Subgroups — #29688887

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456341	C₂	1	1	yes	maximal, characteristic
2	#21454760	C₂	1	1	yes	maximal, characteristic
3	#21454756	C₂	1	1	yes	maximal, characteristic
4	#21446796	C₂	1	1	yes	maximal, characteristic
5	#21446571	C₂	1	2	yes	maximal
6	#21446566	C₂	1	2	yes	maximal
7	#21445993	C₂	1	1	yes	maximal, characteristic
8	#21445702	C₂	1	2	yes	maximal
9	#21445697	C₂	1	2	yes	maximal
10	C₄×#14519688	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1994	C₂	1	1	yes	maximal, characteristic
12	#16840319	C₂²	1	1	yes	characteristic
13	#16840172	C₄	1	4	yes	complemented
14	#16840168	C₄	1	4	yes	complemented
15	#16839893	—	2	4	no	—
16	#16839891	—	2	4	no	—
17	C₂×#14519712	C₂²	1	2	yes	—
18	C₂×#14519712	C₂²	1	2	yes	—
19	#16839798	C₂²	1	2	yes	—
20	#16839798	C₂²	1	2	yes	—
21	#16839790	C₂²	1	2	yes	—
22	#16839789	C₂²	1	2	yes	—
23	#16839786	C₂²	1	1	yes	characteristic
24	#16839720	C₂²	1	2	yes	—
25	#16839718	C₂²	1	2	yes	—
26	C₂×#14519688	C₂²	1	1	yes	characteristic
27	C₂×#14519688	C₂²	1	1	yes	characteristic
28	C₂×#14519688	C₂²	1	1	yes	characteristic
29	C₂×#14519688	C₂²	1	1	yes	characteristic
30	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
31	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
32	#16839522	C₂²	1	2	yes	—
33	#16839522	C₂²	1	2	yes	—
34	C₆₁×#533	C₂²	1	1	yes	characteristic
35	C₆₁×#423	C₂²	1	1	yes	characteristic
36	C₄³×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#339	C₂²	1	2	yes	—
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	C₂²×#92418	C₂³	1	1	yes	characteristic
43	C₂²×#92418	C₂³	1	1	yes	characteristic
44	#14519777	C₂×C₄	1	2	yes	—
45	#14519769	C₂×C₄	1	2	yes	—
46	#14519756	—	2	2	no	—
47	C₄×Dic₁₂₂	—	2	2	no	—
48	#14519736	C₂³	1	2	yes	—
49	#14519712	—	2	4	no	—
50	#14519712	—	2	4	no	—
51	#14519710	C₂×C₄	1	4	yes	—
52	#14519710	C₂×C₄	1	4	yes	—
53	#14519708	C₂×C₄	1	4	yes	—
54	#14519708	C₂×C₄	1	4	yes	—
55	C₄⋊#92418	—	2	2	no	—
56	C₄⋊#92418	—	2	4	no	—
57	#14519706	—	2	4	no	—
58	#14519688	—	2	2	no	—
59	#14519688	—	2	2	no	—
60	#14519688	C₂×C₄	1	2	yes	—
61	#14519688	C₂×C₄	1	2	yes	—
62	#14519688	D₄	1	4	yes	—
63	#14519688	Q₈	1	4	yes	complemented
64	#14519687	—	2	2	no	—
65	C₄×#92418	—	2	2	no	—
66	C₄×#92418	—	2	2	no	—
67	C₄×#92418	Q₈	1	4	yes	complemented
68	C₈×Dic₆₁	—	2	2	no	—
69	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
71	#125×C₆₁	C₂×C₄	1	2	yes	—
72	C₂²⋊Q₈×C₆₁	—	2	2	no	—
73	C₄×Q₈×C₆₁	—	2	2	no	—
74	#117×C₆₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₆₁	—	2	2	no	—
76	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	—	2	2	no	—
81	C₂²×C₄×C₆₁	—	2	1	no	—
82	C₂²×C₄×C₆₁	—	2	1	no	—
83	C₂²×C₄×C₆₁	#41	1	2	yes	—
84	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₁₂₂	—	2	2	no	—
86	M₁₆×C₆₁	—	2	2	no	—
87	M₁₆×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₄⋊C₄×C₆₁	—	2	4	no	—
90	C₄⋊C₄×C₆₁	—	2	2	no	—
91	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₆₁	—	2	2	no	—
93	C₄²×C₆₁	—	2	2	no	—
94	C₄²×C₆₁	—	2	1	no	—
95	C₄²×C₆₁	—	2	1	no	—
96	C₄²×C₆₁	—	4	4	no	—
