Subgroups — C₂×C₆₁×#3046 #29563854

Order 15616.

2742 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#3046	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₆₁×#464	C₂	1	2	yes	maximal, complemented
7	C₂×C₆₁×#462	C₂	1	2	yes	maximal, complemented
8	C₂×C₆₁×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₆₁×#461	C₂	1	2	yes	maximal, complemented
10	C₂×C₆₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#103×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₆₁	C₂²	1	1	yes	characteristic
15	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#137×C₆₁	—	2	8	no	—
17	C₂×#137×C₆₁	C₂²	1	8	yes	complemented
18	C₂×#135×C₆₁	—	2	8	no	—
19	C₂×#135×C₆₁	C₂²	1	8	yes	—
20	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×Q₁₆×C₆₁	C₂²	1	2	yes	—
22	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
23	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#536	—	2	4	no	—
29	C₆₁×#536	C₂²	1	8	yes	complemented
30	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
31	C₂×#124×C₆₁	C₂²	1	2	yes	—
32	C₂×#123×C₆₁	—	2	2	no	—
33	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
34	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
35	C₆₁×#520	—	2	4	no	—
36	C₆₁×#520	C₂²	1	8	yes	complemented
37	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
38	C₂×C₄×D₄×C₆₁	—	2	2	no	—
39	C₂×C₄×D₄×C₆₁	C₂²	1	2	yes	—
40	C₂×#117×C₆₁	—	2	1	no	—
41	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
42	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
43	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#464	C₂²	1	16	yes	complemented
45	C₆₁×#463	C₂²	1	16	yes	complemented
46	C₆₁×#462	C₂²	1	16	yes	complemented
47	C₆₁×#461	C₂²	1	16	yes	complemented
48	C₆₁×#424	C₂²	1	8	yes	complemented
49	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
50	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
51	C₆₁×#417	C₂²	1	8	yes	complemented
52	C₂×#103×C₆₁	C₂²	1	2	yes	—
53	C₂×#103×C₆₁	C₂²	1	2	yes	—
54	C₂×#103×C₆₁	C₂²	1	2	yes	—
55	C₂×#103×C₆₁	C₂²	1	8	yes	complemented
56	C₂×#102×C₆₁	C₂²	1	2	yes	—
57	C₂×#41×C₆₁	—	2	2	no	—
58	C₂×#41×C₆₁	—	2	2	no	—
59	C₂×#41×C₆₁	—	2	4	no	—
60	C₂×#41×C₆₁	—	2	4	no	—
61	C₂×#41×C₆₁	—	2	4	no	—
62	C₂×#41×C₆₁	D₄	1	8	yes	—
63	C₂×#41×C₆₁	C₂³	1	4	yes	—
64	C₂×#41×C₆₁	C₂³	1	4	yes	—
65	C₂²×Q₈×C₆₁	—	2	2	no	—
66	C₂²×Q₈×C₆₁	—	2	1	no	—
67	C₂²×Q₈×C₆₁	—	2	4	no	—
68	C₂²×Q₈×C₆₁	—	2	4	no	—
69	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
71	C₂²×Q₈×C₆₁	C₂³	1	4	yes	—
72	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₆₁	—	2	1	no	—
74	C₂²×D₄×C₆₁	—	2	1	no	—
75	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
76	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₁	—	2	1	no	—
78	C₂³×C₄×C₆₁	—	2	1	no	—
79	C₂³×C₄×C₆₁	—	2	1	no	—
80	C₂³×C₄×C₆₁	—	2	1	no	—
81	C₂³×C₄×C₆₁	—	2	2	no	—
82	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
83	#137×C₆₁	—	2	32	no	—
84	#135×C₆₁	—	2	32	no	—
85	C₂×Q₁₆×C₆₁	—	2	4	no	—
86	C₂×Q₁₆×C₆₁	—	2	8	no	—
87	C₂×Q₁₆×C₆₁	C₂³	1	4	yes	—
88	C₂×Q₁₆×C₆₁	C₂³	1	8	yes	—
89	C₂×SD₁₆×C₆₁	—	2	4	no	—
90	C₂×SD₁₆×C₆₁	—	2	4	no	—
91	C₂×SD₁₆×C₆₁	—	2	8	no	—
92	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₆₁	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₆₁	C₂³	1	8	yes	—
95	C₂×D₈×C₆₁	—	2	4	no	—
96	C₂×D₈×C₆₁	C₂³	1	4	yes	—
97	C₂×M₁₆×C₆₁	—	2	4	no	—
98	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
99	C₂²×C₈×C₆₁	—	2	4	no	—
100	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
101	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
102	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
103	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
104	C₄⋊Q₈×C₆₁	—	2	4	no	—
105	C₄⋊Q₈×C₆₁	C₂³	1	8	yes	—
106	#124×C₆₁	—	2	4	no	—
107	#124×C₆₁	C₂³	1	8	yes	—
108	#123×C₆₁	—	2	16	no	—
