Subgroups — C₆₁×#19986 #29523419

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3362	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3287	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1678	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1631	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1623	C₂	1	2	yes	maximal, complemented
10	C₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₁	—	2	1	no	—
13	Q₈²×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#550	C₂²	1	2	yes	—
15	C₆₁×#500	—	2	1	no	—
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#493	C₂²	1	1	yes	characteristic
18	C₆₁×#492	C₂²	1	2	yes	—
19	C₆₁×#486	—	2	2	no	—
20	C₆₁×#486	C₂²	1	2	yes	—
21	C₆₁×#477	—	2	2	no	—
22	C₆₁×#477	C₂²	1	2	yes	—
23	C₆₁×#476	—	2	2	no	—
24	C₆₁×#476	C₂²	1	2	yes	—
25	C₆₁×#466	—	2	1	no	—
26	C₆₁×#460	—	2	2	no	—
27	C₆₁×#460	C₂²	1	2	yes	—
28	C₆₁×#460	C₂²	1	2	yes	—
29	C₆₁×#449	—	2	2	no	—
30	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
31	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆×C₆₁	—	2	1	no	—
33	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₆₁	—	2	2	no	—
35	C₄×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#335	C₂²	1	1	yes	characteristic
38	C₆₁×#334	C₂²	1	1	yes	characteristic
39	C₆₁×#333	C₂²	1	2	yes	—
40	C₆₁×#331	C₂²	1	2	yes	—
41	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₆₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₆₁	—	2	2	no	—
46	#135×C₆₁	—	4	4	no	—
47	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
48	C₂×SD₁₆×C₆₁	—	2	2	no	—
49	C₂×SD₁₆×C₆₁	—	2	2	no	—
50	C₂×SD₁₆×C₆₁	—	4	2	no	—
51	C₂×SD₁₆×C₆₁	D₄	1	1	yes	characteristic
52	C₂×D₈×C₆₁	—	2	1	no	—
53	C₂×D₈×C₆₁	—	2	1	no	—
54	C₂²×C₈×C₆₁	—	2	1	no	—
55	C₄⋊Q₈×C₆₁	—	2	2	no	—
56	C₄⋊Q₈×C₆₁	—	2	2	no	—
57	C₄⋊Q₈×C₆₁	—	4	2	no	—
58	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
59	#127×C₆₁	—	2	2	no	—
60	#127×C₆₁	C₂³	1	2	yes	—
61	#121×C₆₁	—	2	2	no	—
62	#121×C₆₁	—	4	2	no	—
63	C₄×Q₈×C₆₁	—	2	2	no	—
64	C₄×Q₈×C₆₁	—	2	2	no	—
65	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
66	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₁	—	2	2	no	—
68	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
69	#107×C₆₁	—	2	1	no	—
70	#107×C₆₁	—	2	1	no	—
71	#107×C₆₁	—	2	2	no	—
72	#107×C₆₁	D₄	1	1	yes	characteristic
73	#106×C₆₁	—	2	2	no	—
74	#106×C₆₁	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₁	—	2	2	no	—
76	C₄⋊C₈×C₆₁	—	2	1	no	—
77	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
80	#103×C₆₁	—	2	2	no	—
81	#103×C₆₁	D₄	1	1	yes	characteristic
82	#103×C₆₁	D₄	1	2	yes	—
83	#102×C₆₁	—	2	2	no	—
84	#102×C₆₁	—	2	2	no	—
85	#102×C₆₁	—	2	2	no	—
86	#102×C₆₁	—	2	2	no	—
87	#102×C₆₁	—	2	2	no	—
88	#102×C₆₁	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₁	—	2	1	no	—
90	C₄×C₈×C₆₁	—	2	2	no	—
91	C₄×C₈×C₆₁	—	2	1	no	—
92	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
95	#41×C₆₁	—	4	4	no	—
96	#41×C₆₁	—	8	2	no	—
97	C₂×Q₈×C₆₁	—	2	2	no	—
98	C₂×Q₈×C₆₁	—	4	2	no	—
99	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×D₄×C₆₁	—	2	2	no	—
