Subgroups — #29833536

Order 15616.

558 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11758224	C₂	1	1	yes	maximal, characteristic
2	#21447727	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447724	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447651	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447647	C₂	1	2	yes	maximal, complemented
6	#21447642	C₂	1	2	yes	maximal, complemented
7	#21447617	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447615	C₂	1	2	yes	maximal, complemented
9	M₃₂×Dic₆₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11758224	C₄	1	1	yes	characteristic
13	C₂×#14519717	C₂²	1	2	yes	—
14	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
17	#16839886	C₄	1	1	yes	characteristic
18	D₄×#92418	C₄	1	1	yes	characteristic
19	C₈×C₂²⋊D₆₁	C₂²	1	2	yes	—
20	#16839825	C₄	1	2	yes	—
21	#16839814	C₄	1	1	yes	characteristic
22	#16839812	C₄	1	2	yes	—
23	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
24	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
25	C₄⋊#11758185	C₂²	1	1	yes	characteristic
26	#16839580	C₂²	1	2	yes	—
27	#16839580	C₂²	1	2	yes	—
28	C₂²⋊#11758185	C₂²	1	2	yes	—
29	#16839541	C₂²	1	2	yes	—
30	#16839537	C₂²	1	2	yes	—
31	#16839537	C₂²	1	2	yes	—
32	C₄×#11758185	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₆₁	C₂²	1	2	yes	—
35	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
36	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
37	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#11758224	C₈	1	1	yes	characteristic
42	C₂²×#92418	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
45	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
46	#14519742	C₈	1	1	yes	characteristic
47	D₄×Dic₆₁	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₆₁	C₂×C₄	1	2	yes	—
49	#14519737	C₈	1	2	yes	—
50	#14519735	C₈	1	1	yes	characteristic
51	#14519733	C₈	1	2	yes	—
52	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
53	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
54	#14519717	—	2	4	no	—
55	C₂×#11758185	C₂³	1	1	yes	characteristic
56	C₂×#11758185	C₂³	1	2	yes	—
57	C₂×#11758185	C₂³	1	1	yes	characteristic
58	C₁₆×D₆₁	—	2	2	no	—
59	C₁₆×D₆₁	—	2	2	no	—
60	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
61	#14519706	C₂×C₄	1	2	yes	—
62	#14519706	C₂×C₄	1	2	yes	—
63	C₂²⋊#92418	C₂×C₄	1	2	yes	—
64	#14519689	C₂×C₄	1	2	yes	—
65	C₄×#92418	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
68	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
69	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
70	C₈×Dic₆₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
72	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
73	M₃₂×C₆₁	—	2	4	no	—
74	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
76	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
82	C₂×C₂²⋊D₆₁	C₂×C₈	1	2	yes	—
83	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
84	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
85	#11758224	—	2	4	no	—
86	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₂₂	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
96	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
97	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
99	#11758198	C₂×C₈	1	2	yes	—
100	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
101	#11758196	C₂×C₈	1	2	yes	—
102	#11758196	C₂×C₈	1	2	yes	—
103	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
106	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
107	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
109	C₂×#92418	—	2	4	no	—
110	C₂×#92418	C₂²×C₄	1	2	yes	—
