Subgroups — C₆₁×#34246 #29537679

Order 15616.

3194 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2364	C₂	1	4	yes	maximal, complemented
3	C₆₁×#2363	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2345	C₂	1	2	yes	maximal, complemented
5	D₄×C₂²⋊C₄×C₆₁	C₂	1	2	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₆₁	C₂	1	4	yes	maximal, complemented
7	C₆₁×#2333	C₂	1	4	yes	maximal, complemented
8	C₆₁×#2312	C₂	1	2	yes	maximal, complemented
9	C₄×#121×C₆₁	C₂	1	4	yes	maximal, complemented
10	C₆₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2273	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₂²⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₁	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₁	C₄	1	16	yes	complemented
17	C₂×#121×C₆₁	C₂²	1	2	yes	—
18	C₂×#121×C₆₁	C₂²	1	2	yes	—
19	C₂×#121×C₆₁	C₂²	1	2	yes	—
20	C₂×#121×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	complemented
22	C₂×C₄×D₄×C₆₁	C₂²	1	8	yes	complemented
23	C₂×#117×C₆₁	C₂²	1	2	yes	complemented
24	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₁	C₂²	1	2	yes	complemented
29	C₆₁×#390	C₂²	1	4	yes	complemented
30	C₆₁×#390	C₂²	1	8	yes	complemented
31	C₆₁×#389	C₂²	1	4	yes	complemented
32	C₆₁×#388	C₂²	1	8	yes	complemented
33	C₆₁×#388	C₂²	1	8	yes	complemented
34	C₆₁×#387	C₂²	1	8	yes	complemented
35	C₆₁×#386	C₂²	1	8	yes	complemented
36	C₆₁×#386	C₂²	1	8	yes	complemented
37	C₆₁×#386	C₂²	1	8	yes	complemented
38	C₆₁×#381	C₂²	1	2	yes	complemented
39	C₆₁×#380	C₂²	1	2	yes	—
40	C₆₁×#380	C₂²	1	4	yes	—
41	C₆₁×#380	C₂²	1	2	yes	complemented
42	C₆₁×#379	C₂²	1	2	yes	—
43	C₆₁×#379	C₂²	1	4	yes	—
44	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	complemented
45	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
47	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
48	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	complemented
51	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	complemented
52	C₄×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	complemented
53	C₄×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	complemented
54	C₄×C₂²⋊C₄×C₆₁	C₂²	1	2	yes	complemented
55	C₂×#95×C₆₁	C₂²	1	2	yes	complemented
56	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
57	C₂⁵×C₆₁	C₂³	1	2	yes	—
58	C₂²×D₄×C₆₁	—	4	4	no	—
59	C₂²×D₄×C₆₁	—	4	8	no	—
60	C₂²×D₄×C₆₁	C₂×C₄	1	8	yes	complemented
61	C₂²×D₄×C₆₁	C₂×C₄	1	16	yes	complemented
62	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
63	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₆₁	—	4	4	no	—
67	C₂³×C₄×C₆₁	C₂×C₄	1	8	yes	complemented
68	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
69	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₁	C₂³	1	8	yes	—
73	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
77	#121×C₆₁	C₂×C₄	1	16	yes	complemented
78	#121×C₆₁	C₂×C₄	1	16	yes	complemented
