Subgroups — #29680374

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454241	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	#21445701	C₂	1	1	yes	maximal, characteristic, complemented
8	#21445700	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	#16839714	C₄	1	2	yes	complemented
25	M₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×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	1	no	—
41	#14519785	—	2	1	no	—
42	#14519785	—	2	1	no	—
43	#14519785	—	2	1	no	—
44	#14519783	—	2	1	no	—
45	#14519783	—	2	1	no	—
46	#14519783	—	2	1	no	—
47	#14519783	—	2	1	no	—
48	C₂×#11758194	C₂³	1	2	yes	—
49	C₂×#11758194	C₂³	1	1	yes	characteristic
50	C₂×#11758189	C₂×C₄	1	2	yes	—
51	C₂×#11758189	C₂×C₄	1	2	yes	—
52	C₂²×#92418	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
55	#14519742	—	2	1	no	—
56	#14519739	—	2	1	no	—
57	#14519737	—	2	2	no	—
58	#14519736	C₂³	1	2	yes	—
59	#14519734	C₂×C₄	1	2	yes	—
60	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
61	C₄²×D₆₁	C₂×C₄	1	2	yes	—
62	#14519713	—	2	1	no	—
63	#14519713	—	2	1	no	—
64	#14519711	—	2	1	no	—
65	#14519709	—	2	1	no	—
66	#14519695	—	2	2	no	—
67	#14519693	—	2	2	no	—
68	#14519691	—	2	4	no	—
69	#14519688	—	2	2	no	—
70	#14519688	C₂×C₄	1	2	yes	—
71	#14519687	—	2	2	no	—
72	#14519687	C₂×C₄	1	2	yes	—
73	C₄×#92418	—	2	2	no	—
74	C₄×#92418	C₂×C₄	1	2	yes	—
75	C₈×Dic₆₁	—	2	2	no	—
76	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
78	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
81	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
82	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
84	#117×C₆₁	C₂³	1	2	yes	—
85	#117×C₆₁	C₂³	1	1	yes	characteristic
86	#108×C₆₁	—	2	1	no	—
87	#108×C₆₁	—	2	1	no	—
88	#107×C₆₁	—	2	1	no	—
89	#106×C₆₁	—	2	1	no	—
90	#97×C₆₁	C₂×C₄	1	2	yes	—
91	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
92	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₂×C₂²⋊D₆₁	#41	1	2	yes	—
94	C₂²×Dic₆₁	—	2	1	no	—
95	#11758222	—	2	1	no	—
96	#11758222	—	2	1	no	—
97	#11758222	—	2	1	no	—
98	#11758222	—	2	1	no	—
99	C₂×D₂₄₄	#41	1	1	yes	characteristic
100	C₂×C₄×D₆₁	—	2	2	no	—
101	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
102	C₂×Dic₁₂₂	#41	1	1	yes	characteristic
103	M₁₆×C₆₁	—	2	1	no	—
104	M₁₆×C₆₁	—	2	1	no	—
105	M₁₆×C₆₁	—	2	2	no	—
106	M₁₆×C₆₁	—	2	2	no	—
107	C₂×C₈×C₆₁	—	2	2	no	—
108	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
112	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
113	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
114	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
115	C₄⋊C₄×C₆₁	—	2	1	no	—
116	C₄⋊C₄×C₆₁	—	2	2	no	—
117	C₂²⋊C₄×C₆₁	—	2	1	no	—
118	C₂²⋊C₄×C₆₁	—	2	2	no	—
119	C₄²×C₆₁	—	2	2	no	—
120	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
121	C₂²⋊Dic₆₁	—	2	2	no	—
122	#11758198	—	2	2	no	—
123	C₄⋊Dic₆₁	—	2	2	no	—
124	#11758196	—	2	2	no	—
