Subgroups — C₆₁×#24803 #29528236

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3055	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₁×#466	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#108×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2063	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2051	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1900	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#525	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₁	C₂²	1	2	yes	—
19	C₆₁×#472	—	2	4	no	—
20	C₆₁×#469	—	2	2	no	—
21	C₆₁×#468	—	2	2	no	—
22	C₆₁×#466	—	2	1	no	—
23	C₆₁×#466	—	2	1	no	—
24	C₆₁×#466	—	2	1	no	—
25	C₆₁×#466	—	2	1	no	—
26	C₆₁×#466	—	2	2	no	—
27	C₆₁×#465	—	2	2	no	—
28	C₆₁×#435	—	2	1	no	—
29	C₆₁×#435	—	2	1	no	—
30	C₈×D₄×C₆₁	—	2	1	no	—
31	C₈×D₄×C₆₁	—	2	1	no	—
32	C₂×#108×C₆₁	—	2	4	no	—
33	C₂×#108×C₆₁	C₂²	1	2	yes	—
34	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#428	C₂²	1	2	yes	—
37	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
38	C₆₁×#424	C₂²	1	1	yes	characteristic
39	C₆₁×#418	C₂²	1	2	yes	—
40	C₆₁×#417	C₂²	1	2	yes	—
41	C₂×#102×C₆₁	C₂²	1	2	yes	—
42	C₂×#100×C₆₁	C₂²	1	2	yes	—
43	C₂×#100×C₆₁	C₂²	1	2	yes	—
44	C₆₁×#408	C₂²	1	1	yes	characteristic
45	C₆₁×#407	C₂²	1	1	yes	characteristic
46	C₆₁×#405	C₂²	1	1	yes	characteristic
47	C₆₁×#343	C₂²	1	2	yes	—
48	C₆₁×#343	C₂²	1	2	yes	—
49	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₆₁	—	2	2	no	—
51	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
53	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
54	AGL(1,ℤ/8)×C₆₁	—	4	2	no	—
55	C₂×SD₁₆×C₆₁	—	2	2	no	—
56	C₂×D₈×C₆₁	—	2	2	no	—
57	C₂×D₈×C₆₁	—	2	1	no	—
58	C₂×D₈×C₆₁	—	2	1	no	—
59	C₂×D₈×C₆₁	—	2	2	no	—
60	C₂×D₈×C₆₁	—	2	2	no	—
61	#131×C₆₁	—	2	2	no	—
62	#131×C₆₁	—	2	2	no	—
63	C₂×M₁₆×C₆₁	—	2	2	no	—
64	C₂×M₁₆×C₆₁	—	2	1	no	—
65	C₂×M₁₆×C₆₁	—	2	4	no	—
66	C₂×M₁₆×C₆₁	—	4	2	no	—
67	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
70	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₆₁	—	2	2	no	—
72	C₂²×C₈×C₆₁	—	2	2	no	—
73	C₂²×C₈×C₆₁	—	2	2	no	—
74	C₂²×C₈×C₆₁	—	2	1	no	—
75	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	#123×C₆₁	—	2	1	no	—
82	C₂²⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
83	C₂²⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
84	#121×C₆₁	—	2	2	no	—
85	#121×C₆₁	—	2	2	no	—
86	#121×C₆₁	—	2	2	no	—
87	#121×C₆₁	—	2	2	no	—
88	#121×C₆₁	D₄	1	1	yes	characteristic
89	#121×C₆₁	D₄	1	1	yes	characteristic
90	#120×C₆₁	—	2	1	no	—
91	C₄×D₄×C₆₁	—	2	1	no	—
92	C₄×D₄×C₆₁	—	2	1	no	—
93	#117×C₆₁	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
96	#108×C₆₁	—	2	4	no	—
