Subgroups — C₆₁×#61406 #29564839

Order 15616.

1986 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3506	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3403	C₂	1	4	yes	maximal, complemented
3	C₆₁×#3397	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3396	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3393	C₂	1	2	yes	maximal, complemented
6	C₆₁×#3391	C₂	1	2	yes	maximal, complemented
7	C₆₁×#3155	C₂	1	2	yes	maximal, complemented
8	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#3045	C₂	1	4	yes	maximal, complemented
10	C₆₁×#3036	C₂	1	2	yes	maximal, complemented
11	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#2951	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	4	yes	—
15	C₂×#135×C₆₁	C₂²	1	2	yes	—
16	C₂×#135×C₆₁	C₂²	1	2	yes	—
17	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#559	C₂²	1	8	yes	complemented
19	C₆₁×#556	C₂²	1	4	yes	complemented
20	C₆₁×#555	—	2	2	no	—
21	C₆₁×#553	—	2	4	no	—
22	C₆₁×#550	C₂²	1	4	yes	complemented
23	C₆₁×#547	—	2	2	no	—
24	C₆₁×#538	—	2	4	no	—
25	C₆₁×#535	C₂²	1	4	yes	—
26	C₆₁×#533	C₂²	1	1	yes	characteristic
27	C₆₁×#532	C₂²	1	2	yes	—
28	C₆₁×#525	—	2	4	no	—
29	C₆₁×#520	C₂²	1	4	yes	—
30	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
31	C₆₁×#496	C₂²	1	8	yes	—
32	C₆₁×#495	C₂²	1	4	yes	—
33	C₆₁×#495	C₂²	1	2	yes	complemented
34	C₆₁×#495	C₂²	1	2	yes	complemented
35	C₆₁×#494	C₂²	1	1	yes	characteristic, complemented
36	C₆₁×#493	C₂²	1	4	yes	—
37	C₆₁×#493	C₂²	1	1	yes	characteristic, complemented
38	C₆₁×#492	C₂²	1	2	yes	complemented
39	C₆₁×#464	C₂²	1	8	yes	complemented
40	C₆₁×#463	C₂²	1	8	yes	complemented
41	C₆₁×#463	C₂²	1	8	yes	complemented
42	C₆₁×#461	C₂²	1	4	yes	complemented
43	C₆₁×#460	C₂²	1	8	yes	complemented
44	C₆₁×#459	C₂²	1	4	yes	complemented
45	C₆₁×#459	C₂²	1	8	yes	complemented
46	C₆₁×#446	C₂²	1	2	yes	complemented
47	C₆₁×#446	C₂²	1	2	yes	complemented
48	C₈×Q₈×C₆₁	C₂²	1	2	yes	complemented
49	C₈×Q₈×C₆₁	C₂²	1	2	yes	complemented
50	C₆₁×#440	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₆₁	C₂²	1	4	yes	complemented
52	C₄×D₈×C₆₁	C₂²	1	2	yes	—
53	C₄×D₈×C₆₁	C₂²	1	2	yes	—
54	C₄×D₈×C₆₁	C₂²	1	2	yes	—
55	C₄×D₈×C₆₁	C₂²	1	2	yes	—
56	C₄×D₈×C₆₁	C₂²	1	4	yes	complemented
57	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
58	C₆₁×#424	C₂²	1	2	yes	—
59	C₆₁×#423	C₂²	1	1	yes	characteristic
60	C₆₁×#417	C₂²	1	4	yes	—
61	C₂×#102×C₆₁	C₂²	1	2	yes	—
62	C₆₁×#405	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
64	C₂×#41×C₆₁	—	2	4	no	—
65	C₂×#41×C₆₁	C₂³	1	4	yes	—
66	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₁	—	2	2	no	—
68	AGL(1,ℤ/8)×C₆₁	—	4	8	no	—
69	#135×C₆₁	—	2	8	no	—
70	#135×C₆₁	—	4	4	no	—
71	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
76	C₂×D₈×C₆₁	—	2	4	no	—
77	C₂×D₈×C₆₁	C₂³	1	2	yes	—
78	C₂×D₈×C₆₁	C₂³	1	2	yes	—
79	C₂×D₈×C₆₁	C₂³	1	2	yes	—
80	C₂×D₈×C₆₁	C₂³	1	2	yes	—
81	C₂×D₈×C₆₁	C₂³	1	8	yes	—
82	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
83	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
86	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
88	#127×C₆₁	C₂³	1	2	yes	—
89	#127×C₆₁	C₂³	1	8	yes	—
90	#126×C₆₁	—	2	8	no	—
