Subgroups — C₆₁×#18547 #29521980

Order 15616.

1068 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3109	C₂	1	2	yes	maximal, complemented
4	C₆₁×#1634	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1624	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1558	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1490	C₂	1	2	yes	maximal, complemented
8	C₂×#137×C₆₁	—	2	2	no	—
9	C₆₁×#555	—	2	2	no	—
10	C₆₁×#555	—	2	4	no	—
11	C₆₁×#555	C₂²	1	4	yes	—
12	C₆₁×#532	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₁	—	2	2	no	—
15	C₂×#121×C₆₁	—	2	2	no	—
16	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
17	C₂×#117×C₆₁	—	2	2	no	—
18	C₆₁×#497	—	2	2	no	—
19	C₆₁×#494	C₂²	1	2	yes	complemented
20	C₆₁×#492	C₂²	1	2	yes	complemented
21	C₆₁×#481	—	2	2	no	—
22	C₆₁×#476	C₂²	1	4	yes	complemented
23	C₆₁×#459	C₂²	1	4	yes	complemented
24	C₆₁×#447	—	2	2	no	—
25	C₆₁×#442	—	2	4	no	—
26	C₆₁×#432	C₂²	1	2	yes	—
27	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#333	C₂²	1	4	yes	complemented
29	C₆₁×D₄⋊C₈	C₂²	1	8	yes	complemented
30	C₂×#41×C₆₁	—	2	1	no	—
31	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
32	C₂²×D₄×C₆₁	—	2	2	no	—
33	C₂²×D₄×C₆₁	D₄	1	2	yes	—
34	C₂³×C₄×C₆₁	—	2	2	no	—
35	#137×C₆₁	—	4	8	no	—
36	C₂×Q₁₆×C₆₁	—	2	4	no	—
37	C₂×SD₁₆×C₆₁	—	2	4	no	—
38	C₂×D₈×C₆₁	—	2	4	no	—
39	C₂×M₁₆×C₆₁	—	2	1	no	—
40	C₄⋊Q₈×C₆₁	—	2	2	no	—
41	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
42	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
44	#127×C₆₁	C₂³	1	1	yes	characteristic
45	#124×C₆₁	—	2	8	no	—
46	#124×C₆₁	—	2	8	no	—
47	#124×C₆₁	—	2	1	no	—
48	#121×C₆₁	—	2	8	no	—
49	#121×C₆₁	—	2	8	no	—
50	#121×C₆₁	—	2	2	no	—
51	C₄×D₄×C₆₁	—	2	2	no	—
52	C₄×D₄×C₆₁	—	2	2	no	—
53	C₄×D₄×C₆₁	—	2	2	no	—
54	C₄×D₄×C₆₁	D₄	1	4	yes	—
55	C₄×D₄×C₆₁	C₂³	1	4	yes	—
56	#117×C₆₁	—	2	8	no	—
57	#117×C₆₁	—	2	8	no	—
58	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
61	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
62	C₂×C₄²×C₆₁	—	2	2	no	—
63	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	#106×C₆₁	—	2	4	no	—
65	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
66	#102×C₆₁	—	2	4	no	—
67	#102×C₆₁	—	2	4	no	—
68	C₂²⋊C₈×C₆₁	—	2	2	no	—
69	#97×C₆₁	—	2	1	no	—
70	#97×C₆₁	—	2	2	no	—
71	C₄×C₈×C₆₁	C₂³	1	2	yes	—
72	C₂⁴×C₆₁	—	2	2	no	—
73	#41×C₆₁	—	4	4	no	—
74	C₂×Q₈×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	—	2	1	no	—
76	C₂×Q₈×C₆₁	—	4	4	no	—
77	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₆₁	—	2	8	no	—
79	C₂×D₄×C₆₁	—	2	2	no	—
80	C₂×D₄×C₆₁	—	2	2	no	—
81	C₂×D₄×C₆₁	—	2	1	no	—
82	C₂×D₄×C₆₁	—	4	4	no	—
83	C₂×D₄×C₆₁	—	4	1	no	—
84	C₂×D₄×C₆₁	—	8	4	no	—
85	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₆₁	—	2	2	no	—
87	C₂²×C₄×C₆₁	—	2	2	no	—
88	C₂²×C₄×C₆₁	—	2	4	no	—
89	C₂²×C₄×C₆₁	—	2	4	no	—
90	C₂²×C₄×C₆₁	—	4	8	no	—
91	C₂²×C₄×C₆₁	—	4	1	no	—
92	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₆₁	—	4	8	no	—
95	SD₁₆×C₆₁	—	4	8	no	—
96	D₈×C₆₁	—	4	8	no	—
97	M₁₆×C₆₁	—	2	4	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	2	4	no	—
100	C₄⋊C₄×C₆₁	—	2	4	no	—
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	—	2	4	no	—
103	C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
