Subgroups — #29833576

Order 15616.

516 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454796	C₂	1	1	yes	maximal, characteristic
2	#21447728	C₂	1	1	yes	maximal, characteristic, complemented
3	#21447652	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447652	C₂	1	2	yes	maximal, complemented
5	#21447643	C₂	1	2	yes	maximal, complemented
6	#21447638	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447625	C₂	1	2	yes	maximal, complemented
8	#21447617	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447414	C₂	1	2	yes	maximal, complemented
10	C₄×#14519716	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840299	C₄	1	1	yes	characteristic
13	C₂×#14519716	C₂²	1	1	yes	characteristic
14	C₂×#14519716	C₂²	1	1	yes	characteristic
15	C₂×#14519716	C₂²	1	2	yes	—
16	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
17	#16839893	C₄	1	1	yes	characteristic
18	#16839839	C₄	1	1	yes	characteristic
19	#16839832	C₄	1	2	yes	—
20	C₈×D₂₄₄	C₂²	1	1	yes	characteristic
21	C₈×D₂₄₄	C₂²	1	2	yes	—
22	#16839822	C₄	1	2	yes	—
23	#16839818	C₄	1	1	yes	characteristic
24	#16839716	C₂²	1	2	yes	—
25	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
26	C₄⋊#11758185	C₂²	1	1	yes	characteristic
27	C₄⋊#11758185	C₂²	1	2	yes	—
28	#16839541	C₂²	1	2	yes	—
29	#16839541	C₂²	1	2	yes	—
30	#16839541	C₂²	1	2	yes	—
31	#16839538	C₂²	1	2	yes	—
32	#16839537	C₂²	1	1	yes	characteristic
33	#16839537	C₂²	1	1	yes	characteristic
34	C₄×#11758185	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₆₁	C₂²	1	2	yes	—
36	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
41	C₂×#11758189	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₆₁	C₂³	1	2	yes	—
45	#14519772	C₈	1	2	yes	—
46	#14519768	C₈	1	1	yes	characteristic
47	#14519766	C₈	1	1	yes	characteristic
48	#14519762	C₈	1	2	yes	—
49	#14519753	C₈	1	1	yes	characteristic
50	#14519752	C₈	1	1	yes	characteristic
51	C₄×D₂₄₄	C₂×C₄	1	2	yes	—
52	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
53	#14519734	C₂×C₄	1	2	yes	—
54	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
55	#14519716	—	2	2	no	—
56	#14519716	—	2	4	no	—
57	#14519716	—	2	2	no	—
58	C₂×#11758185	C₂³	1	2	yes	—
59	C₂×#11758185	C₂³	1	1	yes	characteristic
60	C₂×#11758185	C₂³	1	1	yes	characteristic
61	C₁₆×D₆₁	—	2	4	no	—
62	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#92418	C₂×C₄	1	2	yes	—
64	#14519689	C₂×C₄	1	2	yes	—
65	#14519689	C₂×C₄	1	2	yes	—
66	#14519689	C₂×C₄	1	2	yes	—
67	#14519688	C₂×C₄	1	2	yes	—
68	#14519687	C₂×C₄	1	2	yes	—
69	C₄×#92418	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
74	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₆₁	C₂³	1	2	yes	—
80	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
81	C₂×D₂₄₄	C₂×C₈	1	1	yes	characteristic
82	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
91	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
92	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
94	#11758198	C₂×C₈	1	2	yes	—
95	#11758198	C₂×C₈	1	2	yes	—
96	#11758198	C₂×C₈	1	2	yes	—
97	#11758198	C₂×C₈	1	2	yes	—
98	#11758198	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
100	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
101	#11758196	C₂×C₈	1	2	yes	—
102	#11758196	C₂×C₈	1	2	yes	—
103	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
107	C₂×#92418	C₂²×C₄	1	2	yes	—
