Subgroups — C₆₁×#33718 #29537151

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#510	C₂	1	4	yes	maximal, complemented
2	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2251	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2248	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2247	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₆₁	—	2	1	no	—
9	C₂×#135×C₆₁	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₆₁	—	2	1	no	—
11	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#532	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
14	C₆₁×#510	—	2	16	no	—
15	C₂×Q₃₂×C₆₁	C₂²	1	8	yes	—
16	C₂×SD₃₂×C₆₁	C₂²	1	8	yes	—
17	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#495	—	2	1	no	—
19	C₆₁×#495	C₂²	1	2	yes	—
20	C₆₁×#494	—	2	1	no	—
21	C₆₁×#494	—	2	1	no	—
22	C₆₁×#494	C₂²	1	2	yes	—
23	C₆₁×#494	C₂²	1	2	yes	—
24	C₆₁×#494	C₂²	1	1	yes	characteristic
25	C₆₁×#493	C₂²	1	1	yes	characteristic
26	C₆₁×#492	—	2	1	no	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#346	C₂²	1	2	yes	—
29	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
30	C₂×#41×C₆₁	—	2	2	no	—
31	C₂²×Q₈×C₆₁	—	2	2	no	—
32	#135×C₆₁	—	2	8	no	—
33	#135×C₆₁	—	4	4	no	—
34	C₂×Q₁₆×C₆₁	—	2	8	no	—
35	C₂×Q₁₆×C₆₁	—	2	2	no	—
36	C₂×Q₁₆×C₆₁	—	2	2	no	—
37	C₂×Q₁₆×C₆₁	—	2	2	no	—
38	C₂×Q₁₆×C₆₁	—	2	1	no	—
39	C₂×Q₁₆×C₆₁	—	4	4	no	—
40	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
41	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₆₁	—	2	2	no	—
43	C₂×SD₁₆×C₆₁	—	2	2	no	—
44	C₂×D₈×C₆₁	—	2	1	no	—
45	C₂×D₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₁	—	2	2	no	—
49	C₄⋊Q₈×C₆₁	—	2	2	no	—
50	C₄⋊Q₈×C₆₁	—	2	1	no	—
51	C₄⋊Q₈×C₆₁	—	4	2	no	—
52	#127×C₆₁	—	2	1	no	—
53	#124×C₆₁	—	2	2	no	—
54	#121×C₆₁	—	4	2	no	—
55	C₄×D₄×C₆₁	—	4	2	no	—
56	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
57	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
58	Q₃₂×C₆₁	—	2	16	no	—
59	SD₃₂×C₆₁	—	2	16	no	—
60	M₃₂×C₆₁	D₄	1	8	yes	complemented
61	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
62	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
64	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
66	#41×C₆₁	—	4	4	no	—
67	#41×C₆₁	—	4	4	no	—
68	C₂×Q₈×C₆₁	—	2	2	no	—
69	C₂×Q₈×C₆₁	—	2	4	no	—
70	C₂×Q₈×C₆₁	—	4	2	no	—
71	C₂×Q₈×C₆₁	—	4	4	no	—
72	C₂×Q₈×C₆₁	—	4	4	no	—
73	C₂×D₄×C₆₁	—	2	2	no	—
74	C₂×D₄×C₆₁	—	4	2	no	—
75	C₂×D₄×C₆₁	—	4	2	no	—
76	C₂²×C₄×C₆₁	—	4	2	no	—
77	C₂²×C₄×C₆₁	—	4	2	no	—
78	C₂²×C₄×C₆₁	D₈	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₆₁	—	2	8	no	—
81	Q₁₆×C₆₁	—	2	4	no	—
82	Q₁₆×C₆₁	—	4	4	no	—
83	Q₁₆×C₆₁	—	4	4	no	—
84	Q₁₆×C₆₁	—	4	4	no	—
85	Q₁₆×C₆₁	—	4	2	no	—
86	SD₁₆×C₆₁	—	4	4	no	—
87	SD₁₆×C₆₁	—	4	4	no	—
88	D₈×C₆₁	—	2	4	no	—
89	D₈×C₆₁	—	4	2	no	—
90	C₂×C₈×C₆₁	D₈	1	4	yes	—
