Subgroups — #29688889

Order 15616.

670 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	#21456342	C₂	1	1	yes	maximal, characteristic
2	#21454758	C₂	1	2	yes	maximal
3	#21446794	C₂	1	2	yes	maximal
4	#21446572	C₂	1	1	yes	maximal, characteristic
5	#21446564	C₂	1	2	yes	maximal
6	#21445992	C₂	1	1	yes	maximal, characteristic
7	#21445702	C₂	1	2	yes	maximal
8	#21445701	C₂	1	1	yes	maximal, characteristic
9	#21445700	C₂	1	1	yes	maximal, characteristic
10	#97×Dic₆₁	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1946	C₂	1	1	yes	maximal, characteristic
12	#16840315	C₂²	1	2	yes	—
13	#16840170	C₄	1	4	yes	complemented
14	#16840169	C₄	1	4	yes	complemented
15	#16839891	—	2	4	no	—
16	#16839889	—	2	4	no	—
17	C₂×#14519713	C₂²	1	1	yes	characteristic
18	C₂×#14519712	C₂²	1	2	yes	—
19	#16839800	C₂²	1	1	yes	characteristic
20	#16839798	C₂²	1	2	yes	—
21	#16839788	C₂²	1	2	yes	—
22	#16839785	C₂²	1	2	yes	—
23	#16839720	C₂²	1	2	yes	—
24	#16839718	C₂²	1	2	yes	—
25	C₂×#14519688	C₂²	1	1	yes	characteristic
26	C₂×#14519687	C₂²	1	1	yes	characteristic
27	C₂×#14519687	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
30	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
31	#16839523	C₂²	1	2	yes	—
32	#16839522	C₂²	1	2	yes	—
33	#16839521	C₂²	1	2	yes	—
34	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#427	C₂²	1	1	yes	characteristic
36	C₆₁×#424	C₂²	1	2	yes	—
37	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#339	C₂²	1	2	yes	—
39	C₂×#11758194	C₂³	1	2	yes	—
40	C₂²×#92418	C₂³	1	1	yes	characteristic
41	C₂²×#92418	C₂³	1	1	yes	characteristic
42	#14519776	C₂×C₄	1	4	yes	—
43	#14519768	C₂×C₄	1	4	yes	—
44	#14519754	—	2	4	no	—
45	C₄×Dic₁₂₂	—	2	4	no	—
46	#14519736	C₂³	1	2	yes	—
47	C₂×C₄×Dic₆₁	—	2	2	no	—
48	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
50	#14519713	—	2	2	no	—
51	#14519712	—	2	4	no	—
52	#14519711	C₂×C₄	1	2	yes	—
53	#14519710	C₂×C₄	1	4	yes	—
54	#14519709	C₂×C₄	1	2	yes	—
55	#14519708	C₂×C₄	1	4	yes	—
56	#14519706	—	2	4	no	—
57	#14519706	—	2	4	no	—
58	#14519688	—	2	2	no	—
59	#14519688	Q₈	1	4	yes	complemented
60	#14519687	—	2	2	no	—
61	#14519687	C₂×C₄	1	2	yes	—
62	#14519687	C₂×C₄	1	2	yes	—
63	#14519687	D₄	1	2	yes	—
64	#14519687	D₄	1	2	yes	—
65	C₄×#92418	—	2	2	no	—
66	C₈×Dic₆₁	—	2	2	no	—
67	C₈×Dic₆₁	—	2	2	no	—
68	C₈×Dic₆₁	Q₈	1	4	yes	complemented
69	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	#117×C₆₁	C₂³	1	2	yes	—
73	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
74	#108×C₆₁	—	2	2	no	—
75	#107×C₆₁	C₂×C₄	1	2	yes	—
76	#106×C₆₁	C₂×C₄	1	2	yes	—
77	C₄⋊C₈×C₆₁	—	2	4	no	—
78	#97×C₆₁	—	2	2	no	—
79	C₂²×C₄×C₆₁	—	2	1	no	—
80	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₆₁	—	2	1	no	—
82	C₂²×Dic₆₁	#41	1	2	yes	—
83	C₂×Dic₁₂₂	—	2	4	no	—
84	M₁₆×C₆₁	—	2	2	no	—
85	M₁₆×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	—	2	1	no	—
87	C₂×C₈×C₆₁	—	2	1	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₆₁	#41	1	2	yes	—
94	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
95	C₂²⋊C₄×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	—	2	2	no	—
97	C₄²×C₆₁	—	2	1	no	—
98	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
99	C₂²⋊Dic₆₁	—	2	2	no	—
100	C₄⋊Dic₆₁	—	2	4	no	—
