Subgroups — C₆₁×#23999 #29527432

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3020	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₁×#454	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2884	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2031	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2008	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1912	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1906	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1904	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1888	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1830	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#575	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#559	—	2	2	no	—
16	C₆₁×#556	—	2	2	no	—
17	C₆₁×#551	—	2	4	no	—
18	C₆₁×#542	—	2	2	no	—
19	C₆₁×#536	C₂²	1	1	yes	characteristic
20	C₆₁×#535	C₂²	1	1	yes	characteristic
21	C₆₁×#533	C₂²	1	1	yes	characteristic
22	C₆₁×#529	—	2	2	no	—
23	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#519	C₂²	1	1	yes	characteristic
25	C₆₁×#519	C₂²	1	1	yes	characteristic
26	C₆₁×#518	C₂²	1	1	yes	characteristic
27	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#456	—	2	2	no	—
29	C₆₁×#454	—	2	2	no	—
30	C₆₁×#454	—	2	2	no	—
31	C₆₁×#454	—	2	2	no	—
32	C₆₁×#421	C₂²	1	2	yes	—
33	C₆₁×#421	C₂²	1	2	yes	—
34	C₂×#104×C₆₁	C₂²	1	2	yes	—
35	C₆₁×#418	C₂²	1	2	yes	—
36	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#101×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#410	—	2	4	no	—
39	C₆₁×#410	C₂²	1	2	yes	—
40	C₆₁×#410	C₂²	1	2	yes	—
41	C₆₁×#344	C₂²	1	2	yes	—
42	C₆₁×#344	C₂²	1	2	yes	—
43	C₆₁×#339	C₂²	1	2	yes	—
44	C₆₁×H(3,ℤ/4)	C₂²	1	2	yes	—
45	2₋¹⁺⁴×C₆₁	—	2	2	no	—
46	2₋¹⁺⁴×C₆₁	—	2	2	no	—
47	2₋¹⁺⁴×C₆₁	—	4	2	no	—
48	C₂×#41×C₆₁	—	2	2	no	—
49	C₂×#41×C₆₁	—	2	2	no	—
50	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×#41×C₆₁	C₂³	1	2	yes	—
52	C₂²×Q₈×C₆₁	—	2	2	no	—
53	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
55	#137×C₆₁	—	2	2	no	—
56	#137×C₆₁	—	2	2	no	—
57	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
58	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
59	AGL(1,ℤ/8)×C₆₁	—	2	2	no	—
60	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
61	#135×C₆₁	—	4	2	no	—
62	C₂×SD₁₆×C₆₁	—	2	2	no	—
63	C₂×D₈×C₆₁	—	2	2	no	—
64	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
65	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₆₁	—	2	1	no	—
67	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
68	#127×C₆₁	—	2	1	no	—
69	#126×C₆₁	—	2	4	no	—
70	#126×C₆₁	—	2	4	no	—
71	#125×C₆₁	—	2	1	no	—
