Subgroups — #29835552

Order 15616.

514 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456400	C₂	1	2	yes	maximal
2	#21448093	C₂	1	1	yes	maximal, characteristic, complemented
3	#21448070	C₂	1	1	yes	maximal, characteristic, complemented
4	#21448045	C₂	1	2	yes	maximal, complemented
5	#21447910	C₂	1	1	yes	maximal, characteristic
6	#21447908	C₂	1	2	yes	maximal
7	Q₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447677	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447675	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#2216	C₂	1	1	yes	maximal, characteristic
12	#16840314	—	2	2	no	—
13	C₂×Dic₄₈₈	C₂²	1	1	yes	characteristic
14	C₂×#14519723	C₂²	1	2	yes	—
15	C₂²×#11758185	C₂²	1	1	yes	characteristic
16	#16840166	C₂²	1	2	yes	—
17	#16840140	C₂²	1	2	yes	—
18	#16840111	—	2	2	no	—
19	#16840003	C₂²	1	2	yes	—
20	#16840001	—	2	2	no	—
21	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
22	#16839856	C₂²	1	2	yes	—
23	#16839854	—	2	2	no	—
24	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
25	#16839588	C₂²	1	1	yes	characteristic
26	#16839587	C₂²	1	2	yes	—
27	#16839567	C₂²	1	2	yes	—
28	#16839566	C₂²	1	2	yes	—
29	#16839565	C₂²	1	2	yes	—
30	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#467	C₂²	1	2	yes	—
37	C₆₁×#366	C₂²	1	1	yes	characteristic
38	C₆₁×#358	C₂²	1	2	yes	—
39	C₂×Dic₂₄₄	C₂³	1	2	yes	—
40	C₂×#11758202	—	2	2	no	—
41	C₂²×#92418	D₄	1	1	yes	characteristic
42	#14519776	—	2	2	no	—
43	#14519775	—	2	2	no	—
44	#14519767	—	4	2	no	—
45	#14519756	—	2	2	no	—
46	#14519754	—	4	2	no	—
47	Q₈×Dic₆₁	—	2	2	no	—
48	#14519742	—	2	2	no	—
49	#14519742	—	4	2	no	—
50	#14519733	—	4	2	no	—
51	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
52	Dic₄₈₈	—	2	2	no	—
53	#14519723	—	2	4	no	—
54	C₂×#11758185	—	2	2	no	—
55	C₂×#11758185	C₂³	1	1	yes	characteristic
56	C₂×#11758185	C₂³	1	1	yes	characteristic
57	#14519711	C₂³	1	2	yes	—
58	#14519710	—	2	2	no	—
59	#14519702	—	2	2	no	—
60	#14519701	—	2	2	no	—
61	#14519700	—	2	2	no	—
62	C₈×Dic₆₁	D₄	1	1	yes	characteristic
63	C₈×Dic₆₁	D₄	1	1	yes	characteristic
64	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
65	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
67	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	C₂²⋊Q₈×C₆₁	—	2	2	no	—
69	Q₃₂×C₆₁	—	2	2	no	—
70	C₂×C₁₆×C₆₁	—	2	2	no	—
71	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
73	#107×C₆₁	C₂³	1	2	yes	—
74	#103×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
78	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
79	C₂×Dic₁₂₂	—	2	2	no	—
80	C₂×Dic₁₂₂	—	4	2	no	—
81	Q₁₆×C₆₁	—	2	2	no	—
82	Q₁₆×C₆₁	—	2	2	no	—
83	C₂×C₈×C₆₁	—	2	1	no	—
84	C₂×C₈×C₆₁	—	2	1	no	—
85	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₄⋊C₄×C₆₁	—	4	2	no	—
89	C₂²⋊C₄×C₆₁	—	4	2	no	—
90	C₂²⋊Dic₆₁	—	4	2	no	—
91	#11758202	—	4	4	no	—
92	C₄⋊Dic₆₁	—	2	2	no	—
93	C₄⋊Dic₆₁	—	4	2	no	—
94	#11758196	—	4	2	no	—
95	#11758196	—	4	2	no	—
96	#11758196	—	4	2	no	—
97	#11758196	—	4	2	no	—
98	#11758196	—	4	2	no	—
99	C₄×Dic₆₁	—	4	2	no	—
100	C₄×Dic₆₁	—	4	2	no	—
