Subgroups — C₆₁×#34206 #29537639

Order 15616.

3170 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#520	C₂	1	2	yes	maximal, complemented
3	C₆₁×#2367	C₂	1	4	yes	maximal, complemented
4	C₆₁×#2366	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2365	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2354	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2352	C₂	1	4	yes	maximal, complemented
8	D₄×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2325	C₂	1	4	yes	maximal, complemented
10	C₆₁×#2311	C₂	1	2	yes	maximal, complemented
11	C₄×#121×C₆₁	C₂	1	4	yes	maximal, complemented
12	C₆₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#532	C₄	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₁	C₂²	1	2	yes	—
19	C₂×#121×C₆₁	C₂²	1	2	yes	—
20	C₂×#121×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#520	—	2	16	no	—
22	C₂×C₄×Q₈×C₆₁	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₆₁	C₂²	1	8	yes	—
24	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
28	C₂×#117×C₆₁	C₂²	1	2	yes	—
29	C₂×#117×C₆₁	C₂²	1	4	yes	—
30	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
32	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
34	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
35	C₆₁×#391	C₂²	1	4	yes	—
36	C₆₁×#390	C₂²	1	4	yes	—
37	C₆₁×#390	C₂²	1	4	yes	—
38	C₆₁×#389	C₂²	1	4	yes	—
39	C₆₁×#388	C₂²	1	8	yes	—
40	C₆₁×#388	C₂²	1	8	yes	—
41	C₆₁×#387	C₂²	1	8	yes	—
42	C₆₁×#386	C₂²	1	8	yes	—
43	C₆₁×#386	C₂²	1	8	yes	—
44	C₆₁×#385	C₂²	1	8	yes	—
45	C₆₁×#384	C₂²	1	1	yes	characteristic
46	C₆₁×#384	C₂²	1	1	yes	characteristic
47	C₆₁×#382	C₂²	1	2	yes	—
48	C₆₁×#380	C₂²	1	2	yes	—
49	C₆₁×#380	C₂²	1	2	yes	—
50	C₆₁×#380	C₂²	1	4	yes	—
51	C₆₁×#380	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
57	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₆₁	C₂²	1	8	yes	—
60	C₂×#95×C₆₁	C₂²	1	2	yes	—
61	C₂×#41×C₆₁	—	2	8	no	—
62	C₂×#41×C₆₁	—	2	8	no	—
63	C₂×#41×C₆₁	C₂×C₄	1	16	yes	complemented
64	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
68	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₆₁	C₂³	1	8	yes	—
73	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₆₁	C₂×C₄	1	8	yes	complemented
77	#127×C₆₁	C₂×C₄	1	8	yes	complemented
78	#124×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×Q₈×C₆₁	—	2	16	no	—
82	C₄×D₄×C₆₁	—	2	32	no	—
83	C₄×D₄×C₆₁	—	2	16	no	—
84	C₄×D₄×C₆₁	—	4	8	no	—
85	C₄×D₄×C₆₁	C₂×C₄	1	16	yes	complemented
86	C₄×D₄×C₆₁	C₂×C₄	1	16	yes	complemented
87	#117×C₆₁	—	2	8	no	—
88	#117×C₆₁	—	2	16	no	—
89	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
90	C₂×C₄⋊C₄×C₆₁	—	2	16	no	—
91	C₂×C₄⋊C₄×C₆₁	D₄	1	8	yes	complemented
92	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
111	C₂×C₄²×C₆₁	—	4	4	no	—
112	C₂×C₄²×C₆₁	C₂×C₄	1	8	yes	complemented
113	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
114	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
115	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
116	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
117	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
121	#95×C₆₁	C₂³	1	4	yes	—
122	#95×C₆₁	C₂³	1	4	yes	—
123	#95×C₆₁	C₂³	1	4	yes	—
124	#95×C₆₁	C₂³	1	8	yes	—
125	#95×C₆₁	C₂³	1	4	yes	—