79	#121×C₆₁	C₂×C₄	1	16	yes	complemented
80	#121×C₆₁	C₂×C₄	1	16	yes	complemented
81	C₄×D₄×C₆₁	—	4	8	no	—
82	C₄×D₄×C₆₁	—	4	16	no	—
83	#117×C₆₁	—	4	4	no	—
84	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	complemented
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₆₁	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₆₁	—	2	16	no	—
91	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	16	yes	complemented
92	C₂×C₂²⋊C₄×C₆₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₆₁	—	4	4	no	—
104	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
105	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
106	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
108	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
109	#95×C₆₁	C₂³	1	2	yes	—
110	#95×C₆₁	C₂³	1	4	yes	—
111	#95×C₆₁	C₂³	1	2	yes	—
112	#95×C₆₁	C₂³	1	4	yes	—
113	#95×C₆₁	C₂³	1	8	yes	—
114	#95×C₆₁	C₂³	1	4	yes	—
115	#95×C₆₁	C₂³	1	8	yes	—
116	C₂⁴×C₆₁	—	2	4	no	—
117	C₂⁴×C₆₁	—	2	4	no	—
118	C₂⁴×C₆₁	—	2	8	no	—
119	C₂⁴×C₆₁	—	4	2	no	—
120	C₂⁴×C₆₁	—	4	4	no	—
121	C₂⁴×C₆₁	—	4	4	no	—
122	C₂⁴×C₆₁	C₂²×C₄	1	4	yes	—
123	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
124	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
125	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
126	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
127	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
128	C₂×D₄×C₆₁	—	2	16	no	—
129	C₂×D₄×C₆₁	—	2	8	no	—
130	C₂×D₄×C₆₁	—	4	16	no	—
131	C₂×D₄×C₆₁	—	4	8	no	—
132	C₂×D₄×C₆₁	—	4	4	no	—
133	C₂×D₄×C₆₁	—	4	32	no	—
134	C₂×D₄×C₆₁	C₂²×C₄	1	16	yes	—
135	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₆₁	C₂²×C₄	1	16	yes	—
137	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
138	C₂²×C₄×C₆₁	—	2	4	no	—
139	C₂²×C₄×C₆₁	—	2	4	no	—
140	C₂²×C₄×C₆₁	—	2	4	no	—
141	C₂²×C₄×C₆₁	—	2	8	no	—
142	C₂²×C₄×C₆₁	—	2	8	no	—
143	C₂²×C₄×C₆₁	—	2	16	no	—
144	C₂²×C₄×C₆₁	—	4	16	no	—
145	C₂²×C₄×C₆₁	—	4	4	no	—
146	C₂²×C₄×C₆₁	—	4	2	no	—
147	C₂²×C₄×C₆₁	—	4	2	no	—
148	C₂²×C₄×C₆₁	—	4	2	no	—
149	C₂²×C₄×C₆₁	—	4	4	no	—
150	C₂²×C₄×C₆₁	—	4	8	no	—
151	C₂²×C₄×C₆₁	—	4	8	no	—
152	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
157	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
159	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
161	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
167	C₄⋊C₄×C₆₁	—	2	8	no	—
168	C₄⋊C₄×C₆₁	—	4	8	no	—
169	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
170	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
171	C₂²⋊C₄×C₆₁	—	2	4	no	—
172	C₂²⋊C₄×C₆₁	—	2	8	no	—
173	C₂²⋊C₄×C₆₁	—	2	8	no	—
174	C₂²⋊C₄×C₆₁	—	2	8	no	—
175	C₂²⋊C₄×C₆₁	—	2	16	no	—
176	C₂²⋊C₄×C₆₁	—	4	4	no	—
177	C₂²⋊C₄×C₆₁	—	4	8	no	—
178	C₂²⋊C₄×C₆₁	—	4	8	no	—
179	C₂²⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
180	C₂²⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
181	C₂²⋊C₄×C₆₁	#41	1	8	yes	—