97	#108×C₆₁	—	2	2	no	—
98	#108×C₆₁	D₄	1	4	yes	complemented
99	#107×C₆₁	—	2	1	no	—
100	#107×C₆₁	—	2	1	no	—
101	#107×C₆₁	—	2	2	no	—
102	#106×C₆₁	—	2	2	no	—
103	C₄⋊C₈×C₆₁	—	2	1	no	—
104	C₄⋊C₈×C₆₁	—	2	1	no	—
105	#103×C₆₁	—	2	2	no	—
106	#102×C₆₁	—	2	2	no	—
107	#102×C₆₁	—	2	2	no	—
108	#102×C₆₁	—	2	2	no	—
109	#102×C₆₁	—	2	2	no	—
110	#102×C₆₁	—	2	2	no	—
111	#100×C₆₁	—	2	4	no	—
112	#100×C₆₁	—	4	2	no	—
113	C₂²⋊C₈×C₆₁	—	2	1	no	—
114	C₂²⋊C₈×C₆₁	—	2	1	no	—
115	C₂²⋊C₈×C₆₁	—	2	1	no	—
116	C₂²⋊C₈×C₆₁	—	2	1	no	—
117	#97×C₆₁	—	2	1	no	—
118	C₄×C₈×C₆₁	—	2	1	no	—
119	C₂⁴×C₆₁	—	2	2	no	—
120	C₂⁴×C₆₁	#41	1	1	yes	characteristic
121	#41×C₆₁	—	2	1	no	—
122	#41×C₆₁	—	2	1	no	—
123	#41×C₆₁	—	2	1	no	—
124	#41×C₆₁	—	2	1	no	—
125	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
126	C₂×D₄×C₆₁	—	2	1	no	—
127	C₂×D₄×C₆₁	—	2	1	no	—
128	C₂×D₄×C₆₁	—	2	1	no	—
129	C₂×D₄×C₆₁	—	2	2	no	—
130	C₂×D₄×C₆₁	—	2	2	no	—
131	C₂×D₄×C₆₁	—	2	2	no	—
132	C₂×D₄×C₆₁	—	2	2	no	—
133	C₂×D₄×C₆₁	—	2	2	no	—
134	C₂×D₄×C₆₁	—	2	2	no	—
135	C₂×D₄×C₆₁	—	4	2	no	—
136	C₂×D₄×C₆₁	—	4	2	no	—
137	C₂×D₄×C₆₁	—	8	4	no	—
138	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₁	—	2	1	no	—
140	C₂²×C₄×C₆₁	—	2	1	no	—
141	C₂²×C₄×C₆₁	—	2	1	no	—
142	C₂²×C₄×C₆₁	—	2	1	no	—
143	C₂²×C₄×C₆₁	—	2	1	no	—
144	C₂²×C₄×C₆₁	—	2	2	no	—
145	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
150	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
151	SD₁₆×C₆₁	—	4	4	no	—
152	D₈×C₆₁	—	4	4	no	—
153	D₈×C₆₁	—	4	2	no	—
154	D₈×C₆₁	—	4	2	no	—
155	M₁₆×C₆₁	—	2	2	no	—
156	M₁₆×C₆₁	—	2	1	no	—
157	M₁₆×C₆₁	—	2	2	no	—
158	M₁₆×C₆₁	—	2	1	no	—
159	M₁₆×C₆₁	—	2	1	no	—
160	M₁₆×C₆₁	—	2	2	no	—
161	M₁₆×C₆₁	—	2	1	no	—
162	M₁₆×C₆₁	—	4	4	no	—
163	M₁₆×C₆₁	—	4	2	no	—
164	M₁₆×C₆₁	C₂×D₄	1	4	yes	—
165	C₂×C₈×C₆₁	—	2	2	no	—
166	C₂×C₈×C₆₁	—	2	1	no	—
167	C₂×C₈×C₆₁	—	2	1	no	—
168	C₂×C₈×C₆₁	—	2	2	no	—
169	C₂×C₈×C₆₁	—	2	2	no	—
170	C₂×C₈×C₆₁	—	2	2	no	—
171	C₂×C₈×C₆₁	—	2	2	no	—
172	C₂×C₈×C₆₁	—	2	1	no	—
173	C₂×C₈×C₆₁	—	2	1	no	—
174	C₂×C₈×C₆₁	—	2	2	no	—
175	C₂×C₈×C₆₁	—	4	4	no	—
176	C₂×C₈×C₆₁	—	4	2	no	—
177	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₆₁	#41	1	2	yes	—
183	C₄⋊C₄×C₆₁	—	2	1	no	—
184	C₄⋊C₄×C₆₁	—	2	1	no	—
185	C₄⋊C₄×C₆₁	—	2	2	no	—
186	C₄⋊C₄×C₆₁	—	2	2	no	—
187	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
189	C₂²⋊C₄×C₆₁	—	2	1	no	—
190	C₂²⋊C₄×C₆₁	—	2	1	no	—
191	C₂²⋊C₄×C₆₁	—	2	1	no	—
192	C₂²⋊C₄×C₆₁	—	2	1	no	—
193	C₂²⋊C₄×C₆₁	—	2	1	no	—
194	C₂²⋊C₄×C₆₁	—	4	2	no	—
195	C₂²⋊C₄×C₆₁	—	4	2	no	—
196	C₄²×C₆₁	—	2	1	no	—
197	C₂³×C₆₁	—	2	1	no	—
198	C₂³×C₆₁	—	2	1	no	—
199	C₂³×C₆₁	—	2	2	no	—
200	C₂³×C₆₁	—	2	1	no	—
201	C₂³×C₆₁	—	4	1	no	—
202	C₂³×C₆₁	—	4	2	no	—
203	C₂³×C₆₁	—	4	2	no	—
