Subgroups — C₆₁×#3008 #21443049

Order 7808.

784 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×AGL(1,ℤ/8)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#449	C₂	1	4	yes	maximal, complemented
6	C₆₁×#447	C₂	1	4	yes	maximal, complemented
7	C₆₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	H(5,𝔽₂)×C₆₁	—	2	4	no	—
11	H(5,𝔽₂)×C₆₁	—	2	4	no	—
12	C₂×#41×C₆₁	—	2	1	no	—
13	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#41×C₆₁	C₂²	1	2	yes	complemented
16	C₂²×D₄×C₆₁	—	2	1	no	—
17	C₂²×D₄×C₆₁	—	2	2	no	—
18	C₂²×D₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×C₆₁	C₂²	1	2	yes	complemented
21	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
22	C₂×SD₁₆×C₆₁	C₂²	1	2	yes	complemented
23	C₂×D₈×C₆₁	—	2	4	no	—
24	C₂×D₈×C₆₁	C₂²	1	4	yes	complemented
25	C₂×D₈×C₆₁	C₂²	1	2	yes	complemented
26	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
28	#127×C₆₁	—	2	1	no	—
29	#124×C₆₁	—	2	1	no	—
30	#121×C₆₁	C₂²	1	4	yes	complemented
31	#120×C₆₁	—	2	2	no	—
32	#117×C₆₁	C₂²	1	1	yes	characteristic, complemented
33	#103×C₆₁	C₂²	1	2	yes	complemented
34	#102×C₆₁	C₂²	1	2	yes	complemented
35	#102×C₆₁	C₂²	1	4	yes	complemented
36	C₂²⋊C₈×C₆₁	C₂²	1	4	yes	complemented
37	C₂⁴×C₆₁	—	2	1	no	—
38	C₂⁴×C₆₁	—	2	2	no	—
39	C₂⁴×C₆₁	—	2	1	no	—
40	#41×C₆₁	—	2	8	no	—
41	#41×C₆₁	—	2	4	no	—
42	#41×C₆₁	—	2	4	no	—
43	#41×C₆₁	—	2	2	no	—
44	#41×C₆₁	—	2	2	no	—
45	#41×C₆₁	—	2	2	no	—
46	#41×C₆₁	D₄	1	4	yes	complemented
47	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₆₁	—	2	16	no	—
50	C₂×D₄×C₆₁	—	2	8	no	—
51	C₂×D₄×C₆₁	—	2	1	no	—
52	C₂×D₄×C₆₁	—	2	2	no	—
53	C₂×D₄×C₆₁	—	2	4	no	—
54	C₂×D₄×C₆₁	—	2	4	no	—
55	C₂×D₄×C₆₁	—	2	4	no	—
56	C₂×D₄×C₆₁	—	2	1	no	—
57	C₂×D₄×C₆₁	—	2	2	no	—
58	C₂×D₄×C₆₁	—	2	2	no	—
59	C₂×D₄×C₆₁	—	2	2	no	—
60	C₂×D₄×C₆₁	—	2	2	no	—
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	D₄	1	2	yes	—
63	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
64	C₂×D₄×C₆₁	D₄	1	4	yes	complemented
65	C₂×D₄×C₆₁	C₂³	1	2	yes	—
66	C₂×D₄×C₆₁	C₂³	1	2	yes	—
67	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	—	2	1	no	—
69	C₂²×C₄×C₆₁	—	2	2	no	—
70	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
72	SD₁₆×C₆₁	—	2	4	no	—
73	D₈×C₆₁	—	2	8	no	—
74	D₈×C₆₁	—	2	4	no	—
75	M₁₆×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	C₂³	1	2	yes	—
78	C₂×C₈×C₆₁	C₂³	1	2	yes	—
