Subgroups — #29588281

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839786	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839526	C₂	1	2	yes	maximal
3	C₂×#16839526	C₂	1	2	yes	maximal
4	C₂×#16839517	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	2	yes	—
7	C₂³×#92418	C₂²	1	2	yes	—
8	#16839786	—	2	4	no	—
9	C₂×C₄⋊#92418	—	2	2	no	—
10	#16839526	—	2	4	no	—
11	#16839526	—	2	4	no	—
12	#16839517	C₄	1	2	yes	—
13	#16839517	C₄	1	2	yes	—
14	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#336	—	2	2	no	—
18	C₂×#11758194	—	2	2	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	—	2	2	no	—
23	C₂×#11758194	D₄	1	4	yes	—
24	C₂²×#92418	—	2	2	no	—
25	C₂²×#92418	—	2	2	no	—
26	C₂²×#92418	—	2	2	no	—
27	C₂²×#92418	—	2	2	no	—
28	C₂²×#92418	—	2	2	no	—
29	C₂²×#92418	—	2	2	no	—
30	C₂²×#92418	—	2	2	no	—
31	C₂²×#92418	C₂×C₄	1	4	yes	—
32	C₄⋊#92418	—	2	8	no	—
33	C₂³×C₄×C₆₁	Q₈	1	2	yes	—
34	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₁	—	2	1	no	—
36	C₂×M₁₆×C₆₁	—	2	1	no	—
37	C₂×M₁₆×C₆₁	—	2	2	no	—
38	C₂×M₁₆×C₆₁	—	2	2	no	—
39	C₂×M₁₆×C₆₁	—	2	2	no	—
40	C₂×M₁₆×C₆₁	—	2	2	no	—
41	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₆₁	Q₈	1	2	yes	—
43	C₂²×C₈×C₆₁	—	2	1	no	—
44	C₂²×C₈×C₆₁	—	2	1	no	—
45	C₂×C₄²×C₆₁	—	2	2	no	—
46	C₂×C₄²×C₆₁	—	2	2	no	—
47	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₁	D₄	1	2	yes	—
50	C₂⁴×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	—	2	4	no	—
52	C₂²×C₄×C₆₁	—	2	4	no	—
53	C₂²×C₄×C₆₁	—	4	2	no	—
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	#41	1	2	yes	—
62	C₂²×C₄×C₆₁	#41	1	2	yes	—
63	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	#41	1	2	yes	—
65	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
66	M₁₆×C₆₁	—	2	4	no	—
67	M₁₆×C₆₁	—	2	4	no	—
68	M₁₆×C₆₁	—	4	2	no	—
69	M₁₆×C₆₁	—	4	2	no	—
70	C₂×C₈×C₆₁	—	2	2	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	—	4	1	no	—
73	C₂×C₈×C₆₁	—	4	1	no	—
74	C₂×C₈×C₆₁	—	4	1	no	—
75	C₂×C₈×C₆₁	—	4	1	no	—
76	C₄²×C₆₁	—	2	2	no	—
77	C₄²×C₆₁	—	2	2	no	—
78	C₄²×C₆₁	—	2	2	no	—
79	C₄²×C₆₁	—	2	2	no	—
80	#11758194	—	2	4	no	—
81	#11758194	—	2	4	no	—
82	#11758194	—	4	2	no	—
83	#11758194	—	4	2	no	—
84	C₂×#92418	—	2	4	no	—
85	C₂×#92418	—	2	4	no	—
86	C₂×#92418	—	2	4	no	—
87	C₂×#92418	—	2	4	no	—
88	C₂×#92418	—	4	2	no	—
89	C₂×#92418	—	4	2	no	—
90	C₂×#92418	—	4	2	no	—
91	C₂×#92418	—	4	2	no	—
92	C₂×#92418	—	4	2	no	—
93	C₂×#92418	—	4	2	no	—
94	C₂³×C₆₁	—	2	2	no	—
95	C₂³×C₆₁	—	2	2	no	—
96	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₆₁	#125	1	2	yes	—
99	C₂³×C₆₁	#125	1	1	yes	characteristic
100	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	—	2	2	no	—
102	C₂×C₄×C₆₁	—	2	2	no	—
103	C₂×C₄×C₆₁	—	2	2	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	4	8	no	—
107	C₂×C₄×C₆₁	#108	1	4	yes	—
108	C₂×C₄×C₆₁	#117	1	2	yes	—
