Subgroups — C₄×C₆₁×#444 #29523944

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#444	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×#1773	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1765	C₂	1	4	yes	maximal, complemented
7	C₄×#104×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₄×#97×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×#444	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×#97×C₆₁	C₂²	1	2	yes	—
25	C₂×#97×C₆₁	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₁	—	2	2	no	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
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	4	no	—
62	#97×C₆₁	—	2	2	no	—
63	#97×C₆₁	C₂×C₄	1	8	yes	complemented
64	C₄×C₈×C₆₁	—	2	4	no	—
65	C₄×C₈×C₆₁	—	2	2	no	—
66	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₁	D₄	1	2	yes	—
69	C₄×C₈×C₆₁	D₄	1	2	yes	—
70	#41×C₆₁	—	2	4	no	—
71	#41×C₆₁	C₂²×C₄	1	8	yes	—
72	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂×D₄×C₆₁	—	2	2	no	—
74	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₁	—	2	2	no	—
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	4	no	—
78	C₂²×C₄×C₆₁	#41	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
80	Q₁₆×C₆₁	C₄²	1	4	yes	complemented
81	SD₁₆×C₆₁	C₄²	1	8	yes	complemented
82	D₈×C₆₁	C₄²	1	4	yes	complemented
83	M₁₆×C₆₁	—	2	4	no	—
84	M₁₆×C₆₁	—	2	16	no	—
85	M₁₆×C₆₁	C₂²×C₄	1	8	yes	—
86	C₂×C₈×C₆₁	—	2	4	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	16	no	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₆₁	#41	1	4	yes	—
96	C₄⋊C₄×C₆₁	—	2	4	no	—
97	C₄⋊C₄×C₆₁	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₆₁	—	2	2	no	—
99	C₂²⋊C₄×C₆₁	—	2	4	no	—
100	C₄²×C₆₁	—	2	4	no	—
101	C₄²×C₆₁	—	2	2	no	—
102	C₄²×C₆₁	—	2	2	no	—
103	C₄²×C₆₁	—	4	4	no	—
104	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
105	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
106	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₆₁	—	2	2	no	—
109	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
110	Q₈×C₆₁	—	2	2	no	—
111	Q₈×C₆₁	C₂×C₄²	1	4	yes	—
112	D₄×C₆₁	—	2	8	no	—
113	D₄×C₆₁	—	2	2	no	—
114	D₄×C₆₁	C₂×C₄²	1	4	yes	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	8	no	—
117	C₂×C₄×C₆₁	—	2	4	no	—
118	C₂×C₄×C₆₁	—	2	4	no	—
119	C₂×C₄×C₆₁	—	2	4	no	—
120	C₂×C₄×C₆₁	—	4	4	no	—
121	C₂×C₄×C₆₁	—	4	8	no	—
122	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
128	C₈×C₆₁	—	2	8	no	—
129	C₈×C₆₁	—	2	2	no	—
130	C₈×C₆₁	C₂×C₄²	1	2	yes	—
131	C₈×C₆₁	C₂×C₄²	1	2	yes	—
132	C₈×C₆₁	C₄×D₄	1	8	yes	—
133	C₄×#444	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₆₁	—	2	2	no	—
135	C₂²×C₆₁	—	2	4	no	—
136	C₂²×C₆₁	—	4	4	no	—
137	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₆₁	C₄×#41	1	2	yes	—
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	2	4	no	—
141	C₄×C₆₁	—	4	4	no	—
142	C₄×C₆₁	—	4	4	no	—
143	C₄×C₆₁	#444	1	4	yes	direct product factor
144	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
145	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
146	C₄×C₆₁	C₄×#41	1	1	yes	characteristic
147	C₂×#444	C₂×C₆₁	1	2	yes	—
148	C₄×#135	C₂×C₆₁	1	1	yes	characteristic
149	C₄×#131	C₂×C₆₁	1	2	yes	complemented
150	C₄×#108	C₂×C₆₁	1	1	yes	characteristic, complemented
151	#1773	C₂×C₆₁	1	2	yes	complemented
152	#1765	C₂×C₆₁	1	4	yes	complemented
153	C₄×#104	C₂×C₆₁	1	2	yes	complemented
154	C₄×#97	C₂×C₆₁	1	1	yes	characteristic, complemented
155	C₂×C₆₁	—	2	2	no	—
156	C₂×C₆₁	—	4	4	no	—
157	C₂×C₆₁	C₄²×D₄	1	1	yes	characteristic
158	C₂×C₆₁	C₂×#444	1	2	yes	—
159	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
160	C₂×#131	C₂²×C₆₁	1	4	yes	—
161	C₄×#41	C₂²×C₆₁	1	2	yes	—
162	#444	C₄×C₆₁	1	16	yes	direct product factor
163	C₄×Q₁₆	C₄×C₆₁	1	2	yes	complemented
164	C₄×SD₁₆	C₄×C₆₁	1	4	yes	complemented
165	C₄×D₈	C₄×C₆₁	1	2	yes	complemented
166	C₂×#108	C₂²×C₆₁	1	2	yes	—
167	#427	C₂²×C₆₁	1	1	yes	characteristic
168	C₂×#104	C₂²×C₆₁	1	4	yes	—
169	#416	C₂²×C₆₁	1	2	yes	—
170	#405	—	2	4	no	—
171	#405	C₂²×C₆₁	1	4	yes	—
172	C₄×M₁₆	—	2	2	no	—
173	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
174	C₂×#97	C₂²×C₆₁	1	2	yes	—
175	C₂×#97	C₂²×C₆₁	1	2	yes	—
176	C₂×C₄×C₈	—	2	2	no	—
177	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
179	C₄³	C₂²×C₆₁	1	1	yes	characteristic
180	#341	C₂²×C₆₁	1	2	yes	—
181	#339	C₂²×C₆₁	1	4	yes	—
182	C₆₁	C₄×#444	1	1	yes	characteristic, direct product factor, Sylow
183	C₂×#41	C₂³×C₆₁	1	2	yes	—
184	#135	C₂×C₄×C₆₁	1	8	yes	—
185	C₂×Q₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
186	C₂×SD₁₆	C₂×C₄×C₆₁	1	2	yes	—
187	C₂×D₈	C₂×C₄×C₆₁	1	1	yes	characteristic
188	#131	—	2	8	no	—
189	#131	C₂×C₄×C₆₁	1	16	yes	complemented
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
192	C₂²×C₈	—	2	4	no	—
193	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
196	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	—
197	C₄×D₄	—	2	4	no	—
198	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	—
199	#117	—	2	2	no	—
200	#117	C₂³×C₆₁	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
204	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
205	#108	C₂×C₄×C₆₁	1	8	yes	complemented
206	#107	C₂×C₄×C₆₁	1	2	yes	—
207	#106	C₂×C₄×C₆₁	1	2	yes	—
208	#104	C₂×C₄×C₆₁	1	16	yes	complemented
209	#103	C₂×C₄×C₆₁	1	4	yes	—
210	#102	C₂×C₄×C₆₁	1	4	yes	—
211	#97	—	2	4	no	—
212	#97	—	2	4	no	—
213	#97	—	2	2	no	—
214	#97	C₂×C₄×C₆₁	1	8	yes	complemented
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
218	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
219	C₄×C₈	D₄×C₆₁	1	2	yes	—
220	C₄×C₈	D₄×C₆₁	1	2	yes	—
221	#41	—	2	4	no	—
222	#41	C₂²×C₄×C₆₁	1	8	yes	—
223	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	#41×C₆₁	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
231	Q₁₆	C₄²×C₆₁	1	4	yes	complemented
232	SD₁₆	C₄²×C₆₁	1	8	yes	complemented
233	D₈	C₄²×C₆₁	1	4	yes	complemented
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	16	no	—
236	M₁₆	C₂²×C₄×C₆₁	1	8	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	16	no	—
240	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
242	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₆₁	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
246	C₂×C₈	#41×C₆₁	1	4	yes	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×C₄×C₆₁	1	4	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	4	no	—
255	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
256	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
257	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
261	Q₈	—	2	2	no	—
262	Q₈	C₂×C₄²×C₆₁	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	2	2	no	—
265	D₄	C₂×C₄²×C₆₁	1	4	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
275	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
276	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
279	C₈	—	2	8	no	—
280	C₈	—	2	2	no	—
281	C₈	C₂×C₄²×C₆₁	1	2	yes	—
282	C₈	C₂×C₄²×C₆₁	1	2	yes	—
283	C₈	C₄×D₄×C₆₁	1	8	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
288	C₂²	C₄×#41×C₆₁	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	4	no	—
293	C₄	C₆₁×#444	1	4	yes	direct product factor
294	C₄	C₂²×C₄²×C₆₁	1	1	yes	characteristic
295	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
296	C₄	C₄×#41×C₆₁	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	4	4	no	—
299	C₂	C₄²×D₄×C₆₁	1	1	yes	characteristic
300	C₂	C₂×C₆₁×#444	1	2	yes	—

