Subgroups — C₂×C₆₁×#3512 #29566242

Order 15616.

7238 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3512	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₆₁×#553	C₂	1	4	yes	maximal, complemented
3	C₂×C₆₁×#553	C₂	1	8	yes	maximal, complemented
4	C₂×C₆₁×#552	C₂	1	4	yes	maximal, complemented
5	C₂×C₆₁×#541	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#540	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#537	C₂	1	2	yes	maximal, complemented
8	C₂×C₆₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₆₁×#529	C₂	1	4	yes	maximal, complemented
10	C₂²×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#520	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×#41×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#553	C₂²	1	64	yes	complemented
16	C₆₁×#553	C₂²	1	128	yes	complemented
17	C₆₁×#552	C₂²	1	64	yes	complemented
18	C₆₁×#541	C₂²	1	32	yes	complemented
19	C₆₁×#540	C₂²	1	32	yes	complemented
20	C₆₁×#537	C₂²	1	32	yes	complemented
21	C₆₁×#532	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#529	C₂²	1	64	yes	complemented
25	C₂×#126×C₆₁	C₂²	1	8	yes	—
26	C₂×#126×C₆₁	C₂²	1	8	yes	—
27	C₂×#125×C₆₁	C₂²	1	2	yes	—
28	C₂×#125×C₆₁	C₂²	1	2	yes	—
29	C₂×#125×C₆₁	C₂²	1	4	yes	—
30	C₂×#125×C₆₁	C₂²	1	8	yes	—
31	C₂×#125×C₆₁	C₂²	1	16	yes	complemented
32	C₂×#124×C₆₁	C₂²	1	2	yes	—
33	C₂×#123×C₆₁	C₂²	1	8	yes	—
34	C₂×#123×C₆₁	C₂²	1	8	yes	—
35	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	4	yes	—
36	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	4	yes	—
37	C₂×#121×C₆₁	C₂²	1	4	yes	—
38	C₂×#121×C₆₁	C₂²	1	4	yes	—
39	C₂×#121×C₆₁	C₂²	1	8	yes	—
40	C₂×#121×C₆₁	C₂²	1	4	yes	—
41	C₂×#121×C₆₁	C₂²	1	8	yes	—
42	C₆₁×#520	C₂²	1	32	yes	complemented
43	C₄×#41×C₆₁	C₂²	1	16	yes	complemented
44	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
45	C₂×C₄×Q₈×C₆₁	C₂²	1	4	yes	—
46	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
47	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
48	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
49	C₂×C₄×D₄×C₆₁	C₂²	1	8	yes	—
50	C₂×C₄×D₄×C₆₁	C₂²	1	8	yes	—
51	C₂×C₄×D₄×C₆₁	C₂²	1	4	yes	—
52	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
53	C₂×#117×C₆₁	C₂²	1	2	yes	—
54	C₂×#117×C₆₁	C₂²	1	4	yes	—
55	C₂×#117×C₆₁	C₂²	1	8	yes	—
56	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
57	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
58	C₂²×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
59	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
60	C₂²×C₄²×C₆₁	C₂²	1	2	yes	—
61	C₂×#41×C₆₁	—	2	8	no	—
62	C₂×#41×C₆₁	—	2	16	no	—
63	C₂×#41×C₆₁	C₂³	1	16	yes	—
64	C₂×#41×C₆₁	C₂³	1	8	yes	—
65	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
66	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
70	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₁	C₂³	1	4	yes	—
72	C₂²×D₄×C₆₁	C₂³	1	8	yes	—
