Subgroups — C₂²×C₆₁×#461 #29563837

Order 15616.

3222 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₆₁	C₂	1	2	yes	maximal
2	C₂²×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₁×#461	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#103×C₆₁	C₂	1	2	yes	maximal, complemented
7	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₆₁	—	2	1	no	—
10	C₂²×SD₁₆×C₆₁	—	2	16	no	—
11	C₂²×SD₁₆×C₆₁	C₂²	1	24	yes	—
12	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₆₁	—	2	6	no	—
14	C₂×C₄⋊Q₈×C₆₁	C₂²	1	12	yes	complemented
15	C₂×C₄×D₄×C₆₁	—	2	6	no	—
16	C₂×C₄×D₄×C₆₁	C₂²	1	12	yes	—
17	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₆₁	—	2	1	no	—
21	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#461	C₂²	1	64	yes	direct product factor
23	C₂×C₄⋊C₈×C₆₁	C₂²	1	12	yes	complemented
24	C₂×#103×C₆₁	C₂²	1	24	yes	complemented
25	C₂⁵×C₆₁	—	2	1	no	—
26	C₂²×Q₈×C₆₁	—	2	6	no	—
27	C₂²×Q₈×C₆₁	—	2	8	no	—
28	C₂²×Q₈×C₆₁	—	2	8	no	—
29	C₂²×Q₈×C₆₁	C₂³	1	12	yes	—
30	C₂²×D₄×C₆₁	—	2	3	no	—
31	C₂²×D₄×C₆₁	—	2	4	no	—
32	C₂²×D₄×C₆₁	D₄	1	8	yes	—
33	C₂²×D₄×C₆₁	C₂³	1	6	yes	—
34	C₂³×C₄×C₆₁	—	2	1	no	—
35	C₂³×C₄×C₆₁	—	2	2	no	—
36	C₂³×C₄×C₆₁	—	2	12	no	—
37	C₂³×C₄×C₆₁	—	4	8	no	—
38	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
41	C₂×SD₁₆×C₆₁	—	2	96	no	—
42	C₂×SD₁₆×C₆₁	C₂³	1	32	yes	—
43	C₂²×C₈×C₆₁	—	2	8	no	—
44	C₂²×C₈×C₆₁	C₂³	1	12	yes	—
45	C₄⋊Q₈×C₆₁	—	2	24	no	—
46	C₄⋊Q₈×C₆₁	C₂³	1	16	yes	complemented
47	C₄×D₄×C₆₁	—	2	24	no	—
48	C₄×D₄×C₆₁	C₂³	1	16	yes	—
49	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
50	C₂×C₄⋊C₄×C₆₁	—	2	12	no	—
51	C₂×C₄⋊C₄×C₆₁	—	2	3	no	—
52	C₂×C₄⋊C₄×C₆₁	C₂³	1	6	yes	—
53	C₂×C₂²⋊C₄×C₆₁	—	2	12	no	—
54	C₂×C₄²×C₆₁	D₄	1	6	yes	—
55	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
56	C₄⋊C₈×C₆₁	C₂³	1	16	yes	complemented
57	#103×C₆₁	C₂³	1	32	yes	complemented
58	C₂⁴×C₆₁	—	2	8	no	—
59	C₂⁴×C₆₁	—	2	6	no	—
60	C₂⁴×C₆₁	—	4	4	no	—
61	C₂⁴×C₆₁	—	4	4	no	—
62	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂×Q₈×C₆₁	—	2	12	no	—
64	C₂×Q₈×C₆₁	—	2	24	no	—
65	C₂×Q₈×C₆₁	—	2	24	no	—
66	C₂×Q₈×C₆₁	—	4	24	no	—
67	C₂×Q₈×C₆₁	C₂⁴	1	8	yes	—
68	C₂×D₄×C₆₁	—	2	24	no	—
69	C₂×D₄×C₆₁	—	2	6	no	—
70	C₂×D₄×C₆₁	—	2	12	no	—
71	C₂×D₄×C₆₁	C₂×D₄	1	24	yes	—
72	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
73	C₂²×C₄×C₆₁	—	2	4	no	—
74	C₂²×C₄×C₆₁	—	2	6	no	—
75	C₂²×C₄×C₆₁	—	2	12	no	—
76	C₂²×C₄×C₆₁	—	2	16	no	—
77	C₂²×C₄×C₆₁	—	4	8	no	—
78	C₂²×C₄×C₆₁	—	4	4	no	—
79	C₂²×C₄×C₆₁	—	4	48	no	—
80	C₂²×C₄×C₆₁	SD₁₆	1	8	yes	complemented
81	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
82	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
85	C₂²×C₄×C₆₁	#41	1	4	yes	—
86	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
