Subgroups — #29837713

Order 15616.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458809	C₂	1	1	yes	maximal, characteristic
2	#21458712	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448052	C₂	1	2	yes	maximal
4	#21447981	C₂	1	2	yes	maximal, complemented
5	#21447953	C₂	1	1	yes	maximal, characteristic
6	#21447884	C₂	1	2	yes	maximal
7	#21447866	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447675	C₂	1	2	yes	maximal, complemented
10	C₄×#14519722	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840487	—	2	1	no	—
13	#16840486	—	2	1	no	—
14	C₂×#14519723	C₂²	1	2	yes	—
15	C₂×#14519722	C₂²	1	1	yes	characteristic
16	#16840163	C₂²	1	2	yes	—
17	#16840147	C₂²	1	2	yes	—
18	#16840068	—	2	2	no	—
19	#16840044	—	2	1	no	—
20	#16839986	—	2	2	no	—
21	#16839964	—	2	1	no	—
22	C₈×Dic₁₂₂	C₂²	1	1	yes	characteristic
23	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
25	#16839856	C₂²	1	2	yes	—
26	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
27	#16839590	C₂²	1	1	yes	characteristic
28	#16839587	C₂²	1	2	yes	—
29	C₄⋊#11758185	C₂²	1	1	yes	characteristic
30	#16839580	C₂²	1	2	yes	—
31	#16839567	C₂²	1	1	yes	characteristic
32	#16839566	C₂²	1	2	yes	—
33	#16839564	C₂²	1	1	yes	characteristic
34	#16839562	C₂²	1	2	yes	—
35	C₁₆×Dic₆₁	C₂²	1	2	yes	—
36	C₆₁×#500	C₂²	1	1	yes	characteristic
37	C₆₁×#494	C₂²	1	1	yes	characteristic
38	C₆₁×#358	C₂²	1	2	yes	—
39	C₆₁×#358	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
42	C₂×#11758201	—	2	2	no	—
43	C₂×D₄₈₈	C₂³	1	1	yes	characteristic
44	#14519775	—	4	1	no	—
45	#14519772	—	4	2	no	—
46	#14519768	—	2	2	no	—
47	#14519767	—	4	2	no	—
48	#14519766	—	2	2	no	—
49	#14519757	—	4	1	no	—
50	Q₈×Dic₆₁	—	2	2	no	—
51	Q₈×Dic₆₁	—	4	1	no	—
52	C₄×Dic₁₂₂	—	2	1	no	—
53	C₄×Dic₁₂₂	D₄	1	1	yes	characteristic
54	#14519742	—	2	2	no	—
55	C₄×D₂₄₄	—	2	1	no	—
56	#14519735	—	4	1	no	—
57	#14519723	—	2	4	no	—
58	#14519722	—	2	2	no	—
59	C₂×#11758185	C₂³	1	2	yes	—
60	#14519711	C₂³	1	1	yes	characteristic
61	#14519711	C₂³	1	1	yes	characteristic
62	#14519708	—	2	2	no	—
63	C₄⋊#92418	D₄	1	1	yes	characteristic
64	#14519706	D₄	1	2	yes	—
65	#14519702	—	2	1	no	—
66	#14519701	—	2	2	no	—
67	#14519700	—	2	2	no	—
68	#14519699	—	2	1	no	—
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₆₁	—	2	1	no	—
71	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₆₁	—	2	1	no	—
73	C₄⋊Q₈×C₆₁	—	2	1	no	—
74	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
76	#107×C₆₁	—	2	1	no	—
77	#107×C₆₁	C₂³	1	2	yes	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×D₂₄₄	—	2	1	no	—
82	C₂×C₄×D₆₁	—	4	1	no	—
83	C₂×Dic₁₂₂	—	2	1	no	—
84	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
85	Q₁₆×C₆₁	—	2	2	no	—
86	Q₁₆×C₆₁	—	2	2	no	—
87	Q₁₆×C₆₁	—	8	1	no	—
88	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₆₁	—	2	2	no	—
91	C₄⋊C₄×C₆₁	—	4	1	no	—
92	C₄⋊C₄×C₆₁	—	4	1	no	—
93	C₄⋊C₄×C₆₁	—	4	1	no	—
94	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
95	#11758201	—	8	2	no	—
96	#11758198	—	4	1	no	—
97	#11758198	—	4	2	no	—
98	#11758198	—	4	2	no	—
99	C₄⋊Dic₆₁	—	2	1	no	—