73	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
75	C₂³×C₄×C₆₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
77	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
78	C₂³×C₄×C₆₁	C₂³	1	8	yes	—
79	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₆₁	C₂³	1	8	yes	—
81	#127×C₆₁	C₂³	1	8	yes	—
82	#126×C₆₁	C₂³	1	64	yes	—
83	#126×C₆₁	C₂³	1	64	yes	—
84	#125×C₆₁	C₂³	1	16	yes	—
85	#125×C₆₁	C₂³	1	64	yes	—
86	#125×C₆₁	C₂³	1	32	yes	—
87	#125×C₆₁	C₂³	1	16	yes	—
88	#124×C₆₁	C₂³	1	16	yes	—
89	#123×C₆₁	C₂³	1	64	yes	—
90	#123×C₆₁	C₂³	1	64	yes	—
91	C₂²⋊Q₈×C₆₁	C₂³	1	32	yes	—
92	C₂²⋊Q₈×C₆₁	C₂³	1	32	yes	—
93	#121×C₆₁	C₂³	1	32	yes	—
94	#121×C₆₁	C₂³	1	32	yes	—
95	#121×C₆₁	C₂³	1	64	yes	—
96	#121×C₆₁	C₂³	1	32	yes	—
97	#121×C₆₁	C₂³	1	64	yes	—
98	C₄×Q₈×C₆₁	C₂³	1	32	yes	—
99	C₄×Q₈×C₆₁	C₂³	1	16	yes	—
100	C₄×D₄×C₆₁	C₂³	1	32	yes	—
101	C₄×D₄×C₆₁	C₂³	1	32	yes	—
102	C₄×D₄×C₆₁	C₂³	1	32	yes	—
103	C₄×D₄×C₆₁	C₂³	1	16	yes	—
104	C₄×D₄×C₆₁	C₂³	1	64	yes	—
105	C₄×D₄×C₆₁	C₂³	1	64	yes	—
106	#117×C₆₁	C₂³	1	16	yes	—
107	#117×C₆₁	C₂³	1	32	yes	—
108	#117×C₆₁	C₂³	1	64	yes	—
109	#117×C₆₁	C₂³	1	8	yes	—
110	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₆₁	C₂³	1	16	yes	—
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	32	yes	—
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	32	yes	—
114	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
117	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
119	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
120	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₆₁	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
126	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
127	C₂×C₂²⋊C₄×C₆₁	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₆₁	C₂³	1	8	yes	—
129	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
131	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
134	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₁	C₂³	1	16	yes	—
136	C₂×C₄²×C₆₁	C₂³	1	8	yes	—
137	C₂⁴×C₆₁	C₂⁴	1	4	yes	—
138	C₂⁴×C₆₁	C₂⁴	1	2	yes	—
139	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
140	#41×C₆₁	—	2	64	no	—
141	#41×C₆₁	—	2	32	no	—
142	C₂×Q₈×C₆₁	—	2	4	no	—
143	C₂×Q₈×C₆₁	—	2	8	no	—
144	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
145	C₂×Q₈×C₆₁	C₂⁴	1	8	yes	—
146	C₂×D₄×C₆₁	—	2	8	no	—
147	C₂×D₄×C₆₁	—	2	16	no	—
148	C₂×D₄×C₆₁	—	2	4	no	—
149	C₂×D₄×C₆₁	—	2	8	no	—
150	C₂×D₄×C₆₁	—	4	16	no	—
151	C₂×D₄×C₆₁	—	4	8	no	—
152	C₂×D₄×C₆₁	C₂⁴	1	8	yes	—
153	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
154	C₂×D₄×C₆₁	C₂⁴	1	4	yes	—
155	C₂×D₄×C₆₁	C₂⁴	1	8	yes	—
156	C₂×D₄×C₆₁	C₂⁴	1	32	yes	—
157	C₂×D₄×C₆₁	C₂⁴	1	16	yes	—
158	C₂²×C₄×C₆₁	—	2	16	no	—
159	C₂²×C₄×C₆₁	—	2	8	no	—
160	C₂²×C₄×C₆₁	—	4	8	no	—
161	C₂²×C₄×C₆₁	—	4	16	no	—
162	C₂²×C₄×C₆₁	—	4	4	no	—
163	C₂²×C₄×C₆₁	—	4	8	no	—
164	C₂²×C₄×C₆₁	#41	1	8	yes	—
165	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
