Subgroups — #29835611

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456461	C₂	1	2	yes	maximal, complemented
2	#21448080	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448058	C₂	1	2	yes	maximal, complemented
4	#21447933	C₂	1	2	yes	maximal, complemented
5	#21447911	C₂	1	1	yes	maximal, characteristic, complemented
6	#21447906	C₂	1	1	yes	maximal, characteristic
7	#21447682	C₂	1	2	yes	maximal, complemented
8	#21447677	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447665	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447414	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840386	—	2	2	no	—
13	C₂×Dic₄₈₈	C₂²	1	1	yes	characteristic
14	C₂×#14519722	C₂²	1	2	yes	—
15	C₂×D₉₇₆	C₂²	1	1	yes	characteristic
16	C₂×#14519716	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
18	#16840146	C₂²	1	2	yes	—
19	#16840124	—	2	2	no	—
20	#16840009	—	2	2	no	—
21	#16840004	C₂²	1	2	yes	—
22	#16839999	C₂²	1	2	yes	—
23	#16839861	—	2	2	no	—
24	#16839856	C₂²	1	2	yes	—
25	#16839844	C₂²	1	2	yes	—
26	#16839716	C₂²	1	1	yes	characteristic
27	#16839565	C₂²	1	2	yes	—
28	#16839565	C₂²	1	2	yes	—
29	#16839562	C₂²	1	2	yes	—
30	#16839562	C₂²	1	2	yes	—
31	#16839538	C₂²	1	1	yes	characteristic
32	#16839537	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₆₁×#500	C₂²	1	2	yes	—
35	C₆₁×#366	C₂²	1	2	yes	—
36	C₆₁×#366	C₂²	1	1	yes	characteristic
37	C₆₁×#366	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₄₄	C₂³	1	2	yes	—
40	C₂×#11758190	—	2	2	no	—
41	C₂×D₄₈₈	C₂³	1	2	yes	—
42	C₂×#11758189	D₄	1	1	yes	characteristic
43	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
44	#14519772	—	4	2	no	—
45	#14519766	—	4	2	no	—
46	#14519765	—	2	2	no	—
47	#14519762	—	4	2	no	—
48	#14519757	—	2	2	no	—
49	#14519753	—	4	2	no	—
50	#14519752	—	2	2	no	—
51	#14519742	—	2	2	no	—
52	#14519739	—	2	2	no	—
53	#14519734	D₄	1	1	yes	characteristic
54	Dic₄₈₈	—	2	2	no	—
55	#14519722	—	2	4	no	—
56	D₉₇₆	—	2	2	no	—
57	#14519716	—	2	2	no	—
58	C₂×#11758185	C₂³	1	1	yes	characteristic
59	C₂×#11758185	C₂³	1	1	yes	characteristic
60	C₁₆×D₆₁	—	2	2	no	—
61	#14519700	—	2	2	no	—
62	#14519700	—	2	2	no	—
63	#14519697	—	2	2	no	—
64	#14519697	—	2	2	no	—
65	#14519688	D₄	1	1	yes	characteristic
66	#14519687	D₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₁	—	2	2	no	—
69	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	#107×C₆₁	—	2	2	no	—
72	#107×C₆₁	C₂³	1	2	yes	—
73	#107×C₆₁	C₂³	1	2	yes	—
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₆₁	—	4	2	no	—
76	C₂×D₂₄₄	—	2	2	no	—
77	C₂×D₂₄₄	—	4	2	no	—
78	C₂×C₄×D₆₁	—	4	2	no	—
79	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₁₂₂	—	2	2	no	—
81	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₆₁	—	2	2	no	—
84	C₄⋊C₄×C₆₁	—	2	2	no	—
85	C₄⋊C₄×C₆₁	—	4	2	no	—
86	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
87	#11758198	—	4	2	no	—
88	#11758198	—	4	2	no	—
89	#11758198	—	4	2	no	—
90	#11758198	—	4	2	no	—
91	#11758198	—	4	2	no	—
92	#11758198	D₈	1	1	yes	characteristic
93	#11758198	D₈	1	1	yes	characteristic
94	C₄⋊Dic₆₁	—	4	2	no	—
95	#11758196	—	4	2	no	—
96	#11758196	D₈	1	1	yes	characteristic
97	#11758196	D₈	1	1	yes	characteristic
98	C₄×Dic₆₁	—	4	2	no	—