111	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
112	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₆₁	—	2	1	no	—
114	C₈×D₆₁	—	2	1	no	—
115	C₈×D₆₁	—	2	2	no	—
116	C₁₆×C₆₁	—	2	2	no	—
117	C₁₆×C₆₁	—	2	1	no	—
118	C₁₆×C₆₁	#41	1	2	yes	complemented
119	#11758185	—	2	1	no	—
120	#11758185	—	2	2	no	—
121	#11758185	#41	1	2	yes	complemented
122	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
123	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
124	D₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
131	C₂²⋊D₆₁	—	2	4	no	—
132	C₂×Dic₆₁	—	2	4	no	—
133	C₂×Dic₆₁	—	2	4	no	—
134	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
136	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
137	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
138	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
139	C₄×D₆₁	—	2	1	no	—
140	C₄×D₆₁	—	2	1	no	—
141	C₄×D₆₁	—	2	2	no	—
142	C₄×D₆₁	M₃₂	1	4	yes	complemented
143	Dic₁₂₂	—	2	2	no	—
144	C₈×C₆₁	—	2	2	no	—
145	C₈×C₆₁	—	2	1	no	—
146	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
148	#92418	—	2	1	no	—
149	#92418	—	2	2	no	—
150	#92418	#131	1	2	yes	—
151	#33479	—	61	1	no	maximal, Sylow
152	C₂²×C₆₁	—	2	2	no	—
153	C₂²×C₆₁	—	2	2	no	—
154	C₂²×C₆₁	—	2	2	no	—
155	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
156	C₂×D₆₁	—	2	1	no	—
157	C₂×D₆₁	—	2	1	no	—
158	C₂×D₆₁	C₂×M₃₂	1	2	yes	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	—	2	2	no	—
161	C₄×C₆₁	C₂×M₃₂	1	2	yes	—
162	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
163	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
164	Dic₆₁	—	2	2	no	—
165	Dic₆₁	—	2	2	no	—
166	Dic₆₁	—	2	1	no	—
167	Dic₆₁	C₂×M₃₂	1	2	yes	—
168	Dic₆₁	#504	1	2	yes	—
169	C₈×#41	—	61	1	no	—
170	#2183	—	61	1	no	—
171	#2183	—	61	1	no	—
172	#2169	—	61	1	no	—
173	#2169	D₆₁	1	1	yes	characteristic, complemented
174	#2168	—	61	2	no	—
175	#2168	—	61	2	no	—
176	#2162	—	61	1	no	—
177	#2162	—	61	2	no	—
178	C₄×M₃₂	—	61	2	no	—
179	C₂×C₄×C₁₆	—	61	1	no	—
180	C₂×C₆₁	—	4	2	no	—
181	C₂×C₆₁	C₈×#41	1	1	yes	characteristic
182	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
183	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
184	D₆₁	—	2	2	no	—
185	C₄×#41	—	61	1	no	—
186	C₂×M₃₂	—	61	2	no	—
187	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
188	C₂²×C₁₆	—	61	1	no	—
189	C₂²×C₁₆	—	61	1	no	—
190	C₈×Q₈	—	61	1	no	—
191	C₈×Q₈	—	61	1	no	—
192	C₈×D₄	—	61	2	no	—
193	C₈×D₄	—	61	1	no	—
194	C₈×D₄	—	61	2	no	—
195	C₈×D₄	C₂×D₆₁	1	1	yes	characteristic
196	#431	—	61	1	no	—
197	#431	—	61	2	no	—
198	C₂×C₄×C₈	—	61	2	no	—
199	C₂×C₄×C₈	—	61	1	no	—
200	C₄⋊C₁₆	—	61	2	no	—
201	C₄⋊C₁₆	—	61	2	no	—
202	C₄⋊C₁₆	—	61	1	no	—
203	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
204	C₂²⋊C₁₆	—	61	2	no	—
205	C₂²⋊C₁₆	—	61	2	no	—
206	C₂²⋊C₁₆	C₂×D₆₁	1	2	yes	—
207	#346	—	61	2	no	—
208	#346	—	61	2	no	—
209	C₄×C₁₆	—	61	1	no	—
210	C₄×C₁₆	—	61	1	no	—
211	C₄×C₁₆	—	61	2	no	—
212	C₄×C₁₆	—	61	1	no	—
213	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
214	C₆₁	#33479	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	61	1	no	—
216	C₂²×C₈	—	61	2	no	—
217	C₂²×C₈	—	61	1	no	—
218	C₂²×C₈	—	61	1	no	—
219	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
220	C₄×Q₈	—	61	1	no	—
221	C₄×Q₈	—	61	1	no	—
222	C₄×D₄	—	61	1	no	—
223	C₄×D₄	—	61	2	no	—
224	C₄×D₄	—	61	2	no	—
225	C₄×D₄	C₄×D₆₁	1	1	yes	characteristic
226	#117	—	61	1	no	—
227	#117	—	61	2	no	—
228	C₂×C₄²	—	61	2	no	—
229	C₂×C₄²	—	61	1	no	—
230	M₃₂	—	2	4	no	—
231	M₃₂	—	122	4	no	—
232	C₂×C₁₆	—	61	1	no	—
233	C₂×C₁₆	—	61	1	no	—
234	C₂×C₁₆	—	61	2	no	—
235	C₂×C₁₆	—	122	2	no	—
236	C₂×C₁₆	—	122	2	no	—