101	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
102	#11758196	—	2	4	no	—
103	#11758196	—	2	4	no	—
104	C₄×Dic₆₁	—	2	2	no	—
105	C₄×Dic₆₁	—	2	1	no	—
106	C₄×Dic₆₁	—	4	4	no	—
107	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
108	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
109	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
110	#11758194	—	2	2	no	—
111	#11758194	—	2	2	no	—
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	C₂²×C₄	1	2	yes	—
114	C₂×#92418	C₂×D₄	1	1	yes	characteristic
115	C₂×#92418	C₂×D₄	1	1	yes	characteristic
116	C₂×#92418	C₂×Q₈	1	2	yes	—
117	C₂×#92418	C₂×Q₈	1	2	yes	—
118	C₂×#92418	C₂×Q₈	1	2	yes	—
119	C₂×#92418	#41	1	2	yes	—
120	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₆₁	—	2	4	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₆₁	—	2	4	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	2	4	no	—
133	C₂×Dic₆₁	—	4	2	no	—
134	C₂×Dic₆₁	C₄×D₄	1	4	yes	—
135	Dic₁₂₂	—	8	4	no	—
136	C₈×C₆₁	—	2	2	no	—
137	C₈×C₆₁	—	4	2	no	—
138	C₈×C₆₁	C₄×Q₈	1	2	yes	—
139	C₈×C₆₁	C₄×Q₈	1	2	yes	—
140	#92418	—	4	2	no	—
141	#92418	C₄×Q₈	1	4	yes	—
142	#92418	C₄⋊Q₈	1	2	yes	—
143	#92418	C₄⋊Q₈	1	2	yes	—
144	#21736	—	61	1	no	maximal, Sylow
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₆₁	#533	1	1	yes	characteristic
148	C₂²×C₆₁	#533	1	1	yes	characteristic
149	C₄×C₆₁	—	4	2	no	—
150	C₄×C₆₁	—	4	4	no	—
151	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₄×C₆₁	#532	1	1	yes	characteristic
155	Dic₆₁	—	2	4	no	—
156	Dic₆₁	—	2	4	no	—
157	Dic₆₁	—	4	4	no	—
158	#3162	—	61	1	no	—
159	#2961	—	61	2	no	—
160	#1994	—	61	2	no	—
161	#1946	—	61	1	no	—
162	#1946	D₆₁	1	1	yes	characteristic
163	#1945	—	61	2	no	—
164	#1849	—	61	1	no	—
165	#1773	—	61	2	no	—
166	#1772	—	61	1	no	—
167	#1772	—	61	1	no	—
168	C₄×#97	—	61	1	no	—
169	C₂×C₆₁	—	2	1	no	—
170	C₂×C₆₁	—	2	1	no	—
171	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
172	C₂×C₆₁	#2956	1	2	yes	—
173	#533	—	61	2	no	—
174	#501	—	61	4	no	—
175	#501	—	61	4	no	—
176	#446	—	122	4	no	—
177	#446	—	122	4	no	—
178	C₂×#108	—	61	2	no	—
179	C₂×#108	—	61	1	no	—
180	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
181	#427	—	61	1	no	—
182	#427	—	61	2	no	—
183	#427	C₂×D₆₁	1	1	yes	characteristic
184	#424	—	61	2	no	—
185	#424	C₂×D₆₁	1	2	yes	—
186	#423	—	61	2	no	—
187	#405	—	61	2	no	—
188	#405	—	61	2	no	—
189	C₂×#97	—	61	1	no	—
190	C₂×#97	—	61	1	no	—
191	C₂×#97	—	61	1	no	—
192	C₂×#97	C₂×D₆₁	1	1	yes	characteristic
193	C₂×C₄×C₈	—	61	1	no	—
194	C₂×C₄×C₈	—	61	1	no	—
195	C₄³	—	61	1	no	—
196	#339	—	61	2	no	—
197	#339	—	61	2	no	—
198	#339	—	61	2	no	—
199	#339	C₂×D₆₁	1	2	yes	—
200	C₆₁	#21736	1	1	yes	characteristic, complemented, Sylow
201	C₂×M₁₆	—	61	2	no	—
202	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
203	C₂²×C₈	—	61	1	no	—
204	C₂²×C₈	—	61	1	no	—
205	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
207	C₄⋊Q₈	—	61	4	no	—
208	#125	—	61	4	no	—
209	C₂²⋊Q₈	—	122	4	no	—
210	C₄×Q₈	—	122	4	no	—
211	#117	—	61	2	no	—
212	#117	C₂²×D₆₁	1	2	yes	—
213	C₂×C₄²	—	61	1	no	—
214	C₂×C₄²	—	61	1	no	—
215	C₂×C₄²	—	122	2	no	—
216	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
217	#108	—	2	2	no	—
218	#108	—	122	4	no	—
219	#108	—	122	2	no	—