72	#124×C₆₁	—	2	4	no	—
73	#124×C₆₁	—	2	1	no	—
74	#124×C₆₁	—	2	2	no	—
75	#124×C₆₁	D₄	1	2	yes	—
76	#124×C₆₁	D₄	1	2	yes	—
77	#124×C₆₁	D₄	1	2	yes	—
78	#123×C₆₁	—	2	4	no	—
79	#123×C₆₁	—	2	4	no	—
80	#123×C₆₁	—	2	2	no	—
81	C₂²⋊Q₈×C₆₁	—	2	4	no	—
82	C₂²⋊Q₈×C₆₁	—	2	2	no	—
83	C₂²⋊Q₈×C₆₁	—	2	2	no	—
84	#121×C₆₁	—	2	2	no	—
85	#120×C₆₁	—	2	2	no	—
86	C₄×Q₈×C₆₁	—	2	2	no	—
87	C₄×Q₈×C₆₁	—	2	1	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	2	no	—
90	C₄×Q₈×C₆₁	—	4	2	no	—
91	C₄×D₄×C₆₁	—	2	4	no	—
92	C₄×D₄×C₆₁	—	2	2	no	—
93	C₄×D₄×C₆₁	—	2	1	no	—
94	C₄×D₄×C₆₁	—	2	2	no	—
95	C₄×D₄×C₆₁	—	2	2	no	—
96	#117×C₆₁	—	2	4	no	—
97	#117×C₆₁	—	2	1	no	—
98	#117×C₆₁	—	2	2	no	—
99	#117×C₆₁	D₄	1	4	yes	—
100	#117×C₆₁	C₂³	1	2	yes	—
101	#117×C₆₁	C₂³	1	1	yes	characteristic
102	#117×C₆₁	C₂³	1	1	yes	characteristic
103	#117×C₆₁	C₂³	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
106	C₂×C₄²×C₆₁	—	2	1	no	—
107	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
108	#104×C₆₁	—	2	2	no	—
109	#104×C₆₁	—	2	2	no	—
110	#104×C₆₁	—	2	4	no	—
111	#104×C₆₁	—	4	2	no	—
112	#102×C₆₁	—	4	2	no	—
113	#101×C₆₁	—	2	2	no	—
114	#101×C₆₁	—	2	2	no	—
115	#99×C₆₁	—	2	4	no	—
116	#99×C₆₁	—	2	4	no	—
117	C₂⁴×C₆₁	—	2	1	no	—
118	#41×C₆₁	—	2	2	no	—
119	#41×C₆₁	—	2	2	no	—
120	#41×C₆₁	—	2	2	no	—
121	#41×C₆₁	—	2	2	no	—
122	#41×C₆₁	—	2	2	no	—
123	#41×C₆₁	—	2	2	no	—
124	#41×C₆₁	—	4	4	no	—
125	#41×C₆₁	—	4	4	no	—
126	#41×C₆₁	—	4	4	no	—
127	#41×C₆₁	—	4	4	no	—
128	C₂×Q₈×C₆₁	—	2	2	no	—
129	C₂×Q₈×C₆₁	—	2	2	no	—
130	C₂×Q₈×C₆₁	—	2	2	no	—
131	C₂×Q₈×C₆₁	—	2	2	no	—
132	C₂×Q₈×C₆₁	—	4	4	no	—
133	C₂×Q₈×C₆₁	—	4	4	no	—
134	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
136	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₆₁	#41	1	2	yes	—
138	C₂×D₄×C₆₁	—	2	2	no	—
139	C₂×D₄×C₆₁	—	2	2	no	—
140	C₂×D₄×C₆₁	—	2	1	no	—
141	C₂×D₄×C₆₁	—	2	4	no	—
142	C₂×D₄×C₆₁	—	2	1	no	—
143	C₂×D₄×C₆₁	—	2	2	no	—
144	C₂×D₄×C₆₁	—	2	2	no	—
145	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₆₁	#41	1	2	yes	—
148	C₂²×C₄×C₆₁	—	2	4	no	—
149	C₂²×C₄×C₆₁	—	2	2	no	—
150	C₂²×C₄×C₆₁	—	2	4	no	—
151	C₂²×C₄×C₆₁	—	2	1	no	—
152	C₂²×C₄×C₆₁	—	2	1	no	—
153	C₂²×C₄×C₆₁	—	2	2	no	—
154	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
156	Q₁₆×C₆₁	—	4	2	no	—
157	SD₁₆×C₆₁	—	4	4	no	—
158	SD₁₆×C₆₁	—	4	2	no	—
159	SD₁₆×C₆₁	—	4	2	no	—
160	D₈×C₆₁	—	4	4	no	—
161	D₈×C₆₁	—	4	2	no	—
162	M₁₆×C₆₁	—	2	2	no	—
163	M₁₆×C₆₁	—	2	2	no	—
164	M₁₆×C₆₁	—	2	2	no	—
165	M₁₆×C₆₁	—	2	2	no	—
166	C₂×C₈×C₆₁	—	2	2	no	—
167	C₂×C₈×C₆₁	—	2	2	no	—
168	C₄⋊C₄×C₆₁	—	2	2	no	—
169	C₄⋊C₄×C₆₁	—	2	2	no	—
170	C₄⋊C₄×C₆₁	—	2	2	no	—
171	C₄⋊C₄×C₆₁	—	2	2	no	—
172	C₄⋊C₄×C₆₁	—	2	4	no	—