125	C₄×Dic₆₁	—	2	1	no	—
126	C₄×Dic₆₁	—	2	2	no	—
127	C₄×Dic₆₁	—	2	2	no	—
128	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
129	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
130	#11758194	—	2	1	no	—
131	#11758194	—	2	1	no	—
132	#11758194	—	2	2	no	—
133	#11758194	—	2	2	no	—
134	C₂×#92418	—	2	2	no	—
135	C₂×#92418	C₂²×C₄	1	2	yes	—
136	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
137	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
138	C₂×#92418	#41	1	1	yes	characteristic
139	C₂×#92418	#41	1	1	yes	characteristic
140	C₂×#92418	#41	1	1	yes	characteristic
141	C₂×#92418	#41	1	1	yes	characteristic
142	#11758189	—	2	2	no	—
143	#11758189	—	2	2	no	—
144	#11758189	—	2	2	no	—
145	#11758189	—	2	2	no	—
146	C₈×D₆₁	—	2	2	no	—
147	C₈×D₆₁	—	2	2	no	—
148	C₈×D₆₁	—	2	2	no	—
149	C₈×D₆₁	—	2	2	no	—
150	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
151	C₂²×D₆₁	#117	1	2	yes	—
152	C₂×C₄×C₆₁	—	2	2	no	—
153	C₂×C₄×C₆₁	—	4	2	no	—
154	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
160	C₂²⋊D₆₁	—	2	2	no	—
161	C₂²⋊D₆₁	—	2	2	no	—
162	C₂×Dic₆₁	—	2	1	no	—
163	C₂×Dic₆₁	—	2	1	no	—
164	C₂×Dic₆₁	—	4	2	no	—
165	C₂×Dic₆₁	—	4	2	no	—
166	C₂×Dic₆₁	#117	1	2	yes	—
167	D₂₄₄	—	2	1	no	—
168	D₂₄₄	—	2	1	no	—
169	C₄×D₆₁	—	2	2	no	—
170	C₄×D₆₁	—	2	2	no	—
171	C₄×D₆₁	—	4	4	no	—
172	Dic₁₂₂	—	2	1	no	—
173	Dic₁₂₂	—	2	1	no	—
174	C₈×C₆₁	—	2	2	no	—
175	C₈×C₆₁	—	4	2	no	—
176	C₈×C₆₁	#117	1	1	yes	characteristic
177	C₈×C₆₁	#117	1	1	yes	characteristic
178	C₈×C₆₁	#117	1	1	yes	characteristic
179	C₈×C₆₁	#117	1	1	yes	characteristic
180	#92418	—	2	2	no	—
181	#92418	—	4	2	no	—
182	#92418	#117	1	1	yes	characteristic
183	#92418	#117	1	1	yes	characteristic
184	#92418	#117	1	1	yes	characteristic
185	#92418	#117	1	1	yes	characteristic
186	#20787	—	61	1	no	maximal, Sylow
187	C₂²×C₆₁	—	4	1	no	—
188	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
189	C₂²×C₆₁	#529	1	1	yes	characteristic
190	C₂²×C₆₁	#529	1	1	yes	characteristic
191	C₂×D₆₁	—	2	2	no	—
192	C₂×D₆₁	—	4	2	no	—
193	C₄×C₆₁	—	4	2	no	—
194	C₄×C₆₁	—	8	2	no	—
195	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
196	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
197	C₄×C₆₁	#529	1	1	yes	characteristic
198	C₄×C₆₁	#529	1	1	yes	characteristic
199	Dic₆₁	—	2	2	no	—
200	Dic₆₁	—	2	2	no	—
201	Dic₆₁	—	4	1	no	—
202	Dic₆₁	—	8	2	no	—
203	C₄×#131	—	61	1	no	—
204	#1841	—	61	2	no	—
205	#1833	—	61	1	no	—
206	#1833	D₆₁	1	1	yes	characteristic, complemented
207	#1827	—	61	2	no	—
208	#1813	—	61	1	no	—
209	#1812	—	61	1	no	—
210	#1772	—	61	1	no	—
211	#1772	—	61	1	no	—
212	#1757	—	61	2	no	—
213	#1747	—	61	2	no	—
214	C₂×C₆₁	—	2	1	no	—
215	C₂×C₆₁	—	2	1	no	—
216	C₂×C₆₁	—	2	1	no	—
217	C₂×C₆₁	#2283	1	1	yes	characteristic
218	D₆₁	—	4	2	no	—
219	C₂×#131	—	61	1	no	—
220	C₂×#131	—	61	1	no	—
221	C₄×#41	—	61	1	no	—
222	C₂×#108	—	61	1	no	—
