Subgroups — C₆₁×#29811 #29533244

Order 15616.

924 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2621	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2619	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2055	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2000	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₁	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
14	C₂×#121×C₆₁	C₂²	1	2	yes	—
15	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#492	—	2	4	no	—
17	C₆₁×#433	—	2	4	no	—
18	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
19	C₂×#103×C₆₁	C₂²	1	4	yes	—
20	C₂×#102×C₆₁	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
22	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#398	—	2	2	no	—
25	C₆₁×#397	—	2	2	no	—
26	C₆₁×#394	—	2	2	no	—
27	C₆₁×#392	—	2	2	no	—
28	C₆₁×#391	C₂²	1	2	yes	—
29	C₆₁×#390	C₂²	1	2	yes	—
30	C₆₁×#382	C₂²	1	1	yes	characteristic
31	C₆₁×#382	C₂²	1	1	yes	characteristic
32	C₆₁×#340	C₂²	1	4	yes	—
33	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
34	C₂²×D₄×C₆₁	—	2	2	no	—
35	C₂²×D₄×C₆₁	—	2	2	no	—
36	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
38	C₂×SD₁₆×C₆₁	—	2	8	no	—
39	C₂×SD₁₆×C₆₁	—	4	4	no	—
40	C₂×SD₁₆×C₆₁	—	4	4	no	—
41	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₆₁	—	2	2	no	—
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	#127×C₆₁	—	2	2	no	—
46	#127×C₆₁	—	2	2	no	—
47	C₂²⋊Q₈×C₆₁	—	4	4	no	—
48	#121×C₆₁	—	4	4	no	—
49	#117×C₆₁	—	2	2	no	—
50	#117×C₆₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₆₁	—	2	4	no	—
63	#103×C₆₁	—	4	4	no	—
64	#102×C₆₁	—	4	4	no	—
65	C₂²⋊C₈×C₆₁	—	2	4	no	—
66	#97×C₆₁	—	2	2	no	—
67	C₄×C₈×C₆₁	D₄	1	4	yes	complemented
68	#95×C₆₁	—	2	2	no	—
69	#95×C₆₁	—	2	2	no	—
70	#95×C₆₁	—	2	4	no	—
71	C₂⁴×C₆₁	—	2	2	no	—
72	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×Q₈×C₆₁	—	2	2	no	—
74	C₂×Q₈×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	—	2	4	no	—
76	C₂×Q₈×C₆₁	—	2	4	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	2	2	no	—
79	C₂×D₄×C₆₁	—	2	4	no	—
80	C₂×D₄×C₆₁	—	2	4	no	—
81	C₂×D₄×C₆₁	—	4	4	no	—
82	C₂×D₄×C₆₁	—	4	4	no	—
83	C₂×D₄×C₆₁	—	8	4	no	—
84	C₂×D₄×C₆₁	—	8	4	no	—
85	C₂²×C₄×C₆₁	—	2	1	no	—
86	C₂²×C₄×C₆₁	—	2	1	no	—
87	C₂²×C₄×C₆₁	—	2	2	no	—
88	C₂²×C₄×C₆₁	—	2	2	no	—
89	C₂²×C₄×C₆₁	—	2	2	no	—
90	C₂²×C₄×C₆₁	—	2	2	no	—
91	C₂²×C₄×C₆₁	—	2	2	no	—
92	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
96	SD₁₆×C₆₁	—	4	16	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	2	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
103	C₄⋊C₄×C₆₁	—	2	4	no	—
104	C₄⋊C₄×C₆₁	—	2	4	no	—
105	C₄⋊C₄×C₆₁	—	2	4	no	—
106	C₄⋊C₄×C₆₁	—	4	4	no	—
107	C₄⋊C₄×C₆₁	—	4	4	no	—
108	C₂²⋊C₄×C₆₁	—	2	4	no	—
109	C₂²⋊C₄×C₆₁	—	4	4	no	—
110	C₂²⋊C₄×C₆₁	—	4	4	no	—
111	C₂²⋊C₄×C₆₁	—	8	4	no	—
112	C₄²×C₆₁	—	2	1	no	—
113	C₄²×C₆₁	—	2	1	no	—
114	C₄²×C₆₁	C₂×D₄	1	2	yes	—
115	C₄²×C₆₁	#41	1	2	yes	—
116	C₂³×C₆₁	—	2	1	no	—
117	C₂³×C₆₁	—	2	1	no	—
118	C₂³×C₆₁	—	2	1	no	—
119	C₂³×C₆₁	—	2	4	no	—
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	—	4	2	no	—
122	C₂³×C₆₁	—	4	2	no	—
123	C₂³×C₆₁	—	4	4	no	—
124	C₂³×C₆₁	—	4	4	no	—
125	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₆₁	—	4	4	no	—
127	Q₈×C₆₁	—	4	4	no	—