102	C₂×D₄×C₆₁	—	2	2	no	—
103	C₂×D₄×C₆₁	—	2	2	no	—
104	C₂×D₄×C₆₁	—	4	2	no	—
105	C₂×D₄×C₆₁	—	4	2	no	—
106	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₁	—	2	1	no	—
108	C₂²×C₄×C₆₁	—	4	2	no	—
109	Q₁₆×C₆₁	SD₁₆	1	4	yes	complemented
110	SD₁₆×C₆₁	—	2	2	no	—
111	SD₁₆×C₆₁	—	4	4	no	—
112	SD₁₆×C₆₁	—	4	4	no	—
113	SD₁₆×C₆₁	—	8	4	no	—
114	D₈×C₆₁	—	4	2	no	—
115	D₈×C₆₁	—	8	1	no	—
116	C₂×C₈×C₆₁	—	2	1	no	—
117	C₂×C₈×C₆₁	—	2	2	no	—
118	C₂×C₈×C₆₁	—	2	1	no	—
119	C₂×C₈×C₆₁	—	2	1	no	—
120	C₂×C₈×C₆₁	—	4	2	no	—
121	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₆₁	—	2	2	no	—
126	C₄⋊C₄×C₆₁	—	2	1	no	—
127	C₄⋊C₄×C₆₁	—	2	2	no	—
128	C₄⋊C₄×C₆₁	—	2	2	no	—
129	C₄⋊C₄×C₆₁	—	4	2	no	—
130	C₄⋊C₄×C₆₁	—	4	2	no	—
131	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
133	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
134	C₂²⋊C₄×C₆₁	—	2	1	no	—
135	C₂²⋊C₄×C₆₁	—	4	2	no	—
136	C₄²×C₆₁	—	2	2	no	—
137	C₄²×C₆₁	—	2	1	no	—
138	C₄²×C₆₁	—	4	2	no	—
139	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
140	C₂³×C₆₁	—	2	1	no	—
141	C₂³×C₆₁	—	4	2	no	—
142	Q₈×C₆₁	—	2	2	no	—
143	Q₈×C₆₁	—	2	1	no	—
144	Q₈×C₆₁	—	8	2	no	—
145	Q₈×C₆₁	—	16	2	no	—
146	Q₈×C₆₁	C₂×SD₁₆	1	4	yes	—
147	Q₈×C₆₁	#135	1	2	yes	complemented
148	D₄×C₆₁	—	2	1	no	—
149	D₄×C₆₁	—	2	1	no	—
150	D₄×C₆₁	—	4	2	no	—
151	D₄×C₆₁	—	4	2	no	—
152	D₄×C₆₁	—	4	2	no	—
153	D₄×C₆₁	—	4	2	no	—
154	D₄×C₆₁	—	8	4	no	—
155	D₄×C₆₁	—	8	2	no	—
156	D₄×C₆₁	—	16	2	no	—
157	C₂×C₄×C₆₁	—	2	1	no	—
158	C₂×C₄×C₆₁	—	2	2	no	—
159	C₂×C₄×C₆₁	—	2	2	no	—
160	C₂×C₄×C₆₁	—	2	1	no	—
161	C₂×C₄×C₆₁	—	2	1	no	—
162	C₂×C₄×C₆₁	—	4	1	no	—
163	C₂×C₄×C₆₁	—	4	1	no	—
164	C₂×C₄×C₆₁	—	4	2	no	—
165	C₂×C₄×C₆₁	—	8	4	no	—
166	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
169	C₈×C₆₁	—	2	2	no	—
170	C₈×C₆₁	—	2	1	no	—
171	C₈×C₆₁	—	2	1	no	—
172	C₈×C₆₁	—	2	1	no	—
173	C₈×C₆₁	—	4	1	no	—
174	C₈×C₆₁	—	4	2	no	—
175	C₈×C₆₁	—	4	1	no	—
176	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
177	#19986	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₆₁	—	4	1	no	—
179	C₂²×C₆₁	—	4	1	no	—
180	C₂²×C₆₁	—	4	1	no	—
181	C₂²×C₆₁	—	8	2	no	—
182	C₂²×C₆₁	—	8	2	no	—
183	C₂²×C₆₁	—	8	2	no	—
184	C₂²×C₆₁	D₄²	1	1	yes	characteristic
185	C₄×C₆₁	—	2	4	no	—
186	C₄×C₆₁	—	2	2	no	—
187	C₄×C₆₁	—	2	1	no	—
188	C₄×C₆₁	—	4	1	no	—
189	C₄×C₆₁	—	4	2	no	—
190	C₄×C₆₁	—	4	1	no	—
191	C₄×C₆₁	—	8	2	no	—
192	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
193	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
194	C₄×C₆₁	#577	1	1	yes	characteristic
195	C₄×C₆₁	#578	1	1	yes	characteristic
196	Q₈×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
197	#3362	C₂×C₆₁	1	2	yes	complemented
198	#3287	C₂×C₆₁	1	1	yes	characteristic
199	#1706	C₂×C₆₁	1	1	yes	characteristic, complemented
200	#1678	C₂×C₆₁	1	2	yes	complemented
201	#1675	C₂×C₆₁	1	1	yes	characteristic, complemented
202	#1668	C₂×C₆₁	1	1	yes	characteristic, complemented
203	#1631	C₂×C₆₁	1	2	yes	complemented
204	#1623	C₂×C₆₁	1	2	yes	complemented
205	C₈×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
206	C₈×SD₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