99	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×#92418	C₂×D₄	1	1	yes	characteristic
101	C₂×#92418	C₂×D₄	1	1	yes	characteristic
102	Dic₂₄₄	—	2	2	no	—
103	Dic₂₄₄	—	2	2	no	—
104	D₄₈₈	—	2	2	no	—
105	D₄₈₈	—	2	2	no	—
106	#11758190	—	4	4	no	—
107	#11758189	—	2	2	no	—
108	C₈×D₆₁	—	2	1	no	—
109	C₈×D₆₁	—	2	1	no	—
110	C₁₆×C₆₁	#41	1	2	yes	—
111	C₁₆×C₆₁	#41	1	2	yes	—
112	#11758185	—	2	1	no	—
113	#11758185	—	2	1	no	—
114	C₂²×D₆₁	—	4	2	no	—
115	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₆₁	—	16	2	no	—
117	C₂×C₄×C₆₁	—	4	2	no	—
118	C₂×C₄×C₆₁	—	4	2	no	—
119	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₆₁	—	4	2	no	—
123	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
124	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
126	D₂₄₄	—	4	2	no	—
127	D₂₄₄	—	4	2	no	—
128	D₂₄₄	—	16	2	no	—
129	C₄×D₆₁	—	2	1	no	—
130	C₄×D₆₁	—	2	1	no	—
131	C₄×D₆₁	—	16	2	no	—
132	Dic₁₂₂	—	4	2	no	—
133	Dic₁₂₂	—	4	2	no	—
134	C₈×C₆₁	#135	1	2	yes	—
135	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
136	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
137	#92418	—	2	1	no	—
138	#92418	—	2	1	no	—
139	#33739	—	61	1	no	maximal, Sylow
140	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₆₁	—	2	1	no	—
142	C₂×D₆₁	—	2	1	no	—
143	C₂×D₆₁	—	2	1	no	—
144	C₂×D₆₁	—	8	2	no	—
145	C₂×D₆₁	—	8	2	no	—
146	C₂×D₆₁	—	8	2	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	8	2	no	—
150	C₄×C₆₁	—	8	2	no	—
151	C₄×C₆₁	#532	1	1	yes	characteristic
152	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
153	Dic₆₁	—	2	1	no	—
154	Dic₆₁	—	2	1	no	—
155	Dic₆₁	—	2	1	no	—
156	Dic₆₁	—	8	2	no	—
157	#3175	—	61	2	no	—
158	#2252	D₆₁	1	1	yes	characteristic, complemented
159	#2249	—	61	1	no	—
160	#2243	—	61	2	no	—
161	#2220	—	61	2	no	—
162	#2216	—	61	1	no	—
163	#2215	—	61	1	no	—
164	#2175	—	61	2	no	—
165	C₄×Q₃₂	—	61	1	no	—
166	C₄×D₁₆	—	61	1	no	—
167	#2108	—	61	1	no	—
168	C₂×C₆₁	#3169	1	1	yes	characteristic
169	C₂×C₆₁	#3415	1	1	yes	characteristic
170	C₂×C₆₁	#3417	1	1	yes	characteristic
171	D₆₁	—	4	1	no	—
172	D₆₁	—	16	2	no	—
173	#542	—	122	2	no	—
174	C₂×Q₃₂	—	61	1	no	—
175	C₂×SD₃₂	—	61	2	no	—
176	C₂×D₁₆	—	61	1	no	—
177	C₂×M₃₂	—	61	1	no	—
178	C₂²×C₁₆	—	61	1	no	—
179	#500	C₂×D₆₁	1	2	yes	—
180	#495	—	61	2	no	—
181	#489	—	122	2	no	—
182	#468	—	122	2	no	—
183	#467	—	61	2	no	—
184	#466	—	61	2	no	—
185	#440	—	122	2	no	—
186	C₄×Q₁₆	—	61	2	no	—
187	C₄×D₈	—	61	2	no	—
188	#405	—	61	1	no	—
189	#366	C₂×D₆₁	1	1	yes	characteristic
190	#366	C₂×D₆₁	1	1	yes	characteristic
191	#366	C₂×D₆₁	1	2	yes	—
192	#358	—	61	2	no	—
193	#358	—	61	2	no	—
194	#357	—	61	2	no	—
195	#357	—	61	2	no	—
196	#346	—	61	1	no	—
197	#346	—	61	1	no	—
198	C₄×C₁₆	—	61	1	no	—
199	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
200	C₆₁	#33739	1	1	yes	characteristic, complemented, Sylow
201	C₂×Q₁₆	—	61	2	no	—
202	C₂×SD₁₆	—	122	2	no	—
203	C₂×D₈	—	61	2	no	—
204	C₂×M₁₆	—	61	1	no	—
205	C₂²×C₈	—	61	1	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	#126	—	244	2	no	—
208	#124	—	122	2	no	—
209	#124	—	244	2	no	—
210	#123	—	244	2	no	—
211	C₂²⋊Q₈	—	122	2	no	—
212	C₂²⋊Q₈	—	244	2	no	—
213	#121	—	122	2	no	—
214	C₄×Q₈	—	122	2	no	—
215	C₄×D₄	—	122	2	no	—
216	#117	—	61	1	no	—
217	Q₃₂	—	122	2	no	—
