Subgroups — C₆₁×#23994 #29527427

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3025	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3018	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2901	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2031	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2010	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1906	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1896	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1891	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1884	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1827	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#585	C₂²	1	1	yes	characteristic
13	C₆₁×#575	C₂²	1	2	yes	—
14	C₆₁×#568	C₂²	1	1	yes	characteristic
15	C₆₁×#533	—	2	1	no	—
16	C₆₁×#533	C₂²	1	1	yes	characteristic, complemented
17	C₆₁×#532	—	2	1	no	—
18	C₆₁×#532	C₂²	1	1	yes	characteristic
19	C₆₁×#529	—	2	1	no	—
20	C₆₁×#529	—	2	1	no	—
21	C₆₁×#529	—	2	2	no	—
22	C₆₁×#529	—	2	2	no	—
23	C₆₁×#525	—	2	2	no	—
24	C₆₁×#525	C₂²	1	1	yes	characteristic
25	C₆₁×#525	C₂²	1	2	yes	—
26	C₆₁×#525	C₂²	1	1	yes	characteristic, complemented
27	C₄×#41×C₆₁	—	2	2	no	—
28	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
29	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic, complemented
30	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic, complemented
31	C₆₁×#458	—	2	2	no	—
32	C₆₁×B(4,𝔽₂)	—	2	2	no	—
33	C₆₁×#456	—	2	1	no	—
34	C₆₁×#455	—	2	1	no	—
35	C₆₁×#454	—	2	1	no	—
36	C₆₁×#453	—	2	1	no	—
37	C₆₁×#421	—	2	2	no	—
38	C₆₁×#421	C₂²	1	2	yes	—
39	C₆₁×#421	C₂²	1	2	yes	complemented
40	C₂×#104×C₆₁	—	2	2	no	—
41	C₂×#104×C₆₁	C₂²	1	2	yes	—
42	C₂×#104×C₆₁	C₂²	1	2	yes	—
43	C₆₁×#413	C₂²	1	1	yes	characteristic
44	C₆₁×#410	C₂²	1	2	yes	—
45	C₆₁×#410	C₂²	1	1	yes	characteristic
46	C₆₁×#407	C₂²	1	2	yes	complemented
47	C₆₁×#344	C₂²	1	2	yes	complemented
48	C₆₁×#339	C₂²	1	2	yes	complemented
49	C₆₁×#339	C₂²	1	2	yes	complemented
50	C₆₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
51	2₋¹⁺⁴×C₆₁	—	2	2	no	—
52	2₋¹⁺⁴×C₆₁	—	2	1	no	—
53	H(5,𝔽₂)×C₆₁	—	2	2	no	—
54	H(5,𝔽₂)×C₆₁	—	2	2	no	—
55	H(5,𝔽₂)×C₆₁	—	2	1	no	—
56	C₂×#41×C₆₁	—	2	2	no	—
57	C₂×#41×C₆₁	—	2	2	no	—
58	C₂×#41×C₆₁	—	4	2	no	—
59	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
60	C₂×#41×C₆₁	C₂³	1	2	yes	—
61	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₁	—	2	1	no	—
63	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
64	#137×C₆₁	—	2	2	no	—
65	#137×C₆₁	—	2	2	no	—
66	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
67	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
68	#135×C₆₁	—	4	2	no	—
69	#131×C₆₁	—	4	2	no	—
70	C₂×M₁₆×C₆₁	—	2	2	no	—
