Subgroups — C₄×C₆₁×#443 #29523942

Order 15616.

1046 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#443	C₂	1	2	yes	maximal
2	C₄×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#131×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1772	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1764	C₂	1	4	yes	maximal, complemented
7	C₄×#104×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₁	C₂²	1	4	yes	—
11	C₄×#41×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#443	C₄	1	16	yes	direct product factor
13	C₄×Q₁₆×C₆₁	C₄	1	2	yes	complemented
14	C₄×SD₁₆×C₆₁	C₄	1	4	yes	complemented
15	C₄×D₈×C₆₁	C₄	1	2	yes	complemented
16	C₂×#108×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#427	C₂²	1	1	yes	characteristic
18	C₂×#104×C₆₁	C₂²	1	4	yes	—
19	C₆₁×#416	C₂²	1	2	yes	—
20	C₆₁×#405	—	2	4	no	—
21	C₆₁×#405	C₂²	1	4	yes	—
22	C₄×M₁₆×C₆₁	—	2	2	no	—
23	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₁	—	2	2	no	—
25	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₄³×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#341	C₂²	1	2	yes	—
31	C₆₁×#339	C₂²	1	4	yes	—
32	C₂×#41×C₆₁	C₂³	1	2	yes	—
33	#135×C₆₁	C₂×C₄	1	8	yes	—
34	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
36	C₂×D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	#131×C₆₁	—	2	8	no	—
38	#131×C₆₁	C₂×C₄	1	16	yes	complemented
39	C₂×M₁₆×C₆₁	—	2	4	no	—
40	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₁	—	2	4	no	—
42	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
45	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₆₁	—	2	4	no	—
47	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	—
48	#117×C₆₁	—	2	2	no	—
49	#117×C₆₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	#108×C₆₁	C₂×C₄	1	8	yes	complemented
55	#107×C₆₁	C₂×C₄	1	2	yes	—
56	#106×C₆₁	C₂×C₄	1	2	yes	—
57	#104×C₆₁	C₂×C₄	1	16	yes	complemented
58	#103×C₆₁	C₂×C₄	1	4	yes	—
59	#102×C₆₁	C₂×C₄	1	4	yes	—
60	#97×C₆₁	—	2	4	no	—
61	#97×C₆₁	—	2	2	no	—
62	C₄×C₈×C₆₁	—	2	4	no	—
63	C₄×C₈×C₆₁	—	2	4	no	—
64	C₄×C₈×C₆₁	—	2	2	no	—
65	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₁	C₂×C₄	1	8	yes	complemented
68	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₁	D₄	1	2	yes	—
70	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
71	#41×C₆₁	—	2	4	no	—
72	#41×C₆₁	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₆₁	—	2	2	no	—
75	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	4	no	—
79	C₂²×C₄×C₆₁	#41	1	2	yes	—
80	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₆₁	C₄²	1	4	yes	complemented
82	SD₁₆×C₆₁	C₄²	1	8	yes	complemented
83	D₈×C₆₁	C₄²	1	4	yes	complemented
84	M₁₆×C₆₁	—	2	4	no	—
85	M₁₆×C₆₁	—	2	16	no	—
86	M₁₆×C₆₁	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₆₁	—	2	4	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	2	16	no	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	#41	1	4	yes	—
98	C₄⋊C₄×C₆₁	—	2	4	no	—
99	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
100	C₂²⋊C₄×C₆₁	—	2	2	no	—
101	C₂²⋊C₄×C₆₁	—	2	4	no	—
102	C₄²×C₆₁	—	2	8	no	—
103	C₄²×C₆₁	—	2	4	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	2	2	no	—
106	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
107	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₂³×C₆₁	—	2	2	no	—
111	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
112	Q₈×C₆₁	—	2	2	no	—
113	Q₈×C₆₁	C₂×C₄²	1	4	yes	—
114	D₄×C₆₁	—	2	8	no	—
