Subgroups — #29651445

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454834	C₂	1	2	yes	maximal, complemented
2	C₄×#14519799	C₂	1	1	yes	maximal, characteristic
3	#21444717	C₂	1	2	yes	maximal, complemented
4	#21444657	C₂	1	4	yes	maximal, complemented
5	C₈×D₄₈₈	C₂	1	2	yes	maximal, complemented
6	#21444582	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444254	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₆₁	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	#16839831	—	2	4	no	—
19	C₈×D₂₄₄	C₂²	1	4	yes	—
20	#16839819	C₂²	1	2	yes	—
21	#16839817	—	2	2	no	—
22	#16839800	C₄	1	1	yes	characteristic
23	#16839759	C₄	1	2	yes	—
24	#97×D₆₁	—	2	2	no	—
25	#16839515	C₂²	1	2	yes	—
26	#16839513	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	C₄×M₁₆×C₆₁	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×C₈×D₆₁	—	2	4	no	—
40	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
41	C₄×Dic₁₂₂	C₂³	1	2	yes	—
42	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
43	C₄×D₂₄₄	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₆₁	—	2	2	no	—
45	C₄×C₂²⋊D₆₁	—	2	2	no	—
46	#14519736	C₂×C₄	1	2	yes	—
47	#14519734	—	2	2	no	—
48	C₄²×D₆₁	—	2	2	no	—
49	#14519711	C₂×C₄	1	1	yes	characteristic
50	#14519711	C₂×C₄	1	1	yes	characteristic
51	#14519709	C₂×C₄	1	2	yes	—
52	C₄⋊#92418	C₂³	1	4	yes	—
53	#14519706	—	2	4	no	—
54	#14519702	C₂×C₄	1	2	yes	—
55	#14519702	C₂×C₄	1	2	yes	—
56	#14519699	C₂×C₄	1	2	yes	—
57	#14519699	C₂×C₄	1	2	yes	—
58	C₂²⋊#92418	—	2	2	no	—
59	#14519689	—	2	4	no	—
60	#14519688	—	2	2	no	—
61	#14519687	—	2	4	no	—
62	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
63	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₆₁	—	2	1	no	—
67	C₂²⋊C₈×C₆₁	—	2	1	no	—
68	#97×C₆₁	D₄	1	1	yes	characteristic
69	#97×C₆₁	D₄	1	1	yes	characteristic
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	—	4	2	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	M₁₆×C₆₁	—	2	4	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	4	1	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	2	yes	—
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₁	#41	1	1	yes	characteristic
100	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₆₁	—	2	2	no	—
102	#11758198	—	2	2	no	—
103	#11758198	—	2	2	no	—
104	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
105	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
106	#11758196	—	2	2	no	—
107	#11758196	—	2	2	no	—
108	C₄×Dic₆₁	—	2	2	no	—
109	C₄×Dic₆₁	—	2	2	no	—
110	C₄×Dic₆₁	—	2	2	no	—
111	C₂×#92418	—	2	4	no	—
112	Dic₂₄₄	—	2	2	no	—
113	D₄₈₈	M₁₆	1	4	yes	complemented
114	#11758190	—	2	4	no	—
115	C₈×D₆₁	—	4	8	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	—	2	2	no	—
121	C₂×C₄×C₆₁	—	4	1	no	—
122	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₆₁	—	4	4	no	—
129	C₂×Dic₆₁	—	2	2	no	—
130	C₂×Dic₆₁	—	2	2	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	D₂₄₄	—	2	2	no	—
133	D₂₄₄	C₂×M₁₆	1	4	yes	—