223	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
224	#427	—	61	1	no	—
225	#427	C₂×D₆₁	1	1	yes	characteristic
226	#413	—	61	2	no	—
227	#410	—	61	2	no	—
228	#405	—	61	1	no	—
229	#405	—	61	2	no	—
230	#405	—	61	2	no	—
231	#405	—	61	2	no	—
232	#405	—	61	2	no	—
233	#405	C₂×D₆₁	1	1	yes	characteristic
234	C₄×M₁₆	—	61	1	no	—
235	C₄×M₁₆	—	61	2	no	—
236	C₂×C₄×C₈	—	61	1	no	—
237	C₂×C₄×C₈	—	61	2	no	—
238	#344	—	61	2	no	—
239	#344	C₂×D₆₁	1	2	yes	—
240	#339	—	61	2	no	—
241	#339	—	61	2	no	—
242	#338	—	61	2	no	—
243	H(3,ℤ/4)	—	61	2	no	—
244	H(3,ℤ/4)	—	61	2	no	—
245	H(3,ℤ/4)	C₂×D₆₁	1	2	yes	—
246	C₆₁	#20787	1	1	yes	characteristic, complemented, Sylow
247	C₂×#41	—	61	1	no	—
248	#131	—	122	1	no	—
249	#131	—	122	1	no	—
250	#131	—	122	1	no	—
251	#131	—	122	1	no	—
252	#131	—	122	1	no	—
253	#131	—	122	1	no	—
254	#131	—	122	1	no	—
255	#131	—	122	1	no	—
256	C₂×M₁₆	—	61	1	no	—
257	C₂×M₁₆	—	61	2	no	—
258	C₂×M₁₆	—	61	2	no	—
259	C₂×M₁₆	—	61	2	no	—
260	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
261	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
262	C₂²×C₈	—	61	1	no	—
263	C₂²×C₈	—	61	2	no	—
264	C₂²×C₈	—	61	2	no	—
265	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
266	C₄×Q₈	—	122	1	no	—
267	C₄×D₄	—	122	2	no	—
268	C₄×D₄	—	122	1	no	—
269	#117	—	61	2	no	—
270	#117	—	61	2	no	—
271	#117	C₂²×D₆₁	1	2	yes	—
272	#117	C₂²×D₆₁	1	1	yes	characteristic
273	C₂×C₄²	—	61	1	no	—
274	C₂×C₄²	—	61	2	no	—
275	#108	—	2	1	no	—
276	#108	—	2	1	no	—
277	#108	—	122	1	no	—
278	#108	—	122	1	no	—
279	#107	—	2	1	no	—
280	#107	—	122	1	no	—
281	#106	—	2	1	no	—
282	#106	—	122	1	no	—
283	#101	—	122	2	no	—
284	#100	—	122	2	no	—
285	#99	—	122	4	no	—
286	#97	—	61	2	no	—
287	#97	—	61	2	no	—
288	#97	—	122	2	no	—
289	#97	—	122	2	no	—
290	#97	C₄×D₆₁	1	2	yes	—
291	C₄×C₈	—	61	1	no	—
292	C₄×C₈	—	61	1	no	—
293	C₄×C₈	—	61	2	no	—
294	C₄×C₈	—	61	1	no	—
295	C₄×C₈	—	61	1	no	—
296	C₄×C₈	—	61	2	no	—
297	C₄×C₈	—	122	2	no	—
298	C₄×C₈	—	122	2	no	—
299	C₄×C₈	C₄×D₆₁	1	2	yes	—
300	#41	—	122	1	no	—
301	#41	—	122	1	no	—
302	#41	—	122	1	no	—
303	#41	—	122	1	no	—
304	C₂×Q₈	—	61	1	no	—
305	C₂×D₄	—	61	2	no	—
306	C₂×D₄	—	61	1	no	—
307	C₂²×C₄	—	61	2	no	—
308	C₂²×C₄	—	122	2	no	—
309	C₂²×C₄	—	122	1	no	—
310	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
311	M₁₆	—	2	1	no	—
312	M₁₆	—	2	1	no	—
313	M₁₆	—	2	2	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	122	2	no	—
316	M₁₆	—	122	2	no	—
317	M₁₆	—	122	1	no	—
318	M₁₆	—	122	1	no	—
319	M₁₆	—	122	2	no	—
320	M₁₆	—	122	2	no	—
321	M₁₆	—	122	2	no	—
322	M₁₆	—	122	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	61	1	no	—
325	C₂×C₈	—	61	1	no	—
326	C₂×C₈	—	61	1	no	—
327	C₂×C₈	—	61	1	no	—
328	C₂×C₈	—	61	1	no	—
329	C₂×C₈	—	61	1	no	—
330	C₂×C₈	—	61	2	no	—
331	C₂×C₈	—	122	2	no	—
332	C₂×C₈	—	122	2	no	—
333	C₂×C₈	—	122	2	no	—
334	C₂×C₈	—	122	2	no	—
335	C₂×C₈	—	122	2	no	—
336	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