166	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
167	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
173	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
175	C₂²×C₄×C₆₁	C₂⁴	1	16	yes	—
176	C₂²×C₄×C₆₁	C₂⁴	1	32	yes	—
177	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
180	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
181	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
182	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
183	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
184	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
185	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
186	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
187	C₄⋊C₄×C₆₁	C₂⁴	1	32	yes	—
188	C₄⋊C₄×C₆₁	C₂⁴	1	16	yes	—
189	C₂²⋊C₄×C₆₁	C₂⁴	1	16	yes	—
190	C₂²⋊C₄×C₆₁	C₂⁴	1	8	yes	—
191	C₂²⋊C₄×C₆₁	C₂⁴	1	16	yes	—
192	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
193	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
194	C₂²⋊C₄×C₆₁	C₂⁴	1	16	yes	—
195	C₂²⋊C₄×C₆₁	C₂⁴	1	32	yes	—
196	C₄²×C₆₁	C₂⁴	1	4	yes	—
197	C₄²×C₆₁	C₂⁴	1	8	yes	—
198	C₄²×C₆₁	C₂⁴	1	16	yes	—
199	C₄²×C₆₁	C₂⁴	1	4	yes	—
200	C₄²×C₆₁	C₂⁴	1	8	yes	—
201	C₄²×C₆₁	C₂⁴	1	32	yes	—
202	C₂³×C₆₁	—	2	4	no	—
203	C₂³×C₆₁	—	2	8	no	—
204	C₂³×C₆₁	—	2	4	no	—
205	C₂³×C₆₁	—	2	8	no	—
206	C₂³×C₆₁	—	2	8	no	—
207	C₂³×C₆₁	—	2	8	no	—
208	C₂³×C₆₁	C₂×#41	1	4	yes	—
209	C₂³×C₆₁	C₂⁵	1	8	yes	—
210	C₂³×C₆₁	C₂⁵	1	4	yes	—
211	C₂³×C₆₁	C₂⁵	1	2	yes	—
212	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
213	Q₈×C₆₁	—	2	8	no	—
214	Q₈×C₆₁	—	2	16	no	—
215	D₄×C₆₁	—	2	16	no	—
216	D₄×C₆₁	—	2	32	no	—
217	D₄×C₆₁	—	2	8	no	—
218	D₄×C₆₁	—	2	16	no	—
219	D₄×C₆₁	—	4	32	no	—
220	D₄×C₆₁	—	4	16	no	—
221	C₂×C₄×C₆₁	—	2	8	no	—
222	C₂×C₄×C₆₁	—	2	4	no	—
223	C₂×C₄×C₆₁	—	2	4	no	—
224	C₂×C₄×C₆₁	—	2	8	no	—
225	C₂×C₄×C₆₁	—	2	16	no	—
226	C₂×C₄×C₆₁	—	2	4	no	—
227	C₂×C₄×C₆₁	—	2	32	no	—
228	C₂×C₄×C₆₁	—	2	16	no	—
229	C₂×C₄×C₆₁	—	4	16	no	—
230	C₂×C₄×C₆₁	—	4	32	no	—
231	C₂×C₄×C₆₁	—	4	8	no	—
232	C₂×C₄×C₆₁	—	4	16	no	—
233	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
234	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
235	C₂×C₄×C₆₁	C₂×#41	1	16	yes	—
236	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
237	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
238	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
239	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
240	C₂×C₄×C₆₁	C₂⁵	1	4	yes	—
241	C₂×C₄×C₆₁	C₂⁵	1	16	yes	—
242	C₂×C₄×C₆₁	C₂⁵	1	2	yes	—
243	C₂×C₄×C₆₁	C₂⁵	1	2	yes	—
244	C₂×#3512	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂²×C₆₁	—	2	4	no	—
246	C₂²×C₆₁	—	2	8	no	—
247	C₂²×C₆₁	—	2	8	no	—
248	C₂²×C₆₁	—	2	4	no	—
249	C₂²×C₆₁	—	2	8	no	—
250	C₂²×C₆₁	—	2	8	no	—
251	C₂²×C₆₁	—	2	8	no	—
252	C₂²×C₆₁	—	4	16	no	—
253	C₂²×C₆₁	—	4	8	no	—
254	C₂²×C₆₁	C₂²×#41	1	4	yes	—