101	C₄×Dic₆₁	D₈	1	1	yes	characteristic
102	C₄×Dic₆₁	D₈	1	1	yes	characteristic
103	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
105	C₂×#92418	—	2	2	no	—
106	C₂×#92418	C₂×D₄	1	1	yes	characteristic
107	C₂×#92418	C₂×D₄	1	1	yes	characteristic
108	Dic₂₄₄	—	2	2	no	—
109	Dic₂₄₄	—	2	2	no	—
110	C₁₆×C₆₁	—	2	1	no	—
111	C₁₆×C₆₁	#41	1	2	yes	—
112	#11758185	—	2	1	no	—
113	#11758185	#41	1	2	yes	—
114	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₆₁	—	4	2	no	—
116	Q₈×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	4	2	no	—
120	C₂×C₄×C₆₁	—	4	2	no	—
121	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₆₁	—	2	2	no	—
124	C₂×Dic₆₁	—	4	2	no	—
125	C₂×Dic₆₁	—	4	2	no	—
126	C₂×Dic₆₁	Q₃₂	1	4	yes	complemented
127	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
128	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
131	Dic₁₂₂	—	4	2	no	—
132	Dic₁₂₂	—	4	2	no	—
133	Dic₁₂₂	—	8	4	no	—
134	C₈×C₆₁	—	2	1	no	—
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	#135	1	2	yes	—
139	#33734	—	61	1	no	maximal, Sylow
140	C₂²×C₆₁	—	2	1	no	—
141	C₂²×C₆₁	—	2	1	no	—
142	C₂²×C₆₁	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₆₁	—	2	1	no	—
145	C₄×C₆₁	—	8	2	no	—
146	C₄×C₆₁	—	8	2	no	—
147	C₄×C₆₁	#532	1	1	yes	characteristic
148	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
149	Dic₆₁	—	2	1	no	—
150	Dic₆₁	—	8	2	no	—
151	Dic₆₁	—	8	2	no	—
152	Dic₆₁	C₂×Q₃₂	1	2	yes	—
153	Dic₆₁	C₂×Q₃₂	1	2	yes	—
154	Dic₆₁	#508	1	2	yes	—
155	#3170	—	61	2	no	—
156	#2252	—	61	1	no	—
157	#2247	—	61	1	no	—
158	#2241	—	61	2	no	—
159	#2216	—	61	1	no	—
160	#2216	—	61	2	no	—
161	#2216	D₆₁	1	1	yes	characteristic
162	C₄×Q₃₂	—	61	1	no	—
163	C₄×Q₃₂	—	61	1	no	—
164	C₄×Q₃₂	—	61	2	no	—
165	C₂×C₄×C₁₆	—	61	1	no	—
166	C₂×C₆₁	—	2	2	no	—
167	C₂×C₆₁	#3169	1	1	yes	characteristic
168	C₂×C₆₁	C₂²×Q₃₂	1	1	yes	characteristic
169	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
170	#533	—	122	2	no	—
171	C₂×Q₃₂	—	61	2	no	—
172	C₂×Q₃₂	—	61	1	no	—
173	C₂×Q₃₂	C₂×D₆₁	1	1	yes	characteristic
174	C₂²×C₁₆	—	61	1	no	—
175	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
176	#500	—	61	2	no	—
177	#494	—	61	2	no	—
178	#487	—	122	2	no	—
179	#467	—	61	2	no	—
180	#467	—	122	2	no	—
181	#467	C₂×D₆₁	1	2	yes	—
182	C₄×Q₁₆	—	61	2	no	—
183	C₄×Q₁₆	—	61	2	no	—
184	C₄×Q₁₆	—	122	2	no	—
185	C₂×C₄×C₈	—	61	1	no	—
186	#366	—	61	1	no	—
187	#366	—	61	2	no	—
188	#366	C₂×D₆₁	1	1	yes	characteristic
189	#358	—	61	2	no	—
190	#358	—	61	2	no	—
191	#358	—	61	2	no	—
192	#358	C₂×D₆₁	1	2	yes	—
193	C₄×C₁₆	—	61	1	no	—
194	C₄×C₁₆	—	61	1	no	—
195	C₄×C₁₆	—	61	1	no	—
196	C₄×C₁₆	—	61	1	no	—
197	C₆₁	#33734	1	1	yes	characteristic, complemented, Sylow
198	C₂×Q₁₆	—	61	2	no	—
199	C₂×Q₁₆	—	122	2	no	—
200	C₂×Q₁₆	C₂²×D₆₁	1	2	yes	—
201	C₂²×C₈	—	61	1	no	—
202	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
203	C₄⋊Q₈	—	122	2	no	—
204	C₄⋊Q₈	—	122	2	no	—
205	#125	—	244	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	122	2	no	—
208	C₂²⋊Q₈	—	244	2	no	—
209	C₄×Q₈	—	122	2	no	—
210	C₄×Q₈	—	122	2	no	—
