Subgroups — C₆₁×#33361 #29536794

Order 15616.

1252 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3500	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2883	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2129	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2127	C₂	1	4	yes	maximal, complemented
5	C₆₁×#2126	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#354	C₂	1	4	yes	maximal, complemented
7	C₂×2₋¹⁺⁴×C₆₁	C₄	1	1	yes	characteristic, complemented
8	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#548	—	2	2	no	—
10	C₆₁×#546	—	2	4	no	—
11	C₆₁×#546	—	2	2	no	—
12	C₆₁×#536	C₄	1	2	yes	complemented
13	C₆₁×#536	C₂²	1	1	yes	characteristic
14	C₆₁×#535	C₄	1	1	yes	characteristic, complemented
15	C₆₁×#535	C₂²	1	2	yes	—
16	C₆₁×#532	C₄	1	2	yes	complemented
17	C₆₁×#532	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₁	C₄	1	1	yes	characteristic, complemented
19	C₂×#124×C₆₁	C₂²	1	2	yes	—
20	C₆₁×#518	—	2	2	no	—
21	C₄×#41×C₆₁	C₄	1	1	yes	characteristic, complemented
22	C₆₁×#410	—	2	2	no	—
23	C₆₁×#410	C₂²	1	4	yes	—
24	C₂×#99×C₆₁	—	2	4	no	—
25	C₂×#99×C₆₁	C₂²	1	8	yes	—
26	C₆₁×#354	C₂²	1	4	yes	complemented
27	C₆₁×#354	C₂²	1	8	yes	complemented
28	C₆₁×#353	C₂²	1	4	yes	complemented
29	2₋¹⁺⁴×C₆₁	—	4	4	no	—
30	H(5,𝔽₂)×C₆₁	—	4	4	no	—
31	C₂×#41×C₆₁	—	2	2	no	—
32	C₂×#41×C₆₁	—	2	1	no	—
33	C₂×#41×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×#41×C₆₁	C₂×C₄	1	2	yes	complemented
36	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×#41×C₆₁	D₄	1	2	yes	—
38	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
39	C₂²×Q₈×C₆₁	—	2	1	no	—
40	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	complemented
41	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
42	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₆₁	D₄	1	4	yes	—
44	C₂²×D₄×C₆₁	D₄	1	2	yes	—
45	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₁	C₂×C₄	1	2	yes	complemented
47	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
48	#127×C₆₁	C₂×C₄	1	2	yes	complemented
49	#127×C₆₁	C₂³	1	1	yes	characteristic
50	#124×C₆₁	—	2	2	no	—
51	#124×C₆₁	—	2	4	no	—
52	#124×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
53	#124×C₆₁	C₂×C₄	1	2	yes	complemented
54	#124×C₆₁	C₂×C₄	1	2	yes	complemented
55	#124×C₆₁	C₂×C₄	1	2	yes	complemented
56	#124×C₆₁	C₂³	1	1	yes	characteristic
57	#124×C₆₁	C₂³	1	2	yes	—
58	#123×C₆₁	—	2	4	no	—
59	#123×C₆₁	—	2	2	no	—
60	C₂²⋊Q₈×C₆₁	—	2	4	no	—
61	C₂²⋊Q₈×C₆₁	—	2	2	no	—
62	#121×C₆₁	—	2	4	no	—
63	#121×C₆₁	—	2	2	no	—
64	#121×C₆₁	—	2	2	no	—
65	#121×C₆₁	—	2	4	no	—
66	#120×C₆₁	—	2	2	no	—
67	#120×C₆₁	—	2	4	no	—
68	C₄×Q₈×C₆₁	—	2	1	no	—
69	C₄×D₄×C₆₁	—	2	2	no	—
70	C₄×D₄×C₆₁	—	2	1	no	—
71	C₄×D₄×C₆₁	—	2	4	no	—
72	C₄×D₄×C₆₁	—	2	2	no	—
73	C₄×D₄×C₆₁	—	4	8	no	—
74	#117×C₆₁	—	2	4	no	—
75	#117×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
76	#117×C₆₁	C₂×C₄	1	2	yes	complemented
77	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
79	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
80	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	complemented
81	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
82	#99×C₆₁	—	2	4	no	—
83	#99×C₆₁	—	2	8	no	—
84	#99×C₆₁	C₂³	1	8	yes	—
