Subgroups — #29680383

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454246	C₂	1	1	yes	maximal, characteristic
2	#21445967	C₂	1	2	yes	maximal, complemented
3	#21445943	C₂	1	1	yes	maximal, characteristic, complemented
4	#21445921	C₂	1	2	yes	maximal
5	#21445866	C₂	1	1	yes	maximal, characteristic
6	#21445862	C₂	1	1	yes	maximal, characteristic
7	#21445706	C₂	1	1	yes	maximal, characteristic, complemented
8	#21445705	C₂	1	1	yes	maximal, characteristic, complemented
9	#21445618	C₂	1	2	yes	maximal
10	#21445573	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14519785	C₂²	1	1	yes	characteristic
13	C₂×#14519783	C₂²	1	1	yes	characteristic
14	#16840230	C₂²	1	1	yes	characteristic
15	C₂×#14519713	C₂²	1	1	yes	characteristic
16	#16839800	C₂²	1	1	yes	characteristic
17	#16839744	C₂²	1	2	yes	—
18	#16839735	C₂²	1	2	yes	—
19	#16839720	C₂²	1	2	yes	—
20	#16839719	C₄	1	2	yes	complemented
21	#16839718	C₂²	1	2	yes	—
22	#16839718	C₂²	1	1	yes	characteristic
23	#16839716	C₄	1	2	yes	complemented
24	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
25	C₄×#11758189	C₄	1	2	yes	complemented
26	C₂×#14519687	C₂²	1	1	yes	characteristic
27	#97×D₆₁	C₄	1	2	yes	complemented
28	#16839533	C₂²	1	2	yes	—
29	#16839523	C₂²	1	2	yes	—
30	#16839521	C₂²	1	2	yes	—
31	#16839520	C₂²	1	2	yes	—
32	#16839518	C₂²	1	2	yes	—
33	#16839518	C₂²	1	2	yes	—
34	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#427	C₂²	1	1	yes	characteristic
36	C₆₁×#405	C₂²	1	1	yes	characteristic
37	C₆₁×#344	C₂²	1	2	yes	—
38	C₆₁×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#11758222	C₂³	1	1	yes	characteristic
40	#14519785	—	2	2	no	—
41	#14519785	—	2	2	no	—
42	#14519783	—	2	1	no	—
43	#14519783	—	2	1	no	—
44	#14519783	—	2	1	no	—
45	#14519783	—	2	1	no	—
46	C₂×#11758194	C₂³	1	2	yes	—
47	C₂×#11758194	C₂³	1	1	yes	characteristic
48	C₂×#11758189	C₂×C₄	1	2	yes	—
49	C₂×#11758189	C₂×C₄	1	2	yes	—
50	C₂²×#92418	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
53	#14519742	—	2	1	no	—
54	#14519739	—	2	1	no	—
55	#14519737	—	2	2	no	—
56	#14519736	C₂³	1	2	yes	—
57	#14519734	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
59	C₄²×D₆₁	C₂×C₄	1	2	yes	—
60	#14519713	—	2	1	no	—
61	#14519713	—	2	1	no	—
62	#14519711	—	2	1	no	—
63	#14519709	—	2	1	no	—
64	#14519695	—	2	2	no	—
65	#14519693	—	2	2	no	—
66	#14519691	—	2	4	no	—
67	#14519688	—	2	2	no	—
68	#14519688	C₂×C₄	1	2	yes	—
69	#14519687	—	2	2	no	—
70	#14519687	C₂×C₄	1	1	yes	characteristic
71	#14519687	C₂×C₄	1	2	yes	—
72	#14519687	C₂×C₄	1	1	yes	characteristic
73	#14519687	C₂×C₄	1	1	yes	characteristic
74	#14519687	C₂×C₄	1	1	yes	characteristic
75	C₄×#92418	—	2	2	no	—
76	C₄×#92418	C₂×C₄	1	2	yes	—
77	C₈×Dic₆₁	—	2	2	no	—
78	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
80	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	#117×C₆₁	C₂³	1	2	yes	—
83	#117×C₆₁	C₂³	1	1	yes	characteristic
84	#108×C₆₁	—	2	1	no	—
85	#108×C₆₁	—	2	1	no	—
86	#107×C₆₁	—	2	1	no	—
87	#106×C₆₁	—	2	1	no	—
88	#97×C₆₁	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