91	#125×C₆₁	D₄	1	2	yes	—
92	#124×C₆₁	—	2	8	no	—
93	#124×C₆₁	C₂³	1	4	yes	—
94	#124×C₆₁	C₂³	1	8	yes	—
95	#123×C₆₁	—	2	8	no	—
96	#123×C₆₁	—	2	8	no	—
97	C₂²⋊Q₈×C₆₁	—	2	4	no	—
98	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
99	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
100	#121×C₆₁	—	2	2	no	—
101	#121×C₆₁	—	2	2	no	—
102	#121×C₆₁	—	2	4	no	—
103	#121×C₆₁	—	2	4	no	—
104	#121×C₆₁	—	2	8	no	—
105	#121×C₆₁	—	2	4	no	—
106	#120×C₆₁	—	2	8	no	—
107	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
108	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
109	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
110	C₄×D₄×C₆₁	—	2	8	no	—
111	C₄×D₄×C₆₁	—	2	8	no	—
112	C₄×D₄×C₆₁	—	2	2	no	—
113	C₄×D₄×C₆₁	—	2	2	no	—
114	C₄×D₄×C₆₁	—	2	2	no	—
115	C₄×D₄×C₆₁	—	2	2	no	—
116	C₄×D₄×C₆₁	C₂³	1	4	yes	—
117	C₄×D₄×C₆₁	C₂³	1	8	yes	—
118	C₄×D₄×C₆₁	C₂³	1	4	yes	—
119	#117×C₆₁	—	2	4	no	—
120	#117×C₆₁	—	2	4	no	—
121	#117×C₆₁	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
123	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
125	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
126	#107×C₆₁	C₂³	1	2	yes	—
127	#107×C₆₁	C₂³	1	2	yes	—
128	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
129	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
130	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
131	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
132	#103×C₆₁	C₂³	1	8	yes	—
133	#102×C₆₁	C₂³	1	4	yes	—
134	#102×C₆₁	C₂³	1	4	yes	—
135	#102×C₆₁	C₂³	1	8	yes	—
136	#97×C₆₁	C₂³	1	4	yes	—
137	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₆₁	C₂³	1	4	yes	—
142	C₂⁴×C₆₁	—	2	2	no	—
143	#41×C₆₁	—	2	8	no	—
144	#41×C₆₁	—	2	8	no	—
145	#41×C₆₁	—	8	8	no	—
146	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
147	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
149	C₂×D₄×C₆₁	—	2	4	no	—
150	C₂×D₄×C₆₁	—	2	2	no	—
151	C₂×D₄×C₆₁	—	2	2	no	—
152	C₂×D₄×C₆₁	—	2	8	no	—
153	C₂×D₄×C₆₁	—	2	2	no	—
154	C₂×D₄×C₆₁	—	2	8	no	—
155	C₂×D₄×C₆₁	—	2	4	no	—
156	C₂×D₄×C₆₁	—	2	4	no	—
157	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
158	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₆₁	—	2	2	no	—
160	C₂²×C₄×C₆₁	—	2	4	no	—
161	C₂²×C₄×C₆₁	—	2	4	no	—
162	C₂²×C₄×C₆₁	—	2	8	no	—
163	C₂²×C₄×C₆₁	—	2	4	no	—
164	C₂²×C₄×C₆₁	—	8	4	no	—
165	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₆₁	—	2	4	no	—
169	Q₁₆×C₆₁	—	4	2	no	—
170	SD₁₆×C₆₁	—	2	8	no	—
171	SD₁₆×C₆₁	—	4	4	no	—
172	SD₁₆×C₆₁	—	4	8	no	—
173	D₈×C₆₁	—	2	4	no	—
174	D₈×C₆₁	—	2	4	no	—
175	D₈×C₆₁	—	2	4	no	—
176	D₈×C₆₁	—	4	2	no	—
177	D₈×C₆₁	—	4	8	no	—
178	M₁₆×C₆₁	—	2	4	no	—
179	C₂×C₈×C₆₁	—	2	2	no	—
180	C₂×C₈×C₆₁	#41	1	4	yes	complemented
181	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
182	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
183	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
184	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₆₁	—	2	2	no	—
187	C₄⋊C₄×C₆₁	—	2	8	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	2	2	no	—
190	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
193	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₆₁	—	2	4	no	—
