Subgroups — C₆₁×#25798 #29529231

Order 15616.

1430 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3154	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3057	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3028	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#3027	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#3022	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#3021	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2009	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2008	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1976	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#585	C₂²	1	1	yes	characteristic
13	C₆₁×#585	C₂²	1	1	yes	characteristic
14	C₆₁×#578	—	2	2	no	—
15	C₆₁×#577	—	2	2	no	—
16	C₆₁×#577	—	2	2	no	—
17	C₆₁×#576	—	2	2	no	—
18	C₆₁×#575	—	2	2	no	—
19	C₆₁×#575	C₂²	1	2	yes	—
20	C₂×#137×C₆₁	C₂²	1	2	yes	complemented
21	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	complemented
22	C₂×#135×C₆₁	—	2	2	no	—
23	C₂×#135×C₆₁	C₂²	1	2	yes	—
24	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
25	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#536	C₂²	1	1	yes	characteristic
27	C₆₁×#536	C₂²	1	2	yes	complemented
28	C₆₁×#535	C₂²	1	1	yes	characteristic
29	C₆₁×#535	C₂²	1	2	yes	complemented
30	C₆₁×#497	—	2	2	no	—
31	C₆₁×#496	—	2	2	no	—
32	C₆₁×#495	—	2	2	no	—
33	C₆₁×#492	—	2	2	no	—
34	C₆₁×#468	—	2	2	no	—
35	C₆₁×#465	—	2	2	no	—
36	C₆₁×#458	—	2	4	no	—
37	C₆₁×B(4,𝔽₂)	—	2	4	no	—
38	C₆₁×#456	—	2	2	no	—
39	C₆₁×#455	—	2	2	no	—
40	C₆₁×#454	—	2	2	no	—
41	C₆₁×#453	—	2	2	no	—
42	C₆₁×#427	C₂²	1	1	yes	characteristic, complemented
43	C₆₁×#421	C₂²	1	2	yes	complemented
44	C₆₁×#421	C₂²	1	2	yes	complemented
45	C₆₁×#419	C₂²	1	1	yes	characteristic, complemented
46	C₆₁×#418	C₂²	1	1	yes	characteristic, complemented
47	C₆₁×#413	C₂²	1	1	yes	characteristic
48	C₆₁×#410	C₂²	1	1	yes	characteristic
49	C₆₁×#410	C₂²	1	2	yes	complemented
50	C₆₁×#410	C₂²	1	2	yes	complemented
51	C₆₁×#405	C₂²	1	2	yes	complemented
52	C₆₁×#344	C₂²	1	2	yes	complemented
53	2₋¹⁺⁴×C₆₁	—	2	2	no	—
54	2₋¹⁺⁴×C₆₁	—	2	2	no	—
55	2₋¹⁺⁴×C₆₁	—	2	1	no	—
56	2₋¹⁺⁴×C₆₁	—	2	1	no	—
57	H(5,𝔽₂)×C₆₁	—	2	2	no	—
58	H(5,𝔽₂)×C₆₁	—	2	2	no	—
59	H(5,𝔽₂)×C₆₁	—	2	2	no	—
60	H(5,𝔽₂)×C₆₁	—	2	2	no	—
61	H(5,𝔽₂)×C₆₁	—	2	1	no	—
62	H(5,𝔽₂)×C₆₁	—	2	1	no	—
63	C₂×#41×C₆₁	—	2	2	no	—
64	C₂×#41×C₆₁	—	2	2	no	—
65	C₂×#41×C₆₁	—	4	2	no	—
66	C₂×#41×C₆₁	—	4	2	no	—
67	C₂×#41×C₆₁	C₂³	1	2	yes	—
68	C₂×#41×C₆₁	C₂³	1	2	yes	—
69	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
72	#137×C₆₁	—	2	4	no	—
73	#137×C₆₁	—	2	4	no	—
74	#137×C₆₁	—	2	2	no	—
75	#137×C₆₁	—	2	2	no	—
76	#137×C₆₁	—	2	4	no	—
77	#137×C₆₁	—	4	2	no	—
78	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
79	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