255	C₂²×C₆₁	C₂²×#41	1	2	yes	—
256	C₂²×C₆₁	#585	1	4	yes	—
257	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
258	C₄×C₆₁	—	2	8	no	—
259	C₄×C₆₁	—	2	4	no	—
260	C₄×C₆₁	—	2	4	no	—
261	C₄×C₆₁	—	2	8	no	—
262	C₄×C₆₁	—	2	16	no	—
263	C₄×C₆₁	—	2	4	no	—
264	C₄×C₆₁	C₂²×#41	1	4	yes	—
265	C₄×C₆₁	C₂²×#41	1	4	yes	—
266	#3512	C₂×C₆₁	1	32	yes	direct product factor
267	C₂×#553	C₂×C₆₁	1	4	yes	complemented
268	C₂×#553	C₂×C₆₁	1	8	yes	complemented
269	C₂×#552	C₂×C₆₁	1	4	yes	complemented
270	C₂×#541	C₂×C₆₁	1	2	yes	complemented
271	C₂×#540	C₂×C₆₁	1	2	yes	complemented
272	C₂×#537	C₂×C₆₁	1	2	yes	complemented
273	C₂×#532	C₂×C₆₁	1	1	yes	characteristic, complemented
274	C₂×#529	C₂×C₆₁	1	4	yes	complemented
275	C₂²×#125	C₂×C₆₁	1	1	yes	characteristic, complemented
276	C₂×#520	C₂×C₆₁	1	2	yes	complemented
277	C₂×C₄×#41	C₂×C₆₁	1	1	yes	characteristic, complemented
278	C₂×C₆₁	—	2	4	no	—
279	C₂×C₆₁	—	4	8	no	—
280	C₂×C₆₁	—	4	4	no	—
281	C₂×C₆₁	#3512	1	4	yes	direct product factor
282	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
283	C₂×C₆₁	C₂×#585	1	2	yes	—
284	C₂²×#41	C₂²×C₆₁	1	1	yes	characteristic
285	C₂²×#41	C₂²×C₆₁	1	2	yes	—
286	#553	C₂²×C₆₁	1	64	yes	complemented
287	#553	C₂²×C₆₁	1	128	yes	complemented
288	#552	C₂²×C₆₁	1	64	yes	complemented
289	#541	C₂²×C₆₁	1	32	yes	complemented
290	#540	C₂²×C₆₁	1	32	yes	complemented
291	#537	C₂²×C₆₁	1	32	yes	complemented
292	#532	C₂²×C₆₁	1	16	yes	complemented
293	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
294	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
295	#529	C₂²×C₆₁	1	64	yes	complemented
296	C₂×#126	C₂²×C₆₁	1	8	yes	—
297	C₂×#126	C₂²×C₆₁	1	8	yes	—
298	C₂×#125	C₂²×C₆₁	1	2	yes	—
299	C₂×#125	C₂²×C₆₁	1	2	yes	—
300	C₂×#125	C₂²×C₆₁	1	4	yes	—
301	C₂×#125	C₂²×C₆₁	1	8	yes	—
302	C₂×#125	C₂²×C₆₁	1	16	yes	complemented
303	C₂×#124	C₂²×C₆₁	1	2	yes	—
304	C₂×#123	C₂²×C₆₁	1	8	yes	—
305	C₂×#123	C₂²×C₆₁	1	8	yes	—
306	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	4	yes	—
307	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	4	yes	—
308	C₂×#121	C₂²×C₆₁	1	4	yes	—
309	C₂×#121	C₂²×C₆₁	1	4	yes	—
310	C₂×#121	C₂²×C₆₁	1	8	yes	—
311	C₂×#121	C₂²×C₆₁	1	4	yes	—
312	C₂×#121	C₂²×C₆₁	1	8	yes	—
313	#520	C₂²×C₆₁	1	32	yes	complemented
314	C₄×#41	C₂²×C₆₁	1	16	yes	complemented
315	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
316	C₂×C₄×Q₈	C₂²×C₆₁	1	4	yes	—
317	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
318	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
319	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
320	C₂×C₄×D₄	C₂²×C₆₁	1	8	yes	—
321	C₂×C₄×D₄	C₂²×C₆₁	1	8	yes	—
322	C₂×C₄×D₄	C₂²×C₆₁	1	4	yes	—
323	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
324	C₂×#117	C₂²×C₆₁	1	2	yes	—
325	C₂×#117	C₂²×C₆₁	1	4	yes	—
326	C₂×#117	C₂²×C₆₁	1	8	yes	—
327	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