126	#95×C₆₁	C₂³	1	8	yes	—
127	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
128	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
129	#41×C₆₁	—	2	64	no	—
130	C₂×Q₈×C₆₁	—	2	4	no	—
131	C₂×Q₈×C₆₁	—	2	4	no	—
132	C₂×Q₈×C₆₁	C₂²×C₄	1	8	yes	—
133	C₂×D₄×C₆₁	—	2	8	no	—
134	C₂×D₄×C₆₁	—	2	8	no	—
135	C₂×D₄×C₆₁	—	2	4	no	—
136	C₂×D₄×C₆₁	—	2	4	no	—
137	C₂×D₄×C₆₁	—	2	4	no	—
138	C₂×D₄×C₆₁	—	2	4	no	—
139	C₂×D₄×C₆₁	—	4	2	no	—
140	C₂×D₄×C₆₁	—	4	2	no	—
141	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₆₁	C₂²×C₄	1	16	yes	—
143	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
144	C₂×D₄×C₆₁	C₂²×C₄	1	8	yes	—
145	C₂²×C₄×C₆₁	—	2	8	no	—
146	C₂²×C₄×C₆₁	—	2	8	no	—
147	C₂²×C₄×C₆₁	—	2	2	no	—
148	C₂²×C₄×C₆₁	—	2	2	no	—
149	C₂²×C₄×C₆₁	—	2	4	no	—
150	C₂²×C₄×C₆₁	—	2	16	no	—
151	C₂²×C₄×C₆₁	—	4	2	no	—
152	C₂²×C₄×C₆₁	—	4	2	no	—
153	C₂²×C₄×C₆₁	—	4	1	no	—
154	C₂²×C₄×C₆₁	—	4	1	no	—
155	C₂²×C₄×C₆₁	—	4	2	no	—
156	C₂²×C₄×C₆₁	—	4	2	no	—
157	C₂²×C₄×C₆₁	—	4	8	no	—
158	C₂²×C₄×C₆₁	—	4	8	no	—
159	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₆₁	C₂²×C₄	1	16	yes	—
161	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
162	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₆₁	C₂×D₄	1	8	yes	—
164	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₆₁	#41	1	4	yes	—
166	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₆₁	—	2	4	no	—
178	C₄⋊C₄×C₆₁	—	2	16	no	—
179	C₄⋊C₄×C₆₁	—	2	4	no	—
180	C₄⋊C₄×C₆₁	—	2	4	no	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	—	2	4	no	—
183	C₄⋊C₄×C₆₁	—	2	4	no	—
184	C₄⋊C₄×C₆₁	—	4	2	no	—
185	C₄⋊C₄×C₆₁	—	4	2	no	—
186	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
187	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₆₁	C₂×D₄	1	8	yes	—
189	C₄⋊C₄×C₆₁	C₂×D₄	1	8	yes	—
190	C₂²⋊C₄×C₆₁	—	2	8	no	—
191	C₂²⋊C₄×C₆₁	—	2	16	no	—
192	C₂²⋊C₄×C₆₁	—	4	8	no	—
193	C₂²⋊C₄×C₆₁	—	4	8	no	—
194	C₂²⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
195	C₂²⋊C₄×C₆₁	C₂²×C₄	1	16	yes	—
196	C₄²×C₆₁	—	2	16	no	—
197	C₄²×C₆₁	—	2	8	no	—
198	C₄²×C₆₁	—	2	8	no	—
199	C₄²×C₆₁	—	4	4	no	—
200	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
201	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
202	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
203	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
204	C₂³×C₆₁	—	2	1	no	—
205	C₂³×C₆₁	—	2	1	no	—
206	C₂³×C₆₁	—	2	1	no	—
207	C₂³×C₆₁	—	2	1	no	—
208	C₂³×C₆₁	—	2	4	no	—
209	C₂³×C₆₁	—	2	4	no	—
210	C₂³×C₆₁	—	2	4	no	—
211	C₂³×C₆₁	—	2	4	no	—
212	C₂³×C₆₁	—	2	4	no	—
213	C₂³×C₆₁	—	2	4	no	—
214	C₂³×C₆₁	C₂³×C₄	1	8	yes	—
215	C₂³×C₆₁	C₂³×C₄	1	2	yes	—
216	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
217	C₂³×C₆₁	C₂×#41	1	2	yes	—
218	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
219	Q₈×C₆₁	—	2	16	no	—
220	D₄×C₆₁	—	2	32	no	—
221	D₄×C₆₁	—	2	16	no	—
222	D₄×C₆₁	—	2	16	no	—
223	D₄×C₆₁	—	4	8	no	—
224	C₂×C₄×C₆₁	—	2	4	no	—
225	C₂×C₄×C₆₁	—	2	4	no	—
226	C₂×C₄×C₆₁	—	2	4	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	4	no	—
232	C₂×C₄×C₆₁	—	2	1	no	—
233	C₂×C₄×C₆₁	—	2	1	no	—
234	C₂×C₄×C₆₁	—	2	32	no	—
235	C₂×C₄×C₆₁	—	2	8	no	—
236	C₂×C₄×C₆₁	—	2	4	no	—
237	C₂×C₄×C₆₁	—	2	4	no	—
238	C₂×C₄×C₆₁	—	2	4	no	—
239	C₂×C₄×C₆₁	—	2	4	no	—
240	C₂×C₄×C₆₁	—	2	4	no	—
