Subgroups — #29649867

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454932	C₂	1	2	yes	maximal, complemented
2	C₄×#14519799	C₂	1	1	yes	maximal, characteristic
3	#21444723	C₂	1	4	yes	maximal, complemented
4	#21444717	C₂	1	2	yes	maximal, complemented
5	#21444705	C₂	1	2	yes	maximal, complemented
6	C₂×#16839515	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444207	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519799	C₄	1	1	yes	characteristic
10	C₄×#11758222	C₂²	1	2	yes	—
11	#16839892	C₂²	1	4	yes	—
12	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic
13	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
14	C₄×#11758190	C₄	1	2	yes	—
15	C₄×#11758190	C₂²	1	2	yes	—
16	C₄×D₄₈₈	C₄	1	1	yes	characteristic
17	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
18	#16839841	C₂²	1	4	yes	—
19	#16839830	—	2	4	no	—
20	#16839818	—	2	2	no	—
21	#16839800	C₄	1	1	yes	characteristic
22	#16839784	—	2	2	no	—
23	C₂×C₄⋊#92418	C₂²	1	2	yes	—
24	#16839759	C₄	1	2	yes	—
25	#16839515	C₂²	1	2	yes	—
26	#16839515	C₂²	1	2	yes	—
27	#16839494	C₂²	1	4	yes	—
28	#16839491	C₂²	1	4	yes	—
29	C₆₁×#431	C₂²	1	1	yes	characteristic
30	C₆₁×#431	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#322	C₂²	1	2	yes	—
33	C₈²×C₆₁	C₂²	1	2	yes	—
34	C₂×#11758222	C₂×C₄	1	2	yes	—
35	#14519799	—	2	4	no	—
36	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
37	C₂×#11758190	C₂×C₄	1	2	yes	—
38	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
39	C₂×#11758189	—	2	4	no	—
40	C₂²×#92418	—	2	2	no	—
41	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₂₂	C₂³	1	2	yes	—
43	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
44	C₄×D₂₄₄	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₆₁	—	2	2	no	—
46	C₄×C₂²⋊D₆₁	—	2	2	no	—
47	#14519736	C₂×C₄	1	2	yes	—
48	#14519734	—	2	2	no	—
49	C₄²×D₆₁	—	2	2	no	—
50	#14519711	C₂×C₄	1	1	yes	characteristic
51	#14519711	C₂×C₄	1	1	yes	characteristic
52	#14519709	C₂×C₄	1	2	yes	—
53	C₄⋊#92418	—	2	2	no	—
54	C₄⋊#92418	C₂³	1	4	yes	—
55	#14519706	—	2	4	no	—
56	#14519702	C₂×C₄	1	2	yes	—
57	#14519702	C₂×C₄	1	2	yes	—
58	#14519699	C₂×C₄	1	2	yes	—
59	#14519699	C₂×C₄	1	2	yes	—
60	#14519689	—	2	4	no	—
61	#14519687	—	2	4	no	—
62	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₆₁	—	2	1	no	—
66	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
67	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₆₁	—	2	1	no	—
69	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
70	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₁	C₂³	1	2	yes	—
73	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₆₁	—	2	2	no	—
80	#11758222	—	2	4	no	—
81	#11758222	—	2	4	no	—
82	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₆₁	—	2	2	no	—
84	C₂×C₄×D₆₁	—	2	2	no	—
85	C₂×C₄×D₆₁	—	2	2	no	—
86	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
96	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₁	#41	1	1	yes	characteristic
99	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₆₁	—	2	2	no	—
101	#11758198	—	2	2	no	—
102	#11758198	—	2	2	no	—
103	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
105	#11758196	—	2	2	no	—
106	#11758196	—	2	2	no	—
107	C₄×Dic₆₁	—	2	2	no	—
108	C₄×Dic₆₁	—	2	2	no	—
109	C₄×Dic₆₁	—	2	2	no	—
110	C₂×#92418	—	2	4	no	—
111	C₂×#92418	—	8	2	no	—
112	Dic₂₄₄	—	2	2	no	—
113	D₄₈₈	—	2	2	no	—
114	#11758190	—	2	4	no	—
115	#11758189	—	8	4	no	—
116	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	4	1	no	—