85	C₂⁴×C₆₁	C₂²⋊C₄	1	2	yes	—
86	C₂⁴×C₆₁	C₂×D₄	1	4	yes	—
87	#41×C₆₁	—	2	4	no	—
88	#41×C₆₁	—	4	2	no	—
89	#41×C₆₁	—	4	4	no	—
90	#41×C₆₁	—	4	4	no	—
91	#41×C₆₁	—	4	4	no	—
92	#41×C₆₁	—	4	2	no	—
93	#41×C₆₁	—	4	8	no	—
94	#41×C₆₁	—	4	4	no	—
95	C₂×Q₈×C₆₁	—	2	2	no	—
96	C₂×Q₈×C₆₁	—	2	1	no	—
97	C₂×Q₈×C₆₁	—	4	2	no	—
98	C₂×Q₈×C₆₁	—	4	4	no	—
99	C₂×Q₈×C₆₁	—	4	4	no	—
100	C₂×Q₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
102	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×D₄×C₆₁	—	2	8	no	—
105	C₂×D₄×C₆₁	—	2	1	no	—
106	C₂×D₄×C₆₁	—	2	2	no	—
107	C₂×D₄×C₆₁	—	2	4	no	—
108	C₂×D₄×C₆₁	—	2	2	no	—
109	C₂×D₄×C₆₁	—	4	2	no	—
110	C₂×D₄×C₆₁	—	4	4	no	—
111	C₂×D₄×C₆₁	—	4	8	no	—
112	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
113	C₂×D₄×C₆₁	C₂²⋊C₄	1	4	yes	—
114	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
115	C₂×D₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
116	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
118	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
120	C₂×D₄×C₆₁	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₆₁	—	2	1	no	—
122	C₂²×C₄×C₆₁	—	2	2	no	—
123	C₂²×C₄×C₆₁	—	2	1	no	—
124	C₂²×C₄×C₆₁	—	2	2	no	—
125	C₂²×C₄×C₆₁	—	2	2	no	—
126	C₂²×C₄×C₆₁	—	2	1	no	—
127	C₂²×C₄×C₆₁	—	4	8	no	—
128	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
129	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₆₁	—	2	2	no	—
133	C₄⋊C₄×C₆₁	—	2	1	no	—
134	C₄⋊C₄×C₆₁	—	2	4	no	—
135	C₄⋊C₄×C₆₁	—	2	2	no	—
136	C₄⋊C₄×C₆₁	—	4	4	no	—
137	C₂²⋊C₄×C₆₁	—	2	4	no	—
138	C₂²⋊C₄×C₆₁	—	2	4	no	—
139	C₂²⋊C₄×C₆₁	—	2	4	no	—
140	C₂²⋊C₄×C₆₁	—	2	1	no	—
141	C₂²⋊C₄×C₆₁	—	2	2	no	—
142	C₂²⋊C₄×C₆₁	—	2	2	no	—
143	C₂²⋊C₄×C₆₁	—	2	4	no	—
144	C₂²⋊C₄×C₆₁	—	2	2	no	—
145	C₂²⋊C₄×C₆₁	—	2	8	no	—
146	C₂²⋊C₄×C₆₁	—	4	8	no	—
147	C₄²×C₆₁	—	2	2	no	—
148	C₄²×C₆₁	—	2	1	no	—
149	C₄²×C₆₁	—	4	4	no	—
150	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
151	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
153	C₂³×C₆₁	—	2	8	no	—
154	C₂³×C₆₁	—	2	1	no	—
155	C₂³×C₆₁	—	4	2	no	—
156	C₂³×C₆₁	—	8	4	no	—
157	C₂³×C₆₁	—	8	2	no	—
158	C₂³×C₆₁	C₂×C₂²⋊C₄	1	4	yes	—
159	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
162	Q₈×C₆₁	—	2	2	no	—
163	Q₈×C₆₁	—	4	1	no	—
164	Q₈×C₆₁	—	4	4	no	—
165	Q₈×C₆₁	—	4	2	no	—
166	Q₈×C₆₁	—	4	1	no	—
167	Q₈×C₆₁	—	4	2	no	—
168	D₄×C₆₁	—	2	8	no	—
169	D₄×C₆₁	—	2	4	no	—
170	D₄×C₆₁	—	2	4	no	—
171	D₄×C₆₁	—	4	2	no	—
172	D₄×C₆₁	—	4	4	no	—
173	D₄×C₆₁	—	4	2	no	—
174	D₄×C₆₁	—	4	1	no	—
175	D₄×C₆₁	—	4	1	no	—
176	D₄×C₆₁	—	4	2	no	—
177	D₄×C₆₁	—	4	4	no	—
178	D₄×C₆₁	—	8	4	no	—
179	D₄×C₆₁	—	8	2	no	—
180	D₄×C₆₁	#99	1	4	yes	—
181	C₂×C₄×C₆₁	—	2	4	no	—
182	C₂×C₄×C₆₁	—	2	4	no	—
183	C₂×C₄×C₆₁	—	2	2	no	—
184	C₂×C₄×C₆₁	—	2	1	no	—
185	C₂×C₄×C₆₁	—	2	1	no	—
186	C₂×C₄×C₆₁	—	2	2	no	—
187	C₂×C₄×C₆₁	—	2	2	no	—
188	C₂×C₄×C₆₁	—	2	1	no	—
189	C₂×C₄×C₆₁	—	4	4	no	—
190	C₂×C₄×C₆₁	—	4	4	no	—
191	C₂×C₄×C₆₁	—	4	2	no	—
192	C₂×C₄×C₆₁	—	4	4	no	—
193	C₂×C₄×C₆₁	—	4	2	no	—
194	C₂×C₄×C₆₁	—	4	8	no	—
195	C₂×C₄×C₆₁	—	8	2	no	—
196	C₂×C₄×C₆₁	—	8	1	no	—
197	C₂×C₄×C₆₁	—	16	8	no	—
198	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
199	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