328	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
329	C₂²×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
330	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
331	C₂²×C₄²	C₂²×C₆₁	1	2	yes	—
332	C₆₁	C₂×#3512	1	1	yes	characteristic, direct product factor, Sylow
333	C₂×#41	—	2	8	no	—
334	C₂×#41	—	2	16	no	—
335	C₂×#41	C₂³×C₆₁	1	8	yes	—
336	C₂×#41	C₂³×C₆₁	1	16	yes	—
337	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
338	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
339	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
340	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
341	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
342	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
343	C₂²×D₄	C₂³×C₆₁	1	8	yes	—
344	C₂²×D₄	C₂³×C₆₁	1	4	yes	—
345	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
346	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
347	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
348	C₂³×C₄	C₂³×C₆₁	1	2	yes	—
349	C₂³×C₄	C₂³×C₆₁	1	8	yes	—
350	C₂³×C₄	C₂³×C₆₁	1	4	yes	—
351	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
352	C₄⋊Q₈	C₂³×C₆₁	1	8	yes	—
353	#127	C₂³×C₆₁	1	8	yes	—
354	#126	C₂³×C₆₁	1	64	yes	—
355	#126	C₂³×C₆₁	1	64	yes	—
356	#125	C₂³×C₆₁	1	16	yes	—
357	#125	C₂³×C₆₁	1	16	yes	—
358	#125	C₂³×C₆₁	1	32	yes	—
359	#125	C₂³×C₆₁	1	64	yes	—
360	#124	C₂³×C₆₁	1	16	yes	—
361	#123	C₂³×C₆₁	1	64	yes	—
362	#123	C₂³×C₆₁	1	64	yes	—
363	C₂²⋊Q₈	C₂³×C₆₁	1	32	yes	—
364	C₂²⋊Q₈	C₂³×C₆₁	1	32	yes	—
365	#121	C₂³×C₆₁	1	32	yes	—
366	#121	C₂³×C₆₁	1	32	yes	—
367	#121	C₂³×C₆₁	1	64	yes	—
368	#121	C₂³×C₆₁	1	32	yes	—
369	#121	C₂³×C₆₁	1	64	yes	—
370	C₄×Q₈	C₂³×C₆₁	1	16	yes	—
371	C₄×Q₈	C₂³×C₆₁	1	32	yes	—
372	C₄×D₄	C₂³×C₆₁	1	32	yes	—
373	C₄×D₄	C₂³×C₆₁	1	16	yes	—
374	C₄×D₄	C₂³×C₆₁	1	32	yes	—
375	C₄×D₄	C₂³×C₆₁	1	64	yes	—
376	C₄×D₄	C₂³×C₆₁	1	64	yes	—
377	C₄×D₄	C₂³×C₆₁	1	32	yes	—
378	#117	C₂³×C₆₁	1	8	yes	—
379	#117	C₂³×C₆₁	1	16	yes	—
380	#117	C₂³×C₆₁	1	32	yes	—
381	#117	C₂³×C₆₁	1	64	yes	—
382	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
387	C₂×C₄⋊C₄	C₂³×C₆₁	1	16	yes	—
388	C₂×C₄⋊C₄	C₂³×C₆₁	1	32	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
390	C₂×C₄⋊C₄	C₂³×C₆₁	1	32	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₆₁	1	2	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₆₁	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₆₁	1	8	yes	—
401	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
402	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
403	C₂×C₄²	C₂³×C₆₁	1	4	yes	—
404	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
406	C₂×C₄²	C₂³×C₆₁	1	8	yes	—
407	C₂×C₄²	C₂³×C₆₁	1	16	yes	—
408	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
409	C₂⁴	C₂⁴×C₆₁	1	4	yes	—
410	C₂⁴	C₂⁴×C₆₁	1	2	yes	—
411	C₂⁴	C₂⁴×C₆₁	1	1	yes	characteristic
412	#41	—	2	32	no	—
413	#41	—	2	64	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	8	no	—