71	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₆₁	—	2	1	no	—
74	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic, complemented
75	#127×C₆₁	D₄	1	1	yes	characteristic, complemented
76	#126×C₆₁	—	2	2	no	—
77	#126×C₆₁	—	2	2	no	—
78	#125×C₆₁	—	2	2	no	—
79	#125×C₆₁	—	2	1	no	—
80	#124×C₆₁	—	2	2	no	—
81	#124×C₆₁	D₄	1	1	yes	characteristic, complemented
82	#124×C₆₁	D₄	1	1	yes	characteristic, complemented
83	#123×C₆₁	—	2	2	no	—
84	#123×C₆₁	—	2	2	no	—
85	#123×C₆₁	—	2	1	no	—
86	#123×C₆₁	—	2	2	no	—
87	#123×C₆₁	D₄	1	2	yes	—
88	C₂²⋊Q₈×C₆₁	—	2	2	no	—
89	C₂²⋊Q₈×C₆₁	—	2	1	no	—
90	C₂²⋊Q₈×C₆₁	—	2	1	no	—
91	C₂²⋊Q₈×C₆₁	—	2	2	no	—
92	C₂²⋊Q₈×C₆₁	—	2	2	no	—
93	#121×C₆₁	—	2	2	no	—
94	#121×C₆₁	—	2	1	no	—
95	#121×C₆₁	—	2	1	no	—
96	#121×C₆₁	—	2	2	no	—
97	#121×C₆₁	—	2	2	no	—
98	#120×C₆₁	—	2	1	no	—
99	#120×C₆₁	—	2	2	no	—
100	#120×C₆₁	D₄	1	2	yes	—
101	C₄×Q₈×C₆₁	—	2	2	no	—
102	C₄×Q₈×C₆₁	—	2	1	no	—
103	C₄×Q₈×C₆₁	—	2	1	no	—
104	C₄×Q₈×C₆₁	—	2	2	no	—
105	C₄×D₄×C₆₁	—	2	2	no	—
106	C₄×D₄×C₆₁	—	2	2	no	—
107	C₄×D₄×C₆₁	—	2	2	no	—
108	C₄×D₄×C₆₁	—	2	1	no	—
109	C₄×D₄×C₆₁	—	2	1	no	—
110	C₄×D₄×C₆₁	—	2	2	no	—
111	C₄×D₄×C₆₁	—	2	2	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	—	4	2	no	—
116	#117×C₆₁	—	2	2	no	—
117	#117×C₆₁	—	2	2	no	—
118	#117×C₆₁	—	2	2	no	—
119	#117×C₆₁	—	2	1	no	—
120	#117×C₆₁	D₄	1	2	yes	complemented
121	#117×C₆₁	C₂³	1	2	yes	—
122	#117×C₆₁	C₂³	1	1	yes	characteristic
123	#117×C₆₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₁	—	2	1	no	—
125	C₂×C₄²×C₆₁	—	2	1	no	—
126	C₂×C₄²×C₆₁	—	2	1	no	—
127	C₂×C₄²×C₆₁	—	2	1	no	—
128	C₂×C₄²×C₆₁	—	2	1	no	—
129	C₂×C₄²×C₆₁	—	2	1	no	—
130	C₂×C₄²×C₆₁	D₄	1	2	yes	complemented
131	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
133	#104×C₆₁	—	2	2	no	—
134	#104×C₆₁	—	2	4	no	—
135	#104×C₆₁	—	2	2	no	—
136	#104×C₆₁	—	2	2	no	—
137	#104×C₆₁	—	2	2	no	—
138	#104×C₆₁	—	4	2	no	—
139	#101×C₆₁	—	2	1	no	—
140	#100×C₆₁	—	2	1	no	—
141	#99×C₆₁	—	2	2	no	—
142	#99×C₆₁	—	2	4	no	—
143	C₂²⋊C₈×C₆₁	—	4	2	no	—
144	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
145	#41×C₆₁	—	2	2	no	—
146	#41×C₆₁	—	2	2	no	—
147	#41×C₆₁	—	2	4	no	—
148	#41×C₆₁	—	2	2	no	—
149	#41×C₆₁	—	2	2	no	—
150	#41×C₆₁	—	2	2	no	—
151	#41×C₆₁	—	2	2	no	—
152	#41×C₆₁	—	2	2	no	—
153	#41×C₆₁	—	2	2	no	—
154	#41×C₆₁	—	4	4	no	—
155	#41×C₆₁	—	4	8	no	—
156	#41×C₆₁	—	4	2	no	—
157	#41×C₆₁	—	4	2	no	—
158	#41×C₆₁	—	4	2	no	—
159	#41×C₆₁	—	4	2	no	—
160	C₂×Q₈×C₆₁	—	2	2	no	—
161	C₂×Q₈×C₆₁	—	2	2	no	—
162	C₂×Q₈×C₆₁	—	4	2	no	—
163	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
165	C₂×Q₈×C₆₁	#41	1	2	yes	complemented
166	C₂×D₄×C₆₁	—	2	2	no	—
167	C₂×D₄×C₆₁	—	2	2	no	—
168	C₂×D₄×C₆₁	—	2	2	no	—
169	C₂×D₄×C₆₁	—	2	2	no	—
170	C₂×D₄×C₆₁	—	2	2	no	—