90	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
91	C₂×C₂²⋊D₆₁	#41	1	2	yes	—
92	C₂²×Dic₆₁	—	2	1	no	—
93	#11758222	—	2	1	no	—
94	#11758222	—	2	1	no	—
95	#11758222	—	2	1	no	—
96	#11758222	—	2	1	no	—
97	C₂×D₂₄₄	#41	1	1	yes	characteristic
98	C₂×C₄×D₆₁	—	2	2	no	—
99	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₂₂	#41	1	1	yes	characteristic
101	M₁₆×C₆₁	—	2	1	no	—
102	M₁₆×C₆₁	—	2	1	no	—
103	M₁₆×C₆₁	—	2	2	no	—
104	M₁₆×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	—	2	2	no	—
106	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
110	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
111	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
112	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
113	C₄⋊C₄×C₆₁	—	2	1	no	—
114	C₄⋊C₄×C₆₁	—	2	2	no	—
115	C₂²⋊C₄×C₆₁	—	2	1	no	—
116	C₂²⋊C₄×C₆₁	—	2	2	no	—
117	C₄²×C₆₁	—	2	2	no	—
118	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
119	C₂²⋊Dic₆₁	—	2	2	no	—
120	#11758198	—	2	2	no	—
121	C₄⋊Dic₆₁	—	2	2	no	—
122	#11758196	—	2	2	no	—
123	C₄×Dic₆₁	—	2	1	no	—
124	C₄×Dic₆₁	—	2	2	no	—
125	C₄×Dic₆₁	—	2	2	no	—
126	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
128	#11758194	—	2	1	no	—
129	#11758194	—	2	1	no	—
130	#11758194	—	2	2	no	—
131	#11758194	—	2	2	no	—
132	C₂×#92418	—	2	2	no	—
133	C₂×#92418	C₂²×C₄	1	2	yes	—
134	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
135	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
136	C₂×#92418	#41	1	2	yes	—
137	C₂×#92418	#41	1	2	yes	—
138	#11758189	—	2	2	no	—
139	#11758189	—	2	2	no	—
140	#11758189	—	2	4	no	—
141	C₈×D₆₁	—	2	2	no	—
142	C₈×D₆₁	—	2	2	no	—
143	C₈×D₆₁	—	2	4	no	—
144	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
145	C₂²×D₆₁	#117	1	2	yes	—
146	C₂×C₄×C₆₁	—	2	2	no	—
147	C₂×C₄×C₆₁	—	4	2	no	—
148	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₆₁	—	2	2	no	—
155	C₂²⋊D₆₁	—	2	2	no	—
156	C₂×Dic₆₁	—	2	1	no	—
157	C₂×Dic₆₁	—	2	1	no	—
158	C₂×Dic₆₁	—	4	2	no	—
159	C₂×Dic₆₁	—	8	1	no	—
160	C₂×Dic₆₁	#117	1	2	yes	—
161	D₂₄₄	—	2	1	no	—
162	D₂₄₄	—	2	1	no	—
163	C₄×D₆₁	—	2	2	no	—
164	C₄×D₆₁	—	2	2	no	—
165	C₄×D₆₁	—	8	2	no	—
166	Dic₁₂₂	—	2	1	no	—
167	Dic₁₂₂	—	2	1	no	—
168	C₈×C₆₁	—	2	2	no	—
169	C₈×C₆₁	—	4	2	no	—
170	C₈×C₆₁	#117	1	1	yes	characteristic
171	C₈×C₆₁	#117	1	1	yes	characteristic
172	C₈×C₆₁	#117	1	1	yes	characteristic
173	C₈×C₆₁	#117	1	1	yes	characteristic
174	#92418	—	2	2	no	—
175	#92418	—	4	2	no	—
176	#92418	#117	1	2	yes	—
177	#92418	#117	1	2	yes	—
178	#20788	—	61	1	no	maximal, Sylow
179	C₂²×C₆₁	—	4	1	no	—
180	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
181	C₂²×C₆₁	#529	1	1	yes	characteristic
182	C₂²×C₆₁	#529	1	1	yes	characteristic
183	C₂×D₆₁	—	2	2	no	—
184	C₂×D₆₁	—	4	2	no	—
185	C₄×C₆₁	—	4	2	no	—
186	C₄×C₆₁	—	8	2	no	—
187	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
188	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
189	C₄×C₆₁	#529	1	1	yes	characteristic
190	C₄×C₆₁	#529	1	1	yes	characteristic
191	Dic₆₁	—	2	2	no	—
192	Dic₆₁	—	4	1	no	—