198	C₂²⋊C₄×C₆₁	—	2	4	no	—
199	C₂²⋊C₄×C₆₁	—	2	8	no	—
200	C₂²⋊C₄×C₆₁	—	2	8	no	—
201	C₂²⋊C₄×C₆₁	—	2	4	no	—
202	C₂²⋊C₄×C₆₁	—	2	8	no	—
203	C₂²⋊C₄×C₆₁	—	2	4	no	—
204	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
205	C₄²×C₆₁	—	2	8	no	—
206	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
209	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
210	C₄²×C₆₁	C₂⁴	1	4	yes	—
211	C₂³×C₆₁	—	2	4	no	—
212	C₂³×C₆₁	—	2	4	no	—
213	C₂³×C₆₁	—	4	4	no	—
214	C₂³×C₆₁	—	8	2	no	—
215	C₂³×C₆₁	—	8	2	no	—
216	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₆₁	—	2	4	no	—
218	Q₈×C₆₁	—	2	4	no	—
219	Q₈×C₆₁	—	2	4	no	—
220	Q₈×C₆₁	—	4	2	no	—
221	D₄×C₆₁	—	2	4	no	—
222	D₄×C₆₁	—	2	4	no	—
223	D₄×C₆₁	—	2	4	no	—
224	D₄×C₆₁	—	2	4	no	—
225	D₄×C₆₁	—	4	8	no	—
226	D₄×C₆₁	—	4	4	no	—
227	D₄×C₆₁	—	4	4	no	—
228	D₄×C₆₁	—	8	2	no	—
229	D₄×C₆₁	—	8	8	no	—
230	D₄×C₆₁	—	8	8	no	—
231	C₂×C₄×C₆₁	—	2	2	no	—
232	C₂×C₄×C₆₁	—	2	4	no	—
233	C₂×C₄×C₆₁	—	2	4	no	—
234	C₂×C₄×C₆₁	—	4	1	no	—
235	C₂×C₄×C₆₁	—	4	8	no	—
236	C₂×C₄×C₆₁	—	4	8	no	—
237	C₂×C₄×C₆₁	—	8	4	no	—
238	C₂×C₄×C₆₁	—	8	8	no	—
239	C₂×C₄×C₆₁	—	8	2	no	—
240	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
248	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
249	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
250	C₈×C₆₁	—	2	1	no	—
251	C₈×C₆₁	—	2	1	no	—
252	C₈×C₆₁	—	2	4	no	—
253	C₈×C₆₁	C₂×#41	1	2	yes	—
254	C₈×C₆₁	C₂×#41	1	2	yes	—
255	#61406	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂²×C₆₁	—	2	1	no	—
257	C₂²×C₆₁	—	2	1	no	—
258	C₂²×C₆₁	—	4	4	no	—
259	C₂²×C₆₁	—	4	4	no	—
260	C₂²×C₆₁	—	4	4	no	—
261	C₂²×C₆₁	—	4	4	no	—
262	C₂²×C₆₁	—	4	4	no	—
263	C₂²×C₆₁	—	4	4	no	—
264	C₂²×C₆₁	—	8	8	no	—
265	C₂²×C₆₁	#550	1	2	yes	—
266	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
267	C₄×C₆₁	—	2	1	no	—
268	C₄×C₆₁	—	2	1	no	—
269	C₄×C₆₁	—	2	4	no	—
270	C₄×C₆₁	—	2	1	no	—
271	C₄×C₆₁	—	2	4	no	—
272	C₄×C₆₁	—	4	4	no	—
273	C₄×C₆₁	—	4	4	no	—
274	C₄×C₆₁	#550	1	2	yes	—
275	C₄×C₆₁	#576	1	2	yes	—
276	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
277	C₄×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
278	#3506	C₂×C₆₁	1	2	yes	complemented
279	#3403	C₂×C₆₁	1	4	yes	complemented
280	#3397	C₂×C₆₁	1	4	yes	complemented
281	#3396	C₂×C₆₁	1	4	yes	complemented
282	#3393	C₂×C₆₁	1	2	yes	complemented
283	#3391	C₂×C₆₁	1	2	yes	complemented
284	#3155	C₂×C₆₁	1	2	yes	complemented
285	#3147	C₂×C₆₁	1	1	yes	characteristic, complemented
286	#3045	C₂×C₆₁	1	4	yes	complemented
287	#3036	C₂×C₆₁	1	2	yes	complemented
288	#2960	C₂×C₆₁	1	1	yes	characteristic, complemented
289	#2951	C₂×C₆₁	1	2	yes	complemented
290	C₂×C₄×D₈	C₂×C₆₁	1	1	yes	characteristic, complemented
291	C₂×C₆₁	—	2	2	no	—
292	C₂×C₆₁	—	8	4	no	—
293	C₂×C₆₁	—	8	4	no	—
294	C₂×C₆₁	C₂×#550	1	1	yes	characteristic
295	C₂×C₆₁	C₂×#576	1	1	yes	characteristic
296	C₂×C₆₁	#3584	1	1	yes	characteristic
297	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	4	yes	—
298	C₂×#135	C₂²×C₆₁	1	2	yes	—
299	C₂×#135	C₂²×C₆₁	1	2	yes	—
300	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
301	#559	C₂²×C₆₁	1	8	yes	complemented
302	#556	C₂²×C₆₁	1	4	yes	complemented
303	#555	—	2	2	no	—
304	#553	—	2	4	no	—
