Subgroups — C₆₁×#12733 #29516166

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#358	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₆₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₆₁	—	2	1	no	—
7	C₂×#135×C₆₁	—	2	4	no	—
8	C₂×#135×C₆₁	—	2	4	no	—
9	C₂×#135×C₆₁	—	2	2	no	—
10	C₂×#135×C₆₁	C₄	1	4	yes	complemented
11	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₆₁	—	2	1	no	—
13	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂×#107×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#380	—	2	1	no	—
18	C₆₁×#358	—	2	4	no	—
19	C₆₁×#357	—	2	4	no	—
20	C₆₁×C₂²⋊C₁₆	—	2	2	no	—
21	C₆₁×#340	—	2	1	no	—
22	C₂×#41×C₆₁	—	2	8	no	—
23	C₂×#41×C₆₁	—	2	4	no	—
24	C₂×#41×C₆₁	—	4	4	no	—
25	C₂×#41×C₆₁	—	4	2	no	—
26	C₂²×Q₈×C₆₁	—	2	1	no	—
27	C₂²×D₄×C₆₁	—	2	1	no	—
28	C₂²×D₄×C₆₁	—	4	1	no	—
29	C₂³×C₄×C₆₁	—	4	1	no	—
30	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
31	#135×C₆₁	—	2	16	no	—
32	#135×C₆₁	—	2	8	no	—
33	#135×C₆₁	—	2	8	no	—
34	C₂×Q₁₆×C₆₁	—	2	2	no	—
35	C₂×Q₁₆×C₆₁	—	2	1	no	—
36	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₆₁	D₄	1	4	yes	—
38	C₂×SD₁₆×C₆₁	—	2	8	no	—
39	C₂×SD₁₆×C₆₁	—	2	4	no	—
40	C₂×D₈×C₆₁	—	2	2	no	—
41	C₂×D₈×C₆₁	—	2	1	no	—
42	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₆₁	D₄	1	4	yes	—
44	C₂²×C₈×C₆₁	—	2	2	no	—
45	C₂²×C₈×C₆₁	—	2	2	no	—
46	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	D₄	1	2	yes	—
49	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
52	C₂×C₁₆×C₆₁	—	2	2	no	—
53	C₂×C₁₆×C₆₁	—	2	2	no	—
54	C₂×C₁₆×C₆₁	—	2	2	no	—
55	#107×C₆₁	—	2	2	no	—
56	#107×C₆₁	—	2	2	no	—
57	#95×C₆₁	—	4	1	no	—
58	C₂⁴×C₆₁	—	4	1	no	—
59	C₂⁴×C₆₁	D₈	1	1	yes	characteristic
60	#41×C₆₁	—	2	16	no	—
61	#41×C₆₁	—	4	8	no	—
62	#41×C₆₁	—	4	16	no	—
63	C₂×Q₈×C₆₁	—	2	2	no	—
64	C₂×Q₈×C₆₁	—	2	4	no	—
65	C₂×Q₈×C₆₁	—	4	1	no	—
66	C₂×Q₈×C₆₁	—	4	2	no	—
67	C₂×D₄×C₆₁	—	2	2	no	—
68	C₂×D₄×C₆₁	—	2	4	no	—
69	C₂×D₄×C₆₁	—	4	8	no	—
70	C₂×D₄×C₆₁	—	4	4	no	—
71	C₂×D₄×C₆₁	—	4	1	no	—
72	C₂×D₄×C₆₁	—	4	2	no	—
73	C₂²×C₄×C₆₁	—	2	2	no	—
74	C₂²×C₄×C₆₁	—	2	2	no	—
75	C₂²×C₄×C₆₁	—	4	8	no	—
76	C₂²×C₄×C₆₁	—	4	1	no	—
77	C₂²×C₄×C₆₁	—	4	4	no	—
78	C₂²×C₄×C₆₁	—	4	2	no	—
79	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
80	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
83	Q₁₆×C₆₁	—	2	2	no	—
84	Q₁₆×C₆₁	—	2	4	no	—
85	Q₁₆×C₆₁	C₂²⋊C₄	1	4	yes	complemented
86	SD₁₆×C₆₁	—	2	16	no	—
87	D₈×C₆₁	—	2	2	no	—
88	D₈×C₆₁	—	2	4	no	—
89	D₈×C₆₁	C₂²⋊C₄	1	4	yes	complemented
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	4	no	—
94	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	—	4	2	no	—
100	C₄⋊C₄×C₆₁	—	4	2	no	—
101	C₂²⋊C₄×C₆₁	—	8	2	no	—
102	C₁₆×C₆₁	—	4	2	no	—
103	C₂³×C₆₁	—	2	2	no	—
104	C₂³×C₆₁	—	2	2	no	—
105	C₂³×C₆₁	—	2	1	no	—
106	C₂³×C₆₁	—	2	1	no	—
107	C₂³×C₆₁	—	4	2	no	—
108	C₂³×C₆₁	—	4	4	no	—
109	C₂³×C₆₁	—	8	2	no	—
110	C₂³×C₆₁	—	8	1	no	—
111	C₂³×C₆₁	—	8	1	no	—
112	C₂³×C₆₁	#102	1	2	yes	—
113	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₆₁	—	2	4	no	—
115	Q₈×C₆₁	—	4	2	no	—