207	C₂×C₆₁	—	8	1	no	—
208	C₂×C₆₁	—	16	2	no	—
209	C₂×C₆₁	D₄×SD₁₆	1	1	yes	characteristic
210	C₂×C₆₁	#3285	1	1	yes	characteristic
211	C₂×C₆₁	#3292	1	1	yes	characteristic
212	C₂×#135	—	2	1	no	—
213	Q₈²	C₂²×C₆₁	1	2	yes	—
214	#550	C₂²×C₆₁	1	2	yes	—
215	#500	—	2	1	no	—
216	#500	C₂²×C₆₁	1	1	yes	characteristic
217	#493	C₂²×C₆₁	1	1	yes	characteristic
218	#492	C₂²×C₆₁	1	2	yes	—
219	#486	—	2	2	no	—
220	#486	C₂²×C₆₁	1	2	yes	—
221	#477	—	2	2	no	—
222	#477	C₂²×C₆₁	1	2	yes	—
223	#476	—	2	2	no	—
224	#476	C₂²×C₆₁	1	2	yes	—
225	#466	—	2	1	no	—
226	#460	—	2	2	no	—
227	#460	C₂²×C₆₁	1	2	yes	—
228	#460	C₂²×C₆₁	1	2	yes	—
229	#449	—	2	2	no	—
230	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
231	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
232	C₄×Q₁₆	—	2	1	no	—
233	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
234	C₄×SD₁₆	—	2	2	no	—
235	C₄×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
236	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
237	#335	C₂²×C₆₁	1	1	yes	characteristic
238	#334	C₂²×C₆₁	1	1	yes	characteristic
239	#333	C₂²×C₆₁	1	2	yes	—
240	#331	C₂²×C₆₁	1	2	yes	—
241	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
242	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
243	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
244	C₈²	C₂²×C₆₁	1	1	yes	characteristic
245	C₆₁	#19986	1	1	yes	characteristic, direct product factor, Sylow
246	C₂×#41	—	2	2	no	—
247	#135	—	4	4	no	—
248	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic
249	C₂×SD₁₆	—	2	2	no	—
250	C₂×SD₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	4	2	no	—
252	C₂×SD₁₆	D₄×C₆₁	1	1	yes	characteristic
253	C₂×D₈	—	2	1	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂²×C₈	—	2	1	no	—
256	C₄⋊Q₈	—	2	2	no	—
257	C₄⋊Q₈	—	2	2	no	—
258	C₄⋊Q₈	—	4	2	no	—
259	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
260	#127	—	2	2	no	—
261	#127	C₂³×C₆₁	1	2	yes	—
262	#121	—	2	2	no	—
263	#121	—	4	2	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
267	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
268	C₄×D₄	—	2	2	no	—
269	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
270	#107	—	2	1	no	—
271	#107	—	2	1	no	—
272	#107	—	2	2	no	—
273	#107	D₄×C₆₁	1	1	yes	characteristic
274	#106	—	2	2	no	—
275	#106	D₄×C₆₁	1	1	yes	characteristic
276	C₄⋊C₈	—	2	2	no	—
277	C₄⋊C₈	—	2	1	no	—
278	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
279	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
280	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
281	#103	—	2	2	no	—
282	#103	D₄×C₆₁	1	1	yes	characteristic
283	#103	D₄×C₆₁	1	2	yes	—
284	#102	—	2	2	no	—
285	#102	—	2	2	no	—
286	#102	—	2	2	no	—
287	#102	—	2	2	no	—
288	#102	—	2	2	no	—
289	#102	D₄×C₆₁	1	1	yes	characteristic
290	C₂²⋊C₈	—	2	1	no	—
291	C₄×C₈	—	2	2	no	—
292	C₄×C₈	—	2	1	no	—
293	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
294	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
295	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
296	#41	—	4	4	no	—
297	#41	—	8	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