87	SD₁₆×C₆₁	—	2	64	no	—
88	C₂×C₈×C₆₁	—	2	24	no	—
89	C₂×C₈×C₆₁	C₂⁴	1	8	yes	—
90	C₄⋊C₄×C₆₁	—	2	16	no	—
91	C₄⋊C₄×C₆₁	—	2	16	no	—
92	C₄⋊C₄×C₆₁	—	2	6	no	—
93	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
94	C₂²⋊C₄×C₆₁	—	2	16	no	—
95	C₄²×C₆₁	C₂×D₄	1	12	yes	—
96	C₄²×C₆₁	C₂⁴	1	4	yes	—
97	C₂³×C₆₁	—	2	24	no	—
98	C₂³×C₆₁	—	2	4	no	—
99	C₂³×C₆₁	—	4	12	no	—
100	C₂³×C₆₁	—	4	12	no	—
101	C₂³×C₆₁	—	4	32	no	—
102	C₂³×C₆₁	#121	1	4	yes	—
103	C₂³×C₆₁	C₂×SD₁₆	1	4	yes	—
104	C₂³×C₆₁	#137	1	4	yes	—
105	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
106	Q₈×C₆₁	—	2	8	no	—
107	Q₈×C₆₁	—	2	8	no	—
108	Q₈×C₆₁	—	4	24	no	—
109	D₄×C₆₁	—	2	24	no	—
110	D₄×C₆₁	—	2	4	no	—
111	D₄×C₆₁	C₂²×D₄	1	8	yes	—
112	C₂×C₄×C₆₁	—	2	12	no	—
113	C₂×C₄×C₆₁	—	2	4	no	—
114	C₂×C₄×C₆₁	—	2	8	no	—
115	C₂×C₄×C₆₁	—	4	24	no	—
116	C₂×C₄×C₆₁	—	4	12	no	—
117	C₂×C₄×C₆₁	—	4	32	no	—
118	C₂×C₄×C₆₁	#121	1	12	yes	—
119	C₂×C₄×C₆₁	C₂×SD₁₆	1	24	yes	complemented
120	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
122	C₂×C₄×C₆₁	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
124	C₂×C₄×C₆₁	C₂×#41	1	6	yes	—
125	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
126	C₈×C₆₁	—	2	8	no	—
127	C₂²×#461	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₆₁	—	2	8	no	—
129	C₂²×C₆₁	—	4	48	no	—
130	C₂²×C₆₁	—	4	4	no	—
131	C₂²×C₆₁	—	4	4	no	—
132	C₂²×C₆₁	#461	1	16	yes	direct product factor
133	C₂²×C₆₁	C₂×#121	1	6	yes	—
134	C₂²×C₆₁	C₂²×SD₁₆	1	6	yes	—
135	C₂²×C₆₁	C₂×#137	1	6	yes	—
136	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
137	C₄×C₆₁	—	2	4	no	—
138	C₄×C₆₁	—	4	8	no	—
139	C₄×C₆₁	—	4	4	no	—
140	C₄×C₆₁	C₂×#121	1	6	yes	—
141	C₄×C₆₁	C₂²×SD₁₆	1	8	yes	complemented
142	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
143	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
144	C₂³×SD₁₆	C₂×C₆₁	1	2	yes	—
145	C₂²×C₄⋊Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
146	C₂²×C₄×D₄	C₂×C₆₁	1	1	yes	characteristic
147	C₂×#461	C₂×C₆₁	1	24	yes	direct product factor
148	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
149	C₂²×#103	C₂×C₆₁	1	2	yes	complemented
150	C₂×C₆₁	—	4	8	no	—
151	C₂×C₆₁	C₂×#461	1	12	yes	direct product factor
152	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
153	C₂×C₆₁	C₂³×SD₁₆	1	1	yes	characteristic
154	C₂×C₆₁	C₂²×#137	1	1	yes	characteristic
155	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
156	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
157	C₂⁴×C₄	—	2	1	no	—
158	C₂²×SD₁₆	—	2	16	no	—
159	C₂²×SD₁₆	C₂²×C₆₁	1	24	yes	—
160	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
161	C₂×C₄⋊Q₈	—	2	6	no	—
162	C₂×C₄⋊Q₈	C₂²×C₆₁	1	12	yes	complemented
163	C₂×C₄×D₄	—	2	6	no	—
164	C₂×C₄×D₄	C₂²×C₆₁	1	12	yes	—
