Subgroups — #29699933

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448296	C₂	1	1	yes	maximal, characteristic
2	#21446759	C₂	1	2	yes	maximal
3	#21446386	C₂	1	2	yes	maximal
4	#21446379	C₂	1	2	yes	maximal
5	#21445971	C₂	1	1	yes	maximal, characteristic
6	#21445938	C₂	1	2	yes	maximal
7	#21445691	C₂	1	2	yes	maximal
8	#21445678	C₂	1	2	yes	maximal
9	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×C₄⋊#92418	C₂²	1	2	yes	—
13	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
15	C₂×#14519687	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
17	#16839668	C₄	1	1	yes	characteristic
18	#16839668	C₄	1	1	yes	characteristic
19	#16839660	C₄	1	1	yes	characteristic
20	#16839660	C₄	1	1	yes	characteristic
21	#16839646	C₄	1	2	yes	—
22	#16839646	C₄	1	2	yes	—
23	C₄×#11758196	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
25	#16839608	C₂²	1	2	yes	—
26	#16839516	C₂²	1	2	yes	—
27	#16839516	C₂²	1	2	yes	—
28	#16839516	C₂²	1	2	yes	—
29	#16839516	C₂²	1	2	yes	—
30	#16839516	C₂²	1	2	yes	—
31	#16839516	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#97×C₆₁	C₂²	1	2	yes	—
35	C₄³×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#336	C₂²	1	2	yes	—
37	C₂²×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
38	C₂²×#92418	C₂³	1	2	yes	—
39	C₂²×#92418	C₂³	1	2	yes	—
40	C₄×Dic₁₂₂	—	2	2	no	—
41	C₄×Dic₁₂₂	—	2	2	no	—
42	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂×#11758196	C₂×C₄	1	2	yes	—
46	C₂×#11758196	C₂×C₄	1	2	yes	—
47	C₂×#11758196	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
49	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
50	C₄⋊#92418	—	2	4	no	—
51	C₄⋊#92418	—	2	4	no	—
52	C₄⋊#92418	—	4	1	no	—
53	C₄⋊#92418	—	4	1	no	—
54	#14519687	—	4	2	no	—
55	C₈×Dic₆₁	—	4	2	no	—
56	#14519684	C₂×C₄	1	2	yes	—
57	#14519684	C₂×C₄	1	2	yes	—
58	#14519684	C₂×C₄	1	2	yes	—
59	#14519684	C₂×C₄	1	2	yes	—
60	#14519684	C₂×C₄	1	2	yes	—
61	#14519684	C₂×C₄	1	2	yes	—
62	#14519684	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
65	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₁	—	2	2	no	—
73	C₄⋊C₈×C₆₁	Q₈	1	4	yes	complemented
74	#97×C₆₁	—	2	4	no	—
75	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
85	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₂₂	—	2	2	no	—
87	C₂×Dic₁₂₂	—	4	1	no	—
88	C₂×Dic₁₂₂	—	4	1	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	—	2	2	no	—
93	C₂×C₈×C₆₁	—	2	2	no	—
94	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	—	2	1	no	—
97	C₄²×C₆₁	—	2	1	no	—
98	C₄²×C₆₁	—	4	1	no	—
99	C₄²×C₆₁	—	4	1	no	—
100	C₄²×C₆₁	—	4	1	no	—
101	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
102	C₄²×C₆₁	#41	1	2	yes	—
103	C₄²×C₆₁	#41	1	2	yes	—
104	C₄²×C₆₁	#41	1	2	yes	—
105	C₄⋊Dic₆₁	—	2	2	no	—
106	C₄⋊Dic₆₁	—	4	1	no	—
107	C₄⋊Dic₆₁	—	4	1	no	—
108	#11758196	—	2	4	no	—
109	#11758196	—	2	4	no	—
110	#11758196	—	4	2	no	—
111	C₄×Dic₆₁	—	2	2	no	—
112	C₄×Dic₆₁	—	2	2	no	—
113	C₄×Dic₆₁	—	2	2	no	—
114	C₄×Dic₆₁	—	2	2	no	—
115	C₂×#92418	—	2	2	no	—
116	C₂×#92418	—	2	2	no	—
117	C₂×#92418	—	2	2	no	—
118	C₂×#92418	—	2	2	no	—
119	C₂×#92418	—	2	2	no	—
120	C₂×#92418	—	2	2	no	—
121	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	2	1	no	—
125	C₂×C₄×C₆₁	—	2	1	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	1	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	2	1	no	—
132	C₂×C₄×C₆₁	—	2	1	no	—
133	C₂×C₄×C₆₁	—	2	1	no	—
134	C₂×C₄×C₆₁	—	2	1	no	—
135	C₂×C₄×C₆₁	#117	1	2	yes	—
136	C₂×C₄×C₆₁	#117	1	2	yes	—