121	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₆₁	#131	1	2	yes	—
126	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₆₁	—	4	4	no	—
130	C₂×Dic₆₁	—	2	2	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	2	2	no	—
133	D₂₄₄	—	2	2	no	—
134	D₂₄₄	—	2	2	no	—
135	C₄×D₆₁	—	4	2	no	—
136	C₄×D₆₁	—	4	2	no	—
137	C₄×D₆₁	—	4	4	no	—
138	C₄×D₆₁	—	8	2	no	—
139	Dic₁₂₂	—	2	2	no	—
140	Dic₁₂₂	—	2	2	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	—	2	2	no	—
144	C₈×C₆₁	—	2	2	no	—
145	C₈×C₆₁	—	4	2	no	—
146	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
147	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
148	#92418	—	4	4	no	—
149	#17898	—	61	1	no	maximal, Sylow
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	—	2	1	no	—
152	C₂²×C₆₁	#436	1	2	yes	—
153	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×D₆₁	—	4	2	no	—
155	C₂×D₆₁	—	4	2	no	—
156	C₂×D₆₁	—	4	2	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	2	1	no	—
159	C₄×C₆₁	—	2	1	no	—
160	C₄×C₆₁	#436	1	2	yes	—
161	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
162	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
163	C₄×C₆₁	#576	1	1	yes	characteristic
164	C₄×C₆₁	#578	1	1	yes	characteristic
165	Dic₆₁	—	4	2	no	—
166	Dic₆₁	—	4	2	no	—
167	Dic₆₁	—	4	2	no	—
168	#2989	—	61	2	no	—
169	C₄×#135	—	61	1	no	—
170	#1592	—	61	4	no	—
171	#1591	—	61	2	no	—
172	#1589	—	61	2	no	—
173	C₂×#335	—	61	1	no	—
174	#1477	—	61	2	no	—
175	#1450	D₆₁	1	1	yes	characteristic, complemented
176	C₂×C₆₁	—	2	2	no	—
177	C₂×C₆₁	C₂×#436	1	1	yes	characteristic
178	C₂×C₆₁	#2951	1	1	yes	characteristic
179	C₂×C₆₁	#2957	1	1	yes	characteristic
180	D₆₁	—	8	2	no	—
181	C₂×#135	—	61	1	no	—
182	C₄×#41	—	61	2	no	—
183	#446	—	61	4	no	—
184	C₄×Q₁₆	—	61	1	no	—
185	C₄×Q₁₆	—	61	1	no	—
186	C₄×SD₁₆	—	61	2	no	—
187	C₄×SD₁₆	—	61	2	no	—
188	C₄×D₈	—	61	1	no	—
189	C₄×D₈	—	61	1	no	—
190	#436	—	61	4	no	—
191	#435	—	122	4	no	—
192	#432	—	122	2	no	—
193	#431	C₂×D₆₁	1	1	yes	characteristic
194	#431	C₂×D₆₁	1	1	yes	characteristic
195	#427	—	61	1	no	—
196	#423	—	122	2	no	—
197	C₂×C₄⋊C₈	—	61	2	no	—
198	#416	—	61	2	no	—
199	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
200	#335	—	61	2	no	—
201	#335	—	61	2	no	—
202	Q₈⋊C₈	—	61	4	no	—
203	D₄⋊C₈	—	61	4	no	—
204	#322	C₂×D₆₁	1	2	yes	—
205	C₈²	C₂×D₆₁	1	2	yes	—
206	C₆₁	#17898	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	—	61	2	no	—
208	#135	—	122	4	no	—
209	C₂×Q₁₆	—	61	1	no	—
210	C₂×SD₁₆	—	61	2	no	—
211	C₂×D₈	—	61	1	no	—
212	C₂×M₁₆	—	122	4	no	—
213	C₂²×C₈	—	122	2	no	—
214	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
216	C₄×Q₈	—	61	2	no	—
217	C₄×Q₈	—	61	2	no	—
218	C₄×D₄	—	61	2	no	—
219	C₄×D₄	—	61	2	no	—
220	C₄×D₄	—	122	2	no	—
221	C₄×D₄	—	122	2	no	—
222	#117	—	61	2	no	—
223	#117	—	122	2	no	—
224	C₂×C₄²	—	122	2	no	—
225	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
226	#107	—	61	1	no	—
227	#107	—	61	1	no	—
228	#106	—	61	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	61	4	no	—
231	C₄⋊C₈	—	122	4	no	—
232	C₄⋊C₈	—	122	2	no	—
233	C₄⋊C₈	D₂₄₄	1	1	yes	characteristic
234	C₄⋊C₈	D₂₄₄	1	1	yes	characteristic
235	#103	—	61	2	no	—
236	#103	—	61	2	no	—
237	#102	—	61	2	no	—
238	#102	—	61	2	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	C₂²⋊C₈	—	122	4	no	—
241	C₂²⋊C₈	D₂₄₄	1	2	yes	—
242	#97	—	122	4	no	—
243	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