204	C₂³×C₆₁	—	4	2	no	—
205	C₂³×C₆₁	—	8	2	no	—
206	C₂³×C₆₁	—	8	2	no	—
207	C₂³×C₆₁	#121	1	1	yes	characteristic
208	C₂³×C₆₁	#121	1	1	yes	characteristic
209	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
210	Q₈×C₆₁	—	2	1	no	—
211	Q₈×C₆₁	—	2	1	no	—
212	D₄×C₆₁	—	2	1	no	—
213	D₄×C₆₁	—	2	1	no	—
214	D₄×C₆₁	—	4	2	no	—
215	D₄×C₆₁	—	4	1	no	—
216	D₄×C₆₁	—	4	1	no	—
217	D₄×C₆₁	—	4	2	no	—
218	D₄×C₆₁	—	4	2	no	—
219	D₄×C₆₁	—	4	2	no	—
220	D₄×C₆₁	—	4	2	no	—
221	D₄×C₆₁	—	4	2	no	—
222	D₄×C₆₁	—	8	4	no	—
223	D₄×C₆₁	—	8	4	no	—
224	C₂×C₄×C₆₁	—	2	1	no	—
225	C₂×C₄×C₆₁	—	2	1	no	—
226	C₂×C₄×C₆₁	—	2	1	no	—
227	C₂×C₄×C₆₁	—	2	1	no	—
228	C₂×C₄×C₆₁	—	2	1	no	—
229	C₂×C₄×C₆₁	—	2	1	no	—
230	C₂×C₄×C₆₁	—	2	1	no	—
231	C₂×C₄×C₆₁	—	2	1	no	—
232	C₂×C₄×C₆₁	—	2	1	no	—
233	C₂×C₄×C₆₁	—	2	1	no	—
234	C₂×C₄×C₆₁	—	2	1	no	—
235	C₂×C₄×C₆₁	—	2	1	no	—
236	C₂×C₄×C₆₁	—	2	1	no	—
237	C₂×C₄×C₆₁	—	4	1	no	—
238	C₂×C₄×C₆₁	—	4	1	no	—
239	C₂×C₄×C₆₁	—	4	2	no	—
240	C₂×C₄×C₆₁	—	8	2	no	—
241	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
242	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
243	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
247	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
249	C₈×C₆₁	—	2	1	no	—
250	C₈×C₆₁	—	2	1	no	—
251	C₈×C₆₁	—	2	4	no	—
252	C₈×C₆₁	—	4	1	no	—
253	C₈×C₆₁	—	4	1	no	—
254	C₈×C₆₁	—	4	2	no	—
255	C₈×C₆₁	#121	1	2	yes	—
256	C₈×C₆₁	#121	1	2	yes	—
257	#24803	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₆₁	—	2	1	no	—
259	C₂²×C₆₁	—	2	1	no	—
260	C₂²×C₆₁	—	2	1	no	—
261	C₂²×C₆₁	—	2	1	no	—
262	C₂²×C₆₁	—	2	1	no	—
263	C₂²×C₆₁	—	2	1	no	—
264	C₂²×C₆₁	—	2	1	no	—
265	C₂²×C₆₁	—	4	2	no	—
266	C₂²×C₆₁	—	4	2	no	—
267	C₂²×C₆₁	—	4	1	no	—
268	C₂²×C₆₁	—	4	1	no	—
269	C₂²×C₆₁	—	4	1	no	—
270	C₂²×C₆₁	—	8	2	no	—
271	C₂²×C₆₁	—	8	2	no	—
272	C₂²×C₆₁	—	8	2	no	—
273	C₂²×C₆₁	—	16	2	no	—
274	C₂²×C₆₁	—	16	2	no	—
275	C₂²×C₆₁	#472	1	2	yes	complemented
276	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
277	C₂²×C₆₁	#548	1	1	yes	characteristic
278	C₂²×C₆₁	#553	1	1	yes	characteristic
279	C₄×C₆₁	—	2	1	no	—
280	C₄×C₆₁	—	2	1	no	—
281	C₄×C₆₁	—	4	1	no	—
282	C₄×C₆₁	—	4	1	no	—
283	C₄×C₆₁	—	4	1	no	—
284	C₄×C₆₁	—	8	2	no	—
285	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
286	C₄×C₆₁	#537	1	1	yes	characteristic
287	C₄×C₆₁	D₄²	1	1	yes	characteristic
288	C₄×C₆₁	#547	1	1	yes	characteristic
289	C₂×#472	C₂×C₆₁	1	1	yes	characteristic, complemented
290	#3055	C₂×C₆₁	1	2	yes	complemented
291	C₂×#466	C₂×C₆₁	1	1	yes	characteristic, complemented
292	#2929	C₂×C₆₁	1	1	yes	characteristic, complemented
293	C₂²×#108	C₂×C₆₁	1	1	yes	characteristic, complemented
294	#2063	C₂×C₆₁	1	2	yes	complemented