97	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	2	yes	—
100	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
101	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
102	C₂²⋊Dic₆₁	—	2	2	no	—
103	C₄⋊Dic₆₁	—	2	2	no	—
104	C₄⋊Dic₆₁	C₂²×C₄	1	4	yes	—
105	#11758196	—	2	4	no	—
106	C₄×Dic₆₁	—	2	2	no	—
107	#11758194	—	2	2	no	—
108	#11758194	—	2	2	no	—
109	C₂×#92418	—	2	1	no	—
110	C₂×#92418	—	2	1	no	—
111	C₂×#92418	—	2	2	no	—
112	C₂×#92418	C₂²×C₄	1	2	yes	—
113	C₂×#92418	C₂²×C₄	1	2	yes	—
114	C₂×#92418	C₂×D₄	1	2	yes	—
115	C₂×#92418	C₂×D₄	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
120	C₂×#92418	#41	1	2	yes	—
121	C₂×#92418	#41	1	2	yes	—
122	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
123	Q₈×C₆₁	—	8	2	no	—
124	C₂×C₄×C₆₁	—	2	4	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	4	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₆₁	—	2	4	no	—
139	Dic₁₂₂	—	8	2	no	—
140	C₈×C₆₁	—	4	2	no	—
141	#92418	—	2	2	no	—
142	#92418	—	4	2	no	—
143	#92418	C₄×Q₈	1	4	yes	—
144	#92418	C₄×Q₈	1	4	yes	—
145	#92418	C₄⋊Q₈	1	4	yes	—
146	#21736	—	61	1	no	maximal, Sylow
147	C₂²×C₆₁	—	2	2	no	—
148	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₆₁	#533	1	1	yes	characteristic
150	C₂²×C₆₁	#533	1	1	yes	characteristic
151	C₄×C₆₁	—	2	4	no	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	—	4	4	no	—
156	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₄×C₆₁	#532	1	1	yes	characteristic
160	Dic₆₁	—	4	4	no	—
161	#3162	—	61	1	no	—
162	#2961	—	61	1	no	—
163	#2961	—	61	1	no	—
164	#1994	—	61	1	no	—
165	#1994	D₆₁	1	1	yes	characteristic
166	#1946	—	61	2	no	—
167	#1945	—	61	2	no	—
168	#1849	—	61	1	no	—
169	#1773	—	61	2	no	—
170	#1772	—	61	2	no	—
171	C₄×#97	—	61	1	no	—
172	C₂×C₆₁	—	2	1	no	—
173	C₂×C₆₁	—	2	1	no	—
174	C₂×C₆₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₆₁	#2956	1	2	yes	—
176	#533	—	61	1	no	—
177	#533	C₂×D₆₁	1	1	yes	characteristic
178	#501	—	61	4	no	—
179	#501	—	61	4	no	—
180	#446	—	122	4	no	—
181	#446	—	122	4	no	—
182	C₂×#108	—	61	2	no	—
183	C₂×#108	—	61	2	no	—
184	#427	—	61	2	no	—
185	#427	—	61	2	no	—
186	#424	—	61	2	no	—
187	#424	—	61	2	no	—
188	#423	—	61	1	no	—
189	#423	C₂×D₆₁	1	1	yes	characteristic
190	#405	—	61	2	no	—
191	#405	—	61	2	no	—
192	C₂×#97	—	61	1	no	—
193	C₂×#97	—	61	1	no	—
194	C₂×#97	—	61	1	no	—
195	C₂×#97	—	61	1	no	—
196	C₂×C₄×C₈	—	61	1	no	—
197	C₂×C₄×C₈	—	61	1	no	—
198	C₄³	C₂×D₆₁	1	1	yes	characteristic
199	#339	—	61	2	no	—
200	#339	—	61	2	no	—
201	#339	C₂×D₆₁	1	2	yes	—
202	#339	C₂×D₆₁	1	2	yes	—
203	C₆₁	#21736	1	1	yes	characteristic, complemented, Sylow
204	C₂×M₁₆	—	61	2	no	—
205	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
206	C₂²×C₈	—	61	1	no	—
207	C₂²×C₈	—	61	1	no	—
208	C₂²×C₈	—	61	1	no	—
209	C₂²×C₈	—	61	1	no	—
210	C₄⋊Q₈	—	61	2	no	—
211	C₄⋊Q₈	C₄×D₆₁	1	2	yes	—
212	#125	—	61	2	no	—
213	#125	C₄×D₆₁	1	2	yes	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	122	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	122	2	no	—
218	#117	—	61	2	no	—
219	#117	C₂²×D₆₁	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