100	C₄⋊Dic₆₁	—	2	1	no	—
101	C₄⋊Dic₆₁	—	4	2	no	—
102	C₄⋊Dic₆₁	—	4	2	no	—
103	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
104	#11758196	—	4	1	no	—
105	#11758196	—	4	2	no	—
106	#11758196	—	4	2	no	—
107	#11758196	D₈	1	2	yes	—
108	C₄×Dic₆₁	—	4	1	no	—
109	C₄×Dic₆₁	—	4	2	no	—
110	C₄×Dic₆₁	—	4	2	no	—
111	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
112	C₂×#92418	C₂×D₄	1	2	yes	—
113	Dic₂₄₄	—	2	1	no	—
114	Dic₂₄₄	#41	1	2	yes	—
115	D₄₈₈	—	2	1	no	—
116	D₄₈₈	—	2	1	no	—
117	C₁₆×C₆₁	—	2	1	no	—
118	C₁₆×C₆₁	—	2	1	no	—
119	#11758185	—	2	2	no	—
120	C₂²×D₆₁	—	4	1	no	—
121	Q₈×C₆₁	—	4	2	no	—
122	Q₈×C₆₁	—	4	2	no	—
123	Q₈×C₆₁	—	8	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	4	2	no	—
126	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₆₁	—	4	1	no	—
130	C₂×Dic₆₁	—	4	1	no	—
131	C₂×Dic₆₁	—	4	1	no	—
132	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
133	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
134	D₂₄₄	—	4	1	no	—
135	D₂₄₄	—	4	1	no	—
136	C₄×D₆₁	—	8	2	no	—
137	Dic₁₂₂	—	2	2	no	—
138	Dic₁₂₂	—	2	2	no	—
139	Dic₁₂₂	—	4	1	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
143	#92418	—	2	2	no	—
144	#33931	—	61	1	no	maximal, Sylow
145	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₆₁	—	8	1	no	—
147	C₂×D₆₁	—	8	1	no	—
148	C₂×D₆₁	—	8	1	no	—
149	C₄×C₆₁	—	2	1	no	—
150	C₄×C₆₁	—	8	2	no	—
151	C₄×C₆₁	—	8	2	no	—
152	C₄×C₆₁	#508	1	2	yes	—
153	C₄×C₆₁	#548	1	1	yes	characteristic
154	C₄×C₆₁	#577	1	1	yes	characteristic
155	Dic₆₁	—	2	2	no	—
156	Dic₆₁	—	2	2	no	—
157	Dic₆₁	—	4	2	no	—
158	Dic₆₁	—	8	1	no	—
159	Dic₆₁	—	8	1	no	—
160	#3393	—	61	1	no	—
161	#3383	—	61	1	no	—
162	#2242	—	61	2	no	—
163	#2241	D₆₁	1	1	yes	characteristic, complemented
164	#2229	—	61	2	no	—
165	#2224	—	61	1	no	—
166	#2211	—	61	2	no	—
167	#2209	—	61	1	no	—
168	Q₈×C₁₆	—	61	1	no	—
169	C₄×Q₃₂	—	61	2	no	—
170	C₄×SD₃₂	—	61	1	no	—
171	C₂×C₆₁	#3358	1	1	yes	characteristic
172	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
173	C₂×C₆₁	#3417	1	1	yes	characteristic
174	D₆₁	—	16	1	no	—
175	#557	—	122	1	no	—
176	#556	—	122	1	no	—
177	C₂×Q₃₂	—	61	2	no	—
178	C₂×SD₃₂	—	61	1	no	—
179	#500	C₂×D₆₁	1	1	yes	characteristic
180	#499	—	61	2	no	—
181	#495	—	61	2	no	—
182	#494	C₂×D₆₁	1	1	yes	characteristic
183	#479	—	122	2	no	—
184	#475	—	122	1	no	—
185	#464	—	122	2	no	—
186	#461	—	122	1	no	—
187	C₈×Q₈	—	61	1	no	—
188	C₄×Q₁₆	—	61	2	no	—
189	C₄×Q₁₆	—	61	2	no	—
190	C₄×Q₁₆	—	61	1	no	—
191	C₄×D₈	—	61	1	no	—
192	#367	—	61	1	no	—
193	#366	—	61	2	no	—
194	C₄⋊C₁₆	—	61	2	no	—
195	C₄⋊C₁₆	—	61	1	no	—
196	#358	—	61	2	no	—
197	#358	—	61	1	no	—
198	#358	C₂×D₆₁	1	2	yes	—
199	#358	C₂×D₆₁	1	2	yes	—
200	#357	—	61	1	no	—
201	#357	—	61	2	no	—
202	C₄×C₁₆	—	61	2	no	—
203	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
204	C₆₁	#33931	1	1	yes	characteristic, complemented, Sylow
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	61	1	no	—
207	C₂×Q₁₆	C₂²×D₆₁	1	2	yes	—
208	C₂×SD₁₆	—	122	2	no	—
209	C₂×D₈	—	61	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	244	1	no	—
213	#126	—	244	2	no	—
214	#125	—	122	2	no	—