173	C₄⋊C₄×C₆₁	—	2	4	no	—
174	C₄⋊C₄×C₆₁	—	2	1	no	—
175	C₄⋊C₄×C₆₁	—	2	1	no	—
176	C₄⋊C₄×C₆₁	—	2	1	no	—
177	C₄⋊C₄×C₆₁	—	2	2	no	—
178	C₄⋊C₄×C₆₁	—	4	2	no	—
179	C₂²⋊C₄×C₆₁	—	2	4	no	—
180	C₂²⋊C₄×C₆₁	—	2	2	no	—
181	C₂²⋊C₄×C₆₁	—	2	4	no	—
182	C₂²⋊C₄×C₆₁	—	2	1	no	—
183	C₂²⋊C₄×C₆₁	—	2	4	no	—
184	C₂²⋊C₄×C₆₁	—	2	4	no	—
185	C₂²⋊C₄×C₆₁	—	2	4	no	—
186	C₂²⋊C₄×C₆₁	—	2	2	no	—
187	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
188	C₂²⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
189	C₄²×C₆₁	—	2	2	no	—
190	C₄²×C₆₁	—	2	4	no	—
191	C₄²×C₆₁	—	2	1	no	—
192	C₄²×C₆₁	—	2	1	no	—
193	C₄²×C₆₁	—	2	1	no	—
194	C₄²×C₆₁	—	2	2	no	—
195	C₄²×C₆₁	—	4	2	no	—
196	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₆₁	C₂×D₄	1	2	yes	—
199	C₂³×C₆₁	—	2	2	no	—
200	C₂³×C₆₁	—	2	1	no	—
201	C₂³×C₆₁	—	2	2	no	—
202	C₂³×C₆₁	—	4	1	no	—
203	C₂³×C₆₁	—	4	1	no	—
204	C₂³×C₆₁	—	8	2	no	—
205	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₆₁	—	2	4	no	—
207	Q₈×C₆₁	—	2	4	no	—
208	Q₈×C₆₁	—	2	2	no	—
209	Q₈×C₆₁	—	2	2	no	—
210	Q₈×C₆₁	—	2	2	no	—
211	Q₈×C₆₁	—	4	4	no	—
212	Q₈×C₆₁	—	8	2	no	—
213	D₄×C₆₁	—	2	2	no	—
214	D₄×C₆₁	—	2	4	no	—
215	D₄×C₆₁	—	2	2	no	—
216	D₄×C₆₁	—	2	2	no	—
217	D₄×C₆₁	—	4	4	no	—
218	D₄×C₆₁	—	4	2	no	—
219	D₄×C₆₁	—	4	1	no	—
220	D₄×C₆₁	—	4	1	no	—
221	D₄×C₆₁	—	4	2	no	—
222	D₄×C₆₁	—	4	2	no	—
223	D₄×C₆₁	—	8	4	no	—
224	C₂×C₄×C₆₁	—	2	2	no	—
225	C₂×C₄×C₆₁	—	2	2	no	—
226	C₂×C₄×C₆₁	—	2	2	no	—
227	C₂×C₄×C₆₁	—	2	1	no	—
228	C₂×C₄×C₆₁	—	2	2	no	—
229	C₂×C₄×C₆₁	—	2	1	no	—
230	C₂×C₄×C₆₁	—	2	1	no	—
231	C₂×C₄×C₆₁	—	2	2	no	—
232	C₂×C₄×C₆₁	—	2	2	no	—
233	C₂×C₄×C₆₁	—	4	8	no	—
234	C₂×C₄×C₆₁	—	4	2	no	—
235	C₂×C₄×C₆₁	—	4	2	no	—
236	C₂×C₄×C₆₁	—	4	2	no	—
237	C₂×C₄×C₆₁	—	4	2	no	—
238	C₂×C₄×C₆₁	—	8	2	no	—
239	C₂×C₄×C₆₁	—	8	4	no	—
240	C₂×C₄×C₆₁	—	8	1	no	—
241	C₂×C₄×C₆₁	#120	1	4	yes	—
242	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
245	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
246	C₈×C₆₁	—	4	2	no	—
247	C₈×C₆₁	—	4	2	no	—
248	#23999	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂²×C₆₁	—	4	1	no	—
250	C₂²×C₆₁	—	4	2	no	—
251	C₂²×C₆₁	—	4	4	no	—
252	C₂²×C₆₁	—	4	2	no	—
253	C₂²×C₆₁	—	4	1	no	—
254	C₂²×C₆₁	—	4	2	no	—
255	C₂²×C₆₁	—	8	2	no	—
256	C₂²×C₆₁	—	16	1	no	—
257	C₂²×C₆₁	—	16	1	no	—
258	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
259	C₂²×C₆₁	#525	1	1	yes	characteristic
260	C₂²×C₆₁	#535	1	1	yes	characteristic
261	C₄×C₆₁	—	2	4	no	—
262	C₄×C₆₁	—	4	2	no	—
263	C₄×C₆₁	—	4	1	no	—
264	C₄×C₆₁	—	4	2	no	—
265	C₄×C₆₁	—	4	1	no	—
266	C₄×C₆₁	—	4	2	no	—
267	C₄×C₆₁	—	4	2	no	—
268	C₄×C₆₁	—	4	1	no	—
269	C₄×C₆₁	—	8	2	no	—