182	C₄²×C₆₁	—	2	8	no	—
183	C₄²×C₆₁	—	4	2	no	—
184	C₄²×C₆₁	—	4	2	no	—
185	C₄²×C₆₁	—	4	4	no	—
186	C₄²×C₆₁	—	4	8	no	—
187	C₂³×C₆₁	—	2	2	no	—
188	C₂³×C₆₁	—	2	2	no	—
189	C₂³×C₆₁	—	2	2	no	—
190	C₂³×C₆₁	—	2	2	no	—
191	C₂³×C₆₁	—	2	8	no	—
192	C₂³×C₆₁	—	2	8	no	—
193	C₂³×C₆₁	—	2	2	no	—
194	C₂³×C₆₁	—	2	4	no	—
195	C₂³×C₆₁	—	2	4	no	—
196	C₂³×C₆₁	—	2	2	no	—
197	C₂³×C₆₁	—	2	8	no	—
198	C₂³×C₆₁	—	2	8	no	—
199	C₂³×C₆₁	—	2	2	no	—
200	C₂³×C₆₁	—	2	16	no	—
201	C₂³×C₆₁	—	4	4	no	—
202	C₂³×C₆₁	—	4	8	no	—
203	C₂³×C₆₁	—	4	8	no	—
204	C₂³×C₆₁	—	4	16	no	—
205	C₂³×C₆₁	—	8	16	no	—
206	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
207	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
208	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
209	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
210	C₂³×C₆₁	C₂×#41	1	4	yes	—
211	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
212	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
213	D₄×C₆₁	—	2	32	no	—
214	D₄×C₆₁	—	2	16	no	—
215	D₄×C₆₁	—	4	8	no	—
216	D₄×C₆₁	—	4	16	no	—
217	C₂×C₄×C₆₁	—	2	4	no	—
218	C₂×C₄×C₆₁	—	2	2	no	—
219	C₂×C₄×C₆₁	—	2	2	no	—
220	C₂×C₄×C₆₁	—	2	4	no	—
221	C₂×C₄×C₆₁	—	2	8	no	—
222	C₂×C₄×C₆₁	—	2	2	no	—
223	C₂×C₄×C₆₁	—	2	8	no	—
224	C₂×C₄×C₆₁	—	2	8	no	—
225	C₂×C₄×C₆₁	—	2	8	no	—
226	C₂×C₄×C₆₁	—	2	4	no	—
227	C₂×C₄×C₆₁	—	2	8	no	—
228	C₂×C₄×C₆₁	—	2	4	no	—
229	C₂×C₄×C₆₁	—	2	8	no	—
230	C₂×C₄×C₆₁	—	2	8	no	—
231	C₂×C₄×C₆₁	—	2	4	no	—
232	C₂×C₄×C₆₁	—	4	4	no	—
233	C₂×C₄×C₆₁	—	4	4	no	—
234	C₂×C₄×C₆₁	—	4	4	no	—
235	C₂×C₄×C₆₁	—	4	8	no	—
236	C₂×C₄×C₆₁	—	4	16	no	—
237	C₂×C₄×C₆₁	—	8	16	no	—
238	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
239	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	—
241	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₆₁	C₂×#41	1	8	yes	—
244	C₂×C₄×C₆₁	H(5,𝔽₂)	1	4	yes	—
245	#34246	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂²×C₆₁	—	2	2	no	—
247	C₂²×C₆₁	—	2	2	no	—
248	C₂²×C₆₁	—	2	2	no	—
249	C₂²×C₆₁	—	2	2	no	—
250	C₂²×C₆₁	—	2	8	no	—
251	C₂²×C₆₁	—	2	8	no	—
252	C₂²×C₆₁	—	2	2	no	—
253	C₂²×C₆₁	—	2	8	no	—
254	C₂²×C₆₁	—	2	2	no	—
255	C₂²×C₆₁	—	2	8	no	—
256	C₂²×C₆₁	—	2	8	no	—
257	C₂²×C₆₁	—	2	2	no	—
258	C₂²×C₆₁	—	4	16	no	—
259	C₂²×C₆₁	—	4	8	no	—
260	C₂²×C₆₁	—	4	4	no	—
261	C₂²×C₆₁	—	4	8	no	—
262	C₂²×C₆₁	—	8	16	no	—
263	C₂²×C₆₁	C₄×#41	1	4	yes	—
264	C₂²×C₆₁	#518	1	4	yes	—
265	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
266	C₂²×C₆₁	C₂²×#41	1	2	yes	—
267	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
268	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
269	C₄×C₆₁	—	2	2	no	—
270	C₄×C₆₁	—	2	2	no	—
271	C₄×C₆₁	—	2	2	no	—
272	C₄×C₆₁	—	2	8	no	—
273	C₄×C₆₁	—	2	8	no	—
274	C₄×C₆₁	—	4	4	no	—
275	C₄×C₆₁	—	4	8	no	—
276	C₄×C₆₁	#518	1	4	yes	—
277	C₂²×#121	C₂×C₆₁	1	1	yes	characteristic
278	#2364	C₂×C₆₁	1	4	yes	complemented
279	#2363	C₂×C₆₁	1	4	yes	complemented
280	#2345	C₂×C₆₁	1	2	yes	complemented