193	Dic₆₁	—	4	1	no	—
194	Dic₆₁	—	8	2	no	—
195	#2875	—	61	1	no	—
196	#1841	—	61	2	no	—
197	#1833	—	61	1	no	—
198	#1833	D₆₁	1	1	yes	characteristic, complemented
199	#1827	—	61	2	no	—
200	#1813	—	61	1	no	—
201	#1812	—	61	1	no	—
202	#1773	—	61	1	no	—
203	#1773	—	61	1	no	—
204	#1757	—	61	2	no	—
205	#1747	—	61	2	no	—
206	C₂×C₆₁	—	2	1	no	—
207	C₂×C₆₁	—	2	1	no	—
208	C₂×C₆₁	—	2	1	no	—
209	C₂×C₆₁	#2283	1	1	yes	characteristic
210	D₆₁	—	4	2	no	—
211	C₂×#131	—	61	1	no	—
212	C₂×#131	—	61	1	no	—
213	C₄×#41	—	61	1	no	—
214	C₂×#108	—	61	1	no	—
215	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
216	#427	—	61	1	no	—
217	#427	C₂×D₆₁	1	1	yes	characteristic
218	#413	—	61	2	no	—
219	#410	—	61	2	no	—
220	#405	—	61	1	no	—
221	#405	—	61	2	no	—
222	#405	—	61	2	no	—
223	#405	—	61	2	no	—
224	#405	—	61	2	no	—
225	#405	C₂×D₆₁	1	1	yes	characteristic
226	C₄×M₁₆	—	61	1	no	—
227	C₄×M₁₆	—	61	2	no	—
228	C₂×#97	—	61	1	no	—
229	C₂×#97	—	61	2	no	—
230	#344	—	61	2	no	—
231	#344	C₂×D₆₁	1	2	yes	—
232	#339	—	61	2	no	—
233	#339	—	61	2	no	—
234	#338	—	61	2	no	—
235	H(3,ℤ/4)	—	61	2	no	—
236	H(3,ℤ/4)	—	61	2	no	—
237	H(3,ℤ/4)	C₂×D₆₁	1	2	yes	—
238	C₆₁	#20788	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	61	1	no	—
240	#131	—	122	2	no	—
241	#131	—	122	2	no	—
242	#131	—	122	1	no	—
243	#131	—	122	1	no	—
244	#131	—	122	1	no	—
245	#131	—	122	1	no	—
246	C₂×M₁₆	—	61	1	no	—
247	C₂×M₁₆	—	61	2	no	—
248	C₂×M₁₆	—	61	2	no	—
249	C₂×M₁₆	—	61	2	no	—
250	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
251	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
252	C₂²×C₈	—	61	1	no	—
253	C₂²×C₈	—	61	2	no	—
254	C₂²×C₈	—	61	2	no	—
255	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
256	C₄×Q₈	—	122	1	no	—
257	C₄×D₄	—	122	2	no	—
258	C₄×D₄	—	122	1	no	—
259	#117	—	61	2	no	—
260	#117	—	61	2	no	—
261	#117	C₂²×D₆₁	1	2	yes	—
262	#117	C₂²×D₆₁	1	1	yes	characteristic
263	C₂×C₄²	—	61	1	no	—
264	C₂×C₄²	—	61	2	no	—
265	#108	—	2	1	no	—
266	#108	—	2	1	no	—
267	#108	—	122	1	no	—
268	#108	—	122	1	no	—
269	#107	—	2	1	no	—
270	#107	—	122	1	no	—
271	#106	—	2	1	no	—
272	#106	—	122	1	no	—
273	#101	—	122	2	no	—
274	#100	—	122	2	no	—
275	#99	—	122	4	no	—
276	#97	—	61	1	no	—
277	#97	—	61	1	no	—
278	#97	—	61	2	no	—
279	#97	—	61	1	no	—
280	#97	—	61	1	no	—
281	#97	—	61	2	no	—
282	#97	—	122	2	no	—
283	#97	—	122	2	no	—
284	#97	C₄×D₆₁	1	2	yes	—
285	C₄×C₈	—	61	2	no	—
286	C₄×C₈	—	61	2	no	—
287	C₄×C₈	—	122	2	no	—
288	C₄×C₈	—	122	2	no	—
289	C₄×C₈	C₄×D₆₁	1	2	yes	—
290	#41	—	122	1	no	—
291	#41	—	122	1	no	—
292	#41	—	122	1	no	—
293	#41	—	122	1	no	—
294	C₂×Q₈	—	61	1	no	—
295	C₂×D₄	—	61	2	no	—
296	C₂×D₄	—	61	1	no	—
297	C₂²×C₄	—	61	2	no	—
298	C₂²×C₄	—	122	2	no	—
299	C₂²×C₄	—	122	1	no	—
300	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	122	4	no	—
306	M₁₆	—	122	1	no	—
307	M₁₆	—	122	1	no	—
308	M₁₆	—	122	2	no	—
309	M₁₆	—	122	2	no	—
310	M₁₆	—	122	2	no	—
311	M₁₆	—	122	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	61	1	no	—
314	C₂×C₈	—	61	1	no	—
315	C₂×C₈	—	61	2	no	—
316	C₂×C₈	—	61	2	no	—
