Subgroups — #29596608

Order 15616.

490 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	C₂×C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839588	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16839564	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839564	C₂	1	1	yes	maximal, characteristic
5	#21446552	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443911	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₈₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₄₄	—	2	1	no	—
10	C₄×D₄₈₈	—	2	2	no	—
11	C₄×D₄₈₈	—	2	2	no	—
12	C₂×#14519711	C₂²	1	1	yes	characteristic
13	C₂×#14519711	C₂²	1	1	yes	characteristic
14	C₂×#14519711	C₂²	1	1	yes	characteristic
15	C₂×#14519699	—	2	1	no	—
16	#16839660	—	2	1	no	—
17	#16839588	—	2	2	no	—
18	#16839564	—	2	2	no	—
19	#16839564	C₄	1	4	yes	complemented
20	#16839525	—	2	1	no	—
21	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×D₂₄₄	—	2	1	no	—
25	C₂²×C₄×D₆₁	—	4	1	no	—
26	C₂×D₄₈₈	—	2	1	no	—
27	C₂×D₄₈₈	—	2	1	no	—
28	C₂×D₄₈₈	—	2	1	no	—
29	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
30	C₂×D₄₈₈	D₄	1	2	yes	—
31	C₂×D₄₈₈	Q₈	1	2	yes	—
32	C₄×D₂₄₄	—	2	4	no	—
33	C₄×D₂₄₄	—	4	2	no	—
34	C₂×C₄⋊Dic₆₁	—	2	1	no	—
35	C₂×C₄⋊Dic₆₁	—	2	1	no	—
36	C₂×C₄⋊Dic₆₁	—	2	1	no	—
37	C₂×#11758196	—	4	1	no	—
38	C₂×#11758198	—	4	1	no	—
39	#14519711	—	2	1	no	—
40	#14519711	—	2	1	no	—
41	#14519711	—	2	1	no	—
42	#14519711	—	2	1	no	—
43	#14519711	C₂×C₄	1	2	yes	complemented
44	#14519711	D₄	1	2	yes	—
45	#14519699	—	2	4	no	—
46	#14519684	—	4	1	no	—
47	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₁	—	2	1	no	—
51	C₂×C₁₆×C₆₁	—	2	1	no	—
52	C₂×C₁₆×C₆₁	—	2	1	no	—
53	C₂×C₁₆×C₆₁	—	2	1	no	—
54	C₂×C₁₆×C₆₁	—	2	1	no	—
55	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
56	C₄×C₈×C₆₁	—	2	1	no	—
57	C₄×C₈×C₆₁	—	2	1	no	—
58	C₂³×D₆₁	—	4	1	no	—
59	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	—	4	1	no	—
63	C₂²×Dic₆₁	—	4	1	no	—
64	C₂²×Dic₆₁	—	4	1	no	—
65	C₂×D₂₄₄	—	2	2	no	—
66	C₂×D₂₄₄	—	2	2	no	—
67	C₂×D₂₄₄	—	2	2	no	—
68	C₂×D₂₄₄	—	4	1	no	—
69	C₂×D₂₄₄	—	4	1	no	—
70	C₂×D₂₄₄	—	4	1	no	—
71	C₂×C₄×D₆₁	—	4	4	no	—
72	C₂×C₄×D₆₁	—	8	2	no	—
73	C₂×C₄×D₆₁	—	8	2	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
83	C₄²×C₆₁	—	2	1	no	—
84	C₄²×C₆₁	—	2	1	no	—
85	#11758198	—	4	4	no	—
86	C₄⋊Dic₆₁	—	2	2	no	—
87	C₄⋊Dic₆₁	—	2	2	no	—
88	C₄⋊Dic₆₁	—	4	1	no	—
89	C₄⋊Dic₆₁	—	4	1	no	—
90	C₄⋊Dic₆₁	—	4	1	no	—
91	C₄⋊Dic₆₁	—	4	1	no	—
92	#11758196	—	8	2	no	—
93	D₄₈₈	—	2	2	no	—
94	D₄₈₈	—	2	2	no	—
95	D₄₈₈	—	2	2	no	—
96	D₄₈₈	C₄⋊C₄	1	4	yes	complemented
97	C₁₆×C₆₁	—	2	2	no	—
98	C₁₆×C₆₁	—	4	1	no	—
99	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₆₁	—	4	2	no	—
101	C₂²×D₆₁	—	4	2	no	—
102	C₂²×D₆₁	—	4	2	no	—
103	C₂²×D₆₁	—	8	1	no	—
104	C₂²×D₆₁	—	8	1	no	—
105	C₂²×D₆₁	—	8	1	no	—
106	C₂²×D₆₁	—	8	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	2	1	no	—
111	C₂×C₄×C₆₁	—	2	1	no	—
112	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
113	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
117	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
119	C₂×Dic₆₁	—	4	2	no	—
120	C₂×Dic₆₁	—	4	2	no	—
121	C₂×Dic₆₁	—	8	1	no	—
122	C₂×Dic₆₁	—	8	1	no	—
123	C₂×Dic₆₁	—	8	1	no	—
124	C₂×Dic₆₁	—	8	1	no	—
125	C₂×Dic₆₁	—	8	1	no	—
126	C₂×Dic₆₁	—	8	1	no	—
127	C₂×Dic₆₁	—	8	1	no	—
128	D₂₄₄	—	2	4	no	—