171	C₂×D₄×C₆₁	—	2	1	no	—
172	C₂×D₄×C₆₁	—	4	2	no	—
173	C₂×D₄×C₆₁	—	4	4	no	—
174	C₂×D₄×C₆₁	—	4	2	no	—
175	C₂×D₄×C₆₁	—	4	2	no	—
176	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
177	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
179	C₂×D₄×C₆₁	#41	1	2	yes	complemented
180	C₂²×C₄×C₆₁	—	2	1	no	—
181	C₂²×C₄×C₆₁	—	2	1	no	—
182	C₂²×C₄×C₆₁	—	2	1	no	—
183	C₂²×C₄×C₆₁	—	2	1	no	—
184	C₂²×C₄×C₆₁	—	2	2	no	—
185	C₂²×C₄×C₆₁	—	2	2	no	—
186	C₂²×C₄×C₆₁	—	2	1	no	—
187	C₂²×C₄×C₆₁	—	2	1	no	—
188	C₂²×C₄×C₆₁	—	2	1	no	—
189	C₂²×C₄×C₆₁	—	2	1	no	—
190	C₂²×C₄×C₆₁	—	2	1	no	—
191	C₂²×C₄×C₆₁	—	2	2	no	—
192	C₂²×C₄×C₆₁	—	2	2	no	—
193	C₂²×C₄×C₆₁	—	2	2	no	—
194	C₂²×C₄×C₆₁	—	4	4	no	—
195	C₂²×C₄×C₆₁	—	4	2	no	—
196	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
198	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
199	Q₁₆×C₆₁	—	4	2	no	—
200	SD₁₆×C₆₁	—	4	2	no	—
201	SD₁₆×C₆₁	—	4	2	no	—
202	D₈×C₆₁	—	4	2	no	—
203	M₁₆×C₆₁	—	2	2	no	—
204	M₁₆×C₆₁	—	2	2	no	—
205	M₁₆×C₆₁	—	2	2	no	—
206	M₁₆×C₆₁	—	2	2	no	—
207	M₁₆×C₆₁	—	4	2	no	—
208	C₂×C₈×C₆₁	—	2	2	no	—
209	C₂×C₈×C₆₁	—	2	2	no	—
210	C₂×C₈×C₆₁	—	4	2	no	—
211	C₄⋊C₄×C₆₁	—	2	2	no	—
212	C₄⋊C₄×C₆₁	—	2	2	no	—
213	C₄⋊C₄×C₆₁	—	2	2	no	—
214	C₄⋊C₄×C₆₁	—	2	2	no	—
215	C₄⋊C₄×C₆₁	—	2	2	no	—
216	C₄⋊C₄×C₆₁	—	2	2	no	—
217	C₄⋊C₄×C₆₁	—	2	2	no	—
218	C₄⋊C₄×C₆₁	—	2	2	no	—
219	C₄⋊C₄×C₆₁	—	2	1	no	—
220	C₄⋊C₄×C₆₁	—	2	1	no	—
221	C₄⋊C₄×C₆₁	—	2	2	no	—
222	C₄⋊C₄×C₆₁	—	4	2	no	—
223	C₂²⋊C₄×C₆₁	—	2	2	no	—
224	C₂²⋊C₄×C₆₁	—	2	2	no	—
225	C₂²⋊C₄×C₆₁	—	2	2	no	—
226	C₂²⋊C₄×C₆₁	—	2	2	no	—
227	C₂²⋊C₄×C₆₁	—	2	2	no	—
228	C₂²⋊C₄×C₆₁	—	2	2	no	—
229	C₂²⋊C₄×C₆₁	—	2	1	no	—
230	C₂²⋊C₄×C₆₁	—	2	2	no	—
231	C₂²⋊C₄×C₆₁	—	2	2	no	—
232	C₂²⋊C₄×C₆₁	—	2	2	no	—
233	C₂²⋊C₄×C₆₁	—	4	2	no	—
234	C₂²⋊C₄×C₆₁	—	4	2	no	—
235	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
236	C₄²×C₆₁	—	2	1	no	—
237	C₄²×C₆₁	—	2	1	no	—
238	C₄²×C₆₁	—	2	1	no	—
239	C₄²×C₆₁	—	2	1	no	—
240	C₄²×C₆₁	—	2	2	no	—
241	C₄²×C₆₁	—	2	2	no	—
242	C₄²×C₆₁	—	2	1	no	—
243	C₄²×C₆₁	—	2	1	no	—
244	C₄²×C₆₁	—	2	1	no	—
245	C₄²×C₆₁	—	2	2	no	—
246	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
247	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
248	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
249	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
250	C₄²×C₆₁	C₂×D₄	1	2	yes	—
251	C₂³×C₆₁	—	2	1	no	—
252	C₂³×C₆₁	—	2	2	no	—
253	C₂³×C₆₁	—	2	1	no	—
254	C₂³×C₆₁	—	2	2	no	—
255	C₂³×C₆₁	—	4	2	no	—
256	C₂³×C₆₁	—	4	2	no	—
257	C₂³×C₆₁	#120	1	2	yes	—
258	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₆₁	—	2	2	no	—
260	Q₈×C₆₁	—	2	2	no	—
261	Q₈×C₆₁	—	2	2	no	—
262	Q₈×C₆₁	—	2	2	no	—
263	Q₈×C₆₁	—	4	2	no	—
264	Q₈×C₆₁	—	4	4	no	—
265	D₄×C₆₁	—	2	4	no	—
266	D₄×C₆₁	—	2	2	no	—