241	C₂×C₄×C₆₁	—	2	4	no	—
242	C₂×C₄×C₆₁	—	2	4	no	—
243	C₂×C₄×C₆₁	—	2	4	no	—
244	C₂×C₄×C₆₁	—	2	4	no	—
245	C₂×C₄×C₆₁	—	4	4	no	—
246	C₂×C₄×C₆₁	—	4	8	no	—
247	C₂×C₄×C₆₁	—	4	8	no	—
248	C₂×C₄×C₆₁	—	8	16	no	—
249	C₂×C₄×C₆₁	C₄×D₄	1	16	yes	complemented
250	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
251	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
252	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
253	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	—
254	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
255	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
256	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₆₁	C₂²×D₄	1	8	yes	—
258	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
261	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
262	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
263	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
264	#34206	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₂²×C₆₁	—	2	2	no	—
266	C₂²×C₆₁	—	2	2	no	—
267	C₂²×C₆₁	—	2	2	no	—
268	C₂²×C₆₁	—	2	2	no	—
269	C₂²×C₆₁	—	2	2	no	—
270	C₂²×C₆₁	—	2	2	no	—
271	C₂²×C₆₁	—	2	8	no	—
272	C₂²×C₆₁	—	2	8	no	—
273	C₂²×C₆₁	—	2	8	no	—
274	C₂²×C₆₁	—	4	4	no	—
275	C₂²×C₆₁	—	4	4	no	—
276	C₂²×C₆₁	—	4	4	no	—
277	C₂²×C₆₁	—	4	4	no	—
278	C₂²×C₆₁	C₂×C₄×D₄	1	4	yes	—
279	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
280	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
281	C₂²×C₆₁	C₂²×#41	1	1	yes	characteristic
282	C₂²×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
284	C₂²×C₆₁	#585	1	1	yes	characteristic
285	C₂²×C₆₁	#585	1	1	yes	characteristic
286	C₄×C₆₁	—	2	8	no	—
287	C₄×C₆₁	—	2	2	no	—
288	C₄×C₆₁	—	2	2	no	—
289	C₄×C₆₁	—	2	2	no	—
290	C₄×C₆₁	—	2	8	no	—
291	C₄×C₆₁	—	4	4	no	—
292	C₄×C₆₁	—	4	4	no	—
293	C₄×C₆₁	—	4	4	no	—
294	C₄×C₆₁	C₂×C₄×D₄	1	4	yes	—
295	C₄×C₆₁	C₂×C₄×D₄	1	4	yes	—
296	C₄×C₆₁	#520	1	4	yes	complemented
297	C₂×#532	C₂×C₆₁	1	1	yes	characteristic
298	C₂×#520	C₂×C₆₁	1	2	yes	complemented
299	#2367	C₂×C₆₁	1	4	yes	complemented
300	#2366	C₂×C₆₁	1	2	yes	complemented
301	#2365	C₂×C₆₁	1	2	yes	complemented
302	#2354	C₂×C₆₁	1	2	yes	complemented
303	#2352	C₂×C₆₁	1	4	yes	complemented
304	D₄×C₄⋊C₄	C₂×C₆₁	1	2	yes	complemented
305	#2325	C₂×C₆₁	1	4	yes	complemented
306	#2311	C₂×C₆₁	1	2	yes	complemented
307	C₄×#121	C₂×C₆₁	1	4	yes	complemented
308	#2285	C₂×C₆₁	1	1	yes	characteristic, complemented
309	C₂×C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
310	C₂×C₆₁	—	2	4	no	—
311	C₂×C₆₁	—	4	8	no	—
312	C₂×C₆₁	C₂²×C₄×D₄	1	1	yes	characteristic
313	C₂×C₆₁	C₂×#520	1	1	yes	characteristic
314	C₂×C₆₁	#3428	1	1	yes	characteristic
315	C₂×C₆₁	#3496	1	1	yes	characteristic
316	C₂×C₆₁	#3497	1	1	yes	characteristic
317	C₂×C₆₁	#3506	1	1	yes	characteristic
318	C₂×C₆₁	#3507	1	1	yes	characteristic
319	C₂²×#41	C₂²×C₆₁	1	2	yes	—
320	#532	C₄×C₆₁	1	16	yes	complemented
321	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
322	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
323	C₂×#124	C₂²×C₆₁	1	2	yes	—
324	C₂×#121	C₂²×C₆₁	1	2	yes	—
325	C₂×#121	C₂²×C₆₁	1	2	yes	—
326	#520	—	2	16	no	—
327	C₂×C₄×Q₈	C₂²×C₆₁	1	4	yes	—
328	C₂×C₄×D₄	C₂²×C₆₁	1	8	yes	—
329	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
