Subgroups — #29649354

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454946	C₂	1	2	yes	maximal
2	#21454785	C₂	1	1	yes	maximal, characteristic
3	#21444605	C₂	1	2	yes	maximal, complemented
4	#21444598	C₂	1	2	yes	maximal, complemented
5	#21444592	C₂	1	2	yes	maximal, complemented
6	#21444583	C₂	1	2	yes	maximal, complemented
7	#21444568	C₂	1	2	yes	maximal, complemented
8	C₂×#16839515	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#11758222	C₂²	1	1	yes	characteristic
11	#16839892	C₄	1	1	yes	characteristic
12	#16839892	C₄	1	1	yes	characteristic
13	#16839890	—	2	2	no	—
14	#16839841	C₄	1	1	yes	characteristic
15	#16839841	C₄	1	1	yes	characteristic
16	#16839835	—	2	2	no	—
17	#16839833	C₄	1	2	yes	—
18	#16839833	C₄	1	2	yes	—
19	#16839830	—	2	2	no	—
20	#16839825	—	2	2	no	—
21	#16839818	C₂²	1	2	yes	—
22	#16839815	C₂²	1	1	yes	characteristic
23	#16839814	C₂²	1	2	yes	—
24	#16839813	C₂²	1	2	yes	—
25	#16839784	C₂²	1	2	yes	—
26	C₂×C₄⋊#92418	C₂²	1	2	yes	—
27	C₄⋊C₈×D₆₁	C₂²	1	2	yes	—
28	C₄×#11758189	C₂²	1	2	yes	—
29	#16839515	C₂²	1	1	yes	characteristic
30	#16839515	C₂²	1	1	yes	characteristic
31	#16839515	C₂²	1	2	yes	—
32	#16839514	C₂²	1	2	yes	—
33	#16839514	C₂²	1	2	yes	—
34	#16839512	C₂²	1	2	yes	—
35	#16839512	C₂²	1	2	yes	—
36	C₆₁×#432	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#335	C₂²	1	1	yes	characteristic
39	C₆₁×#335	C₂²	1	1	yes	characteristic
40	C₆₁×#334	C₂²	1	2	yes	—
41	C₂×#11758222	Q₈	1	1	yes	characteristic
42	C₂×#11758189	—	2	2	no	—
43	C₂×#11758189	C₂×C₄	1	2	yes	—
44	C₂×#11758189	C₂×C₄	1	2	yes	—
45	C₂²×#92418	—	2	2	no	—
46	C₂×C₈×D₆₁	—	2	2	no	—
47	C₄×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
49	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₄₄	D₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₆₁	C₂×C₄	1	2	yes	—
52	C₄×C₂²⋊D₆₁	D₄	1	2	yes	—
53	#14519736	Q₈	1	1	yes	characteristic
54	#14519734	Q₈	1	2	yes	—
55	C₄²×D₆₁	C₂³	1	2	yes	—
56	C₄⋊#92418	—	2	2	no	—
57	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#92418	C₂³	1	2	yes	—
60	C₄⋊#92418	C₂³	1	2	yes	—
61	#14519706	—	2	2	no	—
62	#14519706	—	2	2	no	—
63	#14519706	C₂×C₄	1	2	yes	—
64	#14519706	C₂×C₄	1	2	yes	—
65	C₂²⋊#92418	C₂×C₄	1	2	yes	—
66	#14519689	—	2	2	no	—
67	#14519689	—	2	2	no	—
68	#14519689	C₂×C₄	1	2	yes	—
69	#14519689	C₂×C₄	1	2	yes	—
70	#14519687	—	2	2	no	—
71	#14519687	C₂×C₄	1	2	yes	—
72	#14519687	C₂×C₄	1	2	yes	—
73	C₄×#92418	C₂³	1	2	yes	—
74	C₈×Dic₆₁	—	2	2	no	—
75	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₆₁	—	2	2	no	—
81	#97×C₆₁	—	2	2	no	—
82	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₂²⋊D₆₁	C₄⋊C₄	1	2	yes	—
90	#11758222	—	4	2	no	—
91	C₂×D₂₄₄	C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
94	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
95	C₂×Dic₁₂₂	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	—	2	2	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	4	1	no	—
100	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
107	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₆₁	C₄⋊C₄	1	2	yes	—
109	#11758198	C₄⋊C₄	1	2	yes	—
110	#11758198	C₄⋊C₄	1	2	yes	—
111	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
112	C₄⋊Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
113	#11758196	C₄⋊C₄	1	2	yes	—
114	#11758196	C₄⋊C₄	1	2	yes	—
115	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
116	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
117	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