267	D₄×C₆₁	—	2	2	no	—
268	D₄×C₆₁	—	2	2	no	—
269	D₄×C₆₁	—	2	2	no	—
270	D₄×C₆₁	—	4	2	no	—
271	D₄×C₆₁	—	4	4	no	—
272	D₄×C₆₁	—	4	4	no	—
273	D₄×C₆₁	—	4	4	no	—
274	D₄×C₆₁	—	4	4	no	—
275	D₄×C₆₁	—	4	2	no	—
276	D₄×C₆₁	—	4	2	no	—
277	D₄×C₆₁	—	4	2	no	—
278	D₄×C₆₁	—	8	4	no	—
279	C₂×C₄×C₆₁	—	2	4	no	—
280	C₂×C₄×C₆₁	—	2	1	no	—
281	C₂×C₄×C₆₁	—	2	1	no	—
282	C₂×C₄×C₆₁	—	2	2	no	—
283	C₂×C₄×C₆₁	—	2	2	no	—
284	C₂×C₄×C₆₁	—	2	1	no	—
285	C₂×C₄×C₆₁	—	2	1	no	—
286	C₂×C₄×C₆₁	—	2	1	no	—
287	C₂×C₄×C₆₁	—	2	1	no	—
288	C₂×C₄×C₆₁	—	2	2	no	—
289	C₂×C₄×C₆₁	—	2	2	no	—
290	C₂×C₄×C₆₁	—	2	2	no	—
291	C₂×C₄×C₆₁	—	2	2	no	—
292	C₂×C₄×C₆₁	—	4	2	no	—
293	C₂×C₄×C₆₁	—	4	2	no	—
294	C₂×C₄×C₆₁	—	4	2	no	—
295	C₂×C₄×C₆₁	—	4	2	no	—
296	C₂×C₄×C₆₁	—	4	4	no	—
297	C₂×C₄×C₆₁	—	4	4	no	—
298	C₂×C₄×C₆₁	—	4	4	no	—
299	C₂×C₄×C₆₁	—	4	2	no	—
300	C₂×C₄×C₆₁	—	4	2	no	—
301	C₂×C₄×C₆₁	—	4	1	no	—
302	C₂×C₄×C₆₁	—	4	1	no	—
303	C₂×C₄×C₆₁	—	4	1	no	—
304	C₂×C₄×C₆₁	—	4	2	no	—
305	C₂×C₄×C₆₁	—	4	1	no	—
306	C₂×C₄×C₆₁	—	4	2	no	—
307	C₂×C₄×C₆₁	—	4	2	no	—
308	C₂×C₄×C₆₁	—	4	2	no	—
309	C₂×C₄×C₆₁	#120	1	2	yes	—
310	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
312	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
313	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
314	C₈×C₆₁	—	4	2	no	—
315	C₈×C₆₁	—	4	2	no	—
316	#23994	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
317	C₂²×C₆₁	—	2	2	no	—
318	C₂²×C₆₁	—	2	4	no	—
319	C₂²×C₆₁	—	2	2	no	—
320	C₂²×C₆₁	—	4	1	no	—
321	C₂²×C₆₁	—	4	2	no	—
322	C₂²×C₆₁	—	4	1	no	—
323	C₂²×C₆₁	—	4	2	no	—
324	C₂²×C₆₁	—	4	2	no	—
325	C₂²×C₆₁	—	4	1	no	—
326	C₂²×C₆₁	—	4	2	no	—
327	C₂²×C₆₁	—	4	1	no	—
328	C₂²×C₆₁	—	8	2	no	—
329	C₂²×C₆₁	—	16	2	no	—
330	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
331	C₂²×C₆₁	#546	1	2	yes	—
332	C₄×C₆₁	—	2	2	no	—
333	C₄×C₆₁	—	2	2	no	—
334	C₄×C₆₁	—	2	2	no	—
335	C₄×C₆₁	—	4	1	no	—
336	C₄×C₆₁	—	4	1	no	—
337	C₄×C₆₁	—	4	2	no	—
338	C₄×C₆₁	—	4	2	no	—
339	C₄×C₆₁	—	4	1	no	—
340	C₄×C₆₁	—	4	2	no	—
341	C₄×C₆₁	—	4	1	no	—
342	C₄×C₆₁	—	8	2	no	—
343	C₄×C₆₁	#525	1	1	yes	characteristic
344	C₄×C₆₁	#535	1	1	yes	characteristic
345	C₄×C₆₁	#546	1	2	yes	—
346	#3444	C₂×C₆₁	1	1	yes	characteristic, complemented
347	#3025	C₂×C₆₁	1	1	yes	characteristic, complemented
348	#3018	C₂×C₆₁	1	1	yes	characteristic, complemented
349	#2901	C₂×C₆₁	1	1	yes	characteristic, complemented
350	#2031	C₂×C₆₁	1	2	yes	complemented
351	#2010	C₂×C₆₁	1	2	yes	complemented
352	#1906	C₂×C₆₁	1	2	yes	complemented
353	#1896	C₂×C₆₁	1	2	yes	complemented
354	#1891	C₂×C₆₁	1	1	yes	characteristic, complemented
355	#1884	C₂×C₆₁	1	1	yes	characteristic, complemented
356	#1827	C₂×C₆₁	1	1	yes	characteristic, complemented
357	C₂×C₆₁	—	2	1	no	—
358	C₂×C₆₁	—	2	2	no	—
359	C₂×C₆₁	—	4	2	no	—