270	C₄×C₆₁	#546	1	2	yes	—
271	C₄×C₆₁	#546	1	2	yes	—
272	#3510	C₂×C₆₁	1	1	yes	characteristic, complemented
273	#3020	C₂×C₆₁	1	1	yes	characteristic, complemented
274	C₂×#454	C₂×C₆₁	1	1	yes	characteristic, complemented
275	#2884	C₂×C₆₁	1	1	yes	characteristic, complemented
276	#2031	C₂×C₆₁	1	2	yes	complemented
277	#2008	C₂×C₆₁	1	2	yes	complemented
278	#1912	C₂×C₆₁	1	1	yes	characteristic, complemented
279	#1906	C₂×C₆₁	1	2	yes	complemented
280	#1904	C₂×C₆₁	1	2	yes	complemented
281	#1888	C₂×C₆₁	1	1	yes	characteristic, complemented
282	#1830	C₂×C₆₁	1	1	yes	characteristic, complemented
283	C₂×C₆₁	—	2	1	no	—
284	C₂×C₆₁	—	2	1	no	—
285	C₂×C₆₁	—	2	1	no	—
286	C₂×C₆₁	—	8	1	no	—
287	C₂×C₆₁	—	8	2	no	—
288	C₂×C₆₁	—	8	2	no	—
289	C₂×C₆₁	#2404	1	1	yes	characteristic
290	C₂×2₋¹⁺⁴	C₂²×C₆₁	1	1	yes	characteristic
291	#575	C₂²×C₆₁	1	2	yes	—
292	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
293	#559	—	2	2	no	—
294	#556	—	2	2	no	—
295	#551	—	2	4	no	—
296	#542	—	2	2	no	—
297	#536	C₂²×C₆₁	1	1	yes	characteristic
298	#535	C₂²×C₆₁	1	1	yes	characteristic
299	#533	C₂²×C₆₁	1	1	yes	characteristic
300	#529	—	2	2	no	—
301	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
302	#519	C₂²×C₆₁	1	1	yes	characteristic
303	#519	C₂²×C₆₁	1	1	yes	characteristic
304	#518	C₂²×C₆₁	1	1	yes	characteristic
305	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
306	#456	—	2	2	no	—
307	#454	—	2	2	no	—
308	#454	—	2	2	no	—
309	#454	—	2	2	no	—
310	#421	C₂²×C₆₁	1	2	yes	—
311	#421	C₂²×C₆₁	1	2	yes	—
312	C₂×#104	C₂²×C₆₁	1	2	yes	—
313	#418	C₂²×C₆₁	1	2	yes	—
314	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
315	C₂×#101	C₂²×C₆₁	1	1	yes	characteristic
316	#410	—	2	4	no	—
317	#410	C₂²×C₆₁	1	2	yes	—
318	#410	C₂²×C₆₁	1	2	yes	—
319	#344	C₂²×C₆₁	1	2	yes	—
320	#344	C₂²×C₆₁	1	2	yes	—
321	#339	C₂²×C₆₁	1	2	yes	—
322	H(3,ℤ/4)	C₂²×C₆₁	1	2	yes	—
323	C₆₁	#23999	1	1	yes	characteristic, direct product factor, Sylow
324	2₋¹⁺⁴	—	2	2	no	—
325	2₋¹⁺⁴	—	2	2	no	—
326	2₋¹⁺⁴	—	4	2	no	—
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	2	2	no	—
329	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
330	C₂×#41	C₂³×C₆₁	1	2	yes	—
331	C₂²×Q₈	—	2	2	no	—
332	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
333	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
334	#137	—	2	2	no	—
335	#137	—	2	2	no	—
336	AGL(1,ℤ/8)	—	2	4	no	—
337	AGL(1,ℤ/8)	—	2	2	no	—
338	AGL(1,ℤ/8)	—	2	2	no	—
339	AGL(1,ℤ/8)	—	4	2	no	—
340	#135	—	4	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×D₈	—	2	2	no	—
343	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
344	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