105	C₂²⋊C₄×C₆₁	—	2	16	no	—
106	C₂²⋊C₄×C₆₁	—	2	4	no	—
107	C₂²⋊C₄×C₆₁	—	2	4	no	—
108	C₂²⋊C₄×C₆₁	—	4	2	no	—
109	C₄²×C₆₁	—	2	8	no	—
110	C₄²×C₆₁	C₂×D₄	1	2	yes	—
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
113	C₂³×C₆₁	—	2	4	no	—
114	C₂³×C₆₁	—	4	4	no	—
115	C₂³×C₆₁	—	4	4	no	—
116	C₂³×C₆₁	—	4	4	no	—
117	C₂³×C₆₁	—	4	1	no	—
118	C₂³×C₆₁	#120	1	1	yes	characteristic
119	Q₈×C₆₁	—	4	4	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	8	2	no	—
122	D₄×C₆₁	—	2	4	no	—
123	D₄×C₆₁	—	2	4	no	—
124	D₄×C₆₁	—	4	4	no	—
125	D₄×C₆₁	—	4	2	no	—
126	D₄×C₆₁	—	4	4	no	—
127	D₄×C₆₁	—	8	8	no	—
128	D₄×C₆₁	—	8	2	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	4	no	—
131	C₂×C₄×C₆₁	—	2	4	no	—
132	C₂×C₄×C₆₁	—	2	4	no	—
133	C₂×C₄×C₆₁	—	4	8	no	—
134	C₂×C₄×C₆₁	—	4	4	no	—
135	C₂×C₄×C₆₁	—	4	4	no	—
136	C₂×C₄×C₆₁	—	4	8	no	—
137	C₂×C₄×C₆₁	—	4	1	no	—
138	C₂×C₄×C₆₁	—	8	2	no	—
139	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₆₁	#120	1	2	yes	—
141	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
143	C₈×C₆₁	—	2	4	no	—
144	C₈×C₆₁	—	4	4	no	—
145	#18547	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₆₁	—	2	2	no	—
147	C₂²×C₆₁	—	4	4	no	—
148	C₂²×C₆₁	—	4	4	no	—
149	C₂²×C₆₁	—	4	4	no	—
150	C₂²×C₆₁	—	8	8	no	—
151	C₂²×C₆₁	—	8	2	no	—
152	C₂²×C₆₁	—	8	1	no	—
153	C₂²×C₆₁	#453	1	2	yes	—
154	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
155	C₄×C₆₁	—	2	2	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	4	4	no	—
159	C₄×C₆₁	—	4	4	no	—
160	C₄×C₆₁	—	8	1	no	—
161	C₄×C₆₁	#575	1	4	yes	—
162	C₂×#555	C₂×C₆₁	1	1	yes	characteristic, complemented
163	#3152	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#3109	C₂×C₆₁	1	2	yes	complemented
165	#1634	C₂×C₆₁	1	4	yes	complemented
166	#1624	C₂×C₆₁	1	4	yes	complemented
167	#1558	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#1490	C₂×C₆₁	1	2	yes	complemented
169	C₂×C₆₁	—	2	2	no	—
170	C₂×C₆₁	—	8	4	no	—
171	C₂×C₆₁	—	16	1	no	—
172	C₂×C₆₁	#3005	1	2	yes	—
173	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
174	C₂×#137	—	2	2	no	—
175	#555	—	2	2	no	—
176	#555	—	2	4	no	—
177	#555	C₂²×C₆₁	1	4	yes	—
178	#532	C₂²×C₆₁	1	1	yes	characteristic
179	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
180	C₂×#124	—	2	2	no	—
181	C₂×#121	—	2	2	no	—
182	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
183	C₂×#117	—	2	2	no	—
184	#497	—	2	2	no	—
185	#494	C₂²×C₆₁	1	2	yes	complemented
186	#492	C₂²×C₆₁	1	2	yes	complemented
187	#481	—	2	2	no	—
188	#476	C₂²×C₆₁	1	4	yes	complemented
189	#459	C₂²×C₆₁	1	4	yes	complemented
190	#447	—	2	2	no	—
191	#442	—	2	4	no	—
192	#432	C₂²×C₆₁	1	2	yes	—
193	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
194	#333	C₂²×C₆₁	1	4	yes	complemented
195	D₄⋊C₈	C₂²×C₆₁	1	8	yes	complemented
196	C₆₁	#18547	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×#41	—	2	1	no	—
198	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	D₄×C₆₁	1	2	yes	—
201	C₂³×C₄	—	2	2	no	—
202	#137	—	4	8	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	2	4	no	—
205	C₂×D₈	—	2	4	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
209	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
210	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