317	C₂×C₈	—	61	2	no	—
318	C₂×C₈	—	122	4	no	—
319	C₂×C₈	—	122	2	no	—
320	C₂×C₈	—	122	2	no	—
321	C₂×C₈	—	122	2	no	—
322	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
325	C₂×C₈	#11758224	1	1	yes	characteristic
326	C₂×C₈	#11758224	1	1	yes	characteristic
327	C₂×C₈	#11758226	1	1	yes	characteristic
328	C₂×C₈	#11758226	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	122	2	no	—
332	C₄⋊C₄	—	122	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	122	2	no	—
336	C₂²⋊C₄	—	122	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	61	1	no	—
339	C₄²	—	61	1	no	—
340	C₄²	—	122	2	no	—
341	C₄²	—	122	1	no	—
342	C₄²	—	122	2	no	—
343	C₄²	C₂×C₄×D₆₁	1	2	yes	—
344	C₂³	—	61	2	no	—
345	C₂³	#41×D₆₁	1	1	yes	characteristic
346	Q₈	—	122	1	no	—
347	Q₈	—	122	1	no	—
348	D₄	—	122	2	no	—
349	D₄	—	122	2	no	—
350	D₄	—	122	1	no	—
351	D₄	—	122	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	61	2	no	—
355	C₂×C₄	—	122	2	no	—
356	C₂×C₄	—	122	2	no	—
357	C₂×C₄	—	122	1	no	—
358	C₂×C₄	—	122	1	no	—
359	C₂×C₄	—	488	2	no	—
360	C₂×C₄	—	488	1	no	—
361	C₂×C₄	—	244	2	no	—
362	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
363	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
364	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
365	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
366	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
367	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	61	2	no	—
371	C₈	—	61	2	no	—
372	C₈	—	122	2	no	—
373	C₈	—	244	2	no	—
374	C₈	#14519735	1	1	yes	characteristic
375	C₈	#14519735	1	1	yes	characteristic
376	C₈	#14519735	1	1	yes	characteristic
377	C₈	#14519735	1	1	yes	characteristic
378	C₂²	—	4	1	no	—
379	C₂²	—	122	2	no	—
380	C₂²	—	244	2	no	—
381	C₂²	#117×D₆₁	1	1	yes	characteristic
382	C₂²	#16840291	1	1	yes	characteristic
383	C₂²	#16840297	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	122	2	no	—
387	C₄	—	488	2	no	—
388	C₄	—	244	1	no	—
389	C₄	—	244	1	no	—
390	C₄	#117×D₆₁	1	1	yes	characteristic
391	C₄	C₂×#14519735	1	1	yes	characteristic
392	C₄	#16840291	1	1	yes	characteristic
393	C₄	#16840297	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	244	2	no	—
398	C₂	#21448212	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1833	C₂
socle	C₂×C₆₁	#2283
prime radical	C₂×#11758222	C₂³
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1833	D₆₁
hypocenter	C₆₁	#20788
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #21445967	C₂	semidirect
3. #21445943	C₂	semidirect
7. #21445706	C₂	semidirect
8. #21445705	C₂	semidirect
10. #21445573	C₂	semidirect
11. C₆₁×#1833	C₂	semidirect
20. #16839719	C₄	semidirect
23. #16839716	C₄	semidirect
25. C₄×#11758189	C₄	semidirect
27. #97×D₆₁	C₄	semidirect
198. #1833	D₆₁	semidirect
238. C₆₁	#20788	semidirect

Maximal subgroups

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

subgroup	index
#21454246	2
#21445967	2
#21445943	2
#21445921	2
#21445866	2
#21445862	2
#21445706	2
#21445705	2
#21445618	2
#21445573	2
C₆₁×#1833	2
#20788	61