80	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
81	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
82	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
83	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
84	#135×C₆₁	—	2	4	no	—
85	#135×C₆₁	—	2	4	no	—
86	#135×C₆₁	—	2	2	no	—
87	#135×C₆₁	—	2	4	no	—
88	#135×C₆₁	—	2	2	no	—
89	#135×C₆₁	—	2	2	no	—
90	#135×C₆₁	—	2	2	no	—
91	C₂×Q₁₆×C₆₁	—	2	2	no	—
92	C₂×Q₁₆×C₆₁	—	2	2	no	—
93	C₂×Q₁₆×C₆₁	D₄	1	2	yes	complemented
94	C₂×SD₁₆×C₆₁	—	2	2	no	—
95	C₂×SD₁₆×C₆₁	—	2	2	no	—
96	C₂×SD₁₆×C₆₁	—	2	2	no	—
97	C₂×SD₁₆×C₆₁	—	2	2	no	—
98	C₂×SD₁₆×C₆₁	—	2	1	no	—
99	C₂×SD₁₆×C₆₁	—	2	1	no	—
100	C₂×SD₁₆×C₆₁	—	2	2	no	—
101	C₂×SD₁₆×C₆₁	—	2	2	no	—
102	C₂×SD₁₆×C₆₁	D₄	1	2	yes	complemented
103	C₂×SD₁₆×C₆₁	D₄	1	2	yes	complemented
104	C₂×D₈×C₆₁	—	2	2	no	—
105	C₂×D₈×C₆₁	—	2	2	no	—
106	C₂×D₈×C₆₁	D₄	1	2	yes	complemented
107	#131×C₆₁	—	2	2	no	—
108	#131×C₆₁	—	2	2	no	—
109	C₂×M₁₆×C₆₁	D₄	1	2	yes	complemented
110	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
111	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
112	C₂²×C₈×C₆₁	D₄	1	2	yes	complemented
113	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
114	C₄⋊Q₈×C₆₁	—	2	1	no	—
115	C₄⋊Q₈×C₆₁	—	2	2	no	—
116	#127×C₆₁	—	2	1	no	—
117	#127×C₆₁	—	2	2	no	—
118	#124×C₆₁	—	2	1	no	—
119	#124×C₆₁	—	2	2	no	—
120	#124×C₆₁	—	2	1	no	—
121	#124×C₆₁	—	2	2	no	—
122	#123×C₆₁	—	2	2	no	—
123	#123×C₆₁	—	2	4	no	—
124	C₂²⋊Q₈×C₆₁	—	2	2	no	—
125	C₂²⋊Q₈×C₆₁	—	4	2	no	—
126	#121×C₆₁	—	2	2	no	—
127	#121×C₆₁	—	4	2	no	—
128	#120×C₆₁	—	2	2	no	—
129	#120×C₆₁	—	2	4	no	—
130	#117×C₆₁	C₂³	1	1	yes	characteristic
131	#117×C₆₁	C₂³	1	1	yes	characteristic
132	#117×C₆₁	C₂³	1	2	yes	—
133	#107×C₆₁	—	2	1	no	—
134	#106×C₆₁	—	2	1	no	—
135	#104×C₆₁	—	2	4	no	—
136	#104×C₆₁	—	2	4	no	—
137	#103×C₆₁	—	2	2	no	—
138	#102×C₆₁	—	2	2	no	—
139	#101×C₆₁	D₄	1	2	yes	complemented
140	#100×C₆₁	D₄	1	2	yes	complemented
141	#99×C₆₁	—	2	4	no	—
142	#99×C₆₁	—	2	4	no	—
143	#99×C₆₁	D₄	1	4	yes	—
144	#97×C₆₁	—	2	2	no	—
145	C₄×C₈×C₆₁	—	2	2	no	—
146	C₂⁴×C₆₁	—	2	1	no	—
147	#41×C₆₁	—	2	2	no	—
148	#41×C₆₁	—	2	2	no	—
149	#41×C₆₁	—	2	4	no	—
150	#41×C₆₁	—	2	4	no	—
151	#41×C₆₁	—	2	2	no	—
152	#41×C₆₁	—	2	2	no	—
153	#41×C₆₁	—	2	2	no	—
154	#41×C₆₁	—	2	2	no	—
155	#41×C₆₁	—	2	2	no	—
156	#41×C₆₁	—	2	2	no	—
157	#41×C₆₁	—	2	2	no	—
158	#41×C₆₁	—	2	2	no	—
159	#41×C₆₁	—	4	4	no	—
160	#41×C₆₁	—	4	4	no	—
161	#41×C₆₁	—	4	2	no	—
162	#41×C₆₁	—	4	2	no	—
163	#41×C₆₁	—	4	2	no	—
164	#41×C₆₁	—	4	2	no	—
165	#41×C₆₁	—	4	2	no	—
166	#41×C₆₁	—	4	2	no	—
167	#41×C₆₁	—	4	2	no	—
168	#41×C₆₁	—	4	2	no	—
169	C₂×Q₈×C₆₁	—	2	2	no	—
170	C₂×Q₈×C₆₁	—	2	2	no	—
171	C₂×Q₈×C₆₁	—	2	1	no	—
172	C₂×Q₈×C₆₁	—	2	2	no	—
173	C₂×Q₈×C₆₁	—	2	1	no	—
174	C₂×Q₈×C₆₁	—	2	2	no	—
175	C₂×Q₈×C₆₁	—	4	2	no	—
176	C₂×Q₈×C₆₁	—	4	2	no	—
177	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