108	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
109	#11758189	—	2	4	no	—
110	C₈×D₆₁	—	2	2	no	—
111	C₈×D₆₁	—	2	1	no	—
112	C₈×D₆₁	—	2	1	no	—
113	C₈×D₆₁	—	2	2	no	—
114	C₈×D₆₁	—	2	2	no	—
115	C₁₆×C₆₁	—	2	2	no	—
116	C₁₆×C₆₁	—	2	1	no	—
117	C₁₆×C₆₁	#41	1	2	yes	—
118	#11758185	—	2	2	no	—
119	#11758185	—	2	1	no	—
120	#11758185	#41	1	2	yes	—
121	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
122	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
133	D₂₄₄	—	2	2	no	—
134	D₂₄₄	—	4	2	no	—
135	C₄×D₆₁	—	2	2	no	—
136	C₄×D₆₁	—	2	1	no	—
137	C₄×D₆₁	—	2	1	no	—
138	C₄×D₆₁	—	2	2	no	—
139	C₄×D₆₁	—	2	2	no	—
140	C₄×D₆₁	—	4	1	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	2	2	no	—
143	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
144	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
145	#92418	—	2	2	no	—
146	#92418	—	2	1	no	—
147	#92418	#131	1	2	yes	—
148	#33483	—	61	1	no	maximal, Sylow
149	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₆₁	—	2	1	no	—
151	C₂×D₆₁	—	2	2	no	—
152	C₂×D₆₁	—	2	1	no	—
153	C₂×D₆₁	—	2	2	no	—
154	C₂×D₆₁	—	2	1	no	—
155	C₂×D₆₁	—	2	2	no	—
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	2	2	no	—
159	C₄×C₆₁	#504	1	2	yes	—
160	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
161	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
162	Dic₆₁	—	2	1	no	—
163	Dic₆₁	—	2	1	no	—
164	Dic₆₁	—	2	1	no	—
165	Dic₆₁	—	2	2	no	—
166	#2966	—	61	1	no	—
167	#2183	—	61	1	no	—
168	Q₈×C₁₆	D₆₁	1	1	yes	characteristic, complemented
169	#2169	—	61	1	no	—
170	#2169	—	61	2	no	—
171	#2168	—	61	2	no	—
172	D₄×C₁₆	—	61	1	no	—
173	#2164	—	61	2	no	—
174	#2162	—	61	1	no	—
175	#2108	—	61	2	no	—
176	C₄×M₃₂	—	61	1	no	—
177	C₂×C₆₁	C₈×#41	1	1	yes	characteristic
178	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
179	C₂×C₆₁	#3407	1	1	yes	characteristic
180	D₆₁	—	4	1	no	—
181	D₆₁	—	4	2	no	—
182	#529	—	61	1	no	—
183	C₂×M₃₂	—	61	1	no	—
184	C₂×M₃₂	—	61	1	no	—
185	C₂×M₃₂	—	61	2	no	—
186	C₂²×C₁₆	—	61	2	no	—
187	#446	—	61	1	no	—
188	C₈×Q₈	C₂×D₆₁	1	1	yes	characteristic
189	#436	—	61	1	no	—
190	#435	—	61	2	no	—
191	C₈×D₄	—	61	1	no	—
192	C₈×D₄	—	61	2	no	—
193	#433	—	61	2	no	—
194	#432	—	61	1	no	—
195	#405	—	61	2	no	—
196	C₂×C₄×C₈	—	61	1	no	—
197	C₄⋊C₁₆	—	61	1	no	—
198	C₄⋊C₁₆	—	61	2	no	—
199	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
201	C₂²⋊C₁₆	—	61	2	no	—
202	C₂²⋊C₁₆	—	61	2	no	—
203	C₂²⋊C₁₆	—	61	2	no	—
204	#346	—	61	1	no	—
205	#346	—	61	1	no	—
206	#346	—	61	2	no	—
207	C₄×C₁₆	—	61	2	no	—
208	C₄×C₁₆	—	61	1	no	—
209	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
210	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
211	C₆₁	#33483	1	1	yes	characteristic, complemented, Sylow
212	C₂×M₁₆	—	61	2	no	—
213	C₂²×C₈	—	61	1	no	—
214	C₂²×C₈	—	61	1	no	—
215	C₂²×C₈	—	61	2	no	—
216	#126	—	61	2	no	—
217	#125	—	61	1	no	—
218	#124	—	61	1	no	—
219	#123	—	61	2	no	—
220	C₂²⋊Q₈	—	61	1	no	—
221	#121	—	61	1	no	—
222	C₄×Q₈	C₄×D₆₁	1	1	yes	characteristic
223	C₄×D₄	—	61	2	no	—
224	C₄×D₄	—	61	1	no	—
225	#117	—	61	2	no	—
226	C₂×C₄²	—	61	1	no	—
227	M₃₂	—	122	2	no	—
228	M₃₂	—	122	2	no	—
229	M₃₂	—	122	4	no	—
230	C₂×C₁₆	—	61	1	no	—
231	C₂×C₁₆	—	61	2	no	—
232	C₂×C₁₆	—	61	1	no	—
233	C₂×C₁₆	—	122	4	no	—
234	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
235	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
237	C₄⋊C₈	—	61	1	no	—
238	C₄⋊C₈	—	61	2	no	—