211	#127	C₂³×C₆₁	1	1	yes	characteristic
212	#124	—	2	8	no	—
213	#124	—	2	8	no	—
214	#124	—	2	1	no	—
215	#121	—	2	8	no	—
216	#121	—	2	8	no	—
217	#121	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	D₄×C₆₁	1	4	yes	—
222	C₄×D₄	C₂³×C₆₁	1	4	yes	—
223	#117	—	2	8	no	—
224	#117	—	2	8	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
227	C₂×C₂²⋊C₄	—	2	4	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
231	#106	—	2	4	no	—
232	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
233	#102	—	2	4	no	—
234	#102	—	2	4	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	#97	—	2	1	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	C₂³×C₆₁	1	2	yes	—
239	C₂⁴	—	2	2	no	—
240	#41	—	4	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
245	C₂×D₄	—	2	8	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	8	4	no	—
252	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	4	8	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
261	Q₁₆	—	4	8	no	—
262	SD₁₆	—	4	8	no	—
263	D₈	—	4	8	no	—
264	M₁₆	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
272	C₂²⋊C₄	—	2	16	no	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	2	8	no	—
277	C₄²	C₂×D₄×C₆₁	1	2	yes	—
278	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
279	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
280	C₂³	—	2	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	4	4	no	—
284	C₂³	—	4	1	no	—
285	C₂³	#120×C₆₁	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	2	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	8	8	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
307	C₂×C₄	#120×C₆₁	1	2	yes	—
308	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
310	C₈	—	2	4	no	—
311	C₈	—	4	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	8	8	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	1	no	—
319	C₂²	C₆₁×#453	1	2	yes	—
320	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	4	no	—
326	C₄	—	8	1	no	—
327	C₄	C₆₁×#575	1	4	yes	—
328	C₂	—	2	2	no	—
329	C₂	—	8	4	no	—
330	C₂	—	16	1	no	—
331	C₂	C₆₁×#3005	1	2	yes	—
332	C₂	C₂×C₆₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#555	C₂	semidirect
2. C₆₁×#3152	C₂	semidirect
3. C₆₁×#3109	C₂	semidirect
4. C₆₁×#1634	C₂	semidirect
5. C₆₁×#1624	C₂	semidirect
6. C₆₁×#1558	C₂	semidirect
7. C₆₁×#1490	C₂	semidirect
19. C₆₁×#494	C₂²	semidirect
20. C₆₁×#492	C₂²	semidirect
22. C₆₁×#476	C₂²	semidirect
23. C₆₁×#459	C₂²	semidirect
28. C₆₁×#333	C₂²	semidirect
29. C₆₁×D₄⋊C₈	C₂²	semidirect
145. #18547	C₆₁	direct (N × H)
162. C₂×#555	C₂×C₆₁	semidirect
163. #3152	C₂×C₆₁	semidirect
164. #3109	C₂×C₆₁	semidirect
165. #1634	C₂×C₆₁	semidirect
166. #1624	C₂×C₆₁	semidirect
167. #1558	C₂×C₆₁	semidirect
168. #1490	C₂×C₆₁	semidirect
185. #494	C₂²×C₆₁	semidirect
186. #492	C₂²×C₆₁	semidirect
188. #476	C₂²×C₆₁	semidirect
189. #459	C₂²×C₆₁	semidirect
194. #333	C₂²×C₆₁	semidirect
195. D₄⋊C₈	C₂²×C₆₁	semidirect
196. C₆₁	#18547	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#555	2
C₆₁×#3152	2
C₆₁×#3109	2
C₆₁×#1634	2
C₆₁×#1624	2
C₆₁×#1558	2
C₆₁×#1490	2
#18547	61

Subgroups — C61×#18547 #29521980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18547 #29521980