305	#550	C₂²×C₆₁	1	4	yes	complemented
306	#547	—	2	2	no	—
307	#538	—	2	4	no	—
308	#535	C₂²×C₆₁	1	4	yes	—
309	#533	C₂²×C₆₁	1	1	yes	characteristic
310	#532	C₂²×C₆₁	1	2	yes	—
311	#525	—	2	4	no	—
312	#520	C₂²×C₆₁	1	4	yes	—
313	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
314	#496	C₂²×C₆₁	1	8	yes	—
315	#495	C₂²×C₆₁	1	4	yes	—
316	#495	C₂²×C₆₁	1	2	yes	complemented
317	#495	C₂²×C₆₁	1	2	yes	complemented
318	#494	C₂²×C₆₁	1	1	yes	characteristic, complemented
319	#493	C₂²×C₆₁	1	4	yes	—
320	#493	C₂²×C₆₁	1	1	yes	characteristic, complemented
321	#492	C₂²×C₆₁	1	2	yes	complemented
322	#464	C₂²×C₆₁	1	8	yes	complemented
323	#463	C₂²×C₆₁	1	8	yes	complemented
324	#463	C₂²×C₆₁	1	8	yes	complemented
325	#461	C₂²×C₆₁	1	4	yes	complemented
326	#460	C₂²×C₆₁	1	8	yes	complemented
327	#459	C₂²×C₆₁	1	4	yes	complemented
328	#459	C₂²×C₆₁	1	8	yes	complemented
329	#446	C₂²×C₆₁	1	2	yes	complemented
330	#446	C₂²×C₆₁	1	2	yes	complemented
331	C₈×Q₈	C₂²×C₆₁	1	2	yes	complemented
332	C₈×Q₈	C₂²×C₆₁	1	2	yes	complemented
333	#440	C₂²×C₆₁	1	8	yes	complemented
334	C₄×Q₁₆	C₂²×C₆₁	1	4	yes	complemented
335	C₄×D₈	C₂²×C₆₁	1	2	yes	—
336	C₄×D₈	C₂²×C₆₁	1	2	yes	—
337	C₄×D₈	C₂²×C₆₁	1	2	yes	—
338	C₄×D₈	C₂²×C₆₁	1	2	yes	—
339	C₄×D₈	C₂²×C₆₁	1	4	yes	complemented
340	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
341	#424	C₂²×C₆₁	1	2	yes	—
342	#423	C₂²×C₆₁	1	1	yes	characteristic
343	#417	C₂²×C₆₁	1	4	yes	—
344	C₂×#102	C₂²×C₆₁	1	2	yes	—
345	#405	C₂²×C₆₁	1	2	yes	—
346	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
347	C₆₁	#61406	1	1	yes	characteristic, direct product factor, Sylow
348	C₂×#41	—	2	4	no	—
349	C₂×#41	C₂³×C₆₁	1	4	yes	—
350	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
351	C₂³×C₄	—	2	2	no	—
352	AGL(1,ℤ/8)	—	4	8	no	—
353	#135	—	2	8	no	—
354	#135	—	4	4	no	—
355	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
356	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
357	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
358	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
359	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
360	C₂×D₈	—	2	4	no	—
361	C₂×D₈	C₂³×C₆₁	1	2	yes	—
362	C₂×D₈	C₂³×C₆₁	1	2	yes	—
363	C₂×D₈	C₂³×C₆₁	1	2	yes	—
364	C₂×D₈	C₂³×C₆₁	1	2	yes	—
365	C₂×D₈	C₂³×C₆₁	1	8	yes	—
366	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
367	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
369	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
370	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
372	#127	C₂³×C₆₁	1	2	yes	—
373	#127	C₂³×C₆₁	1	8	yes	—
374	#126	—	2	8	no	—
375	#125	D₄×C₆₁	1	2	yes	—
376	#124	—	2	8	no	—
377	#124	C₂³×C₆₁	1	4	yes	—
378	#124	C₂³×C₆₁	1	8	yes	—
379	#123	—	2	8	no	—
380	#123	—	2	8	no	—
381	C₂²⋊Q₈	—	2	4	no	—
382	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
383	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
384	#121	—	2	2	no	—
385	#121	—	2	2	no	—
386	#121	—	2	4	no	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	#121	—	2	8	no	—
390	#120	—	2	8	no	—
391	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
392	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