116	Q₈×C₆₁	—	4	4	no	—
117	D₄×C₆₁	—	2	4	no	—
118	D₄×C₆₁	—	4	16	no	—
119	D₄×C₆₁	—	4	2	no	—
120	D₄×C₆₁	—	4	4	no	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	4	no	—
125	C₂×C₄×C₆₁	—	4	16	no	—
126	C₂×C₄×C₆₁	—	4	2	no	—
127	C₂×C₄×C₆₁	—	4	4	no	—
128	C₂×C₄×C₆₁	—	8	2	no	—
129	C₂×C₄×C₆₁	—	8	2	no	—
130	C₂×C₄×C₆₁	—	8	2	no	—
131	C₂×C₄×C₆₁	#102	1	2	yes	—
132	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	#120	1	2	yes	—
134	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
136	C₈×C₆₁	—	2	2	no	—
137	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₆₁	#120	1	2	yes	—
140	#12733	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₆₁	—	2	2	no	—
142	C₂²×C₆₁	—	2	2	no	—
143	C₂²×C₆₁	—	2	2	no	—
144	C₂²×C₆₁	—	4	2	no	—
145	C₂²×C₆₁	—	4	2	no	—
146	C₂²×C₆₁	—	4	4	no	—
147	C₂²×C₆₁	—	8	2	no	—
148	C₂²×C₆₁	—	8	2	no	—
149	C₂²×C₆₁	—	8	4	no	—
150	C₂²×C₆₁	—	8	4	no	—
151	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
152	C₂²×C₆₁	#447	1	2	yes	—
153	C₂²×C₆₁	#508	1	2	yes	—
154	C₂²×C₆₁	#509	1	1	yes	characteristic
155	C₂²×C₆₁	#510	1	1	yes	characteristic
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	—	4	4	no	—
158	C₄×C₆₁	—	16	2	no	—
159	C₄×C₆₁	#379	1	1	yes	characteristic
160	C₄×C₆₁	#418	1	1	yes	characteristic
161	C₄×C₆₁	#450	1	2	yes	—
162	C₂²×#135	C₂×C₆₁	1	1	yes	characteristic
163	C₂×#358	C₂×C₆₁	1	2	yes	complemented
164	C₂×#357	C₂×C₆₁	1	2	yes	complemented
165	C₂×C₂²⋊C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#1808	C₂×C₆₁	1	1	yes	characteristic, complemented
167	C₂×C₆₁	—	4	2	no	—
168	C₂×C₆₁	—	8	4	no	—
169	C₂×C₆₁	#1786	1	1	yes	characteristic
170	C₂×C₆₁	#2138	1	1	yes	characteristic
171	C₂×C₆₁	#2139	1	1	yes	characteristic
172	C₂×C₆₁	#2186	1	2	yes	—
173	C₂×C₆₁	#2189	1	2	yes	—
174	C₂²×#41	—	2	1	no	—
175	C₂×#135	—	2	4	no	—
176	C₂×#135	—	2	4	no	—
177	C₂×#135	—	2	2	no	—
178	C₂×#135	C₄×C₆₁	1	4	yes	complemented
179	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
180	C₂²×SD₁₆	—	2	1	no	—
181	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
182	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
183	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
184	C₂×#107	C₂²×C₆₁	1	2	yes	—
185	#380	—	2	1	no	—
186	#358	—	2	4	no	—
187	#357	—	2	4	no	—
188	C₂²⋊C₁₆	—	2	2	no	—
189	#340	—	2	1	no	—
190	C₆₁	#12733	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×#41	—	2	8	no	—
192	C₂×#41	—	2	4	no	—
193	C₂×#41	—	4	4	no	—
194	C₂×#41	—	4	2	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	4	1	no	—
198	C₂³×C₄	—	4	1	no	—
199	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
200	#135	—	2	16	no	—
201	#135	—	2	8	no	—
202	#135	—	2	8	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	C₂×C₄×C₆₁	1	2	yes	—
206	C₂×Q₁₆	D₄×C₆₁	1	4	yes	—
207	C₂×SD₁₆	—	2	8	no	—
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂×C₄×C₆₁	1	2	yes	—
212	C₂×D₈	D₄×C₆₁	1	4	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
216	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
217	C₂²×C₈	D₄×C₆₁	1	2	yes	—