330	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
331	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
332	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
333	C₂×#117	C₂²×C₆₁	1	2	yes	—
334	C₂×#117	C₂²×C₆₁	1	4	yes	—
335	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
336	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
337	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
338	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
339	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
340	#391	C₂²×C₆₁	1	4	yes	—
341	#390	C₂²×C₆₁	1	4	yes	—
342	#390	C₂²×C₆₁	1	4	yes	—
343	#389	C₂²×C₆₁	1	4	yes	—
344	#388	C₂²×C₆₁	1	8	yes	—
345	#388	C₂²×C₆₁	1	8	yes	—
346	#387	C₂²×C₆₁	1	8	yes	—
347	#386	C₂²×C₆₁	1	8	yes	—
348	#386	C₂²×C₆₁	1	8	yes	—
349	#385	C₂²×C₆₁	1	8	yes	—
350	#384	C₂²×C₆₁	1	1	yes	characteristic
351	#384	C₂²×C₆₁	1	1	yes	characteristic
352	#382	C₂²×C₆₁	1	2	yes	—
353	#380	C₂²×C₆₁	1	4	yes	—
354	#380	C₂²×C₆₁	1	4	yes	—
355	#380	C₂²×C₆₁	1	2	yes	—
356	#380	C₂²×C₆₁	1	2	yes	—
357	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
358	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
359	C₄×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
360	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
361	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
362	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
363	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₆₁	1	8	yes	—
365	C₂×#95	C₂²×C₆₁	1	2	yes	—
366	C₆₁	#34206	1	1	yes	characteristic, direct product factor, Sylow
367	C₂×#41	—	2	8	no	—
368	C₂×#41	—	2	8	no	—
369	C₂×#41	C₂×C₄×C₆₁	1	16	yes	complemented
370	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
371	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
372	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
373	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
374	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
375	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
376	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₆₁	1	8	yes	—
378	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
379	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
380	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
381	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
382	C₄⋊Q₈	C₂×C₄×C₆₁	1	8	yes	complemented
383	#127	C₂×C₄×C₆₁	1	8	yes	complemented
384	#124	C₂×C₄×C₆₁	1	16	yes	complemented
385	#121	C₂×C₄×C₆₁	1	16	yes	complemented
386	#121	C₂×C₄×C₆₁	1	16	yes	complemented
387	C₄×Q₈	—	2	16	no	—
388	C₄×D₄	—	2	32	no	—
389	C₄×D₄	—	2	16	no	—
390	C₄×D₄	—	4	8	no	—
391	C₄×D₄	C₂×C₄×C₆₁	1	16	yes	complemented
392	C₄×D₄	C₂×C₄×C₆₁	1	16	yes	complemented
393	#117	—	2	8	no	—
394	#117	—	2	16	no	—
395	C₂×C₄⋊C₄	—	2	4	no	—
396	C₂×C₄⋊C₄	—	2	16	no	—
397	C₂×C₄⋊C₄	D₄×C₆₁	1	8	yes	complemented
398	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
400	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
417	C₂×C₄²	—	4	4	no	—
418	C₂×C₄²	C₂×C₄×C₆₁	1	8	yes	complemented
419	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
420	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
421	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