118	C₂×#92418	—	2	2	no	—
119	C₂×#92418	—	2	2	no	—
120	C₂×#92418	—	8	2	no	—
121	C₂×#92418	C₂²×C₄	1	2	yes	—
122	C₂×#92418	C₂²×C₄	1	2	yes	—
123	#11758189	—	2	4	no	—
124	#11758189	—	4	2	no	—
125	#11758189	—	8	2	no	—
126	C₈×D₆₁	—	8	2	no	—
127	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×D₆₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	2	1	no	—
132	C₂×C₄×C₆₁	—	2	1	no	—
133	C₂×C₄×C₆₁	—	2	1	no	—
134	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
140	C₂²⋊D₆₁	—	2	4	no	—
141	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×Dic₆₁	C₂×C₄⋊C₄	1	2	yes	—
144	D₂₄₄	—	4	1	no	—
145	C₄×D₆₁	—	2	2	no	—
146	C₄×D₆₁	—	2	2	no	—
147	C₄×D₆₁	—	2	2	no	—
148	C₄×D₆₁	—	2	2	no	—
149	C₄×D₆₁	—	4	2	no	—
150	Dic₁₂₂	—	4	1	no	—
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	—	2	1	no	—
153	C₈×C₆₁	—	2	1	no	—
154	C₈×C₆₁	—	4	2	no	—
155	C₈×C₆₁	—	4	2	no	—
156	C₈×C₆₁	#131	1	2	yes	—
157	#92418	—	2	2	no	—
158	#92418	—	2	2	no	—
159	#92418	—	4	2	no	—
160	#92418	—	4	2	no	—
161	#17840	—	61	1	no	maximal, Sylow
162	C₂²×C₆₁	—	2	1	no	—
163	C₂²×C₆₁	—	2	1	no	—
164	C₂²×C₆₁	#430	1	2	yes	—
165	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂×D₆₁	—	2	2	no	—
167	C₂×D₆₁	—	2	2	no	—
168	C₂×D₆₁	—	2	2	no	—
169	C₄×C₆₁	—	2	1	no	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	—	2	1	no	—
172	C₄×C₆₁	—	2	1	no	—
173	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
174	C₄×C₆₁	#568	1	1	yes	characteristic
175	C₄×C₆₁	#576	1	1	yes	characteristic
176	C₄×C₆₁	#578	1	1	yes	characteristic
177	Dic₆₁	—	2	2	no	—
178	Dic₆₁	—	2	2	no	—
179	Dic₆₁	—	2	2	no	—
180	#2990	—	61	2	no	—
181	#2965	—	61	1	no	—
182	#1573	—	61	2	no	—
183	#1572	—	61	2	no	—
184	#1571	—	61	2	no	—
185	#1569	—	61	2	no	—
186	#1568	D₆₁	1	1	yes	characteristic, complemented
187	#1565	—	61	2	no	—
188	C₂×#335	—	61	1	no	—
189	C₂×C₆₁	—	2	2	no	—
190	C₂×C₆₁	#2906	1	1	yes	characteristic
191	C₂×C₆₁	#2913	1	1	yes	characteristic
192	C₂×C₆₁	C₂×#430	1	1	yes	characteristic
193	D₆₁	—	4	2	no	—
194	C₄×#41	—	61	1	no	—
195	#446	—	61	1	no	—
196	#446	—	61	1	no	—
197	#446	—	122	2	no	—
198	#436	—	61	1	no	—
199	#436	—	61	1	no	—
200	#435	—	61	2	no	—
201	#435	—	61	2	no	—
202	#435	—	122	2	no	—
203	#435	—	122	2	no	—
204	C₈×D₄	—	122	2	no	—
205	#432	—	61	2	no	—
206	#432	C₂×D₆₁	1	2	yes	—
207	#431	—	61	1	no	—
208	#431	—	61	2	no	—
209	#431	—	61	2	no	—
210	#423	—	61	2	no	—
211	C₂×C₄⋊C₈	—	61	2	no	—
212	C₂×C₄⋊C₈	—	61	2	no	—
213	C₄×M₁₆	—	61	2	no	—
214	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
215	#335	—	61	2	no	—
216	#335	—	61	2	no	—
217	#335	—	61	1	no	—
218	#335	—	61	1	no	—
219	#335	—	61	2	no	—
220	#335	C₂×D₆₁	1	1	yes	characteristic
221	#335	C₂×D₆₁	1	1	yes	characteristic
222	#334	—	61	2	no	—
223	#334	—	61	2	no	—
224	#334	C₂×D₆₁	1	2	yes	—
225	C₆₁	#17840	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	61	1	no	—
227	C₂×M₁₆	—	61	2	no	—
228	C₂×M₁₆	—	61	2	no	—
229	C₂×M₁₆	—	122	2	no	—
230	C₂²×C₈	—	122	2	no	—
231	C₂²×C₈	—	122	2	no	—
232	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
234	C₄×Q₈	—	61	1	no	—
235	C₄×Q₈	—	61	1	no	—
236	C₄×D₄	—	61	2	no	—
237	C₄×D₄	—	61	2	no	—
238	C₄×D₄	—	61	1	no	—
239	C₄×D₄	—	61	1	no	—
240	#117	—	61	1	no	—
241	#117	—	61	2	no	—
242	C₂×C₄²	—	61	2	no	—
243	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
244	C₄⋊C₈	—	61	2	no	—
245	C₄⋊C₈	—	61	2	no	—
246	C₄⋊C₈	—	61	1	no	—
247	C₄⋊C₈	—	61	1	no	—
248	C₄⋊C₈	—	61	2	no	—
249	C₄⋊C₈	—	61	2	no	—
250	C₄⋊C₈	—	122	2	no	—
251	C₄⋊C₈	—	122	2	no	—