200	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
202	#33361	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂²×C₆₁	—	2	2	no	—
204	C₂²×C₆₁	—	2	1	no	—
205	C₂²×C₆₁	—	2	4	no	—
206	C₂²×C₆₁	—	4	2	no	—
207	C₂²×C₆₁	—	4	1	no	—
208	C₂²×C₆₁	—	4	4	no	—
209	C₂²×C₆₁	—	4	2	no	—
210	C₂²×C₆₁	—	4	2	no	—
211	C₂²×C₆₁	—	4	1	no	—
212	C₂²×C₆₁	—	8	8	no	—
213	C₂²×C₆₁	—	16	2	no	—
214	C₂²×C₆₁	C₂×#99	1	4	yes	—
215	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
216	C₄×C₆₁	—	2	1	no	—
217	C₄×C₆₁	—	2	2	no	—
218	C₄×C₆₁	—	2	2	no	—
219	C₄×C₆₁	—	4	1	no	—
220	C₄×C₆₁	—	4	1	no	—
221	C₄×C₆₁	—	4	2	no	—
222	C₄×C₆₁	—	4	2	no	—
223	C₄×C₆₁	—	4	1	no	—
224	C₄×C₆₁	—	8	8	no	—
225	C₄×C₆₁	C₂×#99	1	2	yes	—
226	#3500	C₂×C₆₁	1	1	yes	characteristic
227	#2883	C₂×C₆₁	1	2	yes	complemented
228	#2129	C₂×C₆₁	1	2	yes	complemented
229	#2127	C₂×C₆₁	1	4	yes	complemented
230	#2126	C₂×C₆₁	1	2	yes	complemented
231	C₂×#354	C₂×C₆₁	1	4	yes	complemented
232	C₂×C₆₁	—	2	1	no	—
233	C₂×C₆₁	—	2	4	no	—
234	C₂×C₆₁	—	4	2	no	—
235	C₂×C₆₁	—	4	4	no	—
236	C₂×C₆₁	—	4	1	no	—
237	C₂×C₆₁	—	8	1	no	—
238	C₂×C₆₁	C₂²×#99	1	1	yes	characteristic
239	C₂×2₋¹⁺⁴	C₄×C₆₁	1	1	yes	characteristic, complemented
240	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	1	yes	characteristic
241	#548	—	2	2	no	—
242	#546	—	2	4	no	—
243	#546	—	2	2	no	—
244	#536	C₄×C₆₁	1	2	yes	complemented
245	#536	C₂²×C₆₁	1	1	yes	characteristic
246	#535	C₄×C₆₁	1	1	yes	characteristic, complemented
247	#535	C₂²×C₆₁	1	2	yes	—
248	#532	C₄×C₆₁	1	2	yes	complemented
249	#532	C₂²×C₆₁	1	1	yes	characteristic
250	C₂×#124	C₄×C₆₁	1	1	yes	characteristic, complemented
251	C₂×#124	C₂²×C₆₁	1	2	yes	—
252	#518	—	2	2	no	—
253	C₄×#41	C₄×C₆₁	1	1	yes	characteristic, complemented
254	#410	—	2	2	no	—
255	#410	C₂²×C₆₁	1	4	yes	—
256	C₂×#99	—	2	4	no	—
257	C₂×#99	C₂²×C₆₁	1	8	yes	—
258	#354	C₂²×C₆₁	1	4	yes	complemented
259	#354	C₂²×C₆₁	1	8	yes	complemented
260	#353	C₂²×C₆₁	1	4	yes	complemented
261	C₆₁	#33361	1	1	yes	characteristic, direct product factor, Sylow
262	2₋¹⁺⁴	—	4	4	no	—
263	H(5,𝔽₂)	—	4	4	no	—
264	C₂×#41	—	2	2	no	—
265	C₂×#41	—	2	1	no	—
266	C₂×#41	C₂×C₄×C₆₁	1	2	yes	—
267	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
268	C₂×#41	C₂×C₄×C₆₁	1	2	yes	complemented
269	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
270	C₂×#41	D₄×C₆₁	1	2	yes	—
271	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
272	C₂²×Q₈	—	2	1	no	—
273	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
274	C₂²×Q₈	C₂×C₄×C₆₁	1	2	yes	complemented
275	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
276	C₂²×D₄	D₄×C₆₁	1	4	yes	—
277	C₂²×D₄	D₄×C₆₁	1	2	yes	—
278	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
279	C₄⋊Q₈	C₂×C₄×C₆₁	1	2	yes	complemented
280	C₄⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
281	#127	C₂×C₄×C₆₁	1	2	yes	complemented
282	#127	C₂³×C₆₁	1	1	yes	characteristic
283	#124	—	2	2	no	—
284	#124	—	2	4	no	—
285	#124	C₂×C₄×C₆₁	1	2	yes	complemented
286	#124	C₂×C₄×C₆₁	1	2	yes	complemented
287	#124	C₂×C₄×C₆₁	1	2	yes	complemented
288	#124	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
289	#124	C₂³×C₆₁	1	1	yes	characteristic
290	#124	C₂³×C₆₁	1	2	yes	—
291	#123	—	2	4	no	—
292	#123	—	2	2	no	—