422	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
423	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
427	#95	C₂³×C₆₁	1	4	yes	—
428	#95	C₂³×C₆₁	1	4	yes	—
429	#95	C₂³×C₆₁	1	4	yes	—
430	#95	C₂³×C₆₁	1	8	yes	—
431	#95	C₂³×C₆₁	1	4	yes	—
432	#95	C₂³×C₆₁	1	8	yes	—
433	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
434	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
435	#41	—	2	64	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	C₂²×C₄×C₆₁	1	8	yes	—
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	2	no	—
447	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
448	C₂×D₄	C₂²×C₄×C₆₁	1	16	yes	—
449	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
450	C₂×D₄	C₂²×C₄×C₆₁	1	8	yes	—
451	C₂²×C₄	—	2	8	no	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	16	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	—	4	1	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	—	4	2	no	—
463	C₂²×C₄	—	4	8	no	—
464	C₂²×C₄	—	4	8	no	—
465	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
466	C₂²×C₄	C₂²×C₄×C₆₁	1	16	yes	—
467	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
468	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
469	C₂²×C₄	C₂×D₄×C₆₁	1	8	yes	—
470	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
471	C₂²×C₄	#41×C₆₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	16	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	—	4	2	no	—
492	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
493	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
494	C₄⋊C₄	C₂×D₄×C₆₁	1	8	yes	—
495	C₄⋊C₄	C₂×D₄×C₆₁	1	8	yes	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	16	no	—
498	C₂²⋊C₄	—	4	8	no	—
499	C₂²⋊C₄	—	4	8	no	—
500	C₂²⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
501	C₂²⋊C₄	C₂²×C₄×C₆₁	1	16	yes	—
502	C₄²	—	2	16	no	—
503	C₄²	—	2	8	no	—
504	C₄²	—	2	8	no	—
505	C₄²	—	4	4	no	—
506	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
507	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
508	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
509	C₄²	C₂²×C₄×C₆₁	1	4	yes	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	4	no	—
520	C₂³	C₂³×C₄×C₆₁	1	8	yes	—
521	C₂³	C₂³×C₄×C₆₁	1	2	yes	—
522	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
523	C₂³	C₂×#41×C₆₁	1	2	yes	—
524	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
525	Q₈	—	2	16	no	—
526	D₄	—	2	32	no	—
527	D₄	—	2	16	no	—
528	D₄	—	2	16	no	—
529	D₄	—	4	8	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	32	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	—	8	16	no	—
555	C₂×C₄	C₄×D₄×C₆₁	1	16	yes	complemented
556	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
557	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
558	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
559	C₂×C₄	C₂³×C₄×C₆₁	1	8	yes	—
560	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
561	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
562	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₆₁	1	8	yes	—
564	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
566	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
567	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