115	D₄×C₆₁	—	2	2	no	—
116	D₄×C₆₁	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₆₁	—	2	8	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	4	no	—
120	C₂×C₄×C₆₁	—	2	8	no	—
121	C₂×C₄×C₆₁	—	2	4	no	—
122	C₂×C₄×C₆₁	—	2	4	no	—
123	C₂×C₄×C₆₁	—	4	8	no	—
124	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
127	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
130	C₈×C₆₁	—	2	8	no	—
131	C₈×C₆₁	—	2	2	no	—
132	C₈×C₆₁	C₂×C₄²	1	2	yes	—
133	C₈×C₆₁	C₂×C₄²	1	2	yes	—
134	C₈×C₆₁	C₄×D₄	1	4	yes	—
135	C₈×C₆₁	C₄×D₄	1	4	yes	—
136	C₄×#443	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₆₁	—	2	2	no	—
138	C₂²×C₆₁	—	2	4	no	—
139	C₂²×C₆₁	—	4	4	no	—
140	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₆₁	C₄×#41	1	2	yes	—
142	C₄×C₆₁	—	2	8	no	—
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	2	4	no	—
145	C₄×C₆₁	—	4	4	no	—
146	C₄×C₆₁	#443	1	4	yes	direct product factor
147	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
148	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
149	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
150	C₂×#443	C₂×C₆₁	1	2	yes	—
151	C₄×#135	C₂×C₆₁	1	1	yes	characteristic
152	C₄×#131	C₂×C₆₁	1	2	yes	complemented
153	C₄×#108	C₂×C₆₁	1	1	yes	characteristic, complemented
154	#1772	C₂×C₆₁	1	2	yes	complemented
155	#1764	C₂×C₆₁	1	4	yes	complemented
156	C₄×#104	C₂×C₆₁	1	2	yes	complemented
157	C₄²×C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
158	C₂×C₆₁	—	2	2	no	—
159	C₂×C₆₁	—	4	4	no	—
160	C₂×C₆₁	C₄²×D₄	1	1	yes	characteristic
161	C₂×C₆₁	C₂×#443	1	2	yes	—
162	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
163	C₂×#131	C₂²×C₆₁	1	4	yes	—
164	C₄×#41	C₂²×C₆₁	1	2	yes	—
165	#443	C₄×C₆₁	1	16	yes	direct product factor
166	C₄×Q₁₆	C₄×C₆₁	1	2	yes	complemented
167	C₄×SD₁₆	C₄×C₆₁	1	4	yes	complemented
168	C₄×D₈	C₄×C₆₁	1	2	yes	complemented
169	C₂×#108	C₂²×C₆₁	1	2	yes	—
170	#427	C₂²×C₆₁	1	1	yes	characteristic
171	C₂×#104	C₂²×C₆₁	1	4	yes	—
172	#416	C₂²×C₆₁	1	2	yes	—
173	#405	—	2	4	no	—
174	#405	C₂²×C₆₁	1	4	yes	—
175	C₄×M₁₆	—	2	2	no	—
176	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
177	C₂×C₄×C₈	—	2	2	no	—
178	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
182	C₄³	C₂²×C₆₁	1	1	yes	characteristic
183	#341	C₂²×C₆₁	1	2	yes	—
184	#339	C₂²×C₆₁	1	4	yes	—
185	C₆₁	C₄×#443	1	1	yes	characteristic, direct product factor, Sylow
186	C₂×#41	C₂³×C₆₁	1	2	yes	—
187	#135	C₂×C₄×C₆₁	1	8	yes	—
188	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
189	C₂×SD₁₆	C₂×C₄×C₆₁	1	2	yes	—
190	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
191	#131	—	2	8	no	—
192	#131	C₂×C₄×C₆₁	1	16	yes	complemented
193	C₂×M₁₆	—	2	4	no	—
194	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
199	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
202	#117	—	2	2	no	—
203	#117	C₂³×C₆₁	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
207	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
208	#108	C₂×C₄×C₆₁	1	8	yes	complemented
209	#107	C₂×C₄×C₆₁	1	2	yes	—
210	#106	C₂×C₄×C₆₁	1	2	yes	—
211	#104	C₂×C₄×C₆₁	1	16	yes	complemented
212	#103	C₂×C₄×C₆₁	1	4	yes	—
213	#102	C₂×C₄×C₆₁	1	4	yes	—
214	#97	—	2	4	no	—
215	#97	—	2	2	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
220	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
221	C₄×C₈	C₂×C₄×C₆₁	1	8	yes	complemented
222	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
