Subgroups — #29698536

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454737	C₂	1	2	yes	maximal
2	C₂×#16839880	C₂	1	1	yes	maximal, characteristic
3	#21454427	C₂	1	2	yes	maximal, complemented
4	#21446609	C₂	1	2	yes	maximal, complemented
5	#21446060	C₂	1	4	yes	maximal, complemented
6	#21445991	C₂	1	1	yes	maximal, characteristic, complemented
7	#21445870	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₂²	1	1	yes	characteristic
10	#16840528	C₂²	1	4	yes	—
11	C₂×#14519783	C₂²	1	2	yes	—
12	#16839885	—	2	4	no	—
13	#16839880	—	2	4	no	—
14	#16839880	C₄	1	2	yes	complemented
15	#16839880	C₄	1	2	yes	complemented
16	#16839880	C₄	1	2	yes	complemented
17	#16839880	C₄	1	2	yes	complemented
18	#16839810	C₂²	1	4	yes	—
19	#16839809	—	2	4	no	—
20	C₂×#14519713	C₂²	1	2	yes	—
21	C₂×#14519713	C₂²	1	2	yes	—
22	C₂×#14519713	C₂²	1	1	yes	characteristic
23	#108×D₆₁	—	2	4	no	—
24	#16839786	C₂²	1	2	yes	—
25	#16839780	C₂²	1	4	yes	—
26	C₂×#14519705	C₂²	1	2	yes	—
27	#16839532	C₂²	1	4	yes	—
28	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#427	C₂²	1	1	yes	characteristic
30	C₆₁×#405	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#339	C₂²	1	2	yes	—
33	C₂×#11758222	C₂³	1	2	yes	—
34	#14519799	C₂×C₄	1	2	yes	—
35	#14519799	C₂×C₄	1	2	yes	—
36	#14519799	D₄	1	2	yes	—
37	#14519799	D₄	1	2	yes	—
38	C₂×Dic₂₄₄	Q₈	1	1	yes	characteristic
39	C₂×#11758190	Q₈	1	2	yes	—
40	C₂×D₄₈₈	Q₈	1	1	yes	characteristic
41	#14519785	—	4	4	no	—
42	#14519783	—	2	2	no	—
43	#14519783	—	2	2	no	—
44	#14519783	—	2	8	no	—
45	#14519783	C₂×C₄	1	4	yes	—
46	#14519783	C₂×C₄	1	4	yes	—
47	C₂×#11758194	C₂³	1	4	yes	—
48	C₂×#11758194	C₂³	1	2	yes	—
49	C₂×#11758189	—	2	2	no	—
50	M₁₆×D₆₁	—	2	8	no	—
51	C₂×C₈×D₆₁	—	2	2	no	—
52	#14519713	—	2	2	no	—
53	#14519713	—	2	2	no	—
54	#14519713	—	2	2	no	—
55	#14519713	—	2	2	no	—
56	#14519713	C₂×C₄	1	2	yes	—
57	#14519713	C₂×C₄	1	2	yes	—
58	#14519712	—	2	8	no	—
59	C₄⋊#92418	—	4	2	no	—
60	#14519705	—	2	8	no	—
61	#14519705	C₂×C₄	1	4	yes	—
62	#14519705	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	#117×C₆₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
68	#108×C₆₁	D₄	1	2	yes	—
69	#108×C₆₁	D₄	1	2	yes	—
70	#107×C₆₁	—	2	1	no	—
71	#106×C₆₁	—	2	1	no	—
72	#97×C₆₁	—	2	2	no	—
73	C₄×C₈×C₆₁	—	2	2	no	—
74	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	—	2	1	no	—
79	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₂²⋊D₆₁	—	2	2	no	—
81	#11758222	—	2	2	no	—
82	#11758222	—	2	2	no	—
83	#11758222	C₂²×C₄	1	4	yes	—
84	#11758222	C₂×D₄	1	4	yes	—
85	C₂×D₂₄₄	C₂×Q₈	1	2	yes	—
86	C₂×C₄×D₆₁	—	2	2	no	—
87	C₂×Dic₁₂₂	C₂×Q₈	1	2	yes	—
88	M₁₆×C₆₁	—	2	2	no	—
89	M₁₆×C₆₁	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
102	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₂²⋊C₄×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	2	1	no	—
106	C₄²×C₆₁	—	2	2	no	—
107	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
109	#11758194	—	2	2	no	—
110	#11758194	—	2	4	no	—
111	#11758194	—	2	4	no	—
112	#11758194	C₂²×C₄	1	4	yes	—
113	C₂×#92418	—	2	2	no	—
114	C₂×#92418	—	2	4	no	—
115	Dic₂₄₄	C₄⋊C₄	1	2	yes	—
116	Dic₂₄₄	C₄⋊C₄	1	2	yes	—
117	D₄₈₈	C₄⋊C₄	1	2	yes	—
118	D₄₈₈	C₄⋊C₄	1	2	yes	—
119	#11758190	C₄⋊C₄	1	4	yes	—
120	#11758190	C₄⋊C₄	1	4	yes	—
121	#11758189	—	2	4	no	—
122	#11758189	—	2	4	no	—
123	#11758189	—	4	8	no	—
124	C₈×D₆₁	—	2	4	no	—
125	C₈×D₆₁	—	2	4	no	—
126	C₈×D₆₁	—	4	8	no	—
127	C₂³×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²×D₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²⋊D₆₁	—	2	4	no	—