293	C₂²⋊Q₈	—	2	4	no	—
294	C₂²⋊Q₈	—	2	2	no	—
295	#121	—	2	4	no	—
296	#121	—	2	2	no	—
297	#121	—	2	2	no	—
298	#121	—	2	4	no	—
299	#120	—	2	2	no	—
300	#120	—	2	4	no	—
301	C₄×Q₈	—	2	1	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	—	2	1	no	—
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	4	8	no	—
307	#117	—	2	4	no	—
308	#117	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
309	#117	C₂×C₄×C₆₁	1	2	yes	complemented
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	4	no	—
312	C₂×C₂²⋊C₄	—	2	8	no	—
313	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic, complemented
314	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	complemented
315	#99	—	2	4	no	—
316	#99	—	2	8	no	—
317	#99	C₂³×C₆₁	1	8	yes	—
318	C₂⁴	C₂²⋊C₄×C₆₁	1	2	yes	—
319	C₂⁴	C₂×D₄×C₆₁	1	4	yes	—
320	#41	—	2	4	no	—
321	#41	—	4	2	no	—
322	#41	—	4	4	no	—
323	#41	—	4	4	no	—
324	#41	—	4	4	no	—
325	#41	—	4	2	no	—
326	#41	—	4	8	no	—
327	#41	—	4	4	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	4	no	—
332	C₂×Q₈	—	4	4	no	—
333	C₂×Q₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
334	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
335	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
336	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	—	4	4	no	—
344	C₂×D₄	—	4	8	no	—
345	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
346	C₂×D₄	C₂²⋊C₄×C₆₁	1	4	yes	—
347	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
348	C₂×D₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
349	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
350	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
351	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
352	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
353	C₂×D₄	C₂⁴×C₆₁	1	1	yes	characteristic
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	4	8	no	—
361	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
362	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	1	no	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	2	2	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	2	8	no	—
379	C₂²⋊C₄	—	4	8	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	4	4	no	—
383	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
384	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
385	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
386	C₂³	—	2	8	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	8	4	no	—
390	C₂³	—	8	2	no	—
391	C₂³	C₂×C₂²⋊C₄×C₆₁	1	4	yes	—
392	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
393	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
394	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
395	Q₈	—	2	2	no	—
396	Q₈	—	4	1	no	—
397	Q₈	—	4	2	no	—
398	Q₈	—	4	1	no	—
399	Q₈	—	4	4	no	—
400	Q₈	—	4	2	no	—
401	D₄	—	2	8	no	—
402	D₄	—	2	4	no	—
403	D₄	—	2	4	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	4	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	1	no	—
408	D₄	—	4	1	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	4	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	2	no	—
413	D₄	#99×C₆₁	1	4	yes	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	8	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	—	8	1	no	—