360	C₂×C₆₁	—	4	2	no	—
361	C₂×C₆₁	—	8	1	no	—
362	C₂×C₆₁	—	8	2	no	—
363	C₂×C₆₁	#2404	1	1	yes	characteristic
364	#585	C₂²×C₆₁	1	1	yes	characteristic
365	#575	C₂²×C₆₁	1	2	yes	—
366	#568	C₂²×C₆₁	1	1	yes	characteristic
367	#533	—	2	1	no	—
368	#533	C₂²×C₆₁	1	1	yes	characteristic, complemented
369	#532	—	2	1	no	—
370	#532	C₂²×C₆₁	1	1	yes	characteristic
371	#529	—	2	1	no	—
372	#529	—	2	1	no	—
373	#529	—	2	2	no	—
374	#529	—	2	2	no	—
375	#525	—	2	2	no	—
376	#525	C₂²×C₆₁	1	1	yes	characteristic
377	#525	C₂²×C₆₁	1	2	yes	—
378	#525	C₂²×C₆₁	1	1	yes	characteristic, complemented
379	C₄×#41	—	2	2	no	—
380	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
381	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic, complemented
382	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic, complemented
383	#458	—	2	2	no	—
384	B(4,𝔽₂)	—	2	2	no	—
385	#456	—	2	1	no	—
386	#455	—	2	1	no	—
387	#454	—	2	1	no	—
388	#453	—	2	1	no	—
389	#421	—	2	2	no	—
390	#421	C₂²×C₆₁	1	2	yes	—
391	#421	C₂²×C₆₁	1	2	yes	complemented
392	C₂×#104	—	2	2	no	—
393	C₂×#104	C₂²×C₆₁	1	2	yes	—
394	C₂×#104	C₂²×C₆₁	1	2	yes	—
395	#413	C₂²×C₆₁	1	1	yes	characteristic
396	#410	C₂²×C₆₁	1	1	yes	characteristic
397	#410	C₂²×C₆₁	1	2	yes	—
398	#407	C₂²×C₆₁	1	2	yes	complemented
399	#344	C₂²×C₆₁	1	2	yes	complemented
400	#339	C₂²×C₆₁	1	2	yes	complemented
401	#339	C₂²×C₆₁	1	2	yes	complemented
402	H(3,ℤ/4)	C₂²×C₆₁	1	2	yes	complemented
403	C₆₁	#23994	1	1	yes	characteristic, direct product factor, Sylow
404	2₋¹⁺⁴	—	2	2	no	—
405	2₋¹⁺⁴	—	2	1	no	—
406	H(5,𝔽₂)	—	2	2	no	—
407	H(5,𝔽₂)	—	2	2	no	—
408	H(5,𝔽₂)	—	2	1	no	—
409	C₂×#41	—	2	2	no	—
410	C₂×#41	—	2	2	no	—
411	C₂×#41	—	4	2	no	—
412	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
413	C₂×#41	C₂³×C₆₁	1	2	yes	—
414	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
415	C₂³×C₄	—	2	1	no	—
416	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
417	#137	—	2	2	no	—
418	#137	—	2	2	no	—
419	AGL(1,ℤ/8)	—	2	2	no	—
420	AGL(1,ℤ/8)	—	2	2	no	—
421	#135	—	4	2	no	—
422	#131	—	4	2	no	—
423	C₂×M₁₆	—	2	2	no	—
424	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
425	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
426	C₄⋊Q₈	—	2	1	no	—
427	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic, complemented
428	#127	D₄×C₆₁	1	1	yes	characteristic, complemented
429	#126	—	2	2	no	—
430	#126	—	2	2	no	—
431	#125	—	2	2	no	—
432	#125	—	2	1	no	—
433	#124	—	2	2	no	—
434	#124	D₄×C₆₁	1	1	yes	characteristic, complemented
435	#124	D₄×C₆₁	1	1	yes	characteristic, complemented
436	#123	—	2	2	no	—
437	#123	—	2	2	no	—
438	#123	—	2	1	no	—
439	#123	—	2	2	no	—
440	#123	D₄×C₆₁	1	2	yes	—
441	C₂²⋊Q₈	—	2	2	no	—
442	C₂²⋊Q₈	—	2	1	no	—