128	D₄×C₆₁	—	4	4	no	—
129	D₄×C₆₁	—	4	4	no	—
130	D₄×C₆₁	—	8	8	no	—
131	D₄×C₆₁	—	8	8	no	—
132	C₂×C₄×C₆₁	—	2	4	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	—	2	1	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	2	4	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	4	2	no	—
140	C₂×C₄×C₆₁	—	4	2	no	—
141	C₂×C₄×C₆₁	—	4	2	no	—
142	C₂×C₄×C₆₁	—	4	2	no	—
143	C₂×C₄×C₆₁	—	4	2	no	—
144	C₂×C₄×C₆₁	—	4	2	no	—
145	C₂×C₄×C₆₁	—	4	4	no	—
146	C₂×C₄×C₆₁	—	4	4	no	—
147	C₂×C₄×C₆₁	#121	1	4	yes	—
148	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
151	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
152	C₈×C₆₁	—	2	4	no	—
153	C₈×C₆₁	—	4	2	no	—
154	#29811	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂²×C₆₁	—	4	1	no	—
156	C₂²×C₆₁	—	4	2	no	—
157	C₂²×C₆₁	—	4	2	no	—
158	C₂²×C₆₁	—	4	2	no	—
159	C₂²×C₆₁	—	8	2	no	—
160	C₂²×C₆₁	—	8	4	no	—
161	C₂²×C₆₁	—	8	2	no	—
162	C₂²×C₆₁	—	8	2	no	—
163	C₂²×C₆₁	—	8	4	no	—
164	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
165	C₂²×C₆₁	#535	1	1	yes	characteristic
166	C₂²×C₆₁	#535	1	1	yes	characteristic
167	C₂²×C₆₁	#577	1	4	yes	—
168	C₄×C₆₁	—	2	4	no	—
169	C₄×C₆₁	—	4	1	no	—
170	C₄×C₆₁	—	4	2	no	—
171	C₄×C₆₁	—	8	2	no	—
172	C₄×C₆₁	—	8	2	no	—
173	C₄×C₆₁	—	8	2	no	—
174	C₄×C₆₁	#535	1	2	yes	—
175	C₄×C₆₁	#540	1	2	yes	—
176	C₂×#492	C₂×C₆₁	1	1	yes	characteristic, complemented
177	C₂×#433	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#2621	C₂×C₆₁	1	1	yes	characteristic, complemented
179	#2619	C₂×C₆₁	1	1	yes	characteristic
180	#2055	C₂×C₆₁	1	4	yes	complemented
181	#2000	C₂×C₆₁	1	4	yes	complemented
182	#1961	C₂×C₆₁	1	1	yes	characteristic, complemented
183	#1960	C₂×C₆₁	1	1	yes	characteristic, complemented
184	#1835	C₂×C₆₁	1	1	yes	characteristic, complemented
185	C₂×C₆₁	—	8	1	no	—
186	C₂×C₆₁	—	16	2	no	—
187	C₂×C₆₁	#2619	1	1	yes	characteristic
188	C₂×C₆₁	#3057	1	2	yes	—
189	C₂×C₆₁	#3221	1	2	yes	—
190	C₂×C₆₁	#3222	1	2	yes	—
191	C₂²×SD₁₆	C₂²×C₆₁	1	4	yes	—
192	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
193	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
194	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
195	C₂×#121	C₂²×C₆₁	1	2	yes	—
196	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
197	#492	—	2	4	no	—
198	#433	—	2	4	no	—
199	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
200	C₂×#103	C₂²×C₆₁	1	4	yes	—
201	C₂×#102	C₂²×C₆₁	1	4	yes	—
202	C₂×C₂²⋊C₈	C₂²×C₆₁	1	2	yes	—
203	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
205	#398	—	2	2	no	—
206	#397	—	2	2	no	—
207	#394	—	2	2	no	—
208	#392	—	2	2	no	—
209	#391	C₂²×C₆₁	1	2	yes	—
210	#390	C₂²×C₆₁	1	2	yes	—
211	#382	C₂²×C₆₁	1	1	yes	characteristic
212	#382	C₂²×C₆₁	1	1	yes	characteristic
213	#340	C₂²×C₆₁	1	4	yes	—
214	C₆₁	#29811	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
216	C₂²×D₄	—	2	2	no	—
217	C₂²×D₄	—	2	2	no	—
218	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
219	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
220	C₂×SD₁₆	—	2	8	no	—
221	C₂×SD₁₆	—	4	4	no	—
222	C₂×SD₁₆	—	4	4	no	—
223	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
224	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
225	C₄⋊Q₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	#127	—	2	2	no	—