345	C₄⋊Q₈	—	2	1	no	—
346	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
347	#127	—	2	1	no	—
348	#126	—	2	4	no	—
349	#126	—	2	4	no	—
350	#125	—	2	1	no	—
351	#124	—	2	4	no	—
352	#124	—	2	2	no	—
353	#124	—	2	1	no	—
354	#124	D₄×C₆₁	1	2	yes	—
355	#124	D₄×C₆₁	1	2	yes	—
356	#124	D₄×C₆₁	1	2	yes	—
357	#123	—	2	4	no	—
358	#123	—	2	4	no	—
359	#123	—	2	2	no	—
360	C₂²⋊Q₈	—	2	4	no	—
361	C₂²⋊Q₈	—	2	2	no	—
362	C₂²⋊Q₈	—	2	2	no	—
363	#121	—	2	2	no	—
364	#120	—	2	2	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	—	2	2	no	—
369	C₄×Q₈	—	4	2	no	—
370	C₄×D₄	—	2	4	no	—
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	#117	—	2	4	no	—
376	#117	—	2	1	no	—
377	#117	—	2	2	no	—
378	#117	D₄×C₆₁	1	4	yes	—
379	#117	C₂³×C₆₁	1	2	yes	—
380	#117	C₂³×C₆₁	1	1	yes	characteristic
381	#117	C₂³×C₆₁	1	1	yes	characteristic
382	#117	C₂³×C₆₁	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
387	#104	—	2	2	no	—
388	#104	—	2	2	no	—
389	#104	—	2	4	no	—
390	#104	—	4	2	no	—
391	#102	—	4	2	no	—
392	#101	—	2	2	no	—
393	#101	—	2	2	no	—
394	#99	—	2	4	no	—
395	#99	—	2	4	no	—
396	C₂⁴	—	2	1	no	—
397	#41	—	2	2	no	—
398	#41	—	2	2	no	—
399	#41	—	2	2	no	—
400	#41	—	2	2	no	—
401	#41	—	2	2	no	—
402	#41	—	2	2	no	—
403	#41	—	4	4	no	—
404	#41	—	4	4	no	—
405	#41	—	4	4	no	—
406	#41	—	4	4	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	—	4	4	no	—
412	C₂×Q₈	—	4	4	no	—
413	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
414	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
415	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
416	C₂×Q₈	#41×C₆₁	1	2	yes	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
425	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
426	C₂×D₄	#41×C₆₁	1	2	yes	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
435	Q₁₆	—	4	2	no	—
436	SD₁₆	—	4	4	no	—
437	SD₁₆	—	4	2	no	—
438	SD₁₆	—	4	2	no	—
439	D₈	—	4	4	no	—
440	D₈	—	4	2	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	2	2	no	—
443	M₁₆	—	2	2	no	—
444	M₁₆	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	4	2	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	2	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	1	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
467	C₂²⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
468	C₄²	—	2	2	no	—
469	C₄²	—	2	4	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	4	2	no	—
475	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
476	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₆₁	1	2	yes	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	4	1	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	8	2	no	—
