Subgroups — C₂×C₆₁×#1994 #29524760

Order 15616.

1422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1994	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₆₁×#339	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
7	C₆₁×#533	—	2	2	no	—
8	C₆₁×#533	—	2	2	no	—
9	C₆₁×#533	C₄	1	4	yes	complemented
10	C₆₁×#533	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₆₁	C₄	1	2	yes	complemented
12	C₂×#125×C₆₁	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
14	C₂×C₄×Q₈×C₆₁	—	2	2	no	—
15	C₂×#117×C₆₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₁	—	2	2	no	—
17	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#423	—	2	2	no	—
20	C₆₁×#423	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
22	C₄³×C₆₁	—	2	2	no	—
23	C₄³×C₆₁	C₂²	1	4	yes	—
24	C₆₁×#339	C₂²	1	16	yes	—
25	C₂²×Q₈×C₆₁	—	2	2	no	—
26	C₂³×C₄×C₆₁	—	2	2	no	—
27	C₂³×C₄×C₆₁	D₄	1	2	yes	—
28	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₆₁	—	2	2	no	—
30	C₂×M₁₆×C₆₁	—	2	8	no	—
31	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₆₁	—	2	2	no	—
33	C₄⋊Q₈×C₆₁	—	2	4	no	—
34	C₄⋊Q₈×C₆₁	C₂×C₄	1	8	yes	complemented
35	#125×C₆₁	—	2	4	no	—
36	#125×C₆₁	C₂×C₄	1	8	yes	complemented
37	C₂²⋊Q₈×C₆₁	—	2	8	no	—
38	C₂²⋊Q₈×C₆₁	—	2	8	no	—
39	C₄×Q₈×C₆₁	—	2	8	no	—
40	C₄×Q₈×C₆₁	—	2	8	no	—
41	#117×C₆₁	—	2	2	no	—
42	#117×C₆₁	—	2	2	no	—
43	#117×C₆₁	C₂×C₄	1	4	yes	—
44	#117×C₆₁	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
49	C₂×C₄²×C₆₁	—	2	16	no	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	—	2	8	no	—
52	C₂×C₄²×C₆₁	—	2	2	no	—
53	C₂×C₄²×C₆₁	—	2	4	no	—
54	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	D₄	1	2	yes	—
58	C₂×C₄²×C₆₁	D₄	1	2	yes	—
59	C₂×C₄²×C₆₁	D₄	1	2	yes	—
60	C₂×C₄²×C₆₁	Q₈	1	8	yes	complemented
61	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
62	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
63	C₄⋊C₈×C₆₁	—	2	8	no	—
64	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₆₁	—	2	8	no	—
66	C₂×Q₈×C₆₁	—	4	8	no	—
67	C₂²×C₄×C₆₁	—	2	16	no	—
68	C₂²×C₄×C₆₁	—	2	8	no	—
69	C₂²×C₄×C₆₁	—	2	4	no	—
70	C₂²×C₄×C₆₁	—	2	8	no	—
71	C₂²×C₄×C₆₁	—	2	4	no	—
72	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
82	C₂²×C₄×C₆₁	#41	1	4	yes	—
83	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₆₁	—	2	8	no	—
85	M₁₆×C₆₁	—	4	4	no	—
86	C₂×C₈×C₆₁	—	2	4	no	—
87	C₂×C₈×C₆₁	—	4	4	no	—
88	C₄⋊C₄×C₆₁	—	2	16	no	—
89	C₄⋊C₄×C₆₁	—	2	8	no	—
90	C₄⋊C₄×C₆₁	—	2	4	no	—
91	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
92	C₂²⋊C₄×C₆₁	—	2	4	no	—
93	C₂²⋊C₄×C₆₁	—	2	4	no	—
94	C₄²×C₆₁	—	2	32	no	—
95	C₄²×C₆₁	—	2	4	no	—
96	C₄²×C₆₁	—	2	8	no	—
97	C₄²×C₆₁	—	2	4	no	—
98	C₄²×C₆₁	—	2	4	no	—
99	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
102	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₆₁	C₂×D₄	1	4	yes	—
105	C₄²×C₆₁	C₂×Q₈	1	8	yes	complemented
106	C₂³×C₆₁	—	2	4	no	—
107	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂³×C₆₁	C₂²⋊Q₈	1	4	yes	—
110	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₆₁	—	4	16	no	—
112	C₂×C₄×C₆₁	—	2	16	no	—
113	C₂×C₄×C₆₁	—	2	32	no	—
114	C₂×C₄×C₆₁	—	2	8	no	—
115	C₂×C₄×C₆₁	—	2	4	no	—
116	C₂×C₄×C₆₁	—	2	8	no	—
117	C₂×C₄×C₆₁	—	2	8	no	—