295	#2051	C₂×C₆₁	1	2	yes	complemented
296	#1978	C₂×C₆₁	1	1	yes	characteristic, complemented
297	#1952	C₂×C₆₁	1	1	yes	characteristic, complemented
298	#1900	C₂×C₆₁	1	2	yes	complemented
299	#1816	C₂×C₆₁	1	1	yes	characteristic, complemented
300	C₂×C₆₁	—	2	1	no	—
301	C₂×C₆₁	—	2	1	no	—
302	C₂×C₆₁	—	2	2	no	—
303	C₂×C₆₁	—	4	1	no	—
304	C₂×C₆₁	—	8	1	no	—
305	C₂×C₆₁	—	16	2	no	—
306	C₂×C₆₁	#2416	1	1	yes	characteristic
307	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
308	C₂×C₆₁	#3067	1	1	yes	characteristic
309	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
310	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
311	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
312	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
313	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
314	#525	C₂²×C₆₁	1	1	yes	characteristic
315	C₂×#121	C₂²×C₆₁	1	2	yes	—
316	#472	—	2	4	no	—
317	#469	—	2	2	no	—
318	#468	—	2	2	no	—
319	#466	—	2	1	no	—
320	#466	—	2	1	no	—
321	#466	—	2	1	no	—
322	#466	—	2	1	no	—
323	#466	—	2	2	no	—
324	#465	—	2	2	no	—
325	#435	—	2	1	no	—
326	#435	—	2	1	no	—
327	C₈×D₄	—	2	1	no	—
328	C₈×D₄	—	2	1	no	—
329	C₂×#108	—	2	4	no	—
330	C₂×#108	C₂²×C₆₁	1	2	yes	—
331	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
332	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
333	#428	C₂²×C₆₁	1	2	yes	—
334	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
335	#424	C₂²×C₆₁	1	1	yes	characteristic
336	#418	C₂²×C₆₁	1	2	yes	—
337	#417	C₂²×C₆₁	1	2	yes	—
338	C₂×#102	C₂²×C₆₁	1	2	yes	—
339	C₂×#100	C₂²×C₆₁	1	2	yes	—
340	C₂×#100	C₂²×C₆₁	1	2	yes	—
341	#408	C₂²×C₆₁	1	1	yes	characteristic
342	#407	C₂²×C₆₁	1	1	yes	characteristic
343	#405	C₂²×C₆₁	1	1	yes	characteristic
344	#343	C₂²×C₆₁	1	2	yes	—
345	#343	C₂²×C₆₁	1	2	yes	—
346	C₆₁	#24803	1	1	yes	characteristic, direct product factor, Sylow
347	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
350	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
351	AGL(1,ℤ/8)	—	2	4	no	—
352	AGL(1,ℤ/8)	—	4	2	no	—
353	C₂×SD₁₆	—	2	2	no	—
354	C₂×D₈	—	2	2	no	—
355	C₂×D₈	—	2	1	no	—
356	C₂×D₈	—	2	1	no	—
357	C₂×D₈	—	2	2	no	—
358	C₂×D₈	—	2	2	no	—
359	#131	—	2	2	no	—
360	#131	—	2	2	no	—
361	C₂×M₁₆	—	2	2	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂×M₁₆	—	2	4	no	—
364	C₂×M₁₆	—	4	2	no	—
365	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
366	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
367	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
368	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	—	2	2	no	—
371	C₂²×C₈	—	2	2	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