416	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
417	C₂×Q₈	C₂⁴×C₆₁	1	8	yes	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	16	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	4	16	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	C₂⁴×C₆₁	1	8	yes	—
425	C₂×D₄	C₂⁴×C₆₁	1	16	yes	—
426	C₂×D₄	C₂⁴×C₆₁	1	4	yes	—
427	C₂×D₄	C₂⁴×C₆₁	1	8	yes	—
428	C₂×D₄	C₂⁴×C₆₁	1	32	yes	—
429	C₂×D₄	C₂⁴×C₆₁	1	16	yes	—
430	C₂²×C₄	—	2	16	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	—	4	16	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	8	no	—
436	C₂²×C₄	#41×C₆₁	1	8	yes	—
437	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
443	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
444	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
445	C₂²×C₄	C₂⁴×C₆₁	1	8	yes	—
446	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
447	C₂²×C₄	C₂⁴×C₆₁	1	16	yes	—
448	C₂²×C₄	C₂⁴×C₆₁	1	32	yes	—
449	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
452	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
453	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
454	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
455	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
456	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
457	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
458	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
459	C₄⋊C₄	C₂⁴×C₆₁	1	32	yes	—
460	C₄⋊C₄	C₂⁴×C₆₁	1	16	yes	—
461	C₂²⋊C₄	C₂⁴×C₆₁	1	16	yes	—
462	C₂²⋊C₄	C₂⁴×C₆₁	1	8	yes	—
463	C₂²⋊C₄	C₂⁴×C₆₁	1	16	yes	—
464	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
465	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
466	C₂²⋊C₄	C₂⁴×C₆₁	1	16	yes	—
467	C₂²⋊C₄	C₂⁴×C₆₁	1	32	yes	—
468	C₄²	C₂⁴×C₆₁	1	4	yes	—
469	C₄²	C₂⁴×C₆₁	1	8	yes	—
470	C₄²	C₂⁴×C₆₁	1	16	yes	—
471	C₄²	C₂⁴×C₆₁	1	4	yes	—
472	C₄²	C₂⁴×C₆₁	1	8	yes	—
473	C₄²	C₂⁴×C₆₁	1	32	yes	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	8	no	—
478	C₂³	—	2	8	no	—
479	C₂³	—	2	8	no	—
480	C₂³	C₂×#41×C₆₁	1	4	yes	—
481	C₂³	C₂⁵×C₆₁	1	8	yes	—
482	C₂³	C₂⁵×C₆₁	1	4	yes	—
483	C₂³	C₂⁵×C₆₁	1	2	yes	—
484	C₂³	C₂⁵×C₆₁	1	1	yes	characteristic
485	Q₈	—	2	8	no	—
486	Q₈	—	2	16	no	—
487	D₄	—	2	16	no	—
488	D₄	—	2	32	no	—
489	D₄	—	2	8	no	—
490	D₄	—	2	16	no	—
491	D₄	—	4	32	no	—
492	D₄	—	4	16	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	8	no	—
497	C₂×C₄	—	2	16	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	32	no	—
500	C₂×C₄	—	2	16	no	—
501	C₂×C₄	—	4	16	no	—
502	C₂×C₄	—	4	32	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	4	16	no	—
505	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
506	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
507	C₂×C₄	C₂×#41×C₆₁	1	16	yes	—
508	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
509	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