218	SD₃₂	—	122	4	no	—
219	D₁₆	—	122	2	no	—
220	M₃₂	—	122	2	no	—
221	C₂×C₁₆	—	61	1	no	—
222	C₂×C₁₆	—	61	1	no	—
223	C₂×C₁₆	—	122	2	no	—
224	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	C₂²×D₆₁	1	2	yes	—
228	#107	C₂²×D₆₁	1	2	yes	—
229	#103	—	122	2	no	—
230	#103	—	122	2	no	—
231	#102	—	122	2	no	—
232	#102	—	122	2	no	—
233	#97	—	61	1	no	—
234	#97	—	61	1	no	—
235	C₄×C₈	—	61	1	no	—
236	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	122	2	no	—
239	C₂×D₄	—	122	2	no	—
240	C₂×D₄	—	244	2	no	—
241	C₂²×C₄	—	61	1	no	—
242	C₂²×C₄	—	244	2	no	—
243	Q₁₆	—	122	2	no	—
244	Q₁₆	—	122	2	no	—
245	SD₁₆	—	244	4	no	—
246	D₈	—	122	2	no	—
247	D₈	—	122	2	no	—
248	M₁₆	—	122	2	no	—
249	C₂×C₈	—	61	1	no	—
250	C₂×C₈	—	61	1	no	—
251	C₂×C₈	—	122	1	no	—
252	C₂×C₈	—	122	1	no	—
253	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	61	1	no	—
259	C₄⋊C₄	—	61	1	no	—
260	C₄⋊C₄	—	244	2	no	—
261	C₄⋊C₄	—	244	2	no	—
262	C₂²⋊C₄	—	61	1	no	—
263	C₂²⋊C₄	—	61	1	no	—
264	C₂²⋊C₄	—	244	2	no	—
265	C₂²⋊C₄	—	244	2	no	—
266	C₂²⋊C₄	—	244	2	no	—
267	C₂²⋊C₄	—	244	2	no	—
268	C₂²⋊C₄	—	244	2	no	—
269	C₄²	—	61	1	no	—
270	C₄²	—	244	2	no	—
271	C₄²	D₄×D₆₁	1	1	yes	characteristic
272	C₁₆	—	122	1	no	—
273	C₁₆	—	122	1	no	—
274	C₁₆	#11758226	1	2	yes	—
275	C₁₆	#11758226	1	2	yes	—
276	C₂³	—	61	1	no	—
277	C₂³	—	244	2	no	—
278	Q₈	—	16	2	no	—
279	Q₈	—	244	2	no	—
280	Q₈	—	244	2	no	—
281	D₄	—	976	2	no	—
282	D₄	—	244	2	no	—
283	D₄	—	244	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	61	1	no	—
287	C₂×C₄	—	61	1	no	—
288	C₂×C₄	—	61	1	no	—
289	C₂×C₄	—	122	1	no	—
290	C₂×C₄	—	122	1	no	—
291	C₂×C₄	—	976	2	no	—
292	C₂×C₄	—	244	2	no	—
293	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
294	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
296	C₈	—	122	1	no	—
297	C₈	—	122	1	no	—
298	C₈	#14519800	1	2	yes	—
299	C₈	C₂×#11758226	1	1	yes	characteristic
300	C₈	C₂×#11758226	1	1	yes	characteristic
301	C₂²	—	122	1	no	—
302	C₂²	—	122	1	no	—
303	C₂²	—	122	1	no	—
304	C₂²	—	488	2	no	—
305	C₂²	—	488	2	no	—
306	C₂²	—	488	2	no	—
307	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	122	1	no	—
313	C₄	—	122	1	no	—
314	C₄	—	122	1	no	—
315	C₄	—	488	2	no	—
316	C₄	#16840313	1	1	yes	characteristic
317	C₄	C₂×#14519800	1	1	yes	characteristic
318	C₂	—	976	2	no	—
319	C₂	—	244	1	no	—
320	C₂	#21456399	1	1	yes	characteristic
321	C₂	#21458956	1	1	yes	characteristic
322	C₂	#21458968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2252	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	#21447906	C₂
hypercenter	#2252	D₆₁
hypocenter	C₆₁	#33739
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
1. #21456461	C₂	semidirect
2. #21448080	C₂	semidirect
3. #21448058	C₂	semidirect
4. #21447933	C₂	semidirect
5. #21447911	C₂	semidirect
7. #21447682	C₂	semidirect
8. #21447677	C₂	semidirect
9. #21447665	C₂	semidirect
10. #21447414	C₂	semidirect
11. C₆₁×#2252	C₂	semidirect
158. #2252	D₆₁	semidirect
200. C₆₁	#33739	semidirect

Maximal subgroups

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

subgroup	index
#21456461	2
#21448080	2
#21448058	2
#21447933	2
#21447911	2
#21447906	2
#21447682	2
#21447677	2
#21447665	2
#21447414	2
C₆₁×#2252	2
#33739	61