Subgroups — C₆₁×#10839 #29514272

Order 15616.

1110 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#339	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
6	C₂×#117×C₆₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₆₁	—	2	2	no	—
8	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
10	C₆₁×#423	—	2	2	no	—
11	C₆₁×#423	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
13	C₆₁×#384	C₄	1	2	yes	complemented
14	C₆₁×#382	C₄	1	2	yes	complemented
15	C₆₁×#380	—	2	2	no	—
16	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
17	C₄³×C₆₁	—	2	2	no	—
18	C₄³×C₆₁	C₄	1	4	yes	complemented
19	C₆₁×#339	—	2	8	no	—
20	C₂³×C₄×C₆₁	—	2	2	no	—
21	C₂³×C₄×C₆₁	D₄	1	2	yes	—
22	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂×M₁₆×C₆₁	—	2	4	no	—
25	C₂×M₁₆×C₆₁	—	2	4	no	—
26	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	#117×C₆₁	—	2	4	no	—
29	#117×C₆₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
33	C₂×C₄²×C₆₁	—	2	16	no	—
34	C₂×C₄²×C₆₁	—	2	8	no	—
35	C₂×C₄²×C₆₁	—	2	2	no	—
36	C₂×C₄²×C₆₁	—	2	2	no	—
37	C₂×C₄²×C₆₁	—	2	2	no	—
38	C₂×C₄²×C₆₁	—	2	2	no	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	D₄	1	4	yes	—
45	C₂×C₄²×C₆₁	D₄	1	2	yes	—
46	C₂×C₄²×C₆₁	D₄	1	2	yes	—
47	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
48	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
49	C₄⋊C₈×C₆₁	—	2	8	no	—
50	#95×C₆₁	—	2	4	no	—
51	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	—	2	16	no	—
53	C₂²×C₄×C₆₁	—	2	4	no	—
54	C₂²×C₄×C₆₁	—	2	4	no	—
55	C₂²×C₄×C₆₁	—	2	4	no	—
56	C₂²×C₄×C₆₁	—	2	8	no	—
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₆₁	#41	1	2	yes	—
71	C₂²×C₄×C₆₁	#41	1	2	yes	—
72	M₁₆×C₆₁	—	2	8	no	—
73	M₁₆×C₆₁	—	4	4	no	—
74	C₂×C₈×C₆₁	—	2	4	no	—
75	C₂×C₈×C₆₁	—	4	2	no	—
76	C₂×C₈×C₆₁	—	4	2	no	—
77	C₄⋊C₄×C₆₁	—	2	8	no	—
78	C₄⋊C₄×C₆₁	—	4	8	no	—
79	C₂²⋊C₄×C₆₁	—	4	4	no	—
80	C₄²×C₆₁	—	2	32	no	—
81	C₄²×C₆₁	—	2	4	no	—
82	C₄²×C₆₁	—	2	4	no	—
83	C₄²×C₆₁	—	2	4	no	—
84	C₄²×C₆₁	—	4	2	no	—
85	C₄²×C₆₁	—	4	2	no	—
86	C₄²×C₆₁	C₄²	1	2	yes	—
87	C₄²×C₆₁	C₄²	1	2	yes	—
88	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
90	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
91	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
92	C₄²×C₆₁	C₄⋊C₄	1	8	yes	complemented
93	C₂³×C₆₁	—	2	4	no	—
94	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₁	#121	1	2	yes	—
98	C₂³×C₆₁	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₆₁	—	2	16	no	—
100	C₂×C₄×C₆₁	—	2	16	no	—
101	C₂×C₄×C₆₁	—	2	16	no	—
102	C₂×C₄×C₆₁	—	2	8	no	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	2	8	no	—
105	C₂×C₄×C₆₁	—	2	8	no	—
106	C₂×C₄×C₆₁	—	4	4	no	—
107	C₂×C₄×C₆₁	—	4	8	no	—
108	C₂×C₄×C₆₁	#95	1	4	yes	—
109	C₂×C₄×C₆₁	#95	1	4	yes	—
110	C₂×C₄×C₆₁	#104	1	8	yes	—
111	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₁	#117	1	2	yes	—
117	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₆₁	#121	1	2	yes	—
122	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
124	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#125	1	2	yes	—
126	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
129	C₈×C₆₁	—	4	4	no	—
130	#10839	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₆₁	—	2	4	no	—
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	—	2	4	no	—
135	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
136	C₂²×C₆₁	#380	1	2	yes	—
137	C₂²×C₆₁	#389	1	2	yes	—
138	C₂²×C₆₁	#390	1	1	yes	characteristic
139	C₂²×C₆₁	#391	1	1	yes	characteristic
140	C₂²×C₆₁	C₂×#104	1	4	yes	—
141	C₄×C₆₁	—	2	16	no	—
142	C₄×C₆₁	—	2	4	no	—
143	C₄×C₆₁	—	8	4	no	—
144	C₄×C₆₁	#339	1	8	yes	complemented
145	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
146	C₄×C₆₁	#382	1	1	yes	characteristic
147	C₄×C₆₁	#384	1	1	yes	characteristic
148	C₄×C₆₁	#389	1	2	yes	—
149	C₄×C₆₁	#390	1	1	yes	characteristic
150	C₄×C₆₁	#391	1	1	yes	characteristic
151	C₂×#423	C₂×C₆₁	1	1	yes	characteristic
152	#2285	C₂×C₆₁	1	1	yes	characteristic
153	C₂×C₄³	C₂×C₆₁	1	1	yes	characteristic
154	C₂×#339	C₂×C₆₁	1	4	yes	—
155	C₂×C₆₁	—	2	2	no	—
156	C₂×C₆₁	—	2	2	no	—
157	C₂×C₆₁	C₄⋊#95	1	1	yes	characteristic
158	C₂×C₆₁	C₂×#339	1	2	yes	—
159	C₂×C₆₁	#1962	1	2	yes	—
160	C₂×C₆₁	#1994	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