118	C₂×C₄×C₆₁	—	2	8	no	—
119	C₂×C₄×C₆₁	—	4	8	no	—
120	C₂×C₄×C₆₁	#104	1	8	yes	—
121	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₆₁	C₄×Q₈	1	4	yes	—
126	C₂×C₄×C₆₁	C₄×Q₈	1	4	yes	—
127	C₂×C₄×C₆₁	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₆₁	#124	1	2	yes	—
129	C₂×C₄×C₆₁	#124	1	2	yes	—
130	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
135	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
136	C₈×C₆₁	—	4	4	no	—
137	C₂×#1994	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₆₁	—	2	2	no	—
139	C₂²×C₆₁	—	2	2	no	—
140	C₂²×C₆₁	—	2	8	no	—
141	C₂²×C₆₁	#391	1	2	yes	—
142	C₂²×C₆₁	#391	1	2	yes	—
143	C₂²×C₆₁	C₂×#104	1	4	yes	—
144	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	2	yes	—
146	C₄×C₆₁	—	2	16	no	—
147	C₄×C₆₁	—	2	4	no	—
148	C₄×C₆₁	—	4	8	no	—
149	C₄×C₆₁	#391	1	2	yes	—
150	C₄×C₆₁	#391	1	2	yes	—
151	C₄×C₆₁	C₂×#104	1	8	yes	—
152	C₄×C₆₁	C₂×C₄×Q₈	1	2	yes	—
153	C₄×C₆₁	C₂×#124	1	1	yes	characteristic
154	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₂×#533	C₂×C₆₁	1	1	yes	characteristic
156	C₂×#423	C₂×C₆₁	1	1	yes	characteristic
157	C₂×C₄³	C₂×C₆₁	1	1	yes	characteristic
158	#1994	C₂×C₆₁	1	8	yes	direct product factor
159	C₂×#339	C₂×C₆₁	1	4	yes	—
160	C₂×C₆₁	—	2	2	no	—
161	C₂×C₆₁	—	2	2	no	—
162	C₂×C₆₁	#1994	1	4	yes	direct product factor
163	C₂×C₆₁	C₂×#391	1	1	yes	characteristic
164	C₂×C₆₁	C₂²×#104	1	2	yes	—
165	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
166	#533	—	2	2	no	—
167	#533	—	2	2	no	—
168	#533	C₄×C₆₁	1	4	yes	complemented
169	#533	C₂²×C₆₁	1	4	yes	—
170	C₂×C₄⋊Q₈	C₄×C₆₁	1	2	yes	complemented
171	C₂×#125	C₄×C₆₁	1	2	yes	complemented
172	C₂×C₂²⋊Q₈	—	2	2	no	—
173	C₂×C₄×Q₈	—	2	2	no	—
174	C₂×#117	C₂²×C₆₁	1	2	yes	—
175	C₂²×C₄²	—	2	2	no	—
176	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
177	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
178	#423	—	2	2	no	—
179	#423	C₂²×C₆₁	1	4	yes	—
180	C₂×C₄⋊C₈	—	2	2	no	—
181	C₄³	—	2	2	no	—
182	C₄³	C₂²×C₆₁	1	4	yes	—
183	#339	C₂²×C₆₁	1	16	yes	—
184	C₆₁	C₂×#1994	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×Q₈	—	2	2	no	—
186	C₂³×C₄	—	2	2	no	—
187	C₂³×C₄	D₄×C₆₁	1	2	yes	—
188	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	8	no	—
191	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₄⋊Q₈	—	2	4	no	—
194	C₄⋊Q₈	C₂×C₄×C₆₁	1	8	yes	complemented
195	#125	—	2	4	no	—
196	#125	C₂×C₄×C₆₁	1	8	yes	complemented
197	C₂²⋊Q₈	—	2	8	no	—
198	C₂²⋊Q₈	—	2	8	no	—
199	C₄×Q₈	—	2	8	no	—
200	C₄×Q₈	—	2	8	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	C₂×C₄×C₆₁	1	4	yes	—
204	#117	C₂³×C₆₁	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	—	2	16	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	8	no	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
215	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
216	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
217	C₂×C₄²	D₄×C₆₁	1	2	yes	—
218	C₂×C₄²	D₄×C₆₁	1	2	yes	—
219	C₂×C₄²	D₄×C₆₁	1	2	yes	—
220	C₂×C₄²	Q₈×C₆₁	1	8	yes	complemented
221	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
222	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
223	C₄⋊C₈	—	2	8	no	—
