	C₂²	—	2	1	no	—
376	C₂²	—	61	1	no	—
377	C₂²	—	61	1	no	—
378	C₂²	—	122	2	no	—
379	C₂²	—	122	1	no	—
380	C₂²	—	244	2	no	—
381	C₂²	M₁₆×#6923	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	61	4	no	—
384	C₄	—	61	1	no	—
385	C₄	—	61	1	no	—
386	C₄	—	122	2	no	—
387	C₄	—	122	1	no	—
388	C₄	—	244	2	no	—
389	C₄	C₂²⋊C₄×#6923	1	1	yes	characteristic
390	C₄	C₂×C₈×#6923	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	4	1	no	—
393	C₂	—	61	2	no	—
394	C₂	—	122	1	no	—
395	C₂	—	244	1	no	—
396	C₂	C₂²⋊C₈×#6923	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	C₄×C₂²⋊C₈
prime radical	#41×D₆₁	C₂×C₄
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#350	#6923
hypocenter	C₆₁	C₄×#350
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
3. M₃₂×#6923	C₂	semidirect
4. C₂×C₁₆×#6923	C₂	semidirect
5. #21460875	C₂	semidirect
6. #21460872	C₂	semidirect
20. #16840661	C₄	semidirect
21. #16840660	C₄	semidirect
29. #16839547	C₄	semidirect
30. #16839546	C₄	semidirect
31. #16839545	C₄	semidirect
32. C₆₁×#350	C₄	semidirect
48. #14519851	C₂×C₄	semidirect
52. C₂×#11758187	C₂×C₄	semidirect
73. #14519717	C₂×C₄	semidirect
75. C₂×#11758185	C₂×C₄	semidirect
80. M₃₂×C₆₁	C₂×C₄	semidirect
81. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
186. #6923	#350	direct (N × H)
226. #350	#6923	direct (N × H)
231. C₆₁	C₄×#350	semidirect
261. M₃₂	C₂×#6923	semidirect
270. C₂×C₁₆	C₂×#6923	semidirect

Maximal subgroups

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

subgroup	index
#131×#6923	2
D₆₁⋊#350	2
M₃₂×#6923	2
C₂×C₁₆×#6923	2
#21460875	2
#21460872	2
#350×D₆₁	2
C₄×#350	61