129	D₂₄₄	—	4	2	no	—
130	D₂₄₄	—	4	2	no	—
131	D₂₄₄	—	4	2	no	—
132	C₄×D₆₁	—	8	8	no	—
133	C₈×C₆₁	—	4	1	no	—
134	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
136	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₆₁	#123	1	1	yes	characteristic
138	#12935	—	61	1	no	maximal, Sylow
139	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
140	C₂²×C₆₁	#447	1	1	yes	characteristic
141	C₂²×C₆₁	#474	1	1	yes	characteristic
142	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
143	C₂²×C₆₁	#508	1	1	yes	characteristic
144	C₂²×C₆₁	#509	1	1	yes	characteristic
145	C₂²×C₆₁	#510	1	1	yes	characteristic
146	C₂×D₆₁	—	4	4	no	—
147	C₂×D₆₁	—	8	2	no	—
148	C₂×D₆₁	—	8	2	no	—
149	C₂×D₆₁	—	8	2	no	—
150	C₂×D₆₁	—	8	2	no	—
151	C₂×D₆₁	—	8	2	no	—
152	C₂×D₆₁	—	8	2	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	4	1	no	—
155	C₄×C₆₁	#385	1	1	yes	characteristic
156	C₄×C₆₁	#440	1	1	yes	characteristic
157	C₄×C₆₁	#452	1	1	yes	characteristic
158	C₄×C₆₁	#478	1	1	yes	characteristic
159	Dic₆₁	—	8	2	no	—
160	Dic₆₁	—	8	2	no	—
161	Dic₆₁	—	16	1	no	—
162	C₂×C₄×D₈	—	61	1	no	—
163	C₂×#366	—	61	1	no	—
164	C₂×#357	—	61	1	no	—
165	C₂×#357	—	61	1	no	—
166	#1942	—	61	1	no	—
167	#1380	—	61	1	no	—
168	#1374	D₆₁	1	1	yes	characteristic, complemented
169	C₂×C₆₁	#1862	1	1	yes	characteristic
170	C₂×C₆₁	C₄×D₁₆	1	1	yes	characteristic
171	C₂×C₆₁	#2175	1	1	yes	characteristic
172	C₂×C₆₁	#2184	1	1	yes	characteristic
173	C₂×C₆₁	#2189	1	1	yes	characteristic
174	C₂×C₆₁	#2224	1	1	yes	characteristic
175	C₂×C₆₁	#2228	1	1	yes	characteristic
176	D₆₁	—	8	4	no	—
177	C₂²×D₈	—	61	1	no	—
178	C₂×C₄×D₄	—	122	1	no	—
179	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
180	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
181	C₄×D₈	—	122	2	no	—
182	C₄×D₈	—	122	2	no	—
183	C₂×#107	—	61	1	no	—
184	C₂×#107	—	61	1	no	—
185	C₂×#107	—	61	1	no	—
186	C₂×#102	—	122	1	no	—
187	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
188	#389	—	122	1	no	—
189	#366	—	122	2	no	—
190	#357	—	61	4	no	—
191	#357	—	122	2	no	—
192	#340	—	122	1	no	—
193	C₆₁	#12935	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	—	122	1	no	—
195	C₂³×C₄	—	244	1	no	—
196	C₂×D₈	—	61	2	no	—
197	C₂×D₈	—	61	2	no	—
198	C₂×D₈	—	61	2	no	—
199	C₂×D₈	—	122	1	no	—
200	C₂×D₈	—	122	1	no	—
201	C₂×D₈	—	122	1	no	—
202	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
204	C₄×D₄	—	122	4	no	—
205	C₄×D₄	—	244	2	no	—
206	C₂×C₄⋊C₄	—	122	1	no	—
207	C₂×C₄⋊C₄	—	122	1	no	—
208	C₂×C₄⋊C₄	—	122	1	no	—
209	C₂×C₄⋊C₄	—	244	1	no	—
210	C₂×C₂²⋊C₄	—	244	1	no	—
211	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	C₄×D₆₁	1	2	yes	complemented
218	#107	—	61	2	no	—
219	#107	—	61	2	no	—
220	#107	—	122	1	no	—
221	#107	—	122	1	no	—
222	#107	—	122	1	no	—
223	#107	—	122	1	no	—
224	#102	—	122	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	244	1	no	—
228	C₂⁴	—	244	1	no	—
229	C₂×D₄	—	122	2	no	—
230	C₂×D₄	—	122	2	no	—
231	C₂×D₄	—	122	2	no	—
232	C₂×D₄	—	244	1	no	—
233	C₂×D₄	—	244	1	no	—
234	C₂×D₄	—	244	1	no	—
235	C₂²×C₄	—	488	2	no	—
236	C₂²×C₄	—	488	2	no	—
237	C₂²×C₄	—	244	4	no	—
238	C₂²×C₄	—	244	1	no	—
239	C₂²×C₄	—	244	1	no	—
240	C₂²×C₄	—	244	1	no	—
241	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
242	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
244	D₈	—	61	4	no	—