109	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
113	C₂×C₄×C₆₁	#121	1	2	yes	—
114	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₆₁	#123	1	2	yes	—
118	C₂×C₄×C₆₁	#123	1	2	yes	—
119	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#124	1	2	yes	—
121	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
123	C₂×C₄×C₆₁	#125	1	2	yes	—
124	C₂×C₄×C₆₁	#126	1	2	yes	—
125	C₂×C₄×C₆₁	#126	1	2	yes	—
126	C₈×C₆₁	—	4	2	no	—
127	C₈×C₆₁	—	4	2	no	—
128	#92418	—	4	4	no	—
129	#92418	—	4	4	no	—
130	#11269	—	61	1	no	maximal, Sylow
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	—	2	4	no	—
134	C₂²×C₆₁	—	2	2	no	—
135	C₂²×C₆₁	—	2	2	no	—
136	C₂²×C₆₁	#382	1	1	yes	characteristic
137	C₂²×C₆₁	#385	1	1	yes	characteristic
138	C₂²×C₆₁	#389	1	1	yes	characteristic
139	C₂²×C₆₁	#398	1	2	yes	—
140	C₂²×C₆₁	#400	1	2	yes	—
141	C₂²×C₆₁	C₂×#108	1	2	yes	—
142	C₂²×C₆₁	#430	1	2	yes	—
143	C₄×C₆₁	—	2	4	no	—
144	C₄×C₆₁	—	4	2	no	—
145	C₄×C₆₁	#387	1	2	yes	—
146	C₄×C₆₁	#388	1	2	yes	—
147	C₄×C₆₁	#393	1	1	yes	characteristic
148	C₄×C₆₁	#394	1	1	yes	characteristic
149	C₄×C₆₁	#397	1	1	yes	characteristic
150	C₄×C₆₁	#400	1	1	yes	characteristic
151	C₂×#423	—	61	1	no	—
152	C₂×#341	—	61	2	no	—
153	C₂×#341	—	61	2	no	—
154	#1728	D₆₁	1	1	yes	characteristic
155	C₂×#336	—	61	1	no	—
156	C₂×C₆₁	—	2	2	no	—
157	C₂×C₆₁	—	2	2	no	—
158	C₂×C₆₁	#1442	1	1	yes	characteristic
159	C₂×C₆₁	#1837	1	1	yes	characteristic
160	C₂×C₆₁	#1838	1	1	yes	characteristic
161	C₂×C₆₁	#1855	1	2	yes	—
162	C₂×C₆₁	#1950	1	2	yes	—
163	C₂²×M₁₆	—	61	2	no	—
164	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
165	C₂²×M₁₆	C₂×D₆₁	1	1	yes	characteristic
166	C₂³×C₈	—	61	2	no	—
167	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
168	#423	—	122	4	no	—
169	C₂×C₄⋊C₈	—	122	2	no	—
170	#341	—	122	4	no	—
171	#341	—	122	4	no	—
172	#336	—	2	2	no	—
173	#336	—	61	2	no	—
174	#336	—	61	2	no	—
175	C₆₁	#11269	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	Dic₁₂₂	1	2	yes	—
177	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	61	4	no	—
185	C₂×M₁₆	—	122	2	no	—
186	C₂×M₁₆	—	122	2	no	—
187	C₂×M₁₆	—	122	2	no	—
188	C₂×M₁₆	—	122	2	no	—
189	C₂×M₁₆	—	122	2	no	—
190	C₂×M₁₆	Dic₁₂₂	1	2	yes	—
191	C₂×M₁₆	D₂₄₄	1	2	yes	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	—	61	4	no	—
195	C₂²×C₈	—	122	2	no	—
196	C₂²×C₈	—	122	2	no	—
197	C₂²×C₈	—	122	2	no	—
198	C₂²×C₈	—	122	2	no	—
199	C₂²×C₈	—	122	2	no	—
200	C₂²×C₈	—	122	2	no	—
201	C₂²×C₈	—	122	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
205	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
207	C₄⋊C₈	—	122	8	no	—
208	C₂⁴	C₂×Dic₁₂₂	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	#11758196	1	2	yes	—
213	C₂²×C₄	#11758196	1	2	yes	—
214	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