239	C₄⋊C₈	C₄×D₆₁	1	1	yes	characteristic
240	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
241	C₂²⋊C₈	—	61	2	no	—
242	C₂²⋊C₈	—	61	2	no	—
243	C₂²⋊C₈	—	61	2	no	—
244	#97	—	61	2	no	—
245	#97	—	61	2	no	—
246	C₄×C₈	—	61	1	no	—
247	C₄×C₈	—	61	1	no	—
248	C₄×C₈	—	61	1	no	—
249	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₆₁	1	2	yes	—
251	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
252	C₂×D₄	—	61	1	no	—
253	C₂×D₄	—	61	2	no	—
254	C₂²×C₄	—	61	1	no	—
255	C₂²×C₄	—	61	2	no	—
256	C₂²×C₄	—	61	1	no	—
257	C₂²×C₄	—	61	1	no	—
258	M₁₆	—	122	4	no	—
259	C₂×C₈	—	61	1	no	—
260	C₂×C₈	—	61	1	no	—
261	C₂×C₈	—	61	2	no	—
262	C₂×C₈	—	122	2	no	—
263	C₂×C₈	—	122	2	no	—
264	C₂×C₈	—	122	1	no	—
265	C₂×C₈	—	122	1	no	—
266	C₂×C₈	—	122	2	no	—
267	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
269	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
270	C₄⋊C₄	—	61	2	no	—
271	C₄⋊C₄	—	61	2	no	—
272	C₄⋊C₄	—	61	1	no	—
273	C₄⋊C₄	—	61	2	no	—
274	C₄⋊C₄	C₈×D₆₁	1	1	yes	characteristic
275	C₄⋊C₄	C₈×D₆₁	1	2	yes	—
276	C₂²⋊C₄	—	61	2	no	—
277	C₂²⋊C₄	—	61	2	no	—
278	C₂²⋊C₄	—	61	2	no	—
279	C₂²⋊C₄	—	61	2	no	—
280	C₂²⋊C₄	—	61	2	no	—
281	C₄²	—	61	1	no	—
282	C₄²	—	61	1	no	—
283	C₄²	—	61	1	no	—
284	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₆₁	1	2	yes	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	61	2	no	—
289	C₁₆	—	122	2	no	—
290	C₁₆	—	122	1	no	—
291	C₁₆	#11758226	1	2	yes	—
292	C₂³	—	61	2	no	—
293	C₂³	—	61	1	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	122	2	no	—
296	D₄	—	244	2	no	—
297	C₂×C₄	—	61	1	no	—
298	C₂×C₄	—	61	1	no	—
299	C₂×C₄	—	61	1	no	—
300	C₂×C₄	—	61	2	no	—
301	C₂×C₄	—	122	2	no	—
302	C₂×C₄	—	122	1	no	—
303	C₂×C₄	—	122	1	no	—
304	C₂×C₄	—	122	2	no	—
305	C₂×C₄	—	122	2	no	—
306	C₂×C₄	—	244	1	no	—
307	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
309	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	61	2	no	—
315	C₈	—	122	1	no	—
316	C₈	—	122	2	no	—
317	C₈	#41×D₆₁	1	1	yes	characteristic
318	C₈	C₂×#11758226	1	1	yes	characteristic
319	C₂²	—	122	1	no	—
320	C₂²	—	122	2	no	—
321	C₂²	—	122	1	no	—
322	C₂²	—	122	2	no	—
323	C₂²	—	122	1	no	—
324	C₂²	—	122	2	no	—
325	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	122	1	no	—
330	C₄	—	122	2	no	—
331	C₄	—	122	1	no	—
332	C₄	—	122	1	no	—
333	C₄	#16840178	1	2	yes	—
334	C₄	C₄×#11758226	1	1	yes	characteristic
335	C₄	#131×D₆₁	1	1	yes	characteristic
336	C₂	—	244	1	no	—
337	C₂	—	244	2	no	—
338	C₂	C₈×#11758226	1	1	yes	characteristic
339	C₂	C₂×#16840178	1	1	yes	characteristic
340	C₂	#21458920	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	Q₈×C₁₆×C₆₁	C₂
prime radical	#14519752	C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	Q₈×C₁₆	D₆₁
hypocenter	C₆₁	#33483
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21447728	C₂	semidirect
3. #21447652	C₂	semidirect
4. #21447652	C₂	semidirect
5. #21447643	C₂	semidirect
6. #21447638	C₂	semidirect
7. #21447625	C₂	semidirect
8. #21447617	C₂	semidirect
9. #21447414	C₂	semidirect
10. C₄×#14519716	C₂	semidirect
11. Q₈×C₁₆×C₆₁	C₂	semidirect
168. Q₈×C₁₆	D₆₁	semidirect
211. C₆₁	#33483	semidirect

Maximal subgroups

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

subgroup	index
#21454796	2
#21447728	2
#21447652	2
#21447643	2
#21447638	2
#21447625	2
#21447617	2
#21447414	2
C₄×#14519716	2
Q₈×C₁₆×C₆₁	2
#33483	61