218	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	4	1	no	—
221	C₂×C₁₆	—	2	2	no	—
222	C₂×C₁₆	—	2	2	no	—
223	C₂×C₁₆	—	2	2	no	—
224	#107	—	2	2	no	—
225	#107	—	2	2	no	—
226	#95	—	4	1	no	—
227	C₂⁴	—	4	1	no	—
228	C₂⁴	D₈×C₆₁	1	1	yes	characteristic
229	#41	—	2	16	no	—
230	#41	—	4	8	no	—
231	#41	—	4	16	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	8	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
249	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
250	C₂²×C₄	SD₁₆×C₆₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	4	no	—
254	Q₁₆	C₂²⋊C₄×C₆₁	1	4	yes	complemented
255	SD₁₆	—	2	16	no	—
256	D₈	—	2	2	no	—
257	D₈	—	2	4	no	—
258	D₈	C₂²⋊C₄×C₆₁	1	4	yes	complemented
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
264	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
267	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	8	2	no	—
271	C₁₆	—	4	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	8	2	no	—
279	C₂³	—	8	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	#102×C₆₁	1	2	yes	—
282	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	16	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	#102×C₆₁	1	2	yes	—
301	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
302	C₂×C₄	#120×C₆₁	1	2	yes	—
303	C₂×C₄	C₂×SD₁₆×C₆₁	1	1	yes	characteristic
304	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
307	C₈	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
308	C₈	#120×C₆₁	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	4	no	—
319	C₂²	C₂×#102×C₆₁	1	1	yes	characteristic
320	C₂²	C₆₁×#447	1	2	yes	—
321	C₂²	C₆₁×#508	1	2	yes	—
322	C₂²	C₆₁×#509	1	1	yes	characteristic
323	C₂²	C₆₁×#510	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	16	2	no	—
327	C₄	C₆₁×#379	1	1	yes	characteristic
328	C₄	C₆₁×#418	1	1	yes	characteristic
329	C₄	C₆₁×#450	1	2	yes	—
330	C₂	—	4	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₆₁×#1786	1	1	yes	characteristic
333	C₂	C₆₁×#2138	1	1	yes	characteristic
334	C₂	C₆₁×#2139	1	1	yes	characteristic
335	C₂	C₆₁×#2186	1	2	yes	—
336	C₂	C₆₁×#2189	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
prime radical	C₂²×#135×C₆₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ C₂³×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₆₁×#358	C₂	semidirect
3. C₂×C₆₁×#357	C₂	semidirect
4. C₂×C₆₁×C₂²⋊C₁₆	C₂	semidirect
5. C₆₁×#1808	C₂	semidirect
10. C₂×#135×C₆₁	C₄	semidirect
85. Q₁₆×C₆₁	C₂²⋊C₄	semidirect
89. D₈×C₆₁	C₂²⋊C₄	semidirect
140. #12733	C₆₁	direct (N × H)
163. C₂×#358	C₂×C₆₁	semidirect
164. C₂×#357	C₂×C₆₁	semidirect
165. C₂×C₂²⋊C₁₆	C₂×C₆₁	semidirect
166. #1808	C₂×C₆₁	semidirect
178. C₂×#135	C₄×C₆₁	semidirect
190. C₆₁	#12733	direct (N × H)
254. Q₁₆	C₂²⋊C₄×C₆₁	semidirect
258. D₈	C₂²⋊C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₆₁	2
C₂×C₆₁×#358	2
C₂×C₆₁×#357	2
C₂×C₆₁×C₂²⋊C₁₆	2
C₆₁×#1808	2
#12733	61

Subgroups — C61×#12733 #29516166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#12733 #29516166