374	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
375	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
376	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
379	#123	—	2	1	no	—
380	C₂²⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
381	C₂²⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
382	#121	—	2	2	no	—
383	#121	—	2	2	no	—
384	#121	—	2	2	no	—
385	#121	—	2	2	no	—
386	#121	D₄×C₆₁	1	1	yes	characteristic
387	#121	D₄×C₆₁	1	1	yes	characteristic
388	#120	—	2	1	no	—
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	#117	C₂³×C₆₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
393	C₂×C₂²⋊C₄	—	2	2	no	—
394	#108	—	2	4	no	—
395	#108	—	2	2	no	—
396	#108	D₄×C₆₁	1	4	yes	complemented
397	#107	—	2	1	no	—
398	#107	—	2	1	no	—
399	#107	—	2	2	no	—
400	#106	—	2	2	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	#103	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#102	—	2	2	no	—
409	#100	—	2	4	no	—
410	#100	—	4	2	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	1	no	—
413	C₂²⋊C₈	—	2	1	no	—
414	C₂²⋊C₈	—	2	1	no	—
415	#97	—	2	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₂⁴	—	2	2	no	—
418	C₂⁴	#41×C₆₁	1	1	yes	characteristic
419	#41	—	2	1	no	—
420	#41	—	2	1	no	—
421	#41	—	2	1	no	—
422	#41	—	2	1	no	—
423	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	4	2	no	—
434	C₂×D₄	—	4	2	no	—
435	C₂×D₄	—	8	4	no	—
436	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
447	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
449	SD₁₆	—	4	4	no	—
450	D₈	—	4	4	no	—
451	D₈	—	4	2	no	—
452	D₈	—	4	2	no	—
453	M₁₆	—	2	2	no	—
454	M₁₆	—	2	1	no	—
455	M₁₆	—	2	2	no	—
456	M₁₆	—	2	1	no	—
457	M₁₆	—	2	1	no	—
458	M₁₆	—	2	2	no	—
459	M₁₆	—	2	1	no	—
460	M₁₆	—	4	4	no	—
461	M₁₆	—	4	2	no	—
462	M₁₆	C₂×D₄×C₆₁	1	4	yes	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	4	4	no	—
474	C₂×C₈	—	4	2	no	—
475	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
476	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
480	C₂×C₈	#41×C₆₁	1	2	yes	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
486	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₂²⋊C₄	—	4	2	no	—
494	C₄²	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	4	1	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	8	2	no	—
504	C₂³	—	8	2	no	—
505	C₂³	#121×C₆₁	1	1	yes	characteristic
506	C₂³	#121×C₆₁	1	1	yes	characteristic
507	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	D₄	—	2	1	no	—
511	D₄	—	2	1	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	1	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	2	no	—
516	D₄	—	4	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	8	4	no	—