443	C₂²⋊Q₈	—	2	1	no	—
444	C₂²⋊Q₈	—	2	2	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	#121	—	2	2	no	—
447	#121	—	2	2	no	—
448	#121	—	2	1	no	—
449	#121	—	2	1	no	—
450	#121	—	2	2	no	—
451	#120	—	2	1	no	—
452	#120	—	2	2	no	—
453	#120	D₄×C₆₁	1	2	yes	—
454	C₄×Q₈	—	2	2	no	—
455	C₄×Q₈	—	2	1	no	—
456	C₄×Q₈	—	2	1	no	—
457	C₄×Q₈	—	2	2	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	2	no	—
460	C₄×D₄	—	2	2	no	—
461	C₄×D₄	—	2	2	no	—
462	C₄×D₄	—	2	1	no	—
463	C₄×D₄	—	2	1	no	—
464	C₄×D₄	—	2	2	no	—
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	—	2	2	no	—
467	C₄×D₄	—	2	2	no	—
468	C₄×D₄	—	4	2	no	—
469	#117	—	2	2	no	—
470	#117	—	2	2	no	—
471	#117	—	2	2	no	—
472	#117	—	2	1	no	—
473	#117	D₄×C₆₁	1	2	yes	complemented
474	#117	C₂³×C₆₁	1	2	yes	—
475	#117	C₂³×C₆₁	1	1	yes	characteristic
476	#117	C₂³×C₆₁	1	1	yes	characteristic
477	C₂×C₄²	—	2	1	no	—
478	C₂×C₄²	—	2	1	no	—
479	C₂×C₄²	—	2	1	no	—
480	C₂×C₄²	—	2	1	no	—
481	C₂×C₄²	—	2	1	no	—
482	C₂×C₄²	—	2	1	no	—
483	C₂×C₄²	D₄×C₆₁	1	2	yes	complemented
484	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
485	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
486	#104	—	2	2	no	—
487	#104	—	2	4	no	—
488	#104	—	2	2	no	—
489	#104	—	2	2	no	—
490	#104	—	2	2	no	—
491	#104	—	4	2	no	—
492	#101	—	2	1	no	—
493	#100	—	2	1	no	—
494	#99	—	2	2	no	—
495	#99	—	2	4	no	—
496	C₂²⋊C₈	—	4	2	no	—
497	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
498	#41	—	2	2	no	—
499	#41	—	2	2	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	—	4	4	no	—
508	#41	—	4	2	no	—
509	#41	—	4	8	no	—
510	#41	—	4	2	no	—
511	#41	—	4	2	no	—
512	#41	—	4	2	no	—
513	C₂×Q₈	—	2	2	no	—
514	C₂×Q₈	—	2	2	no	—
515	C₂×Q₈	—	4	2	no	—
516	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
517	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
518	C₂×Q₈	#41×C₆₁	1	2	yes	complemented
519	C₂×D₄	—	2	2	no	—
520	C₂×D₄	—	2	2	no	—
521	C₂×D₄	—	2	2	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	2	1	no	—
525	C₂×D₄	—	4	2	no	—
526	C₂×D₄	—	4	4	no	—
527	C₂×D₄	—	4	2	no	—
528	C₂×D₄	—	4	2	no	—
529	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
530	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
531	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
532	C₂×D₄	#41×C₆₁	1	2	yes	complemented
533	C₂²×C₄	—	2	1	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	1	no	—
536	C₂²×C₄	—	2	1	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	2	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	2	no	—
546	C₂²×C₄	—	2	2	no	—
547	C₂²×C₄	—	4	4	no	—
548	C₂²×C₄	—	4	2	no	—
549	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
550	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
551	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