281	D₄×C₂²⋊C₄	C₂×C₆₁	1	4	yes	complemented
282	D₄×C₂²⋊C₄	C₂×C₆₁	1	2	yes	complemented
283	#2333	C₂×C₆₁	1	4	yes	complemented
284	#2312	C₂×C₆₁	1	2	yes	complemented
285	C₄×#121	C₂×C₆₁	1	4	yes	complemented
286	#2280	C₂×C₆₁	1	1	yes	characteristic, complemented
287	#2273	C₂×C₆₁	1	2	yes	complemented
288	C₂×C₄×C₂²⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
289	C₂×C₆₁	—	2	4	no	—
290	C₂×C₆₁	—	2	4	no	—
291	C₂×C₆₁	—	4	2	no	—
292	C₂×C₆₁	—	4	8	no	—
293	C₂×C₆₁	C₂×C₄×#41	1	1	yes	characteristic
294	C₂×C₆₁	C₂×#518	1	1	yes	characteristic
295	C₂×C₆₁	C₂×#518	1	1	yes	characteristic
296	C₂×C₆₁	#3519	1	4	yes	—
297	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
298	C₂³×D₄	C₂²×C₆₁	1	2	yes	—
299	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
300	C₂×#121	C₄×C₆₁	1	16	yes	complemented
301	C₂×#121	C₂²×C₆₁	1	2	yes	—
302	C₂×#121	C₂²×C₆₁	1	2	yes	—
303	C₂×#121	C₂²×C₆₁	1	2	yes	—
304	C₂×#121	C₂²×C₆₁	1	2	yes	—
305	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	complemented
306	C₂×C₄×D₄	C₂²×C₆₁	1	8	yes	complemented
307	C₂×#117	C₂²×C₆₁	1	2	yes	complemented
308	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
309	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
310	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
311	C₂²×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
312	C₂²×C₄²	C₂²×C₆₁	1	2	yes	complemented
313	#390	C₂²×C₆₁	1	8	yes	complemented
314	#390	C₂²×C₆₁	1	4	yes	complemented
315	#389	C₂²×C₆₁	1	4	yes	complemented
316	#388	C₂²×C₆₁	1	8	yes	complemented
317	#388	C₂²×C₆₁	1	8	yes	complemented
318	#387	C₂²×C₆₁	1	8	yes	complemented
319	#386	C₂²×C₆₁	1	8	yes	complemented
320	#386	C₂²×C₆₁	1	8	yes	complemented
321	#386	C₂²×C₆₁	1	8	yes	complemented
322	#381	C₂²×C₆₁	1	2	yes	complemented
323	#380	C₂²×C₆₁	1	4	yes	—
324	#380	C₂²×C₆₁	1	2	yes	—
325	#380	C₂²×C₆₁	1	2	yes	complemented
326	#379	C₂²×C₆₁	1	4	yes	—
327	#379	C₂²×C₆₁	1	2	yes	—
328	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	complemented
329	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
330	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
331	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	—
332	C₄×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
334	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	complemented
335	C₄×C₂²⋊C₄	C₂²×C₆₁	1	8	yes	complemented
336	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	complemented
337	C₄×C₂²⋊C₄	C₂²×C₆₁	1	8	yes	complemented
338	C₄×C₂²⋊C₄	C₂²×C₆₁	1	2	yes	complemented
339	C₂×#95	C₂²×C₆₁	1	2	yes	complemented
340	C₆₁	#34246	1	1	yes	characteristic, direct product factor, Sylow
341	C₂⁵	C₂³×C₆₁	1	1	yes	characteristic
342	C₂⁵	C₂³×C₆₁	1	2	yes	—
343	C₂²×D₄	—	4	4	no	—
344	C₂²×D₄	—	4	8	no	—
345	C₂²×D₄	C₂×C₄×C₆₁	1	8	yes	complemented
346	C₂²×D₄	C₂×C₄×C₆₁	1	16	yes	complemented
347	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