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	4	4	no	—
97	C₄⋊C₄×C₆₁	—	4	4	no	—
98	C₄⋊C₄×C₆₁	—	4	2	no	—
99	C₂²⋊C₄×C₆₁	—	4	4	no	—
100	C₄²×C₆₁	D₈	1	2	yes	—
101	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₁₆×C₆₁	C₂×D₄	1	8	yes	—
104	C₂³×C₆₁	—	4	2	no	—
105	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
106	Q₈×C₆₁	—	4	2	no	—
107	Q₈×C₆₁	—	4	4	no	—
108	Q₈×C₆₁	—	4	2	no	—
109	Q₈×C₆₁	—	4	4	no	—
110	Q₈×C₆₁	—	8	4	no	—
111	D₄×C₆₁	—	4	2	no	—
112	D₄×C₆₁	—	4	2	no	—
113	D₄×C₆₁	—	8	4	no	—
114	D₄×C₆₁	—	8	4	no	—
115	C₂×C₄×C₆₁	—	2	4	no	—
116	C₂×C₄×C₆₁	—	4	2	no	—
117	C₂×C₄×C₆₁	—	4	4	no	—
118	C₂×C₄×C₆₁	—	8	4	no	—
119	C₂×C₄×C₆₁	—	8	4	no	—
120	C₂×C₄×C₆₁	#127	1	2	yes	—
121	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₆₁	#127	1	2	yes	—
127	C₈×C₆₁	C₂×D₈	1	2	yes	—
128	C₈×C₆₁	C₂×D₈	1	2	yes	—
129	C₈×C₆₁	C₂²×D₄	1	2	yes	—
130	#33718	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₆₁	—	2	2	no	—
132	C₂²×C₆₁	—	8	2	no	—
133	C₂²×C₆₁	—	8	4	no	—
134	C₂²×C₆₁	#493	1	2	yes	—
135	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₆₁	—	2	2	no	—
137	C₄×C₆₁	—	8	2	no	—
138	C₄×C₆₁	—	8	4	no	—
139	C₄×C₆₁	#493	1	2	yes	—
140	C₄×C₆₁	#510	1	4	yes	complemented
141	C₄×C₆₁	C₂×#127	1	1	yes	characteristic
142	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
143	C₂×#510	C₂×C₆₁	1	4	yes	complemented
144	#3147	C₂×C₆₁	1	1	yes	characteristic
145	C₂×#494	C₂×C₆₁	1	1	yes	characteristic, complemented
146	#2251	C₂×C₆₁	1	4	yes	complemented
147	#2248	C₂×C₆₁	1	2	yes	complemented
148	#2247	C₂×C₆₁	1	2	yes	complemented
149	C₄×M₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
150	C₂×C₆₁	—	2	2	no	—
151	C₂×C₆₁	—	16	2	no	—
152	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
153	C₂×C₆₁	C₂×#510	1	2	yes	—
154	C₂×#135	—	2	1	no	—
155	C₂×#135	C₂²×C₆₁	1	2	yes	—
156	C₂²×Q₁₆	—	2	1	no	—
157	C₂²×Q₁₆	C₂²×C₆₁	1	2	yes	—
158	#532	—	2	1	no	—
159	C₂×C₄⋊Q₈	—	2	1	no	—
160	#510	—	2	16	no	—
161	C₂×Q₃₂	C₂²×C₆₁	1	8	yes	—
162	C₂×SD₃₂	C₂²×C₆₁	1	8	yes	—
163	C₂×M₃₂	C₂²×C₆₁	1	2	yes	—
164	#495	—	2	1	no	—
165	#495	C₂²×C₆₁	1	2	yes	—
166	#494	—	2	1	no	—
167	#494	—	2	1	no	—
168	#494	C₂²×C₆₁	1	2	yes	—
169	#494	C₂²×C₆₁	1	2	yes	—
170	#494	C₂²×C₆₁	1	1	yes	characteristic
171	#493	C₂²×C₆₁	1	1	yes	characteristic
172	#492	—	2	1	no	—
173	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
174	#346	C₂²×C₆₁	1	2	yes	—
175	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
176	C₆₁	#33718	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×#41	—	2	2	no	—
178	C₂²×Q₈	—	2	2	no	—
179	#135	—	2	8	no	—
180	#135	—	4	4	no	—
181	C₂×Q₁₆	—	2	8	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	4	4	no	—
187	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