137	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₆₁	#131	1	2	yes	—
140	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×Dic₆₁	—	2	2	no	—
147	C₂×Dic₆₁	—	2	2	no	—
148	C₂×Dic₆₁	—	2	2	no	—
149	C₂×Dic₆₁	—	2	2	no	—
150	C₂×Dic₆₁	—	2	2	no	—
151	C₂×Dic₆₁	—	2	2	no	—
152	C₂×Dic₆₁	—	2	2	no	—
153	C₂×Dic₆₁	—	2	2	no	—
154	C₂×Dic₆₁	—	2	2	no	—
155	C₂×Dic₆₁	—	2	2	no	—
156	C₂×Dic₆₁	—	2	2	no	—
157	C₂×Dic₆₁	—	2	2	no	—
158	Dic₁₂₂	—	4	4	no	—
159	C₈×C₆₁	—	2	4	no	—
160	C₈×C₆₁	—	4	2	no	—
161	#92418	—	4	2	no	—
162	#92418	—	4	2	no	—
163	#22824	—	61	1	no	maximal, Sylow
164	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
165	C₂²×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₂²×C₆₁	#520	1	1	yes	characteristic
167	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
168	C₂²×C₆₁	#568	1	1	yes	characteristic
169	C₂²×C₆₁	#568	1	1	yes	characteristic
170	C₂²×C₆₁	#568	1	1	yes	characteristic
171	C₄×C₆₁	—	2	2	no	—
172	C₄×C₆₁	—	2	2	no	—
173	C₄×C₆₁	—	2	2	no	—
174	C₄×C₆₁	—	2	2	no	—
175	C₄×C₆₁	—	2	2	no	—
176	C₄×C₆₁	—	4	1	no	—
177	C₄×C₆₁	#554	1	1	yes	characteristic
178	C₄×C₆₁	#556	1	1	yes	characteristic
179	C₄×C₆₁	#557	1	1	yes	characteristic
180	C₄×C₆₁	#559	1	1	yes	characteristic
181	Dic₆₁	—	4	2	no	—
182	Dic₆₁	—	4	2	no	—
183	Dic₆₁	—	4	2	no	—
184	Dic₆₁	—	4	2	no	—
185	#2295	—	61	1	no	—
186	#1985	—	61	2	no	—
187	#1917	—	61	2	no	—
188	#1916	—	61	2	no	—
189	#1842	—	61	1	no	—
190	#1832	—	61	2	no	—
191	#1771	—	61	2	no	—
192	#1768	—	61	2	no	—
193	#1767	D₆₁	1	1	yes	characteristic
194	C₂×C₆₁	#2355	1	1	yes	characteristic
195	C₂×C₆₁	#2921	1	1	yes	characteristic
196	C₂×C₆₁	#2925	1	1	yes	characteristic
197	C₂×C₆₁	#2961	1	1	yes	characteristic
198	C₂×C₆₁	#2962	1	1	yes	characteristic
199	C₂×C₆₁	#2992	1	1	yes	characteristic
200	C₂×C₆₁	#2995	1	1	yes	characteristic
201	C₂×C₄×Q₈	—	61	1	no	—
202	C₂×C₄⋊C₈	—	61	2	no	—
203	C₂×C₄⋊C₈	—	61	2	no	—
204	C₂×C₄⋊C₈	—	61	1	no	—
205	C₂×C₄⋊C₈	—	61	1	no	—
206	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
207	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
208	C₂×#97	—	61	2	no	—
209	C₂×#97	C₂×D₆₁	1	2	yes	—
210	C₂×C₄×C₈	—	61	2	no	—
211	#391	—	61	1	no	—
212	#391	—	61	1	no	—
213	#389	—	61	1	no	—
214	#389	—	61	1	no	—
215	#387	—	61	2	no	—
216	#387	—	61	2	no	—
217	C₄×C₄⋊C₄	—	61	2	no	—
218	C₄×C₄⋊C₄	—	61	1	no	—
219	#376	—	61	2	no	—
220	C₄³	C₂×D₆₁	1	1	yes	characteristic
221	#336	—	61	2	no	—
222	#336	—	61	2	no	—
223	#336	—	61	2	no	—
224	#336	—	61	2	no	—
225	#336	—	61	2	no	—
226	#336	—	61	2	no	—
227	#336	C₂×D₆₁	1	2	yes	—
228	C₆₁	#22824	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	—	61	1	no	—
230	C₂²×C₈	—	61	2	no	—
231	C₂²×C₈	—	61	2	no	—
232	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
233	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
234	C₄×Q₈	—	122	2	no	—
235	C₄×Q₈	—	122	2	no	—
236	C₂×C₄⋊C₄	—	61	2	no	—
237	C₂×C₄⋊C₄	—	61	1	no	—
238	C₂×C₄⋊C₄	—	61	2	no	—
239	C₂×C₄⋊C₄	—	61	2	no	—
240	C₂×C₄⋊C₄	—	61	1	no	—
241	C₂×C₄⋊C₄	—	61	1	no	—
242	C₂×C₄²	—	61	2	no	—
243	C₂×C₄²	—	61	2	no	—
244	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
245	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
246	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
247	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
248	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	122	4	no	—
253	C₄⋊C₈	—	122	4	no	—
254	C₄⋊C₈	—	244	1	no	—
255	C₄⋊C₈	—	244	1	no	—
256	C₄⋊C₈	Dic₁₂₂	1	4	yes	complemented