217	C₂²×C₄	#11758222	1	1	yes	characteristic
218	C₂²×C₄	#11758222	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₆₁	1	2	yes	—
220	C₂²×C₄	#11758224	1	2	yes	—
221	C₂²×C₄	#11758224	1	2	yes	—
222	C₂²×C₄	#11758226	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	122	4	no	—
229	M₁₆	—	122	4	no	—
230	M₁₆	—	244	2	no	—
231	M₁₆	—	244	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	122	4	no	—
239	C₂×C₈	—	122	4	no	—
240	C₂×C₈	—	122	4	no	—
241	C₂×C₈	—	122	4	no	—
242	C₂×C₈	—	244	2	no	—
243	C₂×C₈	—	244	2	no	—
244	C₂×C₈	—	244	2	no	—
245	C₂×C₈	—	244	2	no	—
246	C₂×C₈	—	244	2	no	—
247	C₂×C₈	—	244	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×#11758196	1	1	yes	characteristic
255	C₂³	#14519754	1	2	yes	—
256	C₂³	#14519768	1	2	yes	—
257	C₂³	#14519769	1	1	yes	characteristic
258	C₂³	#14519777	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#14519712	1	4	yes	—
266	C₂×C₄	#14519735	1	2	yes	—
267	C₂×C₄	#14519737	1	2	yes	—
268	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
269	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
270	C₂×C₄	#14519745	1	1	yes	characteristic
271	C₂×C₄	#14519751	1	1	yes	characteristic
272	C₂×C₄	#14519752	1	2	yes	—
273	C₂×C₄	#14519754	1	2	yes	—
274	C₂×C₄	#14519756	1	1	yes	characteristic
275	C₂×C₄	#14519758	1	1	yes	characteristic
276	C₂×C₄	#14519761	1	2	yes	—
277	C₂×C₄	#14519762	1	2	yes	—
278	C₂×C₄	#14519763	1	2	yes	—
279	C₂×C₄	#14519764	1	1	yes	characteristic
280	C₂×C₄	#14519766	1	1	yes	characteristic
281	C₂×C₄	#14519768	1	2	yes	—
282	C₂×C₄	#14519770	1	2	yes	—
283	C₂×C₄	#14519772	1	2	yes	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	—	244	4	no	—
287	C₈	—	244	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#16839627	1	1	yes	characteristic
294	C₂²	#16839634	1	1	yes	characteristic
295	C₂²	#16839660	1	1	yes	characteristic
296	C₂²	#16839694	1	2	yes	—
297	C₂²	#16839704	1	2	yes	—
298	C₂²	C₂×#14519712	1	2	yes	—
299	C₂²	#16839810	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#16839649	1	2	yes	—
303	C₄	#16839654	1	2	yes	—
304	C₄	#16839674	1	1	yes	characteristic
305	C₄	#16839679	1	1	yes	characteristic
306	C₄	#16839687	1	1	yes	characteristic
307	C₄	#16839701	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#21444075	1	1	yes	characteristic
311	C₂	#21445956	1	1	yes	characteristic
312	C₂	#21445959	1	1	yes	characteristic
313	C₂	#21446023	1	2	yes	—
314	C₂	#21446594	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₁
Fitting	C₆₁×#1728	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₂×Q₈
cosocle	C₂³×C₄×C₆₁	C₂³
hypercenter	#1728	D₆₁
hypocenter	C₆₁	#11269
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
175. C₆₁	#11269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839786	2
C₂×#16839526	2
C₂×#16839517	2
C₆₁×#1728	2
#11269	61