	subgroup	G/H
center	C₄²×C₆₁	C₂×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 ◁ C₄²×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×#1773	C₂	semidirect
6. C₆₁×#1765	C₂	semidirect
7. C₄×#104×C₆₁	C₂	semidirect
8. C₄×#97×C₆₁	C₂	semidirect
12. C₆₁×#444	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
63. #97×C₆₁	C₂×C₄	semidirect
80. Q₁₆×C₆₁	C₄²	semidirect
81. SD₁₆×C₆₁	C₄²	semidirect
82. D₈×C₆₁	C₄²	semidirect
133. C₄×#444	C₆₁	direct (N × H)
143. C₄×C₆₁	#444	direct (N × H)
149. C₄×#131	C₂×C₆₁	semidirect
150. C₄×#108	C₂×C₆₁	semidirect
151. #1773	C₂×C₆₁	semidirect
152. #1765	C₂×C₆₁	semidirect
153. C₄×#104	C₂×C₆₁	semidirect
154. C₄×#97	C₂×C₆₁	semidirect
162. #444	C₄×C₆₁	direct (N × H)
163. C₄×Q₁₆	C₄×C₆₁	semidirect
164. C₄×SD₁₆	C₄×C₆₁	semidirect
165. C₄×D₈	C₄×C₆₁	semidirect
182. C₆₁	C₄×#444	direct (N × H)
189. #131	C₂×C₄×C₆₁	semidirect
205. #108	C₂×C₄×C₆₁	semidirect
208. #104	C₂×C₄×C₆₁	semidirect
214. #97	C₂×C₄×C₆₁	semidirect
231. Q₁₆	C₄²×C₆₁	semidirect
232. SD₁₆	C₄²×C₆₁	semidirect
233. D₈	C₄²×C₆₁	semidirect
293. C₄	C₆₁×#444	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×#444	2
C₄×#135×C₆₁	2
C₄×#131×C₆₁	2
C₄×#108×C₆₁	2
C₆₁×#1773	2
C₆₁×#1765	2
C₄×#104×C₆₁	2
C₄×#97×C₆₁	2
C₄×#444	61

Subgroups — C4×C61×#444 #29523944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#444 #29523944