178	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
179	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
180	C₂×D₄×C₆₁	—	2	2	no	—
181	C₂×D₄×C₆₁	—	2	2	no	—
182	C₂×D₄×C₆₁	—	2	2	no	—
183	C₂×D₄×C₆₁	—	2	2	no	—
184	C₂×D₄×C₆₁	—	2	1	no	—
185	C₂×D₄×C₆₁	—	2	2	no	—
186	C₂×D₄×C₆₁	—	2	1	no	—
187	C₂×D₄×C₆₁	—	2	2	no	—
188	C₂×D₄×C₆₁	—	4	4	no	—
189	C₂×D₄×C₆₁	—	4	2	no	—
190	C₂×D₄×C₆₁	—	4	2	no	—
191	C₂×D₄×C₆₁	—	4	4	no	—
192	C₂×D₄×C₆₁	—	4	4	no	—
193	C₂×D₄×C₆₁	—	4	2	no	—
194	C₂×D₄×C₆₁	—	4	2	no	—
195	C₂×D₄×C₆₁	—	4	2	no	—
196	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
197	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
198	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
199	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₆₁	—	2	2	no	—
201	C₂²×C₄×C₆₁	—	2	2	no	—
202	C₂²×C₄×C₆₁	—	2	1	no	—
203	C₂²×C₄×C₆₁	—	4	2	no	—
204	C₂²×C₄×C₆₁	—	4	2	no	—
205	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
206	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₆₁	—	2	4	no	—
208	Q₁₆×C₆₁	—	2	2	no	—
209	Q₁₆×C₆₁	—	2	2	no	—
210	Q₁₆×C₆₁	—	4	4	no	—
211	SD₁₆×C₆₁	—	2	4	no	—
212	SD₁₆×C₆₁	—	2	4	no	—
213	SD₁₆×C₆₁	—	2	2	no	—
214	SD₁₆×C₆₁	—	2	2	no	—
215	SD₁₆×C₆₁	—	2	2	no	—
216	SD₁₆×C₆₁	—	2	2	no	—
217	SD₁₆×C₆₁	—	4	4	no	—
218	SD₁₆×C₆₁	—	4	4	no	—
219	SD₁₆×C₆₁	—	4	1	no	—
220	SD₁₆×C₆₁	—	4	1	no	—
221	SD₁₆×C₆₁	—	4	2	no	—
222	D₈×C₆₁	—	2	4	no	—
223	D₈×C₆₁	—	2	2	no	—
224	D₈×C₆₁	—	2	2	no	—
225	D₈×C₆₁	—	4	4	no	—
226	M₁₆×C₆₁	—	2	2	no	—
227	M₁₆×C₆₁	—	2	2	no	—
228	M₁₆×C₆₁	—	2	4	no	—
229	M₁₆×C₆₁	C₂×D₄	1	4	yes	—
230	C₂×C₈×C₆₁	—	2	2	no	—
231	C₂×C₈×C₆₁	—	2	2	no	—
232	C₂×C₈×C₆₁	—	2	2	no	—
233	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
234	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
235	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
236	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
237	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
238	C₄⋊C₄×C₆₁	—	2	2	no	—
239	C₄⋊C₄×C₆₁	—	2	1	no	—
240	C₄⋊C₄×C₆₁	—	2	1	no	—
241	C₄⋊C₄×C₆₁	—	4	4	no	—
242	C₄⋊C₄×C₆₁	—	4	2	no	—
243	C₂²⋊C₄×C₆₁	—	2	1	no	—
244	C₂²⋊C₄×C₆₁	—	2	1	no	—
245	C₂²⋊C₄×C₆₁	—	4	4	no	—
246	C₂²⋊C₄×C₆₁	—	4	4	no	—
247	C₂²⋊C₄×C₆₁	—	4	2	no	—
248	C₂²⋊C₄×C₆₁	—	4	2	no	—
249	C₂²⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
250	C₄²×C₆₁	—	2	2	no	—
251	C₄²×C₆₁	—	2	1	no	—
252	C₄²×C₆₁	—	2	1	no	—
253	C₂³×C₆₁	—	2	2	no	—
254	C₂³×C₆₁	—	2	2	no	—
255	C₂³×C₆₁	—	4	2	no	—
256	C₂³×C₆₁	—	4	2	no	—
257	C₂³×C₆₁	—	4	1	no	—
258	C₂³×C₆₁	—	4	2	no	—
259	C₂³×C₆₁	—	16	1	no	—
260	C₂³×C₆₁	#127	1	2	yes	—
261	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
262	Q₈×C₆₁	—	2	2	no	—
263	Q₈×C₆₁	—	2	2	no	—
264	Q₈×C₆₁	—	2	2	no	—
265	Q₈×C₆₁	—	2	2	no	—
266	Q₈×C₆₁	—	2	2	no	—
267	Q₈×C₆₁	—	4	2	no	—
268	Q₈×C₆₁	—	4	2	no	—
269	Q₈×C₆₁	—	4	1	no	—