224	#108	—	122	4	no	—
225	#108	—	122	4	no	—
226	#107	—	61	4	no	—
227	#107	—	61	4	no	—
228	#106	—	61	4	no	—
229	#106	—	61	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	122	4	no	—
232	C₄⋊C₈	—	122	4	no	—
233	C₄⋊C₈	—	122	2	no	—
234	#97	—	61	2	no	—
235	#97	—	61	2	no	—
236	#97	—	61	4	no	—
237	#97	—	61	4	no	—
238	#97	—	122	2	no	—
239	#97	—	122	2	no	—
240	#97	—	122	2	no	—
241	C₄×C₈	—	61	4	no	—
242	C₄×C₈	—	122	2	no	—
243	C₄×C₈	—	122	2	no	—
244	C₄×C₈	—	122	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	122	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	#11758224	1	2	yes	—
250	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	122	2	no	—
254	M₁₆	—	122	2	no	—
255	C₂×C₈	—	2	2	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	—	61	2	no	—
261	C₂×C₈	—	61	2	no	—
262	C₂×C₈	—	61	2	no	—
263	C₂×C₈	—	61	2	no	—
264	C₂×C₈	—	61	2	no	—
265	C₂×C₈	—	61	2	no	—
266	C₂×C₈	—	122	1	no	—
267	C₂×C₈	—	122	1	no	—
268	C₂×C₈	—	122	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	61	4	no	—
272	C₄⋊C₄	—	122	4	no	—
273	C₄⋊C₄	—	122	2	no	—
274	C₄⋊C₄	C₂×C₄×D₆₁	1	4	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	122	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	—	122	2	no	—
282	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
284	C₄²	D₄×D₆₁	1	2	yes	—
285	C₄²	Q₈×D₆₁	1	2	yes	—
286	C₄²	Q₈×D₆₁	1	2	yes	—
287	C₂³	C₂×#11758224	1	1	yes	characteristic
288	Q₈	—	8	2	no	—
289	Q₈	—	488	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	122	4	no	—
297	C₂×C₄	#14519737	1	2	yes	—
298	C₂×C₄	#14519737	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
303	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
304	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	61	4	no	—
307	C₈	—	61	4	no	—
308	C₈	—	61	4	no	—
309	C₈	—	122	2	no	—
310	C₈	—	244	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	C₂×#14519737	1	1	yes	characteristic
313	C₂²	#16840314	1	1	yes	characteristic
314	C₂²	#16840314	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	244	4	no	—
321	C₄	C₄×Q₈×D₆₁	1	1	yes	characteristic
322	C₄	C₄×Q₈×D₆₁	1	1	yes	characteristic
323	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
324	C₄	#16840311	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	#21448388	1	1	yes	characteristic
328	C₂	#21454725	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×#1994	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₂×#41
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1994	D₆₁
hypocenter	C₆₁	#21736
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

Complements

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

N	H ≅ G/N	type
13. #16840172	C₄	semidirect
14. #16840168	C₄	semidirect
63. #14519688	Q₈	semidirect
67. C₄×#92418	Q₈	semidirect
203. C₆₁	#21736	semidirect

Maximal subgroups

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

subgroup	index
#21456341	2
#21454760	2
#21454756	2
#21446796	2
#21446571	2
#21446566	2
#21445993	2
#21445702	2
#21445697	2
C₄×#14519688	2
C₆₁×#1994	2
#21736	61