521	D₄	—	8	4	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
540	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
543	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
544	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
545	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
546	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
547	C₈	—	2	4	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	2	no	—
552	C₈	—	4	1	no	—
553	C₈	#121×C₆₁	1	2	yes	—
554	C₈	#121×C₆₁	1	2	yes	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	1	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	1	no	—
561	C₂²	—	2	1	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	8	2	no	—
568	C₂²	—	8	2	no	—
569	C₂²	—	8	2	no	—
570	C₂²	—	16	2	no	—
571	C₂²	—	16	2	no	—
572	C₂²	C₆₁×#472	1	2	yes	complemented
573	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
574	C₂²	C₆₁×#548	1	1	yes	characteristic
575	C₂²	C₆₁×#553	1	1	yes	characteristic
576	C₄	—	2	1	no	—
577	C₄	—	2	1	no	—
578	C₄	—	4	1	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	8	2	no	—
582	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
583	C₄	C₆₁×#537	1	1	yes	characteristic
584	C₄	D₄²×C₆₁	1	1	yes	characteristic
585	C₄	C₆₁×#547	1	1	yes	characteristic
586	C₂	—	2	1	no	—
587	C₂	—	2	1	no	—
588	C₂	—	2	2	no	—
589	C₂	—	4	1	no	—
590	C₂	—	8	1	no	—
591	C₂	—	16	2	no	—
592	C₂	C₆₁×#2416	1	1	yes	characteristic
593	C₂	C₂×C₆₁×#472	1	1	yes	characteristic
594	C₂	C₆₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#472	C₂	semidirect
2. C₆₁×#3055	C₂	semidirect
3. C₂×C₆₁×#466	C₂	semidirect
4. C₆₁×#2929	C₂	semidirect
5. C₂²×#108×C₆₁	C₂	semidirect
6. C₆₁×#2063	C₂	semidirect
7. C₆₁×#2051	C₂	semidirect
8. C₆₁×#1978	C₂	semidirect
9. C₆₁×#1952	C₂	semidirect
10. C₆₁×#1900	C₂	semidirect
11. C₆₁×#1816	C₂	semidirect
98. #108×C₆₁	D₄	semidirect
257. #24803	C₆₁	direct (N × H)
275. C₂²×C₆₁	#472	semidirect
289. C₂×#472	C₂×C₆₁	semidirect
290. #3055	C₂×C₆₁	semidirect
291. C₂×#466	C₂×C₆₁	semidirect
292. #2929	C₂×C₆₁	semidirect
293. C₂²×#108	C₂×C₆₁	semidirect
294. #2063	C₂×C₆₁	semidirect
295. #2051	C₂×C₆₁	semidirect
296. #1978	C₂×C₆₁	semidirect
297. #1952	C₂×C₆₁	semidirect
298. #1900	C₂×C₆₁	semidirect
299. #1816	C₂×C₆₁	semidirect
346. C₆₁	#24803	direct (N × H)
396. #108	D₄×C₆₁	semidirect
572. C₂²	C₆₁×#472	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#472	2
C₆₁×#3055	2
C₂×C₆₁×#466	2
C₆₁×#2929	2
C₂²×#108×C₆₁	2
C₆₁×#2063	2
C₆₁×#2051	2
C₆₁×#1978	2
C₆₁×#1952	2
C₆₁×#1900	2
C₆₁×#1816	2
#24803	61

Subgroups — C61×#24803 #29528236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24803 #29528236