215	#125	—	244	2	no	—
216	#124	—	122	2	no	—
217	C₂²⋊Q₈	—	244	1	no	—
218	C₄×Q₈	—	61	1	no	—
219	C₄×Q₈	—	122	2	no	—
220	C₄×Q₈	—	122	2	no	—
221	C₄×Q₈	—	122	1	no	—
222	C₄×Q₈	—	244	1	no	—
223	C₄×D₄	—	122	1	no	—
224	#117	—	244	1	no	—
225	Q₃₂	—	122	4	no	—
226	SD₃₂	—	122	2	no	—
227	C₂×C₁₆	—	61	2	no	—
228	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	—	61	1	no	—
232	#107	—	61	1	no	—
233	#107	C₂²×D₆₁	1	2	yes	—
234	#106	—	122	2	no	—
235	C₄⋊C₈	—	61	2	no	—
236	C₄⋊C₈	—	61	1	no	—
237	#103	—	122	2	no	—
238	#103	—	122	2	no	—
239	#103	—	122	1	no	—
240	#102	—	122	1	no	—
241	C₄×C₈	—	61	2	no	—
242	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	61	1	no	—
246	C₂×Q₈	—	122	1	no	—
247	C₂×D₄	—	122	1	no	—
248	C₂²×C₄	—	244	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	8	1	no	—
252	Q₁₆	—	61	2	no	—
253	Q₁₆	—	122	1	no	—
254	SD₁₆	—	488	2	no	—
255	D₈	—	122	1	no	—
256	D₈	—	122	1	no	—
257	C₂×C₈	—	61	2	no	—
258	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	61	2	no	—
265	C₄⋊C₄	—	61	1	no	—
266	C₄⋊C₄	—	122	1	no	—
267	C₄⋊C₄	—	122	1	no	—
268	C₄⋊C₄	—	244	2	no	—
269	C₄⋊C₄	—	244	2	no	—
270	C₄⋊C₄	—	244	2	no	—
271	C₄⋊C₄	—	244	2	no	—
272	C₄⋊C₄	—	244	1	no	—
273	C₂²⋊C₄	—	244	2	no	—
274	C₂²⋊C₄	—	244	2	no	—
275	C₂²⋊C₄	—	244	1	no	—
276	C₄²	—	61	2	no	—
277	C₄²	—	244	2	no	—
278	C₄²	—	244	2	no	—
279	C₄²	—	244	1	no	—
280	C₄²	D₄×D₆₁	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	122	2	no	—
284	C₂³	—	244	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	122	2	no	—
289	Q₈	—	122	2	no	—
290	Q₈	—	244	1	no	—
291	D₄	—	244	1	no	—
292	D₄	—	244	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	2	no	—
297	C₂×C₄	—	488	2	no	—
298	C₂×C₄	—	244	1	no	—
299	C₂×C₄	—	244	1	no	—
300	C₂×C₄	—	244	1	no	—
301	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
302	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	122	2	no	—
306	C₈	#41×D₆₁	1	1	yes	characteristic
307	C₈	#14519824	1	1	yes	characteristic
308	C₂²	—	488	1	no	—
309	C₂²	—	488	1	no	—
310	C₂²	—	488	1	no	—
311	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	122	2	no	—
316	C₄	—	122	2	no	—
317	C₄	—	488	1	no	—
318	C₄	—	488	1	no	—
319	C₄	—	244	2	no	—
320	C₄	#16840193	1	2	yes	—
321	C₄	#16840420	1	1	yes	characteristic
322	C₄	#16840568	1	1	yes	characteristic
323	C₂	—	976	1	no	—
324	C₂	#21458451	1	1	yes	characteristic
325	C₂	C₂×#16840193	1	1	yes	characteristic
326	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×#2241	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂×D₄₈₈	C₂³
hypercenter	#2241	D₆₁
hypocenter	C₆₁	#33931
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
2. #21458712	C₂	semidirect
4. #21447981	C₂	semidirect
7. #21447866	C₂	semidirect
8. C₁₆×Dic₁₂₂	C₂	semidirect
9. #21447675	C₂	semidirect
11. C₆₁×#2241	C₂	semidirect
163. #2241	D₆₁	semidirect
204. C₆₁	#33931	semidirect

Maximal subgroups

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

subgroup	index
#21458809	2
#21458712	2
#21448052	2
#21447981	2
#21447953	2
#21447884	2
#21447866	2
C₁₆×Dic₁₂₂	2
#21447675	2
C₄×#14519722	2
C₆₁×#2241	2
#33931	61