393	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	C₂³×C₆₁	1	4	yes	—
401	C₄×D₄	C₂³×C₆₁	1	4	yes	—
402	C₄×D₄	C₂³×C₆₁	1	8	yes	—
403	#117	—	2	4	no	—
404	#117	—	2	4	no	—
405	#117	C₂³×C₆₁	1	2	yes	—
406	C₂×C₄⋊C₄	—	2	4	no	—
407	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
410	#107	C₂³×C₆₁	1	2	yes	—
411	#107	C₂³×C₆₁	1	2	yes	—
412	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
413	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
414	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
415	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
416	#103	C₂³×C₆₁	1	8	yes	—
417	#102	C₂³×C₆₁	1	4	yes	—
418	#102	C₂³×C₆₁	1	4	yes	—
419	#102	C₂³×C₆₁	1	8	yes	—
420	#97	C₂³×C₆₁	1	4	yes	—
421	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₆₁	1	4	yes	—
426	C₂⁴	—	2	2	no	—
427	#41	—	2	8	no	—
428	#41	—	2	8	no	—
429	#41	—	8	8	no	—
430	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
432	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
442	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	8	4	no	—
449	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
452	Q₁₆	—	2	4	no	—
453	Q₁₆	—	4	2	no	—
454	SD₁₆	—	2	8	no	—
455	SD₁₆	—	4	4	no	—
456	SD₁₆	—	4	8	no	—
457	D₈	—	2	4	no	—
458	D₈	—	2	4	no	—
459	D₈	—	2	4	no	—
460	D₈	—	4	2	no	—
461	D₈	—	4	8	no	—
462	M₁₆	—	2	4	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	#41×C₆₁	1	4	yes	complemented
465	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
466	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
467	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
468	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
469	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
475	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
476	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
478	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
479	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
480	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
489	C₄²	—	2	8	no	—
490	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
491	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
492	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
493	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
494	C₄²	C₂⁴×C₆₁	1	4	yes	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	4	4	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
501	Q₈	—	2	4	no	—
502	Q₈	—	2	4	no	—
503	Q₈	—	2	4	no	—
504	Q₈	—	4	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	4	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	4	no	—
509	D₄	—	4	8	no	—
510	D₄	—	4	4	no	—
511	D₄	—	4	4	no	—
512	D₄	—	8	2	no	—
513	D₄	—	8	8	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	4	no	—
522	C₂×C₄	—	8	8	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
529	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
531	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
532	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
533	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	2	4	no	—
537	C₈	C₂×#41×C₆₁	1	2	yes	—
538	C₈	C₂×#41×C₆₁	1	2	yes	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	—	8	8	no	—