237	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
238	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
239	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
240	C₄⋊C₈	—	61	2	no	—
241	C₄⋊C₈	—	61	2	no	—
242	C₄⋊C₈	—	61	1	no	—
243	C₄⋊C₈	C₄×D₆₁	1	1	yes	characteristic
244	C₂²⋊C₈	—	61	2	no	—
245	C₂²⋊C₈	—	61	2	no	—
246	C₂²⋊C₈	C₄×D₆₁	1	2	yes	—
247	C₄×C₈	—	61	2	no	—
248	C₄×C₈	—	61	1	no	—
249	C₄×C₈	—	61	2	no	—
250	C₄×C₈	—	61	1	no	—
251	C₄×C₈	—	61	2	no	—
252	C₄×C₈	—	61	1	no	—
253	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
254	#41	—	122	4	no	—
255	C₂×Q₈	—	61	1	no	—
256	C₂×D₄	—	61	2	no	—
257	C₂×D₄	C₈×D₆₁	1	1	yes	characteristic
258	C₂²×C₄	—	61	1	no	—
259	C₂²×C₄	—	61	2	no	—
260	C₂²×C₄	—	61	2	no	—
261	C₂²×C₄	—	61	1	no	—
262	C₂²×C₄	—	61	1	no	—
263	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	61	1	no	—
267	C₂×C₈	—	61	1	no	—
268	C₂×C₈	—	61	2	no	—
269	C₂×C₈	—	122	2	no	—
270	C₂×C₈	—	122	4	no	—
271	C₂×C₈	—	122	1	no	—
272	C₂×C₈	—	122	1	no	—
273	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
274	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
276	C₄⋊C₄	—	61	2	no	—
277	C₄⋊C₄	—	61	2	no	—
278	C₄⋊C₄	—	61	1	no	—
279	C₄⋊C₄	C₈×D₆₁	1	1	yes	characteristic
280	C₂²⋊C₄	—	61	2	no	—
281	C₂²⋊C₄	—	61	2	no	—
282	C₂²⋊C₄	C₈×D₆₁	1	2	yes	—
283	C₄²	—	61	1	no	—
284	C₄²	—	61	2	no	—
285	C₄²	—	61	2	no	—
286	C₄²	—	61	1	no	—
287	C₄²	—	61	1	no	—
288	C₄²	—	61	2	no	—
289	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	2	no	—
292	C₁₆	—	61	2	no	—
293	C₁₆	—	122	2	no	—
294	C₁₆	—	122	1	no	—
295	C₁₆	#11758224	1	2	yes	complemented
296	C₂³	—	61	1	no	—
297	C₂³	C₂×C₈×D₆₁	1	2	yes	—
298	Q₈	—	122	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	122	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	61	1	no	—
304	C₂×C₄	—	61	1	no	—
305	C₂×C₄	—	61	4	no	—
306	C₂×C₄	—	61	1	no	—
307	C₂×C₄	—	61	2	no	—
308	C₂×C₄	—	61	2	no	—
309	C₂×C₄	—	122	2	no	—
310	C₂×C₄	—	122	4	no	—
311	C₂×C₄	—	122	4	no	—
312	C₂×C₄	—	122	1	no	—
313	C₂×C₄	—	122	1	no	—
314	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
316	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	—	2	1	no	—
320	C₈	—	61	2	no	—
321	C₈	—	122	1	no	—
322	C₈	—	122	2	no	—
323	C₈	#41×D₆₁	1	1	yes	characteristic
324	C₈	C₂×#11758224	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	61	2	no	—
329	C₂²	—	122	1	no	—
330	C₂²	—	122	1	no	—
331	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	61	2	no	—
335	C₄	—	61	2	no	—
336	C₄	—	122	1	no	—
337	C₄	—	122	2	no	—
338	C₄	—	122	2	no	—
339	C₄	M₃₂×D₆₁	1	2	yes	—
340	C₄	C₄×#11758224	1	1	yes	characteristic
341	C₄	#131×D₆₁	1	1	yes	characteristic
342	C₂	—	4	2	no	—
343	C₂	—	122	2	no	—
344	C₂	C₈×#11758224	1	1	yes	characteristic
345	C₂	C₂×M₃₂×D₆₁	1	1	yes	characteristic
346	C₂	#504×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2169	C₂
prime radical	C₂×#11758224	C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2169	D₆₁
hypocenter	C₆₁	#33479
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
2. #21447727	C₂	semidirect
3. #21447724	C₂	semidirect
4. #21447651	C₂	semidirect
5. #21447647	C₂	semidirect
6. #21447642	C₂	semidirect
7. #21447617	C₂	semidirect
8. #21447615	C₂	semidirect
9. M₃₂×Dic₆₁	C₂	semidirect
10. C₄×C₁₆×D₆₁	C₂	semidirect
11. C₆₁×#2169	C₂	semidirect
118. C₁₆×C₆₁	#41	semidirect
121. #11758185	#41	semidirect
142. C₄×D₆₁	M₃₂	semidirect
173. #2169	D₆₁	semidirect
214. C₆₁	#33479	semidirect
295. C₁₆	#11758224	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11758224	2
#21447727	2
#21447724	2
#21447651	2
#21447647	2
#21447642	2
#21447617	2
#21447615	2
M₃₂×Dic₆₁	2
C₄×C₁₆×D₆₁	2
C₆₁×#2169	2
#33479	61