252	C₄⋊C₈	—	122	2	no	—
253	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
254	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	61	2	no	—
257	C₂²⋊C₈	—	61	2	no	—
258	C₂²⋊C₈	—	61	2	no	—
259	C₂²⋊C₈	—	122	2	no	—
260	C₂²⋊C₈	—	122	2	no	—
261	#97	—	2	2	no	—
262	#97	—	61	2	no	—
263	#97	—	61	2	no	—
264	#97	—	122	2	no	—
265	C₄×C₈	—	61	2	no	—
266	C₄×C₈	—	122	2	no	—
267	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
268	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
271	#41	—	244	2	no	—
272	C₂×Q₈	—	61	1	no	—
273	C₂×D₄	—	61	2	no	—
274	C₂×D₄	—	61	1	no	—
275	C₂²×C₄	—	61	2	no	—
276	C₂²×C₄	—	61	2	no	—
277	C₂²×C₄	—	61	2	no	—
278	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
280	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
281	M₁₆	—	122	4	no	—
282	M₁₆	—	488	2	no	—
283	M₁₆	—	244	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	61	2	no	—
289	C₂×C₈	—	61	2	no	—
290	C₂×C₈	—	122	2	no	—
291	C₂×C₈	—	122	2	no	—
292	C₂×C₈	—	488	2	no	—
293	C₂×C₈	—	488	2	no	—
294	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
298	C₄⋊C₄	—	61	2	no	—
299	C₄⋊C₄	—	61	2	no	—
300	C₄⋊C₄	—	61	1	no	—
301	C₄⋊C₄	—	61	1	no	—
302	C₂²⋊C₄	—	61	2	no	—
303	C₂²⋊C₄	—	61	2	no	—
304	C₂²⋊C₄	—	61	2	no	—
305	C₄²	—	61	2	no	—
306	C₄²	—	61	2	no	—
307	C₄²	—	61	2	no	—
308	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
309	C₄²	D₄×D₆₁	1	1	yes	characteristic
310	C₄²	Q₈×D₆₁	1	1	yes	characteristic
311	C₄²	C₂³×D₆₁	1	1	yes	characteristic
312	C₂³	—	61	2	no	—
313	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
314	Q₈	—	244	1	no	—
315	D₄	—	122	4	no	—
316	D₄	—	244	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	61	2	no	—
323	C₂×C₄	—	61	2	no	—
324	C₂×C₄	—	61	2	no	—
325	C₂×C₄	—	122	2	no	—
326	C₂×C₄	—	122	2	no	—
327	C₂×C₄	—	122	2	no	—
328	C₂×C₄	—	122	2	no	—
329	C₂×C₄	—	244	2	no	—
330	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
331	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
332	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
335	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	122	2	no	—
342	C₈	—	122	2	no	—
343	C₈	—	244	2	no	—
344	C₈	—	244	2	no	—
345	C₈	#14519785	1	2	yes	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	122	2	no	—
349	C₂²	—	122	2	no	—
350	C₂²	—	122	2	no	—
351	C₂²	#16839809	1	2	yes	—
352	C₂²	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	122	2	no	—
358	C₄	—	122	2	no	—
359	C₄	—	122	2	no	—
360	C₄	C₂×#14519785	1	1	yes	characteristic
361	C₄	#16840528	1	1	yes	characteristic
362	C₄	#16840564	1	1	yes	characteristic
363	C₄	#16840576	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	244	2	no	—
366	C₂	#21454388	1	1	yes	characteristic
367	C₂	#21454416	1	1	yes	characteristic
368	C₂	C₂×#16839809	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1568	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#11758222	Q₈
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1568	D₆₁
hypocenter	C₆₁	#17840
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #21444605	C₂	semidirect
4. #21444598	C₂	semidirect
5. #21444592	C₂	semidirect
6. #21444583	C₂	semidirect
7. #21444568	C₂	semidirect
8. C₂×#16839515	C₂	semidirect
9. C₆₁×#1568	C₂	semidirect
186. #1568	D₆₁	semidirect
225. C₆₁	#17840	semidirect

Maximal subgroups

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

subgroup	index
#21454946	2
#21454785	2
#21444605	2
#21444598	2
#21444592	2
#21444583	2
#21444568	2
C₂×#16839515	2
C₆₁×#1568	2
#17840	61