484	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
485	Q₈	—	2	4	no	—
486	Q₈	—	2	4	no	—
487	Q₈	—	2	2	no	—
488	Q₈	—	2	2	no	—
489	Q₈	—	2	2	no	—
490	Q₈	—	4	4	no	—
491	Q₈	—	8	2	no	—
492	D₄	—	2	2	no	—
493	D₄	—	2	4	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	2	no	—
496	D₄	—	4	4	no	—
497	D₄	—	4	1	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	1	no	—
501	D₄	—	4	2	no	—
502	D₄	—	8	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	4	8	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	8	2	no	—
518	C₂×C₄	—	8	4	no	—
519	C₂×C₄	—	8	1	no	—
520	C₂×C₄	#120×C₆₁	1	4	yes	—
521	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
523	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
524	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
525	C₈	—	4	2	no	—
526	C₈	—	4	2	no	—
527	C₂²	—	4	1	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	8	2	no	—
534	C₂²	—	16	1	no	—
535	C₂²	—	16	1	no	—
536	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
537	C₂²	C₆₁×#525	1	1	yes	characteristic
538	C₂²	C₆₁×#535	1	1	yes	characteristic
539	C₄	—	2	4	no	—
540	C₄	—	4	2	no	—
541	C₄	—	4	1	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	1	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	1	no	—
547	C₄	—	8	2	no	—
548	C₄	C₆₁×#546	1	2	yes	—
549	C₄	C₆₁×#546	1	2	yes	—
550	C₂	—	2	1	no	—
551	C₂	—	2	1	no	—
552	C₂	—	2	1	no	—
553	C₂	—	8	2	no	—
554	C₂	—	8	1	no	—
555	C₂	—	8	2	no	—
556	C₂	C₆₁×#2404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₁	#2404
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×#3510	C₂	semidirect
2. C₆₁×#3020	C₂	semidirect
3. C₂×C₆₁×#454	C₂	semidirect
4. C₆₁×#2884	C₂	semidirect
5. C₆₁×#2031	C₂	semidirect
6. C₆₁×#2008	C₂	semidirect
7. C₆₁×#1912	C₂	semidirect
8. C₆₁×#1906	C₂	semidirect
9. C₆₁×#1904	C₂	semidirect
10. C₆₁×#1888	C₂	semidirect
11. C₆₁×#1830	C₂	semidirect
248. #23999	C₆₁	direct (N × H)
272. #3510	C₂×C₆₁	semidirect
273. #3020	C₂×C₆₁	semidirect
274. C₂×#454	C₂×C₆₁	semidirect
275. #2884	C₂×C₆₁	semidirect
276. #2031	C₂×C₆₁	semidirect
277. #2008	C₂×C₆₁	semidirect
278. #1912	C₂×C₆₁	semidirect
279. #1906	C₂×C₆₁	semidirect
280. #1904	C₂×C₆₁	semidirect
281. #1888	C₂×C₆₁	semidirect
282. #1830	C₂×C₆₁	semidirect
323. C₆₁	#23999	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3510	2
C₆₁×#3020	2
C₂×C₆₁×#454	2
C₆₁×#2884	2
C₆₁×#2031	2
C₆₁×#2008	2
C₆₁×#1912	2
C₆₁×#1906	2
C₆₁×#1904	2
C₆₁×#1888	2
C₆₁×#1830	2
#23999	61

Subgroups — C61×#23999 #29527432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23999 #29527432