270	Q₈×C₆₁	—	4	1	no	—
271	Q₈×C₆₁	—	4	2	no	—
272	D₄×C₆₁	—	2	4	no	—
273	D₄×C₆₁	—	2	2	no	—
274	D₄×C₆₁	—	2	2	no	—
275	D₄×C₆₁	—	2	2	no	—
276	D₄×C₆₁	—	2	2	no	—
277	D₄×C₆₁	—	2	2	no	—
278	D₄×C₆₁	—	4	2	no	—
279	D₄×C₆₁	—	4	2	no	—
280	D₄×C₆₁	—	4	4	no	—
281	D₄×C₆₁	—	4	4	no	—
282	D₄×C₆₁	—	4	2	no	—
283	D₄×C₆₁	—	4	2	no	—
284	D₄×C₆₁	—	4	2	no	—
285	D₄×C₆₁	—	4	2	no	—
286	D₄×C₆₁	—	4	1	no	—
287	D₄×C₆₁	—	4	1	no	—
288	D₄×C₆₁	—	4	2	no	—
289	D₄×C₆₁	—	16	4	no	—
290	D₄×C₆₁	—	16	2	no	—
291	C₂×C₄×C₆₁	—	2	2	no	—
292	C₂×C₄×C₆₁	—	2	2	no	—
293	C₂×C₄×C₆₁	—	2	4	no	—
294	C₂×C₄×C₆₁	—	2	2	no	—
295	C₂×C₄×C₆₁	—	2	2	no	—
296	C₂×C₄×C₆₁	—	4	4	no	—
297	C₂×C₄×C₆₁	—	4	4	no	—
298	C₂×C₄×C₆₁	—	4	2	no	—
299	C₂×C₄×C₆₁	—	4	2	no	—
300	C₂×C₄×C₆₁	—	4	2	no	—
301	C₂×C₄×C₆₁	—	4	2	no	—
302	C₂×C₄×C₆₁	—	4	2	no	—
303	C₂×C₄×C₆₁	—	4	2	no	—
304	C₂×C₄×C₆₁	—	4	1	no	—
305	C₂×C₄×C₆₁	—	4	1	no	—
306	C₂×C₄×C₆₁	—	4	1	no	—
307	C₂×C₄×C₆₁	—	4	2	no	—
308	C₂×C₄×C₆₁	#127	1	2	yes	—
309	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
310	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
311	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
313	C₈×C₆₁	—	2	2	no	—
314	C₈×C₆₁	—	2	4	no	—
315	C₈×C₆₁	#120	1	1	yes	characteristic
316	C₈×C₆₁	#120	1	1	yes	characteristic
317	C₈×C₆₁	#120	1	2	yes	—
318	#25798	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
319	C₂²×C₆₁	—	2	2	no	—
320	C₂²×C₆₁	—	4	1	no	—
321	C₂²×C₆₁	—	4	2	no	—
322	C₂²×C₆₁	—	4	2	no	—
323	C₂²×C₆₁	—	4	2	no	—
324	C₂²×C₆₁	—	8	2	no	—
325	C₂²×C₆₁	—	8	2	no	—
326	C₂²×C₆₁	—	8	4	no	—
327	C₂²×C₆₁	—	8	2	no	—
328	C₂²×C₆₁	—	8	1	no	—
329	C₂²×C₆₁	—	16	2	no	—
330	C₂²×C₆₁	—	16	2	no	—
331	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
332	C₂²×C₆₁	D₄²	1	2	yes	—
333	C₄×C₆₁	—	2	2	no	—
334	C₄×C₆₁	—	4	4	no	—
335	C₄×C₆₁	—	4	2	no	—
336	C₄×C₆₁	—	4	2	no	—
337	C₄×C₆₁	—	8	1	no	—
338	C₄×C₆₁	—	8	1	no	—
339	C₄×C₆₁	—	8	1	no	—
340	C₄×C₆₁	C₂×#120	1	1	yes	characteristic
341	C₄×C₆₁	#535	1	1	yes	characteristic
342	C₄×C₆₁	D₄²	1	2	yes	—
343	#3584	C₂×C₆₁	1	1	yes	characteristic, complemented
344	#3154	C₂×C₆₁	1	2	yes	complemented
345	#3057	C₂×C₆₁	1	1	yes	characteristic, complemented
346	#3028	C₂×C₆₁	1	1	yes	characteristic, complemented
347	#3027	C₂×C₆₁	1	1	yes	characteristic, complemented
348	#3022	C₂×C₆₁	1	1	yes	characteristic, complemented
349	#3021	C₂×C₆₁	1	1	yes	characteristic, complemented
350	#2009	C₂×C₆₁	1	2	yes	complemented
351	#2008	C₂×C₆₁	1	2	yes	complemented
352	#1976	C₂×C₆₁	1	2	yes	complemented
353	#1757	C₂×C₆₁	1	1	yes	characteristic, complemented
354	C₂×C₆₁	—	2	2	no	—
355	C₂×C₆₁	—	2	1	no	—
356	C₂×C₆₁	—	4	2	no	—
357	C₂×C₆₁	—	8	2	no	—
358	C₂×C₆₁	—	8	2	no	—
359	C₂×C₆₁	—	16	1	no	—
360	C₂×C₆₁	#2433	1	1	yes	characteristic
361	#585	C₂²×C₆₁	1	1	yes	characteristic
362	#585	C₂²×C₆₁	1	1	yes	characteristic
363	#578	—	2	2	no	—