552	Q₁₆	—	4	2	no	—
553	SD₁₆	—	4	2	no	—
554	SD₁₆	—	4	2	no	—
555	D₈	—	4	2	no	—
556	M₁₆	—	2	2	no	—
557	M₁₆	—	2	2	no	—
558	M₁₆	—	2	2	no	—
559	M₁₆	—	2	2	no	—
560	M₁₆	—	4	2	no	—
561	C₂×C₈	—	2	2	no	—
562	C₂×C₈	—	2	2	no	—
563	C₂×C₈	—	4	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	4	2	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	2	no	—
578	C₂²⋊C₄	—	2	2	no	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	—	2	2	no	—
581	C₂²⋊C₄	—	2	2	no	—
582	C₂²⋊C₄	—	2	1	no	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	2	2	no	—
585	C₂²⋊C₄	—	2	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₂²⋊C₄	—	4	2	no	—
588	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
589	C₄²	—	2	1	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₄²	—	2	1	no	—
593	C₄²	—	2	2	no	—
594	C₄²	—	2	2	no	—
595	C₄²	—	2	1	no	—
596	C₄²	—	2	1	no	—
597	C₄²	—	2	1	no	—
598	C₄²	—	2	2	no	—
599	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
600	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
601	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
602	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
603	C₄²	C₂×D₄×C₆₁	1	2	yes	—
604	C₂³	—	2	2	no	—
605	C₂³	—	2	1	no	—
606	C₂³	—	2	1	no	—
607	C₂³	—	2	2	no	—
608	C₂³	—	4	2	no	—
609	C₂³	—	4	2	no	—
610	C₂³	#120×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	—	4	2	no	—
617	Q₈	—	4	4	no	—
618	D₄	—	2	4	no	—
619	D₄	—	2	2	no	—
620	D₄	—	2	2	no	—
621	D₄	—	2	2	no	—
622	D₄	—	2	2	no	—
623	D₄	—	4	2	no	—
624	D₄	—	4	4	no	—
625	D₄	—	4	4	no	—
626	D₄	—	4	4	no	—
627	D₄	—	4	4	no	—
628	D₄	—	4	2	no	—
629	D₄	—	4	2	no	—
630	D₄	—	4	2	no	—
631	D₄	—	8	4	no	—
632	C₂×C₄	—	2	4	no	—
633	C₂×C₄	—	2	1	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	—	4	2	no	—
647	C₂×C₄	—	4	2	no	—
648	C₂×C₄	—	4	2	no	—
649	C₂×C₄	—	4	4	no	—
650	C₂×C₄	—	4	4	no	—
651	C₂×C₄	—	4	4	no	—
652	C₂×C₄	—	4	2	no	—
653	C₂×C₄	—	4	2	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	2	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	—	4	2	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	2	no	—
662	C₂×C₄	#120×C₆₁	1	2	yes	—
663	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
664	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
665	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
666	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
667	C₈	—	4	2	no	—
668	C₈	—	4	2	no	—
669	C₂²	—	2	2	no	—
670	C₂²	—	2	4	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	4	1	no	—
673	C₂²	—	4	2	no	—
674	C₂²	—	4	1	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	4	1	no	—
678	C₂²	—	4	2	no	—
679	C₂²	—	4	1	no	—
680	C₂²	—	8	2	no	—
681	C₂²	—	16	2	no	—
682	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