220	#107	—	61	2	no	—
221	#107	—	61	4	no	—
222	#107	C₄×D₆₁	1	2	yes	—
223	#106	—	61	2	no	—
224	#106	—	61	4	no	—
225	#106	C₄×D₆₁	1	2	yes	—
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	122	4	no	—
228	C₄⋊C₈	—	122	4	no	—
229	#97	—	2	2	no	—
230	#97	—	61	2	no	—
231	#97	—	61	2	no	—
232	#97	—	61	2	no	—
233	#97	—	61	2	no	—
234	#97	—	61	4	no	—
235	#97	—	122	2	no	—
236	#97	—	122	2	no	—
237	C₄×C₈	—	61	4	no	—
238	C₄×C₈	—	122	2	no	—
239	C₄×C₈	—	122	2	no	—
240	C₄×C₈	—	122	2	no	—
241	C₂×Q₈	—	122	4	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	61	2	no	—
244	C₂²×C₄	—	122	1	no	—
245	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	122	2	no	—
249	M₁₆	—	122	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	61	2	no	—
254	C₂×C₈	—	61	1	no	—
255	C₂×C₈	—	61	1	no	—
256	C₂×C₈	—	61	2	no	—
257	C₂×C₈	—	61	2	no	—
258	C₂×C₈	—	61	2	no	—
259	C₂×C₈	—	61	2	no	—
260	C₂×C₈	—	122	2	no	—
261	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
263	C₂×C₈	D₄×D₆₁	1	2	yes	—
264	C₂×C₈	#11758224	1	2	yes	—
265	C₂×C₈	Q₈×D₆₁	1	2	yes	—
266	C₄⋊C₄	—	61	4	no	—
267	C₄⋊C₄	—	122	4	no	—
268	C₄⋊C₄	—	122	4	no	—
269	C₄⋊C₄	—	122	4	no	—
270	C₄⋊C₄	C₂×C₄×D₆₁	1	4	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	122	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	61	2	no	—
276	C₄²	—	61	2	no	—
277	C₄²	—	61	2	no	—
278	C₄²	—	122	2	no	—
279	C₄²	—	122	1	no	—
280	C₄²	—	244	4	no	—
281	C₄²	Q₈×D₆₁	1	2	yes	—
282	C₂³	#41×D₆₁	1	1	yes	characteristic
283	Q₈	—	488	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	61	4	no	—
288	C₂×C₄	—	122	4	no	—
289	C₂×C₄	—	122	2	no	—
290	C₂×C₄	—	122	4	no	—
291	C₂×C₄	—	244	2	no	—
292	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
296	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
297	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	61	4	no	—
301	C₈	—	61	2	no	—
302	C₈	—	61	2	no	—
303	C₈	—	244	2	no	—
304	C₈	#14519742	1	2	yes	—
305	C₈	#14519742	1	2	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	C₄×D₄×D₆₁	1	1	yes	characteristic
308	C₂²	#16840314	1	1	yes	characteristic
309	C₂²	#16840318	1	1	yes	characteristic
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	122	4	no	—
313	C₄	—	122	4	no	—
314	C₄	—	244	4	no	—
315	C₄	C₄×Q₈×D₆₁	1	1	yes	characteristic
316	C₄	C₂×#14519742	1	1	yes	characteristic
317	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
318	C₄	#16840307	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	#21448390	1	1	yes	characteristic
322	C₂	#21454727	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1946	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₂×#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1946	D₆₁
hypocenter	C₆₁	#21736
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1946

Complements

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

N	H ≅ G/N	type
13. #16840170	C₄	semidirect
14. #16840169	C₄	semidirect
59. #14519688	Q₈	semidirect
68. C₈×Dic₆₁	Q₈	semidirect
200. C₆₁	#21736	semidirect

Maximal subgroups

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

subgroup	index
#21456342	2
#21454758	2
#21446794	2
#21446572	2
#21446564	2
#21445992	2
#21445702	2
#21445701	2
#21445700	2
#97×Dic₆₁	2
C₆₁×#1946	2
#21736	61