364	#577	—	2	2	no	—
365	#577	—	2	2	no	—
366	#576	—	2	2	no	—
367	#575	—	2	2	no	—
368	#575	C₂²×C₆₁	1	2	yes	—
369	C₂×#137	C₂²×C₆₁	1	2	yes	complemented
370	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	complemented
371	C₂×#135	—	2	2	no	—
372	C₂×#135	C₂²×C₆₁	1	2	yes	—
373	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic, complemented
374	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
375	#536	C₂²×C₆₁	1	1	yes	characteristic
376	#536	C₂²×C₆₁	1	2	yes	complemented
377	#535	C₂²×C₆₁	1	1	yes	characteristic
378	#535	C₂²×C₆₁	1	2	yes	complemented
379	#497	—	2	2	no	—
380	#496	—	2	2	no	—
381	#495	—	2	2	no	—
382	#492	—	2	2	no	—
383	#468	—	2	2	no	—
384	#465	—	2	2	no	—
385	#458	—	2	4	no	—
386	B(4,𝔽₂)	—	2	4	no	—
387	#456	—	2	2	no	—
388	#455	—	2	2	no	—
389	#454	—	2	2	no	—
390	#453	—	2	2	no	—
391	#427	C₂²×C₆₁	1	1	yes	characteristic, complemented
392	#421	C₂²×C₆₁	1	2	yes	complemented
393	#421	C₂²×C₆₁	1	2	yes	complemented
394	#419	C₂²×C₆₁	1	1	yes	characteristic, complemented
395	#418	C₂²×C₆₁	1	1	yes	characteristic, complemented
396	#413	C₂²×C₆₁	1	1	yes	characteristic
397	#410	C₂²×C₆₁	1	1	yes	characteristic
398	#410	C₂²×C₆₁	1	2	yes	complemented
399	#410	C₂²×C₆₁	1	2	yes	complemented
400	#405	C₂²×C₆₁	1	2	yes	complemented
401	#344	C₂²×C₆₁	1	2	yes	complemented
402	C₆₁	#25798	1	1	yes	characteristic, direct product factor, Sylow
403	2₋¹⁺⁴	—	2	2	no	—
404	2₋¹⁺⁴	—	2	2	no	—
405	2₋¹⁺⁴	—	2	1	no	—
406	2₋¹⁺⁴	—	2	1	no	—
407	H(5,𝔽₂)	—	2	2	no	—
408	H(5,𝔽₂)	—	2	2	no	—
409	H(5,𝔽₂)	—	2	2	no	—
410	H(5,𝔽₂)	—	2	2	no	—
411	H(5,𝔽₂)	—	2	1	no	—
412	H(5,𝔽₂)	—	2	1	no	—
413	C₂×#41	—	2	2	no	—
414	C₂×#41	—	2	2	no	—
415	C₂×#41	—	4	2	no	—
416	C₂×#41	—	4	2	no	—
417	C₂×#41	C₂³×C₆₁	1	2	yes	—
418	C₂×#41	C₂³×C₆₁	1	2	yes	—
419	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
420	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
421	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
422	#137	—	2	4	no	—
423	#137	—	2	4	no	—
424	#137	—	2	2	no	—
425	#137	—	2	2	no	—
426	#137	—	2	4	no	—
427	#137	—	4	2	no	—
428	AGL(1,ℤ/8)	—	2	4	no	—
429	AGL(1,ℤ/8)	—	2	4	no	—
430	AGL(1,ℤ/8)	—	2	2	no	—
431	AGL(1,ℤ/8)	—	2	2	no	—
432	AGL(1,ℤ/8)	—	2	4	no	—
433	AGL(1,ℤ/8)	—	4	2	no	—
434	#135	—	2	4	no	—
435	#135	—	2	4	no	—
436	#135	—	2	2	no	—
437	#135	—	2	4	no	—
438	#135	—	2	2	no	—
439	#135	—	2	2	no	—
440	#135	—	2	2	no	—
441	C₂×Q₁₆	—	2	2	no	—
442	C₂×Q₁₆	—	2	2	no	—
443	C₂×Q₁₆	D₄×C₆₁	1	2	yes	complemented
444	C₂×SD₁₆	—	2	2	no	—
445	C₂×SD₁₆	—	2	2	no	—
446	C₂×SD₁₆	—	2	2	no	—
447	C₂×SD₁₆	—	2	2	no	—
448	C₂×SD₁₆	—	2	1	no	—
449	C₂×SD₁₆	—	2	1	no	—
450	C₂×SD₁₆	—	2	2	no	—
451	C₂×SD₁₆	—	2	2	no	—
452	C₂×SD₁₆	D₄×C₆₁	1	2	yes	complemented
453	C₂×SD₁₆	D₄×C₆₁	1	2	yes	complemented
454	C₂×D₈	—	2	2	no	—
455	C₂×D₈	—	2	2	no	—
456	C₂×D₈	D₄×C₆₁	1	2	yes	complemented