228	#127	—	2	2	no	—
229	C₂²⋊Q₈	—	4	4	no	—
230	#121	—	4	4	no	—
231	#117	—	2	2	no	—
232	#117	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
237	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₂²⋊C₄	—	2	2	no	—
240	C₂×C₂²⋊C₄	—	2	4	no	—
241	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
242	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
244	C₄⋊C₈	—	2	4	no	—
245	#103	—	4	4	no	—
246	#102	—	4	4	no	—
247	C₂²⋊C₈	—	2	4	no	—
248	#97	—	2	2	no	—
249	C₄×C₈	D₄×C₆₁	1	4	yes	complemented
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	4	no	—
253	C₂⁴	—	2	2	no	—
254	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	4	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	4	4	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	—	8	4	no	—
266	C₂×D₄	—	8	4	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
277	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
278	SD₁₆	—	4	16	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	4	4	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	2	4	no	—
291	C₂²⋊C₄	—	4	4	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	8	4	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	C₂×D₄×C₆₁	1	2	yes	—
297	C₄²	#41×C₆₁	1	2	yes	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	4	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	4	no	—
305	C₂³	—	4	4	no	—
306	C₂³	—	4	2	no	—
307	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	Q₈	—	4	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	8	8	no	—
313	D₄	—	8	8	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	#121×C₆₁	1	4	yes	—
330	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
332	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
333	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
334	C₈	—	2	4	no	—
335	C₈	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
346	C₂²	C₆₁×#535	1	1	yes	characteristic
347	C₂²	C₆₁×#535	1	1	yes	characteristic
348	C₂²	C₆₁×#577	1	4	yes	—
349	C₄	—	2	4	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	C₆₁×#535	1	2	yes	—
356	C₄	C₆₁×#540	1	2	yes	—
357	C₂	—	8	1	no	—
358	C₂	—	16	2	no	—
359	C₂	C₆₁×#2619	1	1	yes	characteristic
360	C₂	C₆₁×#3057	1	2	yes	—
361	C₂	C₆₁×#3221	1	2	yes	—
362	C₂	C₆₁×#3222	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×#117×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#492	C₂	semidirect
2. C₂×C₆₁×#433	C₂	semidirect
3. C₆₁×#2621	C₂	semidirect
5. C₆₁×#2055	C₂	semidirect
6. C₆₁×#2000	C₂	semidirect
7. C₆₁×#1961	C₂	semidirect
8. C₆₁×#1960	C₂	semidirect
9. C₆₁×#1835	C₂	semidirect
67. C₄×C₈×C₆₁	D₄	semidirect
154. #29811	C₆₁	direct (N × H)
176. C₂×#492	C₂×C₆₁	semidirect
177. C₂×#433	C₂×C₆₁	semidirect
178. #2621	C₂×C₆₁	semidirect
180. #2055	C₂×C₆₁	semidirect
181. #2000	C₂×C₆₁	semidirect
182. #1961	C₂×C₆₁	semidirect
183. #1960	C₂×C₆₁	semidirect
184. #1835	C₂×C₆₁	semidirect
214. C₆₁	#29811	direct (N × H)
249. C₄×C₈	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#492	2
C₂×C₆₁×#433	2
C₆₁×#2621	2
C₆₁×#2619	2
C₆₁×#2055	2
C₆₁×#2000	2
C₆₁×#1961	2
C₆₁×#1960	2
C₆₁×#1835	2
#29811	61

Subgroups — C61×#29811 #29533244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29811 #29533244