109	C₂²⋊Q₈×C₆₁	—	2	8	no	—
110	C₂²⋊Q₈×C₆₁	—	2	8	no	—
111	C₄×Q₈×C₆₁	—	2	4	no	—
112	C₄×Q₈×C₆₁	C₂³	1	8	yes	—
113	C₄×D₄×C₆₁	—	2	16	no	—
114	C₄×D₄×C₆₁	—	2	4	no	—
115	C₄×D₄×C₆₁	C₂³	1	8	yes	—
116	#117×C₆₁	—	2	8	no	—
117	#117×C₆₁	D₄	1	4	yes	—
118	#117×C₆₁	C₂³	1	4	yes	—
119	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
121	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
122	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
123	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
124	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
125	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
126	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
128	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
129	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
131	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
132	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
133	C₂×C₄²×C₆₁	—	2	2	no	—
134	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
135	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
136	C₄⋊C₈×C₆₁	C₂³	1	8	yes	—
137	#103×C₆₁	C₂³	1	8	yes	—
138	#103×C₆₁	C₂³	1	8	yes	—
139	#103×C₆₁	C₂³	1	8	yes	—
140	#102×C₆₁	C₂³	1	8	yes	—
141	C₂⁴×C₆₁	—	2	1	no	—
142	C₂⁴×C₆₁	—	2	1	no	—
143	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
144	#41×C₆₁	—	2	16	no	—
145	#41×C₆₁	—	2	8	no	—
146	#41×C₆₁	—	2	8	no	—
147	#41×C₆₁	—	2	8	no	—
148	#41×C₆₁	—	4	8	no	—
149	#41×C₆₁	C₂×D₄	1	16	yes	—
150	C₂×Q₈×C₆₁	—	2	1	no	—
151	C₂×Q₈×C₆₁	—	2	4	no	—
152	C₂×Q₈×C₆₁	—	2	2	no	—
153	C₂×Q₈×C₆₁	—	2	1	no	—
154	C₂×Q₈×C₆₁	—	2	2	no	—
155	C₂×Q₈×C₆₁	—	2	4	no	—
156	C₂×Q₈×C₆₁	—	2	2	no	—
157	C₂×Q₈×C₆₁	—	2	8	no	—
158	C₂×Q₈×C₆₁	—	2	4	no	—
159	C₂×Q₈×C₆₁	—	2	8	no	—
160	C₂×Q₈×C₆₁	—	2	2	no	—
161	C₂×Q₈×C₆₁	—	4	4	no	—
162	C₂×Q₈×C₆₁	—	4	8	no	—
163	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
164	C₂×Q₈×C₆₁	C₂⁴	1	4	yes	—
165	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
166	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
167	C₂×D₄×C₆₁	—	2	8	no	—
168	C₂×D₄×C₆₁	—	2	4	no	—
169	C₂×D₄×C₆₁	—	2	4	no	—
170	C₂×D₄×C₆₁	—	2	1	no	—
171	C₂×D₄×C₆₁	—	2	2	no	—
172	C₂×D₄×C₆₁	—	2	1	no	—
173	C₂×D₄×C₆₁	—	2	2	no	—
174	C₂×D₄×C₆₁	—	2	2	no	—
175	C₂×D₄×C₆₁	—	4	4	no	—
176	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
177	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
178	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₁	—	2	1	no	—
180	C₂²×C₄×C₆₁	—	2	8	no	—
181	C₂²×C₄×C₆₁	—	2	4	no	—
182	C₂²×C₄×C₆₁	—	2	4	no	—
183	C₂²×C₄×C₆₁	—	2	2	no	—
184	C₂²×C₄×C₆₁	—	2	2	no	—
185	C₂²×C₄×C₆₁	—	2	1	no	—
186	C₂²×C₄×C₆₁	—	2	4	no	—
187	C₂²×C₄×C₆₁	—	2	4	no	—
188	C₂²×C₄×C₆₁	—	2	8	no	—
189	C₂²×C₄×C₆₁	—	4	4	no	—
190	C₂²×C₄×C₆₁	—	4	4	no	—
191	C₂²×C₄×C₆₁	—	8	4	no	—
192	C₂²×C₄×C₆₁	—	8	2	no	—
193	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
198	C₂²×C₄×C₆₁	#41	1	4	yes	—
199	C₂²×C₄×C₆₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
201	Q₁₆×C₆₁	—	2	8	no	—
202	Q₁₆×C₆₁	—	2	16	no	—
203	SD₁₆×C₆₁	—	2	8	no	—
204	SD₁₆×C₆₁	—	2	8	no	—
205	SD₁₆×C₆₁	—	2	16	no	—
206	D₈×C₆₁	—	2	8	no	—
207	M₁₆×C₆₁	—	2	8	no	—
208	C₂×C₈×C₆₁	—	2	2	no	—
209	C₂×C₈×C₆₁	—	2	2	no	—
210	C₂×C₈×C₆₁	—	2	8	no	—
211	C₂×C₈×C₆₁	—	2	4	no	—
212	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
213	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
214	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
215	C₄⋊C₄×C₆₁	—	2	4	no	—
216	C₄⋊C₄×C₆₁	—	2	8	no	—
217	C₄⋊C₄×C₆₁	—	2	2	no	—
218	C₄⋊C₄×C₆₁	—	2	8	no	—
219	C₄⋊C₄×C₆₁	—	2	8	no	—
220	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
221	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
222	C₄⋊C₄×C₆₁	C₂⁴	1	4	yes	—
223	C₂²⋊C₄×C₆₁	—	2	8	no	—
224	C₂²⋊C₄×C₆₁	—	2	8	no	—
225	C₂²⋊C₄×C₆₁	—	2	8	no	—
226	C₂²⋊C₄×C₆₁	C₂×D₄	1	8	yes	—
227	C₄²×C₆₁