348	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
349	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
350	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
351	C₂³×C₄	—	4	4	no	—
352	C₂³×C₄	C₂×C₄×C₆₁	1	8	yes	complemented
353	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
354	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₆₁	1	8	yes	—
358	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
360	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
362	#121	C₂×C₄×C₆₁	1	16	yes	complemented
363	#121	C₂×C₄×C₆₁	1	16	yes	complemented
364	#121	C₂×C₄×C₆₁	1	16	yes	complemented
365	#121	C₂×C₄×C₆₁	1	16	yes	complemented
366	C₄×D₄	—	4	8	no	—
367	C₄×D₄	—	4	16	no	—
368	#117	—	4	4	no	—
369	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	complemented
370	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
374	C₂×C₂²⋊C₄	—	2	8	no	—
375	C₂×C₂²⋊C₄	—	2	16	no	—
376	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	16	yes	complemented
377	C₂×C₂²⋊C₄	C₂³×C₆₁	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₆₁	1	2	yes	—
388	C₂×C₄²	—	4	4	no	—
389	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
390	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
391	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
392	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
393	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
394	#95	C₂³×C₆₁	1	2	yes	—
395	#95	C₂³×C₆₁	1	4	yes	—
396	#95	C₂³×C₆₁	1	2	yes	—
397	#95	C₂³×C₆₁	1	4	yes	—
398	#95	C₂³×C₆₁	1	8	yes	—
399	#95	C₂³×C₆₁	1	4	yes	—
400	#95	C₂³×C₆₁	1	8	yes	—
401	C₂⁴	—	2	4	no	—
402	C₂⁴	—	2	4	no	—
403	C₂⁴	—	2	8	no	—
404	C₂⁴	—	4	2	no	—
405	C₂⁴	—	4	4	no	—
406	C₂⁴	—	4	4	no	—
407	C₂⁴	C₂²×C₄×C₆₁	1	4	yes	—
408	C₂⁴	C₂²×C₄×C₆₁	1	8	yes	—
409	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
410	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
411	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
412	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
413	C₂×D₄	—	2	16	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	4	16	no	—
416	C₂×D₄	—	4	8	no	—
417	C₂×D₄	—	4	4	no	—
418	C₂×D₄	—	4	32	no	—
419	C₂×D₄	C₂²×C₄×C₆₁	1	16	yes	—
420	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
421	C₂×D₄	C₂²×C₄×C₆₁	1	16	yes	—
422	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	8	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	16	no	—
429	C₂²×C₄	—	4	16	no	—
430	C₂²×C₄	—	4	4	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	8	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
438	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
439	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
440	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
441	C₂²×C₄	C₂²×C₄×C₆₁	1	8	yes	—
442	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
443	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