548	C₂²	C₆₁×#550	1	2	yes	—
549	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
550	C₄	—	2	1	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	4	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	4	no	—
557	C₄	C₆₁×#550	1	2	yes	—
558	C₄	C₆₁×#576	1	2	yes	—
559	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
560	C₄	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
561	C₂	—	2	2	no	—
562	C₂	—	8	4	no	—
563	C₂	—	8	4	no	—
564	C₂	C₂×C₆₁×#550	1	1	yes	characteristic
565	C₂	C₂×C₆₁×#576	1	1	yes	characteristic
566	C₂	C₆₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₆₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3506	C₂	semidirect
2. C₆₁×#3403	C₂	semidirect
3. C₆₁×#3397	C₂	semidirect
4. C₆₁×#3396	C₂	semidirect
5. C₆₁×#3393	C₂	semidirect
6. C₆₁×#3391	C₂	semidirect
7. C₆₁×#3155	C₂	semidirect
8. C₆₁×#3147	C₂	semidirect
9. C₆₁×#3045	C₂	semidirect
10. C₆₁×#3036	C₂	semidirect
11. C₆₁×#2960	C₂	semidirect
12. C₆₁×#2951	C₂	semidirect
13. C₂×C₄×D₈×C₆₁	C₂	semidirect
18. C₆₁×#559	C₂²	semidirect
19. C₆₁×#556	C₂²	semidirect
22. C₆₁×#550	C₂²	semidirect
33. C₆₁×#495	C₂²	semidirect
34. C₆₁×#495	C₂²	semidirect
35. C₆₁×#494	C₂²	semidirect
37. C₆₁×#493	C₂²	semidirect
38. C₆₁×#492	C₂²	semidirect
39. C₆₁×#464	C₂²	semidirect
40. C₆₁×#463	C₂²	semidirect
41. C₆₁×#463	C₂²	semidirect
42. C₆₁×#461	C₂²	semidirect
43. C₆₁×#460	C₂²	semidirect
44. C₆₁×#459	C₂²	semidirect
45. C₆₁×#459	C₂²	semidirect
46. C₆₁×#446	C₂²	semidirect
47. C₆₁×#446	C₂²	semidirect
48. C₈×Q₈×C₆₁	C₂²	semidirect
49. C₈×Q₈×C₆₁	C₂²	semidirect
50. C₆₁×#440	C₂²	semidirect
51. C₄×Q₁₆×C₆₁	C₂²	semidirect
56. C₄×D₈×C₆₁	C₂²	semidirect
180. C₂×C₈×C₆₁	#41	semidirect
255. #61406	C₆₁	direct (N × H)
278. #3506	C₂×C₆₁	semidirect
279. #3403	C₂×C₆₁	semidirect
280. #3397	C₂×C₆₁	semidirect
281. #3396	C₂×C₆₁	semidirect
282. #3393	C₂×C₆₁	semidirect
283. #3391	C₂×C₆₁	semidirect
284. #3155	C₂×C₆₁	semidirect
285. #3147	C₂×C₆₁	semidirect
286. #3045	C₂×C₆₁	semidirect
287. #3036	C₂×C₆₁	semidirect
288. #2960	C₂×C₆₁	semidirect
289. #2951	C₂×C₆₁	semidirect
290. C₂×C₄×D₈	C₂×C₆₁	semidirect
301. #559	C₂²×C₆₁	semidirect
302. #556	C₂²×C₆₁	semidirect
305. #550	C₂²×C₆₁	semidirect
316. #495	C₂²×C₆₁	semidirect
317. #495	C₂²×C₆₁	semidirect
318. #494	C₂²×C₆₁	semidirect
320. #493	C₂²×C₆₁	semidirect
321. #492	C₂²×C₆₁	semidirect
322. #464	C₂²×C₆₁	semidirect
323. #463	C₂²×C₆₁	semidirect
324. #463	C₂²×C₆₁	semidirect
325. #461	C₂²×C₆₁	semidirect
326. #460	C₂²×C₆₁	semidirect
327. #459	C₂²×C₆₁	semidirect
328. #459	C₂²×C₆₁	semidirect
329. #446	C₂²×C₆₁	semidirect
330. #446	C₂²×C₆₁	semidirect
331. C₈×Q₈	C₂²×C₆₁	semidirect
332. C₈×Q₈	C₂²×C₆₁	semidirect
333. #440	C₂²×C₆₁	semidirect
334. C₄×Q₁₆	C₂²×C₆₁	semidirect
339. C₄×D₈	C₂²×C₆₁	semidirect
347. C₆₁	#61406	direct (N × H)
464. C₂×C₈	#41×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3506	2
C₆₁×#3403	2
C₆₁×#3397	2
C₆₁×#3396	2
C₆₁×#3393	2
C₆₁×#3391	2
C₆₁×#3155	2
C₆₁×#3147	2
C₆₁×#3045	2
C₆₁×#3036	2
C₆₁×#2960	2
C₆₁×#2951	2
C₂×C₄×D₈×C₆₁	2
#61406	61

Subgroups — C61×#61406 #29564839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61406 #29564839