337	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
338	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
339	C₂×C₈	#11758224	1	1	yes	characteristic
340	C₂×C₈	#11758224	1	1	yes	characteristic
341	C₂×C₈	#11758226	1	1	yes	characteristic
342	C₂×C₈	#11758226	1	1	yes	characteristic
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	122	2	no	—
346	C₄⋊C₄	—	122	2	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₂²⋊C₄	—	122	2	no	—
350	C₂²⋊C₄	—	122	2	no	—
351	C₄²	—	2	2	no	—
352	C₄²	—	61	1	no	—
353	C₄²	—	61	1	no	—
354	C₄²	—	122	2	no	—
355	C₄²	—	122	1	no	—
356	C₄²	—	122	2	no	—
357	C₄²	C₂×C₄×D₆₁	1	2	yes	—
358	C₂³	—	61	2	no	—
359	C₂³	#41×D₆₁	1	1	yes	characteristic
360	Q₈	—	122	1	no	—
361	Q₈	—	122	1	no	—
362	D₄	—	122	2	no	—
363	D₄	—	122	2	no	—
364	D₄	—	122	1	no	—
365	D₄	—	122	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	61	2	no	—
369	C₂×C₄	—	122	1	no	—
370	C₂×C₄	—	122	2	no	—
371	C₂×C₄	—	122	2	no	—
372	C₂×C₄	—	122	1	no	—
373	C₂×C₄	—	244	4	no	—
374	C₂×C₄	—	244	2	no	—
375	C₂×C₄	—	244	2	no	—
376	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
377	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
378	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
379	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
380	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
381	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	—	61	1	no	—
385	C₈	—	61	1	no	—
386	C₈	—	61	1	no	—
387	C₈	—	61	1	no	—
388	C₈	—	122	2	no	—
389	C₈	—	244	2	no	—
390	C₈	#14519735	1	1	yes	characteristic
391	C₈	#14519735	1	1	yes	characteristic
392	C₈	#14519735	1	1	yes	characteristic
393	C₈	#14519735	1	1	yes	characteristic
394	C₂²	—	4	1	no	—
395	C₂²	—	122	2	no	—
396	C₂²	—	244	2	no	—
397	C₂²	#117×D₆₁	1	1	yes	characteristic
398	C₂²	#16840291	1	1	yes	characteristic
399	C₂²	#16840297	1	1	yes	characteristic
400	C₄	—	4	2	no	—
401	C₄	—	8	2	no	—
402	C₄	—	122	2	no	—
403	C₄	—	122	2	no	—
404	C₄	—	488	2	no	—
405	C₄	—	244	1	no	—
406	C₄	#117×D₆₁	1	1	yes	characteristic
407	C₄	C₂×#14519735	1	1	yes	characteristic
408	C₄	#16840291	1	1	yes	characteristic
409	C₄	#16840297	1	1	yes	characteristic
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	244	2	no	—
414	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	#20787
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. #21445701	C₂	semidirect
8. #21445700	C₂	semidirect
10. #21445573	C₂	semidirect
11. C₆₁×#1833	C₂	semidirect
20. #16839719	C₄	semidirect
23. #16839716	C₄	semidirect
24. #16839714	C₄	semidirect
27. C₄×C₈×D₆₁	C₄	semidirect
206. #1833	D₆₁	semidirect
246. C₆₁	#20787	semidirect

Maximal subgroups

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

subgroup	index
#21454241	2
#21445967	2
#21445943	2
#21445921	2
#21445866	2
#21445862	2
#21445701	2
#21445700	2
#21445618	2
#21445573	2
C₆₁×#1833	2
#20787	61