510	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
511	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
512	C₂×C₄	C₂⁵×C₆₁	1	4	yes	—
513	C₂×C₄	C₂⁵×C₆₁	1	16	yes	—
514	C₂×C₄	C₂⁵×C₆₁	1	2	yes	—
515	C₂×C₄	C₂⁵×C₆₁	1	2	yes	—
516	C₂²	—	2	4	no	—
517	C₂²	—	2	8	no	—
518	C₂²	—	2	8	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	8	no	—
521	C₂²	—	2	8	no	—
522	C₂²	—	2	8	no	—
523	C₂²	—	4	16	no	—
524	C₂²	—	4	8	no	—
525	C₂²	C₂²×#41×C₆₁	1	4	yes	—
526	C₂²	C₂²×#41×C₆₁	1	2	yes	—
527	C₂²	C₆₁×#585	1	4	yes	—
528	C₂²	C₂⁶×C₆₁	1	1	yes	characteristic
529	C₄	—	2	8	no	—
530	C₄	—	2	4	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	8	no	—
533	C₄	—	2	16	no	—
534	C₄	—	2	4	no	—
535	C₄	C₂²×#41×C₆₁	1	4	yes	—
536	C₄	C₂²×#41×C₆₁	1	4	yes	—
537	C₂	—	2	4	no	—
538	C₂	—	4	8	no	—
539	C₂	—	4	4	no	—
540	C₂	C₆₁×#3512	1	4	yes	direct product factor
541	C₂	C₂³×#41×C₆₁	1	1	yes	characteristic
542	C₂	C₂×C₆₁×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₁
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
1. C₆₁×#3512	C₂	direct (N × H)
2. C₂×C₆₁×#553	C₂	semidirect
3. C₂×C₆₁×#553	C₂	semidirect
4. C₂×C₆₁×#552	C₂	semidirect
5. C₂×C₆₁×#541	C₂	semidirect
6. C₂×C₆₁×#540	C₂	semidirect
7. C₂×C₆₁×#537	C₂	semidirect
8. C₂×C₆₁×#532	C₂	semidirect
9. C₂×C₆₁×#529	C₂	semidirect
10. C₂²×#125×C₆₁	C₂	semidirect
11. C₂×C₆₁×#520	C₂	semidirect
12. C₂×C₄×#41×C₆₁	C₂	semidirect
15. C₆₁×#553	C₂²	semidirect
16. C₆₁×#553	C₂²	semidirect
17. C₆₁×#552	C₂²	semidirect
18. C₆₁×#541	C₂²	semidirect
19. C₆₁×#540	C₂²	semidirect
20. C₆₁×#537	C₂²	semidirect
21. C₆₁×#532	C₂²	semidirect
24. C₆₁×#529	C₂²	semidirect
31. C₂×#125×C₆₁	C₂²	semidirect
42. C₆₁×#520	C₂²	semidirect
43. C₄×#41×C₆₁	C₂²	semidirect
244. C₂×#3512	C₆₁	direct (N × H)
266. #3512	C₂×C₆₁	direct (N × H)
267. C₂×#553	C₂×C₆₁	semidirect
268. C₂×#553	C₂×C₆₁	semidirect
269. C₂×#552	C₂×C₆₁	semidirect
270. C₂×#541	C₂×C₆₁	semidirect
271. C₂×#540	C₂×C₆₁	semidirect
272. C₂×#537	C₂×C₆₁	semidirect
273. C₂×#532	C₂×C₆₁	semidirect
274. C₂×#529	C₂×C₆₁	semidirect
275. C₂²×#125	C₂×C₆₁	semidirect
276. C₂×#520	C₂×C₆₁	semidirect
277. C₂×C₄×#41	C₂×C₆₁	semidirect
281. C₂×C₆₁	#3512	direct (N × H)
286. #553	C₂²×C₆₁	semidirect
287. #553	C₂²×C₆₁	semidirect
288. #552	C₂²×C₆₁	semidirect
289. #541	C₂²×C₆₁	semidirect
290. #540	C₂²×C₆₁	semidirect
291. #537	C₂²×C₆₁	semidirect
292. #532	C₂²×C₆₁	semidirect
295. #529	C₂²×C₆₁	semidirect
302. C₂×#125	C₂²×C₆₁	semidirect
313. #520	C₂²×C₆₁	semidirect
314. C₄×#41	C₂²×C₆₁	semidirect
332. C₆₁	C₂×#3512	direct (N × H)
540. C₂	C₆₁×#3512	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×#3512	2
C₂×C₆₁×#553	2
C₂×C₆₁×#552	2
C₂×C₆₁×#541	2
C₂×C₆₁×#540	2
C₂×C₆₁×#537	2
C₂×C₆₁×#532	2
C₂×C₆₁×#529	2
C₂²×#125×C₆₁	2
C₂×C₆₁×#520	2
C₂×C₄×#41×C₆₁	2
C₂×#3512	61

Subgroups — C2×C61×#3512 #29566242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3512 #29566242