683	C₂²	C₆₁×#546	1	2	yes	—
684	C₄	—	2	2	no	—
685	C₄	—	2	2	no	—
686	C₄	—	2	2	no	—
687	C₄	—	4	1	no	—
688	C₄	—	4	1	no	—
689	C₄	—	4	2	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	2	no	—
693	C₄	—	4	1	no	—
694	C₄	—	8	2	no	—
695	C₄	C₆₁×#525	1	1	yes	characteristic
696	C₄	C₆₁×#535	1	1	yes	characteristic
697	C₄	C₆₁×#546	1	2	yes	—
698	C₂	—	2	1	no	—
699	C₂	—	2	2	no	—
700	C₂	—	4	2	no	—
701	C₂	—	4	2	no	—
702	C₂	—	8	1	no	—
703	C₂	—	8	2	no	—
704	C₂	C₆₁×#2404	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₁	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂×C₆₁	#2404
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3444	C₂	semidirect
2. C₆₁×#3025	C₂	semidirect
3. C₆₁×#3018	C₂	semidirect
4. C₆₁×#2901	C₂	semidirect
5. C₆₁×#2031	C₂	semidirect
6. C₆₁×#2010	C₂	semidirect
7. C₆₁×#1906	C₂	semidirect
8. C₆₁×#1896	C₂	semidirect
9. C₆₁×#1891	C₂	semidirect
10. C₆₁×#1884	C₂	semidirect
11. C₆₁×#1827	C₂	semidirect
16. C₆₁×#533	C₂²	semidirect
26. C₆₁×#525	C₂²	semidirect
29. C₄×#41×C₆₁	C₂²	semidirect
30. C₂²×C₄²×C₆₁	C₂²	semidirect
39. C₆₁×#421	C₂²	semidirect
46. C₆₁×#407	C₂²	semidirect
47. C₆₁×#344	C₂²	semidirect
48. C₆₁×#339	C₂²	semidirect
49. C₆₁×#339	C₂²	semidirect
50. C₆₁×H(3,ℤ/4)	C₂²	semidirect
74. C₄⋊Q₈×C₆₁	D₄	semidirect
75. #127×C₆₁	D₄	semidirect
81. #124×C₆₁	D₄	semidirect
82. #124×C₆₁	D₄	semidirect
120. #117×C₆₁	D₄	semidirect
130. C₂×C₄²×C₆₁	D₄	semidirect
165. C₂×Q₈×C₆₁	#41	semidirect
179. C₂×D₄×C₆₁	#41	semidirect
316. #23994	C₆₁	direct (N × H)
346. #3444	C₂×C₆₁	semidirect
347. #3025	C₂×C₆₁	semidirect
348. #3018	C₂×C₆₁	semidirect
349. #2901	C₂×C₆₁	semidirect
350. #2031	C₂×C₆₁	semidirect
351. #2010	C₂×C₆₁	semidirect
352. #1906	C₂×C₆₁	semidirect
353. #1896	C₂×C₆₁	semidirect
354. #1891	C₂×C₆₁	semidirect
355. #1884	C₂×C₆₁	semidirect
356. #1827	C₂×C₆₁	semidirect
368. #533	C₂²×C₆₁	semidirect
378. #525	C₂²×C₆₁	semidirect
381. C₄×#41	C₂²×C₆₁	semidirect
382. C₂²×C₄²	C₂²×C₆₁	semidirect
391. #421	C₂²×C₆₁	semidirect
398. #407	C₂²×C₆₁	semidirect
399. #344	C₂²×C₆₁	semidirect
400. #339	C₂²×C₆₁	semidirect
401. #339	C₂²×C₆₁	semidirect
402. H(3,ℤ/4)	C₂²×C₆₁	semidirect
403. C₆₁	#23994	direct (N × H)
427. C₄⋊Q₈	D₄×C₆₁	semidirect
428. #127	D₄×C₆₁	semidirect
434. #124	D₄×C₆₁	semidirect
435. #124	D₄×C₆₁	semidirect
473. #117	D₄×C₆₁	semidirect
483. C₂×C₄²	D₄×C₆₁	semidirect
518. C₂×Q₈	#41×C₆₁	semidirect
532. C₂×D₄	#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×#3444	2
C₆₁×#3025	2
C₆₁×#3018	2
C₆₁×#2901	2
C₆₁×#2031	2
C₆₁×#2010	2
C₆₁×#1906	2
C₆₁×#1896	2
C₆₁×#1891	2
C₆₁×#1884	2
C₆₁×#1827	2
#23994	61

Subgroups — C61×#23994 #29527427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23994 #29527427