165	C₂²×C₄⋊C₄	—	2	1	no	—
166	C₂²×C₄⋊C₄	—	2	1	no	—
167	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	—	2	1	no	—
169	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
170	#461	C₂²×C₆₁	1	64	yes	direct product factor
171	C₂×C₄⋊C₈	C₂²×C₆₁	1	12	yes	complemented
172	C₂×#103	C₂²×C₆₁	1	24	yes	complemented
173	C₆₁	C₂²×#461	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁵	—	2	1	no	—
175	C₂²×Q₈	—	2	6	no	—
176	C₂²×Q₈	—	2	8	no	—
177	C₂²×Q₈	—	2	8	no	—
178	C₂²×Q₈	C₂³×C₆₁	1	12	yes	—
179	C₂²×D₄	—	2	3	no	—
180	C₂²×D₄	—	2	4	no	—
181	C₂²×D₄	D₄×C₆₁	1	8	yes	—
182	C₂²×D₄	C₂³×C₆₁	1	6	yes	—
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	—	2	12	no	—
186	C₂³×C₄	—	4	8	no	—
187	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
188	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
190	C₂×SD₁₆	—	2	96	no	—
191	C₂×SD₁₆	C₂³×C₆₁	1	32	yes	—
192	C₂²×C₈	—	2	8	no	—
193	C₂²×C₈	C₂³×C₆₁	1	12	yes	—
194	C₄⋊Q₈	—	2	24	no	—
195	C₄⋊Q₈	C₂³×C₆₁	1	16	yes	complemented
196	C₄×D₄	—	2	24	no	—
197	C₄×D₄	C₂³×C₆₁	1	16	yes	—
198	C₂×C₄⋊C₄	—	2	12	no	—
199	C₂×C₄⋊C₄	—	2	12	no	—
200	C₂×C₄⋊C₄	—	2	3	no	—
201	C₂×C₄⋊C₄	C₂³×C₆₁	1	6	yes	—
202	C₂×C₂²⋊C₄	—	2	12	no	—
203	C₂×C₄²	D₄×C₆₁	1	6	yes	—
204	C₂×C₄²	C₂³×C₆₁	1	6	yes	—
205	C₄⋊C₈	C₂³×C₆₁	1	16	yes	complemented
206	#103	C₂³×C₆₁	1	32	yes	complemented
207	C₂⁴	—	2	8	no	—
208	C₂⁴	—	2	6	no	—
209	C₂⁴	—	4	4	no	—
210	C₂⁴	—	4	4	no	—
211	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
212	C₂×Q₈	—	2	12	no	—
213	C₂×Q₈	—	2	24	no	—
214	C₂×Q₈	—	2	24	no	—
215	C₂×Q₈	—	4	24	no	—
216	C₂×Q₈	C₂⁴×C₆₁	1	8	yes	—
217	C₂×D₄	—	2	24	no	—
218	C₂×D₄	—	2	6	no	—
219	C₂×D₄	—	2	12	no	—
220	C₂×D₄	C₂×D₄×C₆₁	1	24	yes	—
221	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	6	no	—
224	C₂²×C₄	—	2	12	no	—
225	C₂²×C₄	—	2	16	no	—
226	C₂²×C₄	—	4	8	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	48	no	—
229	C₂²×C₄	SD₁₆×C₆₁	1	8	yes	complemented
230	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
231	C₂²×C₄	C₂×D₄×C₆₁	1	3	yes	—
232	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
233	C₂²×C₄	C₂×D₄×C₆₁	1	6	yes	—
234	C₂²×C₄	#41×C₆₁	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₆₁	1	3	yes	—
236	SD₁₆	—	2	64	no	—
237	C₂×C₈	—	2	24	no	—
238	C₂×C₈	C₂⁴×C₆₁	1	8	yes	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	16	no	—
241	C₄⋊C₄	—	2	6	no	—
242	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
243	C₂²⋊C₄	—	2	16	no	—
244	C₄²	C₂×D₄×C₆₁	1	12	yes	—
245	C₄²	C₂⁴×C₆₁	1	4	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	24	no	—
248	C₂³	—	4	12	no	—
249	C₂³	—	4	12	no	—
250	C₂³	—	4	32	no	—
251	C₂³	#121×C₆₁	1	4	yes	—
252	C₂³	C₂×SD₁₆×C₆₁	1	4	yes	—
253	C₂³	#137×C₆₁	1	4	yes	—
254	C₂³	C₂²×D₄×C₆₁	1	3	yes	—