446	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	4	8	no	—
454	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
455	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	16	no	—
461	C₂²⋊C₄	—	4	4	no	—
462	C₂²⋊C₄	—	4	8	no	—
463	C₂²⋊C₄	—	4	8	no	—
464	C₂²⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
465	C₂²⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
466	C₂²⋊C₄	#41×C₆₁	1	8	yes	—
467	C₄²	—	2	8	no	—
468	C₄²	—	4	2	no	—
469	C₄²	—	4	2	no	—
470	C₄²	—	4	4	no	—
471	C₄²	—	4	8	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	8	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	2	8	no	—
485	C₂³	—	2	16	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	16	no	—
490	C₂³	—	8	16	no	—
491	C₂³	C₂³×C₄×C₆₁	1	8	yes	—
492	C₂³	C₂³×C₄×C₆₁	1	2	yes	—
493	C₂³	C₂³×C₄×C₆₁	1	2	yes	—
494	C₂³	C₂³×C₄×C₆₁	1	2	yes	—
495	C₂³	C₂×#41×C₆₁	1	4	yes	—
496	C₂³	H(5,𝔽₂)×C₆₁	1	4	yes	—
497	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
498	D₄	—	2	32	no	—
499	D₄	—	2	16	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	16	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	8	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	4	4	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	4	16	no	—
522	C₂×C₄	—	8	16	no	—
523	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
524	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
525	C₂×C₄	C₂³×C₄×C₆₁	1	8	yes	—
526	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
527	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
528	C₂×C₄	C₂×#41×C₆₁	1	8	yes	—
529	C₂×C₄	H(5,𝔽₂)×C₆₁	1	4	yes	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	8	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	8	no	—
540	C₂²	—	2	8	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	4	8	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	8	no	—
546	C₂²	—	8	16	no	—
547	C₂²	C₄×#41×C₆₁	1	4	yes	—
548	C₂²	C₆₁×#518	1	4	yes	—
549	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
550	C₂²	C₂²×#41×C₆₁	1	2	yes	—
551	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
552	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	8	no	—
557	C₄	—	2	8	no	—
558	C₄	—	4	4	no	—
559	C₄	—	4	8	no	—
560	C₄	C₆₁×#518	1	4	yes	—
561	C₂	—	2	4	no	—
562	C₂	—	2	4	no	—
563	C₂	—	4	2	no	—
564	C₂	—	4	8	no	—
565	C₂	C₂×C₄×#41×C₆₁	1	1	yes	characteristic
566	C₂	C₂×C₆₁×#518	1	1	yes	characteristic
567	C₂	C₂×C₆₁×#518	1	1	yes	characteristic
568	C₂	C₆₁×#3519	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₁
Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂²×#121×C₆₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2364	C₂	semidirect
3. C₆₁×#2363	C₂	semidirect
4. C₆₁×#2345	C₂	semidirect
5. D₄×C₂²⋊C₄×C₆₁	C₂	semidirect
6. D₄×C₂²⋊C₄×C₆₁	C₂	semidirect
7. C₆₁×#2333	C₂	semidirect
8. C₆₁×#2312	C₂	semidirect
9. C₄×#121×C₆₁	C₂	semidirect
