Subgroups — #29650567

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456402	C₂	1	1	yes	maximal, characteristic
2	#21454748	C₂	1	2	yes	maximal
3	#21444728	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444727	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444725	C₂	1	2	yes	maximal, complemented
6	#21444639	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444636	C₂	1	2	yes	maximal, complemented
8	#21444573	C₂	1	1	yes	maximal, characteristic
9	#21444272	C₂	1	2	yes	maximal
10	#322×D₆₁	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840316	C₂²	1	2	yes	—
13	#16840166	C₄	1	1	yes	characteristic
14	#16840140	C₄	1	1	yes	characteristic
15	#16840111	C₄	1	2	yes	—
16	#16840004	C₄	1	1	yes	characteristic
17	#16840002	C₄	1	2	yes	—
18	#16840000	C₄	1	1	yes	characteristic
19	#16839893	C₂²	1	2	yes	—
20	#16839891	—	2	2	no	—
21	#16839889	C₂²	1	2	yes	—
22	C₈×Dic₁₂₂	—	2	2	no	—
23	C₈×Dic₁₂₂	C₂²	1	2	yes	—
24	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
25	C₄×#11758202	C₂²	1	2	yes	—
26	#16839787	—	2	2	no	—
27	#16839784	C₂²	1	2	yes	—
28	#16839716	—	2	2	no	—
29	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	#16839513	C₂²	1	1	yes	characteristic
33	#16839512	C₂²	1	2	yes	—
34	#16839494	C₂²	1	2	yes	—
35	Q₈⋊#92418	C₂²	1	2	yes	—
36	#16839492	C₂²	1	2	yes	—
37	#16839484	C₂²	1	1	yes	characteristic
38	#16839484	C₂²	1	1	yes	characteristic
39	#16839483	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#334	C₂²	1	1	yes	characteristic
43	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₆₁×#322	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#11758202	C₂×C₄	1	2	yes	—
47	C₂×#11758189	—	2	2	no	—
48	C₂×C₈×D₆₁	—	2	1	no	—
49	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₆₁	D₄	1	2	yes	—
52	#14519776	C₂×C₄	1	2	yes	—
53	#14519775	C₂×C₄	1	2	yes	—
54	#14519767	—	2	2	no	—
55	#14519757	C₂×C₄	1	2	yes	—
56	#14519755	—	2	2	no	—
57	#14519753	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₂₂	—	2	2	no	—
59	C₄×Dic₁₂₂	C₂³	1	2	yes	—
60	#14519734	—	2	2	no	—
61	C₄²×D₆₁	C₂³	1	1	yes	characteristic
62	#14519711	C₂×C₄	1	1	yes	characteristic
63	#14519710	C₂×C₄	1	2	yes	—
64	C₄⋊#92418	—	2	2	no	—
65	C₄⋊#92418	C₂³	1	2	yes	—
66	#14519706	—	2	2	no	—
67	#14519706	—	2	2	no	—
68	#14519706	—	2	2	no	—
69	#14519702	C₂×C₄	1	2	yes	—
70	#14519701	C₂×C₄	1	2	yes	—
71	#14519700	C₂×C₄	1	2	yes	—
72	#14519688	—	2	2	no	—
73	#14519687	—	2	2	no	—
74	C₄×#92418	C₂³	1	1	yes	characteristic
75	C₄×#92418	C₂³	1	1	yes	characteristic
76	C₄×#92418	C₂³	1	1	yes	characteristic
77	C₈×Dic₆₁	—	2	2	no	—
78	C₈×Dic₆₁	—	2	1	no	—
79	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₁	D₄	1	2	yes	—
82	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
84	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₆₁	—	2	2	no	—
86	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
87	#103×C₆₁	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₆₁	—	2	2	no	—
89	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
96	C₂×Dic₁₂₂	—	2	2	no	—
97	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	1	no	—
101	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
106	C₄²×C₆₁	—	2	2	no	—
107	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
108	#11758202	—	2	4	no	—
109	#11758198	—	2	2	no	—
110	#11758198	—	2	2	no	—
111	C₄⋊Dic₆₁	—	2	2	no	—
112	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
113	#11758196	—	2	2	no	—
114	#11758196	—	2	2	no	—