211	C₄×Q₈	—	244	2	no	—
212	#117	—	244	2	no	—
213	C₂×C₄²	—	61	1	no	—
214	Q₃₂	—	2	2	no	—
215	Q₃₂	—	122	4	no	—
216	Q₃₂	—	122	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	61	1	no	—
219	C₂×C₁₆	—	61	1	no	—
220	C₂×C₁₆	—	122	2	no	—
221	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
223	#107	—	61	2	no	—
224	#107	—	122	2	no	—
225	#107	C₂²×D₆₁	1	2	yes	—
226	#103	—	2	2	no	—
227	#103	—	122	2	no	—
228	#103	—	122	2	no	—
229	#103	—	122	2	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	61	1	no	—
232	C₄×C₈	—	61	1	no	—
233	C₄×C₈	—	61	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	122	2	no	—
236	C₂×Q₈	—	244	2	no	—
237	C₂²×C₄	—	61	1	no	—
238	C₂²×C₄	—	61	1	no	—
239	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	2	2	no	—
242	Q₁₆	—	122	2	no	—
243	Q₁₆	—	122	2	no	—
244	Q₁₆	—	244	4	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	61	1	no	—
248	C₂×C₈	—	61	1	no	—
249	C₂×C₈	—	122	2	no	—
250	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	122	2	no	—
255	C₄⋊C₄	—	244	2	no	—
256	C₄⋊C₄	—	244	2	no	—
257	C₄⋊C₄	—	244	2	no	—
258	C₄⋊C₄	—	244	2	no	—
259	C₄⋊C₄	—	244	2	no	—
260	C₄⋊C₄	—	244	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	244	2	no	—
263	C₄²	—	61	1	no	—
264	C₄²	—	61	1	no	—
265	C₄²	—	61	1	no	—
266	C₄²	—	61	1	no	—
267	C₄²	—	244	2	no	—
268	C₄²	—	244	2	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	61	2	no	—
271	C₁₆	—	122	1	no	—
272	C₁₆	#11758224	1	2	yes	—
273	C₂³	D₈×D₆₁	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	488	4	no	—
277	Q₈	—	244	2	no	—
278	Q₈	—	244	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	61	1	no	—
284	C₂×C₄	—	61	1	no	—
285	C₂×C₄	—	61	1	no	—
286	C₂×C₄	—	61	4	no	—
287	C₂×C₄	—	61	1	no	—
288	C₂×C₄	—	122	2	no	—
289	C₂×C₄	—	244	2	no	—
290	C₂×C₄	—	244	2	no	—
291	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	61	2	no	—
295	C₈	—	122	1	no	—
296	C₈	#41×D₆₁	1	1	yes	characteristic
297	C₈	C₂×#11758224	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	Q₃₂×D₆₁	1	2	yes	—
301	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	61	2	no	—
306	C₄	—	61	2	no	—
307	C₄	—	61	2	no	—
308	C₄	—	122	1	no	—
309	C₄	—	488	2	no	—
310	C₄	—	488	2	no	—
311	C₄	#16840307	1	1	yes	characteristic
312	C₄	#135×D₆₁	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#21456393	1	1	yes	characteristic
315	C₂	C₂×Q₃₂×D₆₁	1	1	yes	characteristic
316	C₂	#508×D₆₁	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×#2216	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#2216	D₆₁
hypocenter	C₆₁	#33734
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
2. #21448093	C₂	semidirect
3. #21448070	C₂	semidirect
4. #21448045	C₂	semidirect
7. Q₃₂×Dic₆₁	C₂	semidirect
8. #21447677	C₂	semidirect
9. #21447675	C₂	semidirect
126. C₂×Dic₆₁	Q₃₂	semidirect
197. C₆₁	#33734	semidirect

Maximal subgroups

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

subgroup	index
#21456400	2
#21448093	2
#21448070	2
#21448045	2
#21447910	2
#21447908	2
Q₃₂×Dic₆₁	2
#21447677	2
#21447675	2
C₂×C₁₆×Dic₆₁	2
C₆₁×#2216	2
#33734	61