568	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
569	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	8	no	—
578	C₂²	—	2	8	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	C₂×C₄×D₄×C₆₁	1	4	yes	—
584	C₂²	C₂⁴×C₄×C₆₁	1	1	yes	characteristic
585	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₆₁	1	1	yes	characteristic
587	C₂²	C₂×H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
588	C₂²	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
589	C₂²	C₆₁×#585	1	1	yes	characteristic
590	C₂²	C₆₁×#585	1	1	yes	characteristic
591	C₄	—	2	8	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	8	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	4	no	—
598	C₄	—	4	4	no	—
599	C₄	C₂×C₄×D₄×C₆₁	1	4	yes	—
600	C₄	C₂×C₄×D₄×C₆₁	1	4	yes	—
601	C₄	C₆₁×#520	1	4	yes	complemented
602	C₂	—	2	4	no	—
603	C₂	—	4	8	no	—
604	C₂	C₂²×C₄×D₄×C₆₁	1	1	yes	characteristic
605	C₂	C₂×C₆₁×#520	1	1	yes	characteristic
606	C₂	C₆₁×#3428	1	1	yes	characteristic
607	C₂	C₆₁×#3496	1	1	yes	characteristic
608	C₂	C₆₁×#3497	1	1	yes	characteristic
609	C₂	C₆₁×#3506	1	1	yes	characteristic
610	C₂	C₆₁×#3507	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₁
Frattini, cosocle	C₂³	C₂⁵×C₆₁
prime radical	C₂×C₆₁×#532	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. C₂×C₆₁×#520	C₂	semidirect
3. C₆₁×#2367	C₂	semidirect
4. C₆₁×#2366	C₂	semidirect
5. C₆₁×#2365	C₂	semidirect
6. C₆₁×#2354	C₂	semidirect
7. C₆₁×#2352	C₂	semidirect
8. D₄×C₄⋊C₄×C₆₁	C₂	semidirect
9. C₆₁×#2325	C₂	semidirect
10. C₆₁×#2311	C₂	semidirect
11. C₄×#121×C₆₁	C₂	semidirect
12. C₆₁×#2285	C₂	semidirect
13. C₂×C₄×C₄⋊C₄×C₆₁	C₂	semidirect
15. C₆₁×#532	C₄	semidirect
63. C₂×#41×C₆₁	C₂×C₄	semidirect
76. C₄⋊Q₈×C₆₁	C₂×C₄	semidirect
77. #127×C₆₁	C₂×C₄	semidirect
78. #124×C₆₁	C₂×C₄	semidirect
79. #121×C₆₁	C₂×C₄	semidirect
80. #121×C₆₁	C₂×C₄	semidirect
85. C₄×D₄×C₆₁	C₂×C₄	semidirect
86. C₄×D₄×C₆₁	C₂×C₄	semidirect
91. C₂×C₄⋊C₄×C₆₁	D₄	semidirect
112. C₂×C₄²×C₆₁	C₂×C₄	semidirect
249. C₂×C₄×C₆₁	C₄×D₄	semidirect
264. #34206	C₆₁	direct (N × H)
296. C₄×C₆₁	#520	semidirect
298. C₂×#520	C₂×C₆₁	semidirect
299. #2367	C₂×C₆₁	semidirect
300. #2366	C₂×C₆₁	semidirect
301. #2365	C₂×C₆₁	semidirect
302. #2354	C₂×C₆₁	semidirect
303. #2352	C₂×C₆₁	semidirect
304. D₄×C₄⋊C₄	C₂×C₆₁	semidirect
305. #2325	C₂×C₆₁	semidirect
306. #2311	C₂×C₆₁	semidirect
307. C₄×#121	C₂×C₆₁	semidirect
308. #2285	C₂×C₆₁	semidirect
309. C₂×C₄×C₄⋊C₄	C₂×C₆₁	semidirect
320. #532	C₄×C₆₁	semidirect
366. C₆₁	#34206	direct (N × H)
369. C₂×#41	C₂×C₄×C₆₁	semidirect
382. C₄⋊Q₈	C₂×C₄×C₆₁	semidirect
383. #127	C₂×C₄×C₆₁	semidirect
384. #124	C₂×C₄×C₆₁	semidirect
385. #121	C₂×C₄×C₆₁	semidirect
386. #121	C₂×C₄×C₆₁	semidirect
391. C₄×D₄	C₂×C₄×C₆₁	semidirect
392. C₄×D₄	C₂×C₄×C₆₁	semidirect
397. C₂×C₄⋊C₄	D₄×C₆₁	semidirect
418. C₂×C₄²	C₂×C₄×C₆₁	semidirect
555. C₂×C₄	C₄×D₄×C₆₁	semidirect
601. C₄	C₆₁×#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#532	2
C₂×C₆₁×#520	2
C₆₁×#2367	2
C₆₁×#2366	2
C₆₁×#2365	2
C₆₁×#2354	2
C₆₁×#2352	2
D₄×C₄⋊C₄×C₆₁	2
C₆₁×#2325	2
C₆₁×#2311	2
C₄×#121×C₆₁	2
C₆₁×#2285	2
C₂×C₄×C₄⋊C₄×C₆₁	2
#34206	61

Subgroups — C61×#34206 #29537639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#34206 #29537639