255	Q₈	—	2	8	no	—
256	Q₈	—	2	8	no	—
257	Q₈	—	4	24	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	24	no	—
260	D₄	C₂²×D₄×C₆₁	1	8	yes	—
261	C₂×C₄	—	2	12	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	12	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	—	4	32	no	—
267	C₂×C₄	#121×C₆₁	1	12	yes	—
268	C₂×C₄	C₂×SD₁₆×C₆₁	1	24	yes	complemented
269	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
270	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
271	C₂×C₄	C₂²×D₄×C₆₁	1	6	yes	—
272	C₂×C₄	C₂²×D₄×C₆₁	1	3	yes	—
273	C₂×C₄	C₂×#41×C₆₁	1	6	yes	—
274	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	48	no	—
280	C₂²	C₆₁×#461	1	16	yes	direct product factor
281	C₂²	C₂×#121×C₆₁	1	6	yes	—
282	C₂²	C₂²×SD₁₆×C₆₁	1	6	yes	—
283	C₂²	C₂×#137×C₆₁	1	6	yes	—
284	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	8	no	—
288	C₄	C₂×#121×C₆₁	1	6	yes	—
289	C₄	C₂²×SD₁₆×C₆₁	1	8	yes	complemented
290	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
291	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
292	C₂	—	4	8	no	—
293	C₂	C₂×C₆₁×#461	1	12	yes	direct product factor
294	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
295	C₂	C₂³×SD₁₆×C₆₁	1	1	yes	characteristic
296	C₂	C₂²×#137×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂³×D₄×C₆₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊Q₈×C₆₁	C₂	semidirect
4. C₂×C₆₁×#461	C₂	direct (N × H)
5. C₂²×C₄⋊C₈×C₆₁	C₂	semidirect
6. C₂²×#103×C₆₁	C₂	semidirect
14. C₂×C₄⋊Q₈×C₆₁	C₂²	semidirect
22. C₆₁×#461	C₂²	direct (N × H)
23. C₂×C₄⋊C₈×C₆₁	C₂²	semidirect
24. C₂×#103×C₆₁	C₂²	semidirect
46. C₄⋊Q₈×C₆₁	C₂³	semidirect
56. C₄⋊C₈×C₆₁	C₂³	semidirect
57. #103×C₆₁	C₂³	semidirect
80. C₂²×C₄×C₆₁	SD₁₆	semidirect
119. C₂×C₄×C₆₁	C₂×SD₁₆	semidirect
127. C₂²×#461	C₆₁	direct (N × H)
132. C₂²×C₆₁	#461	direct (N × H)
141. C₄×C₆₁	C₂²×SD₁₆	semidirect
145. C₂²×C₄⋊Q₈	C₂×C₆₁	semidirect
147. C₂×#461	C₂×C₆₁	direct (N × H)
148. C₂²×C₄⋊C₈	C₂×C₆₁	semidirect
149. C₂²×#103	C₂×C₆₁	semidirect
151. C₂×C₆₁	C₂×#461	direct (N × H)
162. C₂×C₄⋊Q₈	C₂²×C₆₁	semidirect
170. #461	C₂²×C₆₁	direct (N × H)
171. C₂×C₄⋊C₈	C₂²×C₆₁	semidirect
172. C₂×#103	C₂²×C₆₁	semidirect
173. C₆₁	C₂²×#461	direct (N × H)
195. C₄⋊Q₈	C₂³×C₆₁	semidirect
205. C₄⋊C₈	C₂³×C₆₁	semidirect
206. #103	C₂³×C₆₁	semidirect
229. C₂²×C₄	SD₁₆×C₆₁	semidirect
268. C₂×C₄	C₂×SD₁₆×C₆₁	semidirect
280. C₂²	C₆₁×#461	direct (N × H)
289. C₄	C₂²×SD₁₆×C₆₁	semidirect
293. C₂	C₂×C₆₁×#461	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₆₁	2
C₂²×C₄⋊Q₈×C₆₁	2
C₂²×C₄×D₄×C₆₁	2
C₂×C₆₁×#461	2
C₂²×C₄⋊C₈×C₆₁	2
C₂²×#103×C₆₁	2
C₂²×#461	61

Subgroups — C22×C61×#461 #29563837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₆₁×#461 #29563837