134	C₄×D₆₁	—	4	4	no	—
135	C₄×D₆₁	—	4	4	no	—
136	C₄×D₆₁	—	8	2	no	—
137	Dic₁₂₂	—	2	2	no	—
138	Dic₁₂₂	—	2	2	no	—
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	2	2	no	—
143	C₈×C₆₁	—	4	2	no	—
144	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
145	C₈×C₆₁	#135	1	4	yes	complemented
146	#92418	—	4	4	no	—
147	#18035	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×D₆₁	—	2	4	no	—
153	C₂×D₆₁	—	4	2	no	—
154	C₂×D₆₁	—	4	2	no	—
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	2	1	no	—
159	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
160	C₄×C₆₁	#568	1	1	yes	characteristic
161	C₄×C₆₁	C₂×#135	1	2	yes	—
162	Dic₆₁	—	4	2	no	—
163	Dic₆₁	—	4	2	no	—
164	Dic₆₁	—	4	2	no	—
165	#2970	—	61	2	no	—
166	C₄×#135	—	61	1	no	—
167	#1591	—	61	2	no	—
168	#1582	—	61	4	no	—
169	C₈×D₈	—	61	2	no	—
170	#1568	—	61	1	no	—
171	#1488	—	61	2	no	—
172	C₈×M₁₆	D₆₁	1	1	yes	characteristic, complemented
173	C₂×C₆₁	—	4	1	no	—
174	C₂×C₆₁	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₆₁	C₄×#135	1	1	yes	characteristic
176	C₂×C₆₁	#2957	1	1	yes	characteristic
177	D₆₁	—	4	4	no	—
178	C₂×#135	—	61	1	no	—
179	C₄×#41	—	61	2	no	—
180	#446	—	61	4	no	—
181	C₄×Q₁₆	—	61	1	no	—
182	C₄×Q₁₆	—	61	1	no	—
183	C₄×SD₁₆	—	61	2	no	—
184	C₄×SD₁₆	—	61	2	no	—
185	C₄×D₈	—	61	1	no	—
186	C₄×D₈	—	61	1	no	—
187	#435	—	122	4	no	—
188	C₈×D₄	—	61	4	no	—
189	#432	—	61	2	no	—
190	#432	—	122	2	no	—
191	#431	C₂×D₆₁	1	1	yes	characteristic
192	#427	—	61	1	no	—
193	#416	—	61	2	no	—
194	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
195	C₂×#97	—	122	2	no	—
196	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
197	#335	—	61	2	no	—
198	#334	—	61	2	no	—
199	Q₈⋊C₈	—	61	4	no	—
200	D₄⋊C₈	—	61	4	no	—
201	#322	C₂×D₆₁	1	2	yes	—
202	C₈²	C₂×D₆₁	1	2	yes	—
203	C₆₁	#18035	1	1	yes	characteristic, complemented, Sylow
204	C₂×#41	—	61	2	no	—
205	#135	—	122	4	no	—
206	C₂×Q₁₆	—	61	1	no	—
207	C₂×SD₁₆	—	61	2	no	—
208	C₂×D₈	—	61	1	no	—
209	C₂×M₁₆	D₂₄₄	1	2	yes	—
210	C₂²×C₈	—	122	4	no	—
211	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
213	C₄×Q₈	—	61	2	no	—
214	C₄×Q₈	—	61	2	no	—
215	C₄×D₄	—	61	2	no	—
216	C₄×D₄	—	61	2	no	—
217	C₄×D₄	—	122	2	no	—
218	C₄×D₄	—	122	2	no	—
219	#117	—	61	2	no	—
220	#117	—	122	2	no	—
221	C₂×C₄²	—	122	2	no	—
222	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
223	#107	—	61	1	no	—
224	#107	—	61	1	no	—
225	#106	—	61	2	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	61	4	no	—
228	C₄⋊C₈	—	122	4	no	—
229	#103	—	61	2	no	—
230	#103	—	61	2	no	—
231	#102	—	61	2	no	—
232	#102	—	61	2	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	122	2	no	—
235	C₂²⋊C₈	—	122	4	no	—
236	#97	—	122	4	no	—
237	#97	—	122	2	no	—
238	#97	D₂₄₄	1	1	yes	characteristic
239	#97	D₂₄₄	1	1	yes	characteristic
240	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
241	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₆₁	1	2	yes	—
243	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₆₁	1	2	yes	—