244	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₆₁	1	2	yes	—
246	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₆₁	1	2	yes	—
249	#41	—	122	4	no	—
250	#41	—	122	4	no	—
251	C₂×Q₈	—	61	2	no	—
252	C₂×D₄	—	61	2	no	—
253	C₂×D₄	—	122	2	no	—
254	C₂²×C₄	—	122	2	no	—
255	C₂²×C₄	—	122	2	no	—
256	C₂²×C₄	—	122	2	no	—
257	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
259	C₂²×C₄	#11758222	1	1	yes	characteristic
260	Q₁₆	—	122	2	no	—
261	SD₁₆	—	122	4	no	—
262	D₈	—	122	2	no	—
263	M₁₆	—	488	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	122	4	no	—
267	C₂×C₈	—	488	2	no	—
268	C₂×C₈	#11758189	1	4	yes	complemented
269	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
273	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
274	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
275	C₄⋊C₄	—	61	2	no	—
276	C₄⋊C₄	—	61	2	no	—
277	C₄⋊C₄	—	122	2	no	—
278	C₄⋊C₄	—	122	2	no	—
279	C₂²⋊C₄	—	122	2	no	—
280	C₂²⋊C₄	—	122	2	no	—
281	C₂²⋊C₄	—	122	2	no	—
282	C₄²	—	122	2	no	—
283	C₄²	—	122	2	no	—
284	C₄²	—	122	2	no	—
285	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
286	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
287	C₄²	#11758222	1	1	yes	characteristic
288	C₄²	C₂³×D₆₁	1	1	yes	characteristic
289	C₂³	—	122	2	no	—
290	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
291	Q₈	—	122	2	no	—
292	Q₈	—	122	2	no	—
293	D₄	—	122	2	no	—
294	D₄	—	122	2	no	—
295	D₄	—	244	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	122	2	no	—
300	C₂×C₄	—	122	2	no	—
301	C₂×C₄	—	122	2	no	—
302	C₂×C₄	—	488	2	no	—
303	C₂×C₄	—	244	2	no	—
304	C₂×C₄	—	244	2	no	—
305	C₂×C₄	—	244	4	no	—
306	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
307	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
308	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
309	C₂×C₄	C₂×#11758189	1	2	yes	—
310	C₂×C₄	#14519783	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
313	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	244	4	no	—
320	C₈	C₂×#11758189	1	2	yes	—
321	C₈	C₂×#11758189	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	244	2	no	—
325	C₂²	—	244	2	no	—
326	C₂²	—	244	2	no	—
327	C₂²	#16839841	1	2	yes	—
328	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	244	2	no	—
333	C₄	—	244	2	no	—
334	C₄	—	244	2	no	—
335	C₄	#16839841	1	2	yes	—
336	C₄	C₂²×#11758189	1	1	yes	characteristic
337	C₄	C₂×#14519783	1	1	yes	characteristic
338	C₄	#16840567	1	1	yes	characteristic
339	C₄	#16840579	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	488	2	no	—
342	C₂	C₂×#16839841	1	1	yes	characteristic
343	C₂	#21454696	1	1	yes	characteristic
344	C₂	#21454737	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1450	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519799	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1450	D₆₁
hypocenter	C₆₁	#17898
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #21454932	C₂	semidirect
3. #21444723	C₂	semidirect
4. #21444717	C₂	semidirect
5. #21444705	C₂	semidirect
6. C₂×#16839515	C₂	semidirect
7. #21444207	C₂	semidirect
8. C₆₁×#1450	C₂	semidirect
89. C₂×C₈×C₆₁	M₁₆	semidirect
175. #1450	D₆₁	semidirect
206. C₆₁	#17898	semidirect
268. C₂×C₈	#11758189	semidirect

Maximal subgroups

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

subgroup	index
#21454932	2
C₄×#14519799	2
#21444723	2
#21444717	2
#21444705	2
C₂×#16839515	2
#21444207	2
C₆₁×#1450	2
#17898	61