79	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₆₁	—	2	1	no	—
81	C₂²⋊C₄×C₆₁	—	2	4	no	—
82	C₄²×C₆₁	—	2	1	no	—
83	C₂³×C₆₁	—	2	8	no	—
84	C₂³×C₆₁	—	2	1	no	—
85	C₂³×C₆₁	—	2	2	no	—
86	C₂³×C₆₁	—	2	4	no	—
87	C₂³×C₆₁	—	2	2	no	—
88	C₂³×C₆₁	—	4	2	no	—
89	C₂³×C₆₁	—	8	2	no	—
90	C₂³×C₆₁	—	8	2	no	—
91	C₂³×C₆₁	—	8	4	no	—
92	C₂³×C₆₁	C₂×D₄	1	2	yes	—
93	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
94	Q₈×C₆₁	—	2	2	no	—
95	Q₈×C₆₁	—	2	1	no	—
96	Q₈×C₆₁	C₂×D₄	1	2	yes	—
97	D₄×C₆₁	—	2	8	no	—
98	D₄×C₆₁	—	2	8	no	—
99	D₄×C₆₁	—	2	4	no	—
100	D₄×C₆₁	—	2	4	no	—
101	D₄×C₆₁	—	2	2	no	—
102	D₄×C₆₁	—	2	4	no	—
103	D₄×C₆₁	—	2	2	no	—
104	D₄×C₆₁	—	2	1	no	—
105	D₄×C₆₁	—	2	2	no	—
106	D₄×C₆₁	—	2	2	no	—
107	D₄×C₆₁	—	4	2	no	—
108	D₄×C₆₁	—	8	4	no	—
109	D₄×C₆₁	C₂×D₄	1	2	yes	—
110	D₄×C₆₁	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₆₁	—	2	8	no	—
112	C₂×C₄×C₆₁	—	2	1	no	—
113	C₂×C₄×C₆₁	—	2	4	no	—
114	C₂×C₄×C₆₁	—	2	2	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
122	C₈×C₆₁	—	2	2	no	—
123	C₈×C₆₁	—	2	2	no	—
124	C₂²×C₆₁	—	2	2	no	—
125	C₂²×C₆₁	—	2	2	no	—
126	C₂²×C₆₁	—	2	4	no	—
127	C₂²×C₆₁	—	2	2	no	—
128	C₂²×C₆₁	—	2	4	no	—
129	C₂²×C₆₁	—	4	4	no	—
130	C₂²×C₆₁	—	4	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	—	4	4	no	—
133	C₂²×C₆₁	—	8	4	no	—
134	C₂²×C₆₁	—	8	4	no	—
135	C₂²×C₆₁	—	8	8	no	—
136	C₂²×C₆₁	#120	1	2	yes	—
137	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₆₁	—	2	2	no	—
139	C₄×C₆₁	—	2	2	no	—
140	C₄×C₆₁	—	4	2	no	—
141	C₄×C₆₁	#120	1	2	yes	—
142	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
144	#3008	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₁	—	2	2	no	—
146	C₂×C₆₁	—	4	2	no	—
147	C₂×C₆₁	—	4	4	no	—
148	C₂×C₆₁	—	4	2	no	—
149	C₂×C₆₁	—	8	2	no	—
150	C₂×C₆₁	C₂×#120	1	1	yes	characteristic
151	C₂×C₆₁	#576	1	2	yes	—
152	C₂×H(5,𝔽₂)	C₂×C₆₁	1	1	yes	characteristic, complemented
153	C₂×AGL(1,ℤ/8)	C₂×C₆₁	1	1	yes	characteristic, complemented
154	C₂²×D₈	C₂×C₆₁	1	1	yes	characteristic, complemented
155	#535	C₂×C₆₁	1	1	yes	characteristic, complemented
156	#449	C₂×C₆₁	1	4	yes	complemented
157	#447	C₂×C₆₁	1	4	yes	complemented
158	#418	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#416	C₂×C₆₁	1	1	yes	characteristic, complemented
160	#408	C₂×C₆₁	1	1	yes	characteristic, complemented
161	C₆₁	#3008	1	1	yes	characteristic, direct product factor, Sylow
162	H(5,𝔽₂)	—	2	4	no	—