10. C₆₁×#2280	C₂	semidirect
11. C₆₁×#2273	C₂	semidirect
12. C₂×C₄×C₂²⋊C₄×C₆₁	C₂	semidirect
16. C₂×#121×C₆₁	C₄	semidirect
21. C₂×C₄×D₄×C₆₁	C₂²	semidirect
22. C₂×C₄×D₄×C₆₁	C₂²	semidirect
23. C₂×#117×C₆₁	C₂²	semidirect
28. C₂²×C₄²×C₆₁	C₂²	semidirect
29. C₆₁×#390	C₂²	semidirect
30. C₆₁×#390	C₂²	semidirect
31. C₆₁×#389	C₂²	semidirect
32. C₆₁×#388	C₂²	semidirect
33. C₆₁×#388	C₂²	semidirect
34. C₆₁×#387	C₂²	semidirect
35. C₆₁×#386	C₂²	semidirect
36. C₆₁×#386	C₂²	semidirect
37. C₆₁×#386	C₂²	semidirect
38. C₆₁×#381	C₂²	semidirect
41. C₆₁×#380	C₂²	semidirect
44. C₄×C₄⋊C₄×C₆₁	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₆₁	C₂²	semidirect
51. C₄×C₂²⋊C₄×C₆₁	C₂²	semidirect
52. C₄×C₂²⋊C₄×C₆₁	C₂²	semidirect
53. C₄×C₂²⋊C₄×C₆₁	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₆₁	C₂²	semidirect
55. C₂×#95×C₆₁	C₂²	semidirect
60. C₂²×D₄×C₆₁	C₂×C₄	semidirect
61. C₂²×D₄×C₆₁	C₂×C₄	semidirect
67. C₂³×C₄×C₆₁	C₂×C₄	semidirect
77. #121×C₆₁	C₂×C₄	semidirect
78. #121×C₆₁	C₂×C₄	semidirect
79. #121×C₆₁	C₂×C₄	semidirect
80. #121×C₆₁	C₂×C₄	semidirect
84. C₂×C₄⋊C₄×C₆₁	C₂×C₄	semidirect
91. C₂×C₂²⋊C₄×C₆₁	C₂×C₄	semidirect
245. #34246	C₆₁	direct (N × H)
278. #2364	C₂×C₆₁	semidirect
279. #2363	C₂×C₆₁	semidirect
280. #2345	C₂×C₆₁	semidirect
281. D₄×C₂²⋊C₄	C₂×C₆₁	semidirect
282. D₄×C₂²⋊C₄	C₂×C₆₁	semidirect
283. #2333	C₂×C₆₁	semidirect
284. #2312	C₂×C₆₁	semidirect
285. C₄×#121	C₂×C₆₁	semidirect
286. #2280	C₂×C₆₁	semidirect
287. #2273	C₂×C₆₁	semidirect
288. C₂×C₄×C₂²⋊C₄	C₂×C₆₁	semidirect
300. C₂×#121	C₄×C₆₁	semidirect
305. C₂×C₄×D₄	C₂²×C₆₁	semidirect
306. C₂×C₄×D₄	C₂²×C₆₁	semidirect
307. C₂×#117	C₂²×C₆₁	semidirect
312. C₂²×C₄²	C₂²×C₆₁	semidirect
313. #390	C₂²×C₆₁	semidirect
314. #390	C₂²×C₆₁	semidirect
315. #389	C₂²×C₆₁	semidirect
316. #388	C₂²×C₆₁	semidirect
317. #388	C₂²×C₆₁	semidirect
318. #387	C₂²×C₆₁	semidirect
319. #386	C₂²×C₆₁	semidirect
320. #386	C₂²×C₆₁	semidirect
321. #386	C₂²×C₆₁	semidirect
322. #381	C₂²×C₆₁	semidirect
325. #380	C₂²×C₆₁	semidirect
328. C₄×C₄⋊C₄	C₂²×C₆₁	semidirect
334. C₄×C₂²⋊C₄	C₂²×C₆₁	semidirect
335. C₄×C₂²⋊C₄	C₂²×C₆₁	semidirect
336. C₄×C₂²⋊C₄	C₂²×C₆₁	semidirect
337. C₄×C₂²⋊C₄	C₂²×C₆₁	semidirect
338. C₄×C₂²⋊C₄	C₂²×C₆₁	semidirect
339. C₂×#95	C₂²×C₆₁	semidirect
340. C₆₁	#34246	direct (N × H)
345. C₂²×D₄	C₂×C₄×C₆₁	semidirect
346. C₂²×D₄	C₂×C₄×C₆₁	semidirect
352. C₂³×C₄	C₂×C₄×C₆₁	semidirect
362. #121	C₂×C₄×C₆₁	semidirect
363. #121	C₂×C₄×C₆₁	semidirect
364. #121	C₂×C₄×C₆₁	semidirect
365. #121	C₂×C₄×C₆₁	semidirect
369. C₂×C₄⋊C₄	C₂×C₄×C₆₁	semidirect
376. C₂×C₂²⋊C₄	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
C₂²×#121×C₆₁	2
C₆₁×#2364	2
C₆₁×#2363	2
C₆₁×#2345	2
D₄×C₂²⋊C₄×C₆₁	2
C₆₁×#2333	2
C₆₁×#2312	2
C₄×#121×C₆₁	2
C₆₁×#2280	2
C₆₁×#2273	2
C₂×C₄×C₂²⋊C₄×C₆₁	2
#34246	61

Subgroups — C61×#34246 #29537679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#34246 #29537679