224	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂²×C₄	—	2	16	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
234	C₂²×C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
238	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₆₁	1	4	yes	—
242	C₂²×C₄	#41×C₆₁	1	4	yes	—
243	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
244	M₁₆	—	2	8	no	—
245	M₁₆	—	4	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	4	4	no	—
248	C₄⋊C₄	—	2	16	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	32	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
260	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
261	C₄²	C₂²⋊C₄×C₆₁	1	2	yes	—
262	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
263	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
264	C₄²	C₂×D₄×C₆₁	1	4	yes	—
265	C₄²	C₂×Q₈×C₆₁	1	8	yes	complemented
266	C₂³	—	2	4	no	—
267	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
269	C₂³	C₂²⋊Q₈×C₆₁	1	4	yes	—
270	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	C₂×C₄	—	2	16	no	—
273	C₂×C₄	—	2	32	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	#104×C₆₁	1	8	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
285	C₂×C₄	C₄×Q₈×C₆₁	1	4	yes	—
286	C₂×C₄	C₄×Q₈×C₆₁	1	4	yes	—
287	C₂×C₄	C₂²⋊Q₈×C₆₁	1	4	yes	—
288	C₂×C₄	#124×C₆₁	1	2	yes	—
289	C₂×C₄	#124×C₆₁	1	2	yes	—
290	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
291	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
292	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₆₁	1	2	yes	—
295	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
296	C₈	—	4	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	8	no	—
300	C₂²	C₆₁×#391	1	2	yes	—
301	C₂²	C₆₁×#391	1	2	yes	—
302	C₂²	C₂×#104×C₆₁	1	4	yes	—
303	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
304	C₂²	C₂×C₂²⋊Q₈×C₆₁	1	2	yes	—
305	C₄	—	2	16	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	8	no	—
308	C₄	C₆₁×#391	1	2	yes	—
309	C₄	C₆₁×#391	1	2	yes	—
310	C₄	C₂×#104×C₆₁	1	8	yes	—
311	C₄	C₂×C₄×Q₈×C₆₁	1	2	yes	—
312	C₄	C₂×#124×C₆₁	1	1	yes	characteristic
313	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	C₆₁×#1994	1	4	yes	direct product factor
317	C₂	C₂×C₆₁×#391	1	1	yes	characteristic
318	C₂	C₂²×#104×C₆₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₆₁×#1994	C₂	direct (N × H)
9. C₆₁×#533	C₄	semidirect
11. C₂×C₄⋊Q₈×C₆₁	C₄	semidirect
12. C₂×#125×C₆₁	C₄	semidirect
34. C₄⋊Q₈×C₆₁	C₂×C₄	semidirect
36. #125×C₆₁	C₂×C₄	semidirect
60. C₂×C₄²×C₆₁	Q₈	semidirect
105. C₄²×C₆₁	C₂×Q₈	semidirect
137. C₂×#1994	C₆₁	direct (N × H)
158. #1994	C₂×C₆₁	direct (N × H)
162. C₂×C₆₁	#1994	direct (N × H)
168. #533	C₄×C₆₁	semidirect
170. C₂×C₄⋊Q₈	C₄×C₆₁	semidirect
171. C₂×#125	C₄×C₆₁	semidirect
184. C₆₁	C₂×#1994	direct (N × H)
194. C₄⋊Q₈	C₂×C₄×C₆₁	semidirect
196. #125	C₂×C₄×C₆₁	semidirect
220. C₂×C₄²	Q₈×C₆₁	semidirect
265. C₄²	C₂×Q₈×C₆₁	semidirect
316. C₂	C₆₁×#1994	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×#533	2
C₂×C₆₁×#423	2
C₂×C₄³×C₆₁	2
C₆₁×#1994	2
C₂×C₆₁×#339	2
C₂×#1994	61

Subgroups — C2×C61×#1994 #29524760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#1994 #29524760