457	#131	—	2	2	no	—
458	#131	—	2	2	no	—
459	C₂×M₁₆	D₄×C₆₁	1	2	yes	complemented
460	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
461	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
462	C₂²×C₈	D₄×C₆₁	1	2	yes	complemented
463	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
464	C₄⋊Q₈	—	2	1	no	—
465	C₄⋊Q₈	—	2	2	no	—
466	#127	—	2	1	no	—
467	#127	—	2	2	no	—
468	#124	—	2	1	no	—
469	#124	—	2	2	no	—
470	#124	—	2	1	no	—
471	#124	—	2	2	no	—
472	#123	—	2	2	no	—
473	#123	—	2	4	no	—
474	C₂²⋊Q₈	—	2	2	no	—
475	C₂²⋊Q₈	—	4	2	no	—
476	#121	—	2	2	no	—
477	#121	—	4	2	no	—
478	#120	—	2	2	no	—
479	#120	—	2	4	no	—
480	#117	C₂³×C₆₁	1	1	yes	characteristic
481	#117	C₂³×C₆₁	1	1	yes	characteristic
482	#117	C₂³×C₆₁	1	2	yes	—
483	#107	—	2	1	no	—
484	#106	—	2	1	no	—
485	#104	—	2	4	no	—
486	#104	—	2	4	no	—
487	#103	—	2	2	no	—
488	#102	—	2	2	no	—
489	#101	D₄×C₆₁	1	2	yes	complemented
490	#100	D₄×C₆₁	1	2	yes	complemented
491	#99	—	2	4	no	—
492	#99	—	2	4	no	—
493	#99	D₄×C₆₁	1	4	yes	—
494	#97	—	2	2	no	—
495	C₄×C₈	—	2	2	no	—
496	C₂⁴	—	2	1	no	—
497	#41	—	2	2	no	—
498	#41	—	2	2	no	—
499	#41	—	2	4	no	—
500	#41	—	2	4	no	—
501	#41	—	2	2	no	—
502	#41	—	2	2	no	—
503	#41	—	2	2	no	—
504	#41	—	2	2	no	—
505	#41	—	2	2	no	—
506	#41	—	2	2	no	—
507	#41	—	2	2	no	—
508	#41	—	2	2	no	—
509	#41	—	4	4	no	—
510	#41	—	4	4	no	—
511	#41	—	4	2	no	—
512	#41	—	4	2	no	—
513	#41	—	4	2	no	—
514	#41	—	4	2	no	—
515	#41	—	4	2	no	—
516	#41	—	4	2	no	—
517	#41	—	4	2	no	—
518	#41	—	4	2	no	—
519	C₂×Q₈	—	2	2	no	—
520	C₂×Q₈	—	2	2	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	2	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	2	no	—
525	C₂×Q₈	—	4	2	no	—
526	C₂×Q₈	—	4	2	no	—
527	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
528	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
529	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
530	C₂×D₄	—	2	2	no	—
531	C₂×D₄	—	2	2	no	—
532	C₂×D₄	—	2	2	no	—
533	C₂×D₄	—	2	2	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	2	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	4	4	no	—
539	C₂×D₄	—	4	2	no	—
540	C₂×D₄	—	4	2	no	—
541	C₂×D₄	—	4	4	no	—
542	C₂×D₄	—	4	4	no	—
543	C₂×D₄	—	4	2	no	—
544	C₂×D₄	—	4	2	no	—
545	C₂×D₄	—	4	2	no	—
546	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
547	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
548	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
549	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
550	C₂²×C₄	—	2	2	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	4	2	no	—
554	C₂²×C₄	—	4	2	no	—
555	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
556	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
557	Q₁₆	—	2	4	no	—
558	Q₁₆	—	2	2	no	—
559	Q₁₆	—	2	2	no	—
560	Q₁₆	—	4	4	no	—
561	SD₁₆	—	2	4	no	—
562	SD₁₆	—	2	4	no	—