115	#11758196	—	2	2	no	—
116	#11758196	—	2	2	no	—
117	#11758196	—	2	2	no	—
118	C₄×Dic₆₁	—	2	2	no	—
119	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₆₁	#41	1	1	yes	characteristic
122	C₂×#92418	—	2	2	no	—
123	C₂×#92418	—	2	1	no	—
124	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
125	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
126	C₂×#92418	C₂×D₄	1	2	yes	—
127	Dic₂₄₄	—	2	2	no	—
128	#11758189	—	4	4	no	—
129	C₈×D₆₁	—	2	4	no	—
130	C₈×D₆₁	—	2	2	no	—
131	C₈×D₆₁	—	2	2	no	—
132	C₈×D₆₁	—	4	2	no	—
133	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₆₁	—	2	2	no	—
135	Q₈×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₆₁	—	2	2	no	—
142	C₂×Dic₆₁	—	2	2	no	—
143	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
146	C₄×D₆₁	—	2	1	no	—
147	C₄×D₆₁	—	2	1	no	—
148	C₄×D₆₁	—	2	2	no	—
149	C₄×D₆₁	#137	1	4	yes	—
150	Dic₁₂₂	—	2	2	no	—
151	Dic₁₂₂	—	2	2	no	—
152	Dic₁₂₂	—	4	4	no	—
153	C₈×C₆₁	—	2	1	no	—
154	C₈×C₆₁	—	2	2	no	—
155	C₈×C₆₁	—	4	1	no	—
156	C₈×C₆₁	—	4	2	no	—
157	C₈×C₆₁	#131	1	2	yes	—
158	#92418	—	2	1	no	—
159	#92418	—	2	2	no	—
160	#92418	—	4	1	no	—
161	#92418	—	4	2	no	—
162	#92418	#131	1	2	yes	—
163	#17961	—	61	1	no	maximal, Sylow
164	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₆₁	—	2	1	no	—
166	C₂×D₆₁	—	2	1	no	—
167	C₂×D₆₁	#441	1	2	yes	—
168	C₄×C₆₁	—	2	1	no	—
169	C₄×C₆₁	—	4	2	no	—
170	C₄×C₆₁	—	4	2	no	—
171	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
172	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
173	C₄×C₆₁	C₂×#137	1	2	yes	—
174	Dic₆₁	—	2	1	no	—
175	Dic₆₁	—	4	2	no	—
176	Dic₆₁	—	4	2	no	—
177	Dic₆₁	#441	1	2	yes	—
178	Dic₆₁	#444	1	2	yes	—
179	#3170	—	61	1	no	—
180	#2960	—	61	2	no	—
181	#1593	—	61	1	no	—
182	#1593	—	61	1	no	—
183	#1593	—	61	2	no	—
184	#1579	—	61	2	no	—
185	#1579	—	61	1	no	—
186	#1579	D₆₁	1	1	yes	characteristic, complemented
187	#1566	—	61	1	no	—
188	#1493	—	61	2	no	—
189	C₂×#322	—	61	1	no	—
190	C₂×C₆₁	#2931	1	1	yes	characteristic
191	C₂×C₆₁	C₂×#441	1	1	yes	characteristic
192	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
193	D₆₁	—	2	2	no	—
194	#533	—	61	2	no	—
195	#500	—	61	1	no	—
196	#494	—	61	1	no	—
197	#487	—	61	2	no	—
198	#467	—	61	1	no	—
199	#467	—	61	1	no	—
200	#467	—	61	2	no	—
201	#446	—	61	2	no	—
202	#446	—	61	2	no	—
203	#446	—	122	2	no	—
204	C₈×Q₈	—	61	2	no	—
205	C₈×Q₈	—	122	2	no	—
206	C₈×Q₈	C₂×D₆₁	1	2	yes	—
207	C₄×Q₁₆	—	61	2	no	—
208	C₄×Q₁₆	—	61	1	no	—
209	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
210	#424	—	122	2	no	—
211	#423	—	61	2	no	—
212	#405	—	122	2	no	—
213	C₂×C₄×C₈	—	61	1	no	—
214	C₂×C₄×C₈	—	61	1	no	—
215	C₂×C₄×C₈	—	61	1	no	—
216	#334	—	61	2	no	—
217	#334	—	61	1	no	—
218	#334	C₂×D₆₁	1	1	yes	characteristic
219	Q₈⋊C₈	—	61	2	no	—
220	Q₈⋊C₈	—	61	2	no	—
221	Q₈⋊C₈	—	61	2	no	—
222	Q₈⋊C₈	C₂×D₆₁	1	2	yes	—
223	#322	—	61	1	no	—
224	#322	—	61	1	no	—
225	#322	—	61	1	no	—
226	#322	C₂×D₆₁	1	1	yes	characteristic
227	C₆₁	#17961	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₁₆	—	61	2	no	—
229	C₂×Q₁₆	—	61	1	no	—
230	C₂×Q₁₆	C₄×D₆₁	1	1	yes	characteristic
231	C₂×M₁₆	—	122	2	no	—
232	C₂²×C₈	—	61	1	no	—
233	C₂²×C₈	—	61	1	no	—
234	C₂²×C₈	—	61	2	no	—
235	C₂²×C₈	—	122	1	no	—
236	C₄⋊Q₈	—	61	2	no	—
237	C₄⋊Q₈	—	61	2	no	—
238	#125	—	122	2	no	—
239	C₂²⋊Q₈	—	61	2	no	—
240	C₂²⋊Q₈	—	61	2	no	—
241	C₂²⋊Q₈	—	122	2	no	—
242	C₄×Q₈	—	61	2	no	—
243	C₄×Q₈	—	122	2	no	—
244	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
245	#117	—	122	2	no	—