223	C₄×C₈	D₄×C₆₁	1	2	yes	—
224	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
225	#41	—	2	4	no	—
226	#41	C₂²×C₄×C₆₁	1	8	yes	—
227	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	#41×C₆₁	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
235	Q₁₆	C₄²×C₆₁	1	4	yes	complemented
236	SD₁₆	C₄²×C₆₁	1	8	yes	complemented
237	D₈	C₄²×C₆₁	1	4	yes	complemented
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	16	no	—
240	M₁₆	C₂²×C₄×C₆₁	1	8	yes	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	16	no	—
244	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
248	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
251	C₂×C₈	#41×C₆₁	1	4	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
261	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
262	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	C₂×C₄²×C₆₁	1	4	yes	—
268	D₄	—	2	8	no	—
269	D₄	—	2	2	no	—
270	D₄	C₂×C₄²×C₆₁	1	4	yes	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
279	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
280	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
281	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
282	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
284	C₈	—	2	8	no	—
285	C₈	—	2	2	no	—
286	C₈	C₂×C₄²×C₆₁	1	2	yes	—
287	C₈	C₂×C₄²×C₆₁	1	2	yes	—
288	C₈	C₄×D₄×C₆₁	1	4	yes	—
289	C₈	C₄×D₄×C₆₁	1	4	yes	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
294	C₂²	C₄×#41×C₆₁	1	2	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	2	4	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	C₆₁×#443	1	4	yes	direct product factor
300	C₄	C₂²×C₄²×C₆₁	1	1	yes	characteristic
301	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
302	C₄	C₄×#41×C₆₁	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	4	4	no	—
305	C₂	C₄²×D₄×C₆₁	1	1	yes	characteristic
306	C₂	C₂×C₆₁×#443	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₆₁	D₄
commutator	C₄	C₂²×C₄²×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#131×C₆₁	C₂	semidirect
4. C₄×#108×C₆₁	C₂	semidirect
5. C₆₁×#1772	C₂	semidirect
6. C₆₁×#1764	C₂	semidirect
7. C₄×#104×C₆₁	C₂	semidirect
8. C₄²×C₈×C₆₁	C₂	semidirect
12. C₆₁×#443	C₄	direct (N × H)
13. C₄×Q₁₆×C₆₁	C₄	semidirect
14. C₄×SD₁₆×C₆₁	C₄	semidirect
15. C₄×D₈×C₆₁	C₄	semidirect
38. #131×C₆₁	C₂×C₄	semidirect
54. #108×C₆₁	C₂×C₄	semidirect
57. #104×C₆₁	C₂×C₄	semidirect
67. C₄×C₈×C₆₁	C₂×C₄	semidirect
81. Q₁₆×C₆₁	C₄²	semidirect
82. SD₁₆×C₆₁	C₄²	semidirect
83. D₈×C₆₁	C₄²	semidirect
136. C₄×#443	C₆₁	direct (N × H)
146. C₄×C₆₁	#443	direct (N × H)
152. C₄×#131	C₂×C₆₁	semidirect
153. C₄×#108	C₂×C₆₁	semidirect
154. #1772	C₂×C₆₁	semidirect
155. #1764	C₂×C₆₁	semidirect
156. C₄×#104	C₂×C₆₁	semidirect
157. C₄²×C₈	C₂×C₆₁	semidirect
165. #443	C₄×C₆₁	direct (N × H)
166. C₄×Q₁₆	C₄×C₆₁	semidirect
167. C₄×SD₁₆	C₄×C₆₁	semidirect
168. C₄×D₈	C₄×C₆₁	semidirect
185. C₆₁	C₄×#443	direct (N × H)
192. #131	C₂×C₄×C₆₁	semidirect
208. #108	C₂×C₄×C₆₁	semidirect
211. #104	C₂×C₄×C₆₁	semidirect
221. C₄×C₈	C₂×C₄×C₆₁	semidirect
235. Q₁₆	C₄²×C₆₁	semidirect
236. SD₁₆	C₄²×C₆₁	semidirect
237. D₈	C₄²×C₆₁	semidirect
299. C₄	C₆₁×#443	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×#443	2
C₄×#135×C₆₁	2
C₄×#131×C₆₁	2
C₄×#108×C₆₁	2
C₆₁×#1772	2
C₆₁×#1764	2
C₄×#104×C₆₁	2
C₄²×C₈×C₆₁	2
C₄×#443	61

Subgroups — C4×C61×#443 #29523942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#443 #29523942