245	D₈	—	122	2	no	—
246	D₈	—	122	2	no	—
247	D₈	—	122	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
252	C₂×C₈	#11758222	1	1	yes	characteristic
253	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
254	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
255	C₂×C₈	#11758224	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
257	C₄⋊C₄	—	122	2	no	—
258	C₄⋊C₄	—	122	2	no	—
259	C₄⋊C₄	—	488	2	no	—
260	C₄⋊C₄	—	244	1	no	—
261	C₄⋊C₄	—	244	1	no	—
262	C₄⋊C₄	—	244	1	no	—
263	C₄⋊C₄	—	244	1	no	—
264	C₂²⋊C₄	—	244	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	4	1	no	—
269	C₂³	—	488	1	no	—
270	C₂³	—	488	1	no	—
271	C₂³	—	488	1	no	—
272	C₂³	—	488	1	no	—
273	C₂³	—	244	2	no	—
274	C₂³	—	244	2	no	—
275	C₂³	—	244	2	no	—
276	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
277	D₄	—	122	4	no	—
278	D₄	—	244	2	no	—
279	D₄	—	244	2	no	—
280	D₄	—	244	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	488	8	no	—
287	C₂×C₄	—	488	1	no	—
288	C₂×C₄	—	488	1	no	—
289	C₂×C₄	—	488	1	no	—
290	C₂×C₄	—	488	1	no	—
291	C₂×C₄	—	488	1	no	—
292	C₂×C₄	—	488	1	no	—
293	C₂×C₄	—	488	1	no	—
294	C₂×C₄	—	244	2	no	—
295	C₂×C₄	—	244	2	no	—
296	C₂×C₄	D₉₇₆	1	2	yes	—
297	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
298	C₂×C₄	#14519747	1	1	yes	characteristic
299	C₂×C₄	#14519757	1	1	yes	characteristic
300	C₂×C₄	#14519799	1	1	yes	characteristic
301	C₂×C₄	#14519805	1	1	yes	characteristic
302	C₂×C₄	#14519810	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
305	C₈	#14519737	1	1	yes	characteristic
306	C₈	#14519755	1	1	yes	characteristic
307	C₈	#14519759	1	1	yes	characteristic
308	C₂²	—	488	2	no	—
309	C₂²	—	488	2	no	—
310	C₂²	—	488	2	no	—
311	C₂²	—	488	2	no	—
312	C₂²	—	488	2	no	—
313	C₂²	—	488	2	no	—
314	C₂²	—	244	4	no	—
315	C₂²	C₄×D₄₈₈	1	1	yes	characteristic
316	C₂²	#16839898	1	1	yes	characteristic
317	C₂²	#16840043	1	1	yes	characteristic
318	C₂²	C₂×D₉₇₆	1	1	yes	characteristic
319	C₂²	#16840194	1	1	yes	characteristic
320	C₂²	#16840200	1	1	yes	characteristic
321	C₂²	#16840205	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	976	1	no	—
325	C₄	—	488	2	no	—
326	C₄	—	488	2	no	—
327	C₄	#16839636	1	1	yes	characteristic
328	C₄	#16839864	1	1	yes	characteristic
329	C₄	#16839926	1	1	yes	characteristic
330	C₄	#16840063	1	1	yes	characteristic
331	C₂	—	488	4	no	—
332	C₂	#21446069	1	1	yes	characteristic
333	C₂	C₄×D₉₇₆	1	1	yes	characteristic
334	C₂	#21447685	1	1	yes	characteristic
335	C₂	#21447733	1	1	yes	characteristic
336	C₂	#21447761	1	1	yes	characteristic
337	C₂	#21447957	1	1	yes	characteristic
338	C₂	#21447977	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1374	C₂
socle	C₂³×C₆₁	C₂×D₈
prime radical	C₂²×D₄₈₈	C₂²
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1374	D₆₁
hypocenter	C₆₁	#12935
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#16839588	C₂	semidirect
5. #21446552	C₂	semidirect
6. #21443911	C₂	semidirect
7. C₆₁×#1374	C₂	semidirect
19. #16839564	C₄	semidirect
43. #14519711	C₂×C₄	semidirect
55. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
96. D₄₈₈	C₄⋊C₄	semidirect
168. #1374	D₆₁	semidirect
193. C₆₁	#12935	semidirect
217. C₂×C₁₆	C₄×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄₈₈	2
C₂×#16839588	2
C₂×#16839564	2
#21446552	2
#21443911	2
C₆₁×#1374	2
#12935	61