246	C₂×C₄²	—	61	1	no	—
247	#107	—	61	1	no	—
248	#107	—	61	2	no	—
249	#107	C₄×D₆₁	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	61	2	no	—
252	C₄⋊C₈	—	122	2	no	—
253	C₄⋊C₈	—	122	2	no	—
254	C₄⋊C₈	—	122	2	no	—
255	C₄⋊C₈	—	122	2	no	—
256	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
257	#103	—	61	2	no	—
258	#103	—	61	2	no	—
259	#103	—	61	2	no	—
260	#103	C₄×D₆₁	1	2	yes	—
261	#97	—	122	2	no	—
262	#97	—	122	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	61	1	no	—
265	C₄×C₈	—	61	1	no	—
266	C₄×C₈	—	61	2	no	—
267	C₄×C₈	—	61	1	no	—
268	C₄×C₈	—	61	1	no	—
269	C₄×C₈	—	61	1	no	—
270	C₄×C₈	—	122	1	no	—
271	C₄×C₈	—	122	2	no	—
272	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
275	C₂×Q₈	—	61	2	no	—
276	C₂×Q₈	—	122	2	no	—
277	C₂×Q₈	C₂×C₄×D₆₁	1	2	yes	—
278	C₂²×C₄	—	61	1	no	—
279	C₂²×C₄	—	61	1	no	—
280	C₂²×C₄	—	61	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	122	4	no	—
283	Q₁₆	—	122	2	no	—
284	M₁₆	—	244	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	61	1	no	—
288	C₂×C₈	—	61	1	no	—
289	C₂×C₈	—	61	2	no	—
290	C₂×C₈	—	122	2	no	—
291	C₂×C₈	—	122	4	no	—
292	C₂×C₈	—	122	2	no	—
293	C₂×C₈	—	122	2	no	—
294	C₂×C₈	—	122	1	no	—
295	C₂×C₈	—	244	2	no	—
296	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
298	C₂×C₈	D₄×D₆₁	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	61	2	no	—
301	C₄⋊C₄	—	122	2	no	—
302	C₄⋊C₄	—	122	2	no	—
303	C₄⋊C₄	—	122	2	no	—
304	C₄⋊C₄	—	122	2	no	—
305	C₄⋊C₄	—	122	2	no	—
306	C₄⋊C₄	—	122	2	no	—
307	C₄⋊C₄	C₂×C₄×D₆₁	1	2	yes	—
308	C₂²⋊C₄	—	122	2	no	—
309	C₂²⋊C₄	—	122	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	61	1	no	—
312	C₄²	—	61	1	no	—
313	C₄²	—	61	1	no	—
314	C₄²	—	122	2	no	—
315	C₄²	C₂³×D₆₁	1	1	yes	characteristic
316	C₂³	—	61	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	122	2	no	—
320	Q₈	—	122	2	no	—
321	Q₈	—	244	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	61	1	no	—
325	C₂×C₄	—	61	1	no	—
326	C₂×C₄	—	61	4	no	—
327	C₂×C₄	—	61	1	no	—
328	C₂×C₄	—	122	2	no	—
329	C₂×C₄	—	122	1	no	—
330	C₂×C₄	—	122	1	no	—
331	C₂×C₄	—	122	2	no	—
332	C₂×C₄	—	122	2	no	—
333	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
335	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	61	2	no	—
341	C₈	—	122	1	no	—
342	C₈	—	122	2	no	—
343	C₈	—	244	1	no	—
344	C₈	—	244	2	no	—
345	C₈	#14519783	1	2	yes	—
346	C₂²	—	61	2	no	—
347	C₂²	—	122	1	no	—
348	C₂²	—	122	1	no	—
349	C₂²	C₄×D₄×D₆₁	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	61	2	no	—
354	C₄	—	61	2	no	—
355	C₄	—	122	1	no	—
356	C₄	—	244	2	no	—
357	C₄	—	244	2	no	—
358	C₄	#131×D₆₁	1	1	yes	characteristic
359	C₄	C₂×#14519783	1	1	yes	characteristic
360	C₄	#137×D₆₁	1	2	yes	—
361	C₂	—	122	2	no	—
362	C₂	#21454526	1	1	yes	characteristic
363	C₂	#441×D₆₁	1	1	yes	characteristic
364	C₂	#444×D₆₁	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×#1579	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1579	D₆₁
hypocenter	C₆₁	#17961
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #21444728	C₂	semidirect
4. #21444727	C₂	semidirect
5. #21444725	C₂	semidirect
6. #21444639	C₂	semidirect
7. #21444636	C₂	semidirect
11. C₆₁×#1579	C₂	semidirect
186. #1579	D₆₁	semidirect
227. C₆₁	#17961	semidirect

Maximal subgroups

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

subgroup	index
#21456402	2
#21454748	2
#21444728	2
#21444727	2
#21444725	2
#21444639	2
#21444636	2
#21444573	2
#21444272	2
#322×D₆₁	2
C₆₁×#1579	2
#17961	61