563	SD₁₆	—	2	2	no	—
564	SD₁₆	—	2	2	no	—
565	SD₁₆	—	2	2	no	—
566	SD₁₆	—	2	2	no	—
567	SD₁₆	—	4	4	no	—
568	SD₁₆	—	4	4	no	—
569	SD₁₆	—	4	1	no	—
570	SD₁₆	—	4	1	no	—
571	SD₁₆	—	4	2	no	—
572	D₈	—	2	4	no	—
573	D₈	—	2	2	no	—
574	D₈	—	2	2	no	—
575	D₈	—	4	4	no	—
576	M₁₆	—	2	2	no	—
577	M₁₆	—	2	2	no	—
578	M₁₆	—	2	4	no	—
579	M₁₆	C₂×D₄×C₆₁	1	4	yes	—
580	C₂×C₈	—	2	2	no	—
581	C₂×C₈	—	2	2	no	—
582	C₂×C₈	—	2	2	no	—
583	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
584	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
585	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
586	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
587	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	4	4	no	—
592	C₄⋊C₄	—	4	2	no	—
593	C₂²⋊C₄	—	2	1	no	—
594	C₂²⋊C₄	—	2	1	no	—
595	C₂²⋊C₄	—	4	4	no	—
596	C₂²⋊C₄	—	4	4	no	—
597	C₂²⋊C₄	—	4	2	no	—
598	C₂²⋊C₄	—	4	2	no	—
599	C₂²⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
600	C₄²	—	2	2	no	—
601	C₄²	—	2	1	no	—
602	C₄²	—	2	1	no	—
603	C₂³	—	2	2	no	—
604	C₂³	—	2	2	no	—
605	C₂³	—	4	2	no	—
606	C₂³	—	4	2	no	—
607	C₂³	—	4	1	no	—
608	C₂³	—	4	2	no	—
609	C₂³	—	16	1	no	—
610	C₂³	#127×C₆₁	1	2	yes	—
611	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
612	Q₈	—	2	2	no	—
613	Q₈	—	2	2	no	—
614	Q₈	—	2	2	no	—
615	Q₈	—	2	2	no	—
616	Q₈	—	2	2	no	—
617	Q₈	—	4	2	no	—
618	Q₈	—	4	2	no	—
619	Q₈	—	4	1	no	—
620	Q₈	—	4	2	no	—
621	Q₈	—	4	1	no	—
622	D₄	—	2	4	no	—
623	D₄	—	2	2	no	—
624	D₄	—	2	2	no	—
625	D₄	—	2	2	no	—
626	D₄	—	2	2	no	—
627	D₄	—	2	2	no	—
628	D₄	—	4	2	no	—
629	D₄	—	4	2	no	—
630	D₄	—	4	4	no	—
631	D₄	—	4	4	no	—
632	D₄	—	4	2	no	—
633	D₄	—	4	2	no	—
634	D₄	—	4	2	no	—
635	D₄	—	4	2	no	—
636	D₄	—	4	1	no	—
637	D₄	—	4	2	no	—
638	D₄	—	4	1	no	—
639	D₄	—	16	4	no	—
640	D₄	—	16	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	4	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	4	4	no	—
647	C₂×C₄	—	4	4	no	—
648	C₂×C₄	—	4	2	no	—
649	C₂×C₄	—	4	2	no	—
650	C₂×C₄	—	4	2	no	—
651	C₂×C₄	—	4	2	no	—
652	C₂×C₄	—	4	2	no	—
653	C₂×C₄	—	4	2	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	#127×C₆₁	1	2	yes	—
659	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
660	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
661	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
662	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
663	C₈	—	2	4	no	—
664	C₈	—	2	2	no	—
665	C₈	#120×C₆₁	1	1	yes	characteristic
666	C₈	#120×C₆₁	1	1	yes	characteristic
667	C₈	#120×C₆₁	1	2	yes	—
668	C₂²	—	2	2	no	—
669	C₂²	—	4	1	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	4	2	no	—
672	C₂²	—	4	2	no	—
673	C₂²	—	8	2	no	—
674	C₂²	—	8	2	no	—
675	C₂²	—	8	4	no	—
676	C₂²	—	8	2	no	—
677	C₂²	—	8	1	no	—
678	C₂²	—	16	2	no	—
679	C₂²	—	16	2	no	—
680	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