430	C₂×C₄	—	16	8	no	—
431	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
432	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
433	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
434	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
435	C₂²	—	2	2	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	8	8	no	—
445	C₂²	—	16	2	no	—
446	C₂²	C₂×#99×C₆₁	1	4	yes	—
447	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
448	C₄	—	2	1	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	4	1	no	—
452	C₄	—	4	1	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	1	no	—
456	C₄	—	8	8	no	—
457	C₄	C₂×#99×C₆₁	1	2	yes	—
458	C₂	—	2	1	no	—
459	C₂	—	2	4	no	—
460	C₂	—	4	2	no	—
461	C₂	—	4	4	no	—
462	C₂	—	4	1	no	—
463	C₂	—	8	1	no	—
464	C₂	C₂²×#99×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	C₂²×#99
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂×D₄	C₂⁴×C₆₁
prime radical	C₆₁×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂×D₄×C₆₁ ◁ C₂×H(5,𝔽₂)×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2883	C₂	semidirect
3. C₆₁×#2129	C₂	semidirect
4. C₆₁×#2127	C₂	semidirect
5. C₆₁×#2126	C₂	semidirect
6. C₂×C₆₁×#354	C₂	semidirect
7. C₂×2₋¹⁺⁴×C₆₁	C₄	semidirect
12. C₆₁×#536	C₄	semidirect
14. C₆₁×#535	C₄	semidirect
16. C₆₁×#532	C₄	semidirect
18. C₂×#124×C₆₁	C₄	semidirect
21. C₄×#41×C₆₁	C₄	semidirect
26. C₆₁×#354	C₂²	semidirect
27. C₆₁×#354	C₂²	semidirect
28. C₆₁×#353	C₂²	semidirect
35. C₂×#41×C₆₁	C₂×C₄	semidirect
36. C₂×#41×C₆₁	C₂×C₄	semidirect
40. C₂²×Q₈×C₆₁	C₂×C₄	semidirect
41. C₂²×Q₈×C₆₁	C₂×C₄	semidirect
46. C₄⋊Q₈×C₆₁	C₂×C₄	semidirect
47. C₄⋊Q₈×C₆₁	C₂×C₄	semidirect
48. #127×C₆₁	C₂×C₄	semidirect
52. #124×C₆₁	C₂×C₄	semidirect
53. #124×C₆₁	C₂×C₄	semidirect
54. #124×C₆₁	C₂×C₄	semidirect
55. #124×C₆₁	C₂×C₄	semidirect
75. #117×C₆₁	C₂×C₄	semidirect
76. #117×C₆₁	C₂×C₄	semidirect
80. C₂×C₄²×C₆₁	C₂×C₄	semidirect
81. C₂×C₄²×C₆₁	C₂×C₄	semidirect
202. #33361	C₆₁	direct (N × H)
227. #2883	C₂×C₆₁	semidirect
228. #2129	C₂×C₆₁	semidirect
229. #2127	C₂×C₆₁	semidirect
230. #2126	C₂×C₆₁	semidirect
231. C₂×#354	C₂×C₆₁	semidirect
239. C₂×2₋¹⁺⁴	C₄×C₆₁	semidirect
244. #536	C₄×C₆₁	semidirect
246. #535	C₄×C₆₁	semidirect
248. #532	C₄×C₆₁	semidirect
250. C₂×#124	C₄×C₆₁	semidirect
253. C₄×#41	C₄×C₆₁	semidirect
258. #354	C₂²×C₆₁	semidirect
259. #354	C₂²×C₆₁	semidirect
260. #353	C₂²×C₆₁	semidirect
261. C₆₁	#33361	direct (N × H)
268. C₂×#41	C₂×C₄×C₆₁	semidirect
269. C₂×#41	C₂×C₄×C₆₁	semidirect
273. C₂²×Q₈	C₂×C₄×C₆₁	semidirect
274. C₂²×Q₈	C₂×C₄×C₆₁	semidirect
279. C₄⋊Q₈	C₂×C₄×C₆₁	semidirect
280. C₄⋊Q₈	C₂×C₄×C₆₁	semidirect
281. #127	C₂×C₄×C₆₁	semidirect
285. #124	C₂×C₄×C₆₁	semidirect
286. #124	C₂×C₄×C₆₁	semidirect
287. #124	C₂×C₄×C₆₁	semidirect
288. #124	C₂×C₄×C₆₁	semidirect
308. #117	C₂×C₄×C₆₁	semidirect
309. #117	C₂×C₄×C₆₁	semidirect
313. C₂×C₄²	C₂×C₄×C₆₁	semidirect
314. C₂×C₄²	C₂×C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3500	2
C₆₁×#2883	2
C₆₁×#2129	2
C₆₁×#2127	2
C₆₁×#2126	2
C₂×C₆₁×#354	2
#33361	61

Subgroups — C61×#33361 #29536794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33361 #29536794