162	C₂×#117	C₂²×C₆₁	1	2	yes	—
163	C₂²×C₄²	—	2	2	no	—
164	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
165	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
166	#423	—	2	2	no	—
167	#423	C₄×C₆₁	1	4	yes	complemented
168	C₂×C₄⋊C₈	—	2	2	no	—
169	#384	C₄×C₆₁	1	2	yes	complemented
170	#382	C₄×C₆₁	1	2	yes	complemented
171	#380	—	2	2	no	—
172	C₄×C₄⋊C₄	—	2	2	no	—
173	C₄³	—	2	2	no	—
174	C₄³	C₄×C₆₁	1	4	yes	complemented
175	#339	—	2	8	no	—
176	C₆₁	#10839	1	1	yes	characteristic, direct product factor, Sylow
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	D₄×C₆₁	1	2	yes	—
179	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	C₂×C₄×C₆₁	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	#117	—	2	4	no	—
186	#117	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	16	no	—
191	C₂×C₄²	—	2	8	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
197	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
200	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₆₁	1	4	yes	—
202	C₂×C₄²	D₄×C₆₁	1	2	yes	—
203	C₂×C₄²	D₄×C₆₁	1	2	yes	—
204	C₂×C₄²	Q₈×C₆₁	1	4	yes	—
205	C₂×C₄²	Q₈×C₆₁	1	2	yes	—
206	C₄⋊C₈	—	2	8	no	—
207	#95	—	2	4	no	—
208	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
218	C₂²×C₄	C₄⋊C₄×C₆₁	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
227	C₂²×C₄	#41×C₆₁	1	2	yes	—
228	C₂²×C₄	#41×C₆₁	1	2	yes	—
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	32	no	—
238	C₄²	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	C₄²×C₆₁	1	2	yes	—
244	C₄²	C₄²×C₆₁	1	2	yes	—
245	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
246	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
247	C₄²	C₄⋊C₄×C₆₁	1	4	yes	—
248	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
249	C₄²	C₄⋊C₄×C₆₁	1	8	yes	complemented
250	C₂³	—	2	4	no	—
251	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
252	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
253	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
254	C₂³	#121×C₆₁	1	2	yes	—
255	C₂³	C₂²⋊Q₈×C₆₁	1	2	yes	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#95×C₆₁	1	4	yes	—
266	C₂×C₄	#95×C₆₁	1	4	yes	—
267	C₂×C₄	#104×C₆₁	1	8	yes	—
268	C₂×C₄	C₂×C₄²×C₆₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
273	C₂×C₄	#117×C₆₁	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
276	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
277	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
278	C₂×C₄	#121×C₆₁	1	2	yes	—
279	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
280	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
281	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
282	C₂×C₄	#125×C₆₁	1	2	yes	—
283	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
285	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₂×#95×C₆₁	1	1	yes	characteristic
292	C₂²	C₆₁×#380	1	2	yes	—
293	C₂²	C₆₁×#389	1	2	yes	—
294	C₂²	C₆₁×#390	1	1	yes	characteristic
295	C₂²	C₆₁×#391	1	1	yes	characteristic
296	C₂²	C₂×#104×C₆₁	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	4	no	—
299	C₄	—	8	4	no	—
300	C₄	C₆₁×#339	1	8	yes	complemented
301	C₄	C₄×C₄⋊C₄×C₆₁	1	2	yes	—
302	C₄	C₆₁×#382	1	1	yes	characteristic
303	C₄	C₆₁×#384	1	1	yes	characteristic
304	C₄	C₆₁×#389	1	2	yes	—
305	C₄	C₆₁×#390	1	1	yes	characteristic
306	C₄	C₆₁×#391	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	C₆₁×C₄⋊#95	1	1	yes	characteristic
310	C₂	C₂×C₆₁×#339	1	2	yes	—
311	C₂	C₆₁×#1962	1	2	yes	—
312	C₂	C₆₁×#1994	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
socle	C₂³×C₆₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₆₁×#423	C₄	semidirect
13. C₆₁×#384	C₄	semidirect
14. C₆₁×#382	C₄	semidirect
18. C₄³×C₆₁	C₄	semidirect
92. C₄²×C₆₁	C₄⋊C₄	semidirect
130. #10839	C₆₁	direct (N × H)
144. C₄×C₆₁	#339	semidirect
167. #423	C₄×C₆₁	semidirect
169. #384	C₄×C₆₁	semidirect
170. #382	C₄×C₆₁	semidirect
174. C₄³	C₄×C₆₁	semidirect
176. C₆₁	#10839	direct (N × H)
249. C₄²	C₄⋊C₄×C₆₁	semidirect
300. C₄	C₆₁×#339	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#423	2
C₆₁×#2285	2
C₂×C₄³×C₆₁	2
C₂×C₆₁×#339	2
#10839	61

Subgroups — C61×#10839 #29514272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10839 #29514272