681	C₂²	D₄²×C₆₁	1	2	yes	—
682	C₄	—	2	2	no	—
683	C₄	—	4	4	no	—
684	C₄	—	4	2	no	—
685	C₄	—	4	2	no	—
686	C₄	—	8	1	no	—
687	C₄	—	8	1	no	—
688	C₄	—	8	1	no	—
689	C₄	C₂×#120×C₆₁	1	1	yes	characteristic
690	C₄	C₆₁×#535	1	1	yes	characteristic
691	C₄	D₄²×C₆₁	1	2	yes	—
692	C₂	—	2	2	no	—
693	C₂	—	2	1	no	—
694	C₂	—	4	2	no	—
695	C₂	—	8	2	no	—
696	C₂	—	8	2	no	—
697	C₂	—	16	1	no	—
698	C₂	C₆₁×#2433	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	#2433
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3584	C₂	semidirect
2. C₆₁×#3154	C₂	semidirect
3. C₆₁×#3057	C₂	semidirect
4. C₆₁×#3028	C₂	semidirect
5. C₆₁×#3027	C₂	semidirect
6. C₆₁×#3022	C₂	semidirect
7. C₆₁×#3021	C₂	semidirect
8. C₆₁×#2009	C₂	semidirect
9. C₆₁×#2008	C₂	semidirect
10. C₆₁×#1976	C₂	semidirect
11. C₆₁×#1757	C₂	semidirect
20. C₂×#137×C₆₁	C₂²	semidirect
21. C₂×AGL(1,ℤ/8)×C₆₁	C₂²	semidirect
24. C₂²×SD₁₆×C₆₁	C₂²	semidirect
27. C₆₁×#536	C₂²	semidirect
29. C₆₁×#535	C₂²	semidirect
42. C₆₁×#427	C₂²	semidirect
43. C₆₁×#421	C₂²	semidirect
44. C₆₁×#421	C₂²	semidirect
45. C₆₁×#419	C₂²	semidirect
46. C₆₁×#418	C₂²	semidirect
49. C₆₁×#410	C₂²	semidirect
50. C₆₁×#410	C₂²	semidirect
51. C₆₁×#405	C₂²	semidirect
52. C₆₁×#344	C₂²	semidirect
93. C₂×Q₁₆×C₆₁	D₄	semidirect
102. C₂×SD₁₆×C₆₁	D₄	semidirect
103. C₂×SD₁₆×C₆₁	D₄	semidirect
106. C₂×D₈×C₆₁	D₄	semidirect
109. C₂×M₁₆×C₆₁	D₄	semidirect
112. C₂²×C₈×C₆₁	D₄	semidirect
139. #101×C₆₁	D₄	semidirect
140. #100×C₆₁	D₄	semidirect
318. #25798	C₆₁	direct (N × H)
343. #3584	C₂×C₆₁	semidirect
344. #3154	C₂×C₆₁	semidirect
345. #3057	C₂×C₆₁	semidirect
346. #3028	C₂×C₆₁	semidirect
347. #3027	C₂×C₆₁	semidirect
348. #3022	C₂×C₆₁	semidirect
349. #3021	C₂×C₆₁	semidirect
350. #2009	C₂×C₆₁	semidirect
351. #2008	C₂×C₆₁	semidirect
352. #1976	C₂×C₆₁	semidirect
353. #1757	C₂×C₆₁	semidirect
369. C₂×#137	C₂²×C₆₁	semidirect
370. C₂×AGL(1,ℤ/8)	C₂²×C₆₁	semidirect
373. C₂²×SD₁₆	C₂²×C₆₁	semidirect
376. #536	C₂²×C₆₁	semidirect
378. #535	C₂²×C₆₁	semidirect
391. #427	C₂²×C₆₁	semidirect
392. #421	C₂²×C₆₁	semidirect
393. #421	C₂²×C₆₁	semidirect
394. #419	C₂²×C₆₁	semidirect
395. #418	C₂²×C₆₁	semidirect
398. #410	C₂²×C₆₁	semidirect
399. #410	C₂²×C₆₁	semidirect
400. #405	C₂²×C₆₁	semidirect
401. #344	C₂²×C₆₁	semidirect
402. C₆₁	#25798	direct (N × H)
443. C₂×Q₁₆	D₄×C₆₁	semidirect
452. C₂×SD₁₆	D₄×C₆₁	semidirect
453. C₂×SD₁₆	D₄×C₆₁	semidirect
456. C₂×D₈	D₄×C₆₁	semidirect
459. C₂×M₁₆	D₄×C₆₁	semidirect
462. C₂²×C₈	D₄×C₆₁	semidirect
489. #101	D₄×C₆₁	semidirect
490. #100	D₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3584	2
C₆₁×#3154	2
C₆₁×#3057	2
C₆₁×#3028	2
C₆₁×#3027	2
C₆₁×#3022	2
C₆₁×#3021	2
C₆₁×#2009	2
C₆₁×#2008	2
C₆₁×#1976	2
C₆₁×#1757	2
#25798	61

Subgroups — C61×#25798 #29529231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25798 #29529231