301	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	4	2	no	—
310	Q₁₆	SD₁₆×C₆₁	1	4	yes	complemented
311	SD₁₆	—	2	2	no	—
312	SD₁₆	—	4	4	no	—
313	SD₁₆	—	4	4	no	—
314	SD₁₆	—	8	4	no	—
315	D₈	—	4	2	no	—
316	D₈	—	8	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
323	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
324	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
325	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
333	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
334	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
335	C₂²⋊C₄	—	2	1	no	—
336	C₂²⋊C₄	—	4	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	4	2	no	—
340	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
341	C₂³	—	2	1	no	—
342	C₂³	—	4	2	no	—
343	Q₈	—	2	2	no	—
344	Q₈	—	2	1	no	—
345	Q₈	—	8	2	no	—
346	Q₈	—	16	2	no	—
347	Q₈	C₂×SD₁₆×C₆₁	1	4	yes	—
348	Q₈	#135×C₆₁	1	2	yes	complemented
349	D₄	—	2	1	no	—
350	D₄	—	2	1	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	8	4	no	—
356	D₄	—	8	2	no	—
357	D₄	—	16	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	4	no	—
367	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
369	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	2	no	—
376	C₈	—	4	1	no	—
377	C₈	C₂×SD₁₆×C₆₁	1	2	yes	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	1	no	—
380	C₂²	—	4	1	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	D₄²×C₆₁	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	2	4	no	—
387	C₄	—	2	2	no	—
388	C₄	—	4	2	no	—
389	C₄	—	4	1	no	—
390	C₄	—	4	1	no	—
391	C₄	—	8	2	no	—
392	C₄	C₂²×SD₁₆×C₆₁	1	1	yes	characteristic
393	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
394	C₄	C₆₁×#577	1	1	yes	characteristic
395	C₄	C₆₁×#578	1	1	yes	characteristic
396	C₂	—	8	1	no	—
397	C₂	—	16	2	no	—
398	C₂	D₄×SD₁₆×C₆₁	1	1	yes	characteristic
399	C₂	C₆₁×#3285	1	1	yes	characteristic
400	C₂	C₆₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3287	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₆₁	C₂	semidirect
2. C₆₁×#3362	C₂	semidirect
4. C₆₁×#1706	C₂	semidirect
5. C₆₁×#1678	C₂	semidirect
6. C₆₁×#1675	C₂	semidirect
7. C₆₁×#1668	C₂	semidirect
8. C₆₁×#1631	C₂	semidirect
9. C₆₁×#1623	C₂	semidirect
10. C₈×Q₁₆×C₆₁	C₂	semidirect
11. C₈×SD₁₆×C₆₁	C₂	semidirect
109. Q₁₆×C₆₁	SD₁₆	semidirect
147. Q₈×C₆₁	#135	semidirect
177. #19986	C₆₁	direct (N × H)
196. Q₈×Q₁₆	C₂×C₆₁	semidirect
197. #3362	C₂×C₆₁	semidirect
199. #1706	C₂×C₆₁	semidirect
200. #1678	C₂×C₆₁	semidirect
201. #1675	C₂×C₆₁	semidirect
202. #1668	C₂×C₆₁	semidirect
203. #1631	C₂×C₆₁	semidirect
204. #1623	C₂×C₆₁	semidirect
205. C₈×Q₁₆	C₂×C₆₁	semidirect
206. C₈×SD₁₆	C₂×C₆₁	semidirect
245. C₆₁	#19986	direct (N × H)
310. Q₁₆	SD₁₆×C₆₁	semidirect
348. Q₈	#135×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆×C₆₁	2
C₆₁×#3362	2
C₆₁×#3287	2
C₆₁×#1706	2
C₆₁×#1678	2
C₆₁×#1675	2
C₆₁×#1668	2
C₆₁×#1631	2
C₆₁×#1623	2
C₈×Q₁₆×C₆₁	2
C₈×SD₁₆×C₆₁	2
#19986	61

Subgroups — C61×#19986 #29523419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19986 #29523419