257	#97	—	2	4	no	—
258	#97	—	244	2	no	—
259	C₄×C₈	—	244	2	no	—
260	#95	—	61	2	no	—
261	#95	—	61	2	no	—
262	#95	—	61	2	no	—
263	#95	—	61	2	no	—
264	#95	—	61	2	no	—
265	#95	—	61	2	no	—
266	#95	—	61	2	no	—
267	C₂×Q₈	—	122	2	no	—
268	C₂×Q₈	—	244	1	no	—
269	C₂×Q₈	—	244	1	no	—
270	C₂²×C₄	—	61	2	no	—
271	C₂²×C₄	—	61	2	no	—
272	C₂²×C₄	—	61	2	no	—
273	C₂²×C₄	—	61	2	no	—
274	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
280	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	122	2	no	—
287	C₂×C₈	—	122	2	no	—
288	C₂×C₈	—	122	2	no	—
289	C₂×C₈	—	122	2	no	—
290	C₂×C₈	—	122	2	no	—
291	C₂×C₈	—	122	2	no	—
292	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
293	C₄⋊C₄	—	122	4	no	—
294	C₄⋊C₄	—	122	2	no	—
295	C₄⋊C₄	—	122	4	no	—
296	C₄⋊C₄	—	244	2	no	—
297	C₄⋊C₄	—	244	1	no	—
298	C₄⋊C₄	—	244	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	122	2	no	—
306	C₄²	—	122	2	no	—
307	C₄²	—	122	2	no	—
308	C₄²	—	122	2	no	—
309	C₄²	C₂×Dic₁₂₂	1	2	yes	—
310	C₄²	#11758222	1	2	yes	—
311	C₄²	#11758224	1	2	yes	—
312	C₄²	#11758226	1	2	yes	—
313	C₂³	C₂²×C₄×D₆₁	1	1	yes	characteristic
314	Q₈	—	244	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	122	2	no	—
329	C₂×C₄	—	122	2	no	—
330	C₂×C₄	—	122	2	no	—
331	C₂×C₄	—	122	2	no	—
332	C₂×C₄	—	122	2	no	—
333	C₂×C₄	—	122	2	no	—
334	C₂×C₄	—	122	2	no	—
335	C₂×C₄	—	122	2	no	—
336	C₂×C₄	—	122	2	no	—
337	C₂×C₄	—	122	2	no	—
338	C₂×C₄	—	122	2	no	—
339	C₂×C₄	—	122	2	no	—
340	C₂×C₄	#14519735	1	2	yes	—
341	C₂×C₄	#14519735	1	2	yes	—
342	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
343	C₂×C₄	C₄×Dic₁₂₂	1	2	yes	—
344	C₂×C₄	#14519785	1	2	yes	—
345	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
346	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
347	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
348	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
349	C₂×C₄	#14519823	1	1	yes	characteristic
350	C₂×C₄	#14519824	1	1	yes	characteristic
351	C₈	—	2	4	no	—
352	C₈	—	4	2	no	—
353	C₈	—	244	2	no	—
354	C₈	—	244	2	no	—
355	C₂²	C₂×#14519735	1	1	yes	characteristic
356	C₂²	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
357	C₂²	#16840242	1	1	yes	characteristic
358	C₂²	C₂×#14519785	1	1	yes	characteristic
359	C₂²	#16840528	1	1	yes	characteristic
360	C₂²	#16840528	1	1	yes	characteristic
361	C₂²	#16840528	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	4	1	no	—
368	C₄	—	244	2	no	—
369	C₄	—	244	2	no	—
370	C₄	—	244	2	no	—
371	C₄	—	244	2	no	—
372	C₄	#16840469	1	1	yes	characteristic
373	C₄	#16840482	1	1	yes	characteristic
374	C₄	#16840491	1	1	yes	characteristic
375	C₄	#16840497	1	1	yes	characteristic
376	C₂	#21448774	1	1	yes	characteristic
377	C₂	#21454456	1	1	yes	characteristic
378	C₂	#21454487	1	1	yes	characteristic
379	C₂	#21454759	1	1	yes	characteristic
380	C₂	#21454766	1	1	yes	characteristic
381	C₂	#21454960	1	1	yes	characteristic
382	C₂	#21454991	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂³×C₄
Fitting	C₆₁×#1767	C₂
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1767	D₆₁
hypocenter	C₆₁	#22824
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1767

Complements

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

N	H ≅ G/N	type
73. C₄⋊C₈×C₆₁	Q₈	semidirect
228. C₆₁	#22824	semidirect
256. C₄⋊C₈	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21448296	2
#21446759	2
#21446386	2
#21446379	2
#21445971	2
#21445938	2
#21445691	2
#21445678	2
C₆₁×#1767	2
#22824	61