140	C₂²⋊D₆₁	—	2	4	no	—
141	C₂×Dic₆₁	—	2	2	no	—
142	D₂₄₄	—	2	2	no	—
143	D₂₄₄	C₂×C₄⋊C₄	1	4	yes	—
144	C₄×D₆₁	—	2	4	no	—
145	C₄×D₆₁	—	2	4	no	—
146	Dic₁₂₂	—	2	2	no	—
147	Dic₁₂₂	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₆₁	—	2	4	no	—
149	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
154	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
155	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
156	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
157	#92418	—	2	4	no	—
158	#92418	—	4	4	no	—
159	#22694	—	61	1	no	maximal, Sylow
160	C₂²×C₆₁	—	4	1	no	—
161	C₂²×C₆₁	#520	1	1	yes	characteristic
162	C₂²×C₆₁	#547	1	1	yes	characteristic
163	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
164	C₂×D₆₁	—	2	4	no	—
165	C₂×D₆₁	—	8	2	no	—
166	C₄×C₆₁	—	2	2	no	—
167	C₄×C₆₁	—	4	1	no	—
168	C₄×C₆₁	—	8	2	no	—
169	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₆₁	D₄²	1	1	yes	characteristic
172	C₄×C₆₁	#554	1	1	yes	characteristic
173	Dic₆₁	—	2	4	no	—
174	#2957	—	61	2	no	—
175	C₂×#443	—	61	1	no	—
176	#2916	—	61	2	no	—
177	#1953	—	61	2	no	—
178	#1861	—	61	4	no	—
179	#1848	—	61	1	no	—
180	#1814	—	61	2	no	—
181	#1772	D₆₁	1	1	yes	characteristic, complemented
182	C₂×C₆₁	—	2	1	no	—
183	C₂×C₆₁	—	2	1	no	—
184	C₂×C₆₁	—	2	1	no	—
185	C₂×C₆₁	D₄×C₄⋊C₄	1	1	yes	characteristic
186	D₆₁	—	4	4	no	—
187	C₂×#135	—	61	1	no	—
188	#568	—	61	4	no	—
189	C₂×#131	—	61	2	no	—
190	#444	—	122	4	no	—
191	#443	—	61	2	no	—
192	#443	—	61	2	no	—
193	#443	—	61	2	no	—
194	#443	—	61	2	no	—
195	#443	—	122	4	no	—
196	#430	—	61	4	no	—
197	#430	—	122	4	no	—
198	C₂×#108	—	61	1	no	—
199	C₂×#108	—	61	2	no	—
200	C₂×#108	—	61	2	no	—
201	C₂×#108	—	122	4	no	—
202	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
203	#427	C₂×D₆₁	1	1	yes	characteristic
204	#423	—	61	2	no	—
205	#421	—	61	4	no	—
206	C₂×#104	—	61	2	no	—
207	#405	C₂×D₆₁	1	2	yes	—
208	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
209	#344	—	61	4	no	—
210	#339	C₂×D₆₁	1	2	yes	—
211	C₆₁	#22694	1	1	yes	characteristic, complemented, Sylow
212	C₂×#41	—	61	2	no	—
213	#135	—	61	2	no	—
214	#135	—	61	2	no	—
215	#135	—	61	2	no	—
216	#135	—	61	2	no	—
217	C₂×Q₁₆	—	61	1	no	—
218	C₂×SD₁₆	—	61	2	no	—
219	C₂×D₈	—	61	1	no	—
220	#131	—	61	4	no	—
221	#131	—	61	4	no	—
222	#131	—	122	8	no	—
223	#131	—	122	2	no	—
224	#131	—	122	2	no	—
225	#131	—	244	4	no	—
226	C₂×M₁₆	—	61	2	no	—
227	C₂×M₁₆	—	61	4	no	—
228	C₂×M₁₆	—	122	2	no	—
229	C₂×M₁₆	—	122	8	no	—
230	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
231	C₂²×C₈	—	122	2	no	—
232	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
234	#117	C₂²×D₆₁	1	2	yes	—
235	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
236	#108	—	61	2	no	—
237	#108	—	61	2	no	—
238	#108	—	122	8	no	—
239	#108	—	122	2	no	—
240	#108	—	122	2	no	—
241	#108	—	122	2	no	—
242	#108	—	122	2	no	—
243	#108	D₂₄₄	1	2	yes	—
244	#108	D₂₄₄	1	2	yes	—
245	#107	—	2	1	no	—
246	#106	—	2	1	no	—
247	C₄⋊C₈	—	244	2	no	—
248	#104	—	61	4	no	—
249	#104	—	61	4	no	—
250	#104	—	122	8	no	—
251	#97	—	2	2	no	—
252	C₄×C₈	—	2	2	no	—
253	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
254	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
255	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
256	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
257	#41	—	61	4	no	—
258	#41	—	61	4	no	—
259	#41	—	122	2	no	—
260	#41	—	122	2	no	—
261	C₂×Q₈	—	61	2	no	—