163	H(5,𝔽₂)	—	2	4	no	—
164	C₂×#41	—	2	1	no	—
165	C₂×#41	C₂²×C₆₁	1	1	yes	characteristic
166	C₂×#41	C₂²×C₆₁	1	1	yes	characteristic
167	C₂×#41	C₂²×C₆₁	1	2	yes	complemented
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	C₂²×C₆₁	1	1	yes	characteristic
171	C₂²×D₄	C₂²×C₆₁	1	1	yes	characteristic, complemented
172	C₂²×D₄	C₂²×C₆₁	1	2	yes	complemented
173	AGL(1,ℤ/8)	—	2	4	no	—
174	C₂×SD₁₆	C₂²×C₆₁	1	2	yes	complemented
175	C₂×D₈	—	2	4	no	—
176	C₂×D₈	C₂²×C₆₁	1	4	yes	complemented
177	C₂×D₈	C₂²×C₆₁	1	2	yes	complemented
178	C₂×M₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
179	C₂²×C₈	C₂²×C₆₁	1	1	yes	characteristic, complemented
180	#127	—	2	1	no	—
181	#124	—	2	1	no	—
182	#121	C₂²×C₆₁	1	4	yes	complemented
183	#120	—	2	2	no	—
184	#117	C₂²×C₆₁	1	1	yes	characteristic, complemented
185	#103	C₂²×C₆₁	1	2	yes	complemented
186	#102	C₂²×C₆₁	1	2	yes	complemented
187	#102	C₂²×C₆₁	1	4	yes	complemented
188	C₂²⋊C₈	C₂²×C₆₁	1	4	yes	complemented
189	C₂⁴	—	2	1	no	—
190	C₂⁴	—	2	2	no	—
191	C₂⁴	—	2	1	no	—
192	#41	—	2	8	no	—
193	#41	—	2	4	no	—
194	#41	—	2	4	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	D₄×C₆₁	1	4	yes	complemented
199	C₂×Q₈	D₄×C₆₁	1	1	yes	characteristic
200	C₂×Q₈	C₂³×C₆₁	1	1	yes	characteristic
201	C₂×D₄	—	2	16	no	—
202	C₂×D₄	—	2	8	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	D₄×C₆₁	1	2	yes	—
215	C₂×D₄	D₄×C₆₁	1	1	yes	characteristic
216	C₂×D₄	D₄×C₆₁	1	4	yes	complemented
217	C₂×D₄	C₂³×C₆₁	1	2	yes	—
218	C₂×D₄	C₂³×C₆₁	1	2	yes	—
219	C₂×D₄	C₂³×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
223	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
224	SD₁₆	—	2	4	no	—
225	D₈	—	2	8	no	—
226	D₈	—	2	4	no	—
227	M₁₆	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂³×C₆₁	1	2	yes	—
230	C₂×C₈	C₂³×C₆₁	1	2	yes	—
231	C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	1	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	—	8	2	no	—
243	C₂³	—	8	4	no	—
244	C₂³	C₂×D₄×C₆₁	1	2	yes	—
245	C₂³	C₂×D₄×C₆₁	1	1	yes	characteristic
246	Q₈	—	2	2	no	—
247	Q₈	—	2	1	no	—
248	Q₈	C₂×D₄×C₆₁	1	2	yes	—
249	D₄	—	2	8	no	—
250	D₄	—	2	8	no	—
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	4	no	—
255	D₄	—	2	2	no	—
256	D₄	—	2	1	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	C₂×D₄×C₆₁	1	2	yes	—
262	D₄	C₂×D₄×C₆₁	1	4	yes	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