188	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×SD₁₆	—	2	2	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	C₂³×C₆₁	1	2	yes	—
193	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	C₄⋊Q₈	—	4	2	no	—
199	#127	—	2	1	no	—
200	#124	—	2	2	no	—
201	#121	—	4	2	no	—
202	C₄×D₄	—	4	2	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
205	Q₃₂	—	2	16	no	—
206	SD₃₂	—	2	16	no	—
207	M₃₂	D₄×C₆₁	1	8	yes	complemented
208	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
209	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
210	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
213	#41	—	4	4	no	—
214	#41	—	4	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	D₈×C₆₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
227	Q₁₆	—	2	8	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	—	4	4	no	—
232	Q₁₆	—	4	2	no	—
233	SD₁₆	—	4	4	no	—
234	SD₁₆	—	4	4	no	—
235	D₈	—	2	4	no	—
236	D₈	—	4	2	no	—
237	C₂×C₈	D₈×C₆₁	1	4	yes	—
238	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
240	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
242	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₄²	D₈×C₆₁	1	2	yes	—
248	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
250	C₁₆	C₂×D₄×C₆₁	1	8	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	—	8	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	#127×C₆₁	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
270	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
271	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
273	C₈	#127×C₆₁	1	2	yes	—
274	C₈	C₂×D₈×C₆₁	1	2	yes	—
275	C₈	C₂×D₈×C₆₁	1	2	yes	—
276	C₈	C₂²×D₄×C₆₁	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	4	no	—
280	C₂²	C₆₁×#493	1	2	yes	—
281	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₆₁×#493	1	2	yes	—
286	C₄	C₆₁×#510	1	4	yes	complemented
287	C₄	C₂×#127×C₆₁	1	1	yes	characteristic
288	C₄	C₂²×D₈×C₆₁	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₆₁×#493	1	1	yes	characteristic
292	C₂	C₂×C₆₁×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#510	C₂	semidirect
3. C₂×C₆₁×#494	C₂	semidirect
4. C₆₁×#2251	C₂	semidirect
5. C₆₁×#2248	C₂	semidirect
6. C₆₁×#2247	C₂	semidirect
7. C₄×M₃₂×C₆₁	C₂	semidirect
60. M₃₂×C₆₁	D₄	semidirect
130. #33718	C₆₁	direct (N × H)
140. C₄×C₆₁	#510	semidirect
143. C₂×#510	C₂×C₆₁	semidirect
145. C₂×#494	C₂×C₆₁	semidirect
146. #2251	C₂×C₆₁	semidirect
147. #2248	C₂×C₆₁	semidirect
148. #2247	C₂×C₆₁	semidirect
149. C₄×M₃₂	C₂×C₆₁	semidirect
176. C₆₁	#33718	direct (N × H)
207. M₃₂	D₄×C₆₁	semidirect
286. C₄	C₆₁×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#510	2
C₆₁×#3147	2
C₂×C₆₁×#494	2
C₆₁×#2251	2
C₆₁×#2248	2
C₆₁×#2247	2
C₄×M₃₂×C₆₁	2
#33718	61

Subgroups — C61×#33718 #29537151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33718 #29537151