262	C₂×D₄	—	61	2	no	—
263	C₂×D₄	—	122	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	122	2	no	—
266	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
267	Q₁₆	—	61	2	no	—
268	Q₁₆	—	61	2	no	—
269	SD₁₆	—	61	4	no	—
270	SD₁₆	—	61	4	no	—
271	D₈	—	61	2	no	—
272	D₈	—	61	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	61	4	no	—
275	M₁₆	—	122	2	no	—
276	M₁₆	—	122	4	no	—
277	M₁₆	—	122	4	no	—
278	M₁₆	—	122	4	no	—
279	M₁₆	—	122	4	no	—
280	M₁₆	—	244	8	no	—
281	M₁₆	C₂×D₂₄₄	1	4	yes	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	122	2	no	—
284	C₂×C₈	—	122	4	no	—
285	C₂×C₈	—	122	4	no	—
286	C₂×C₈	—	122	4	no	—
287	C₂×C₈	—	244	8	no	—
288	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
292	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
293	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
294	C₂×C₈	#11758222	1	1	yes	characteristic
295	C₂×C₈	#11758222	1	1	yes	characteristic
296	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
297	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
298	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
299	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	2	no	—
304	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
305	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
306	C₂³	—	122	2	no	—
307	C₂³	#14519824	1	1	yes	characteristic
308	Q₈	—	61	4	no	—
309	Q₈	—	122	2	no	—
310	D₄	—	61	4	no	—
311	D₄	—	122	4	no	—
312	D₄	—	122	4	no	—
313	D₄	—	122	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	122	2	no	—
319	C₂×C₄	—	122	4	no	—
320	C₂×C₄	—	122	4	no	—
321	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
325	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
326	C₂×C₄	#14519823	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	122	4	no	—
329	C₈	—	244	4	no	—
330	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
331	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
332	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
333	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
334	C₈	C₄×D₂₄₄	1	1	yes	characteristic
335	C₈	C₄×D₂₄₄	1	1	yes	characteristic
336	C₈	C₄×D₂₄₄	1	1	yes	characteristic
337	C₈	C₄×D₂₄₄	1	1	yes	characteristic
338	C₂²	—	4	1	no	—
339	C₂²	—	122	4	no	—
340	C₂²	—	488	2	no	—
341	C₂²	#16840242	1	1	yes	characteristic
342	C₂²	#16840417	1	1	yes	characteristic
343	C₂²	Q₈×D₂₄₄	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	122	4	no	—
348	C₄	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
349	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
350	C₄	D₄×D₂₄₄	1	1	yes	characteristic
351	C₄	#16840472	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	244	4	no	—
356	C₂	C₄⋊C₄×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1772	C₂
socle	C₂×C₆₁	D₄×C₄⋊C₄
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1772	D₆₁
hypocenter	C₆₁	#22694
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #21454427	C₂	semidirect
4. #21446609	C₂	semidirect
5. #21446060	C₂	semidirect
6. #21445991	C₂	semidirect
7. #21445870	C₂	semidirect
8. C₆₁×#1772	C₂	semidirect
14. #16839880	C₄	semidirect
15. #16839880	C₄	semidirect
16. #16839880	C₄	semidirect
17. #16839880	C₄	semidirect
181. #1772	D₆₁	semidirect
211. C₆₁	#22694	semidirect

Maximal subgroups

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

subgroup	index
#21454737	2
C₂×#16839880	2
#21454427	2
#21446609	2
#21446060	2
#21445991	2
#21445870	2
C₆₁×#1772	2
#22694	61