246	#41	—	122	4	no	—
247	#41	—	244	2	no	—
248	C₂×Q₈	—	61	2	no	—
249	C₂×D₄	—	61	2	no	—
250	C₂×D₄	—	122	2	no	—
251	C₂²×C₄	—	122	2	no	—
252	C₂²×C₄	—	122	2	no	—
253	C₂²×C₄	—	122	2	no	—
254	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
256	C₂²×C₄	#11758222	1	1	yes	characteristic
257	Q₁₆	—	122	2	no	—
258	SD₁₆	—	122	4	no	—
259	D₈	—	61	4	no	—
260	M₁₆	—	2	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	122	4	no	—
265	C₂×C₈	—	244	8	no	—
266	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
271	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
272	C₄⋊C₄	—	61	2	no	—
273	C₄⋊C₄	—	61	2	no	—
274	C₄⋊C₄	—	122	2	no	—
275	C₄⋊C₄	—	122	2	no	—
276	C₂²⋊C₄	—	122	2	no	—
277	C₂²⋊C₄	—	122	2	no	—
278	C₂²⋊C₄	—	122	2	no	—
279	C₄²	—	122	2	no	—
280	C₄²	—	122	2	no	—
281	C₄²	—	122	2	no	—
282	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
283	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
284	C₄²	#11758222	1	1	yes	characteristic
285	C₄²	C₂³×D₆₁	1	1	yes	characteristic
286	C₂³	—	122	2	no	—
287	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
288	Q₈	—	122	2	no	—
289	Q₈	—	122	2	no	—
290	D₄	—	61	4	no	—
291	D₄	—	122	2	no	—
292	D₄	—	244	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	122	2	no	—
298	C₂×C₄	—	122	2	no	—
299	C₂×C₄	—	122	2	no	—
300	C₂×C₄	—	488	2	no	—
301	C₂×C₄	—	244	4	no	—
302	C₂×C₄	—	244	4	no	—
303	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
304	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
305	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
308	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	244	4	no	—
315	C₈	M₁₆×D₆₁	1	2	yes	—
316	C₈	#14519799	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	122	4	no	—
321	C₂²	—	244	2	no	—
322	C₂²	—	244	2	no	—
323	C₂²	C₂×C₄×D₂₄₄	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	244	2	no	—
329	C₄	—	244	2	no	—
330	C₄	—	244	2	no	—
331	C₄	C₂×M₁₆×D₆₁	1	1	yes	characteristic
332	C₄	#16840528	1	1	yes	characteristic
333	C₄	C₂×#14519799	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	244	4	no	—
336	C₂	M₁₆×D₂₄₄	1	1	yes	characteristic
337	C₂	C₄×#14519799	1	1	yes	characteristic
338	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	C₈×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#14519799	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₈×M₁₆	D₆₁
hypocenter	C₆₁	#18035
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #21454834	C₂	semidirect
3. #21444717	C₂	semidirect
4. #21444657	C₂	semidirect
5. C₈×D₄₈₈	C₂	semidirect
6. #21444582	C₂	semidirect
7. #21444254	C₂	semidirect
8. C₈×M₁₆×C₆₁	C₂	semidirect
113. D₄₈₈	M₁₆	semidirect
145. C₈×C₆₁	#135	semidirect
172. C₈×M₁₆	D₆₁	semidirect
203. C₆₁	#18035	semidirect
316. C₈	#14519799	semidirect

Maximal subgroups

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

subgroup	index
#21454834	2
C₄×#14519799	2
#21444717	2
#21444657	2
C₈×D₄₈₈	2
#21444582	2
#21444254	2
C₈×M₁₆×C₆₁	2
#18035	61