270	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
273	C₂×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	8	8	no	—
288	C₂²	#120×C₆₁	1	2	yes	—
289	C₂²	C₂²×D₄×C₆₁	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	2	no	—
293	C₄	#120×C₆₁	1	2	yes	—
294	C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
295	C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	4	4	no	—
299	C₂	—	4	2	no	—
300	C₂	—	8	2	no	—
301	C₂	C₂×#120×C₆₁	1	1	yes	characteristic
302	C₂	C₆₁×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)×C₆₁	C₂	semidirect
2. C₂×AGL(1,ℤ/8)×C₆₁	C₂	semidirect
3. C₂²×D₈×C₆₁	C₂	semidirect
4. C₆₁×#535	C₂	semidirect
5. C₆₁×#449	C₂	semidirect
6. C₆₁×#447	C₂	semidirect
7. C₆₁×#418	C₂	semidirect
8. C₆₁×#416	C₂	semidirect
9. C₆₁×#408	C₂	semidirect
15. C₂×#41×C₆₁	C₂²	semidirect
19. C₂²×D₄×C₆₁	C₂²	semidirect
20. C₂²×D₄×C₆₁	C₂²	semidirect
22. C₂×SD₁₆×C₆₁	C₂²	semidirect
24. C₂×D₈×C₆₁	C₂²	semidirect
25. C₂×D₈×C₆₁	C₂²	semidirect
26. C₂×M₁₆×C₆₁	C₂²	semidirect
27. C₂²×C₈×C₆₁	C₂²	semidirect
30. #121×C₆₁	C₂²	semidirect
32. #117×C₆₁	C₂²	semidirect
33. #103×C₆₁	C₂²	semidirect
34. #102×C₆₁	C₂²	semidirect
35. #102×C₆₁	C₂²	semidirect
36. C₂²⋊C₈×C₆₁	C₂²	semidirect
46. #41×C₆₁	D₄	semidirect
64. C₂×D₄×C₆₁	D₄	semidirect
144. #3008	C₆₁	direct (N × H)
152. C₂×H(5,𝔽₂)	C₂×C₆₁	semidirect
153. C₂×AGL(1,ℤ/8)	C₂×C₆₁	semidirect
154. C₂²×D₈	C₂×C₆₁	semidirect
155. #535	C₂×C₆₁	semidirect
156. #449	C₂×C₆₁	semidirect
157. #447	C₂×C₆₁	semidirect
158. #418	C₂×C₆₁	semidirect
159. #416	C₂×C₆₁	semidirect
160. #408	C₂×C₆₁	semidirect
161. C₆₁	#3008	direct (N × H)
167. C₂×#41	C₂²×C₆₁	semidirect
171. C₂²×D₄	C₂²×C₆₁	semidirect
172. C₂²×D₄	C₂²×C₆₁	semidirect
174. C₂×SD₁₆	C₂²×C₆₁	semidirect
176. C₂×D₈	C₂²×C₆₁	semidirect
177. C₂×D₈	C₂²×C₆₁	semidirect
178. C₂×M₁₆	C₂²×C₆₁	semidirect
179. C₂²×C₈	C₂²×C₆₁	semidirect
182. #121	C₂²×C₆₁	semidirect
184. #117	C₂²×C₆₁	semidirect
185. #103	C₂²×C₆₁	semidirect
186. #102	C₂²×C₆₁	semidirect
187. #102	C₂²×C₆₁	semidirect
188. C₂²⋊C₈	C₂²×C₆₁	semidirect
198. #41	D₄×C₆₁	semidirect
216. C₂×D₄	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)×C₆₁	2
C₂×AGL(1,ℤ/8)×C₆₁	2
C₂²×D₈×C₆₁	2
C₆₁×#535	2
C₆₁×#449	2
C₆₁×#447	2
C₆₁×#418	2
C₆₁×#416	2
C₆₁×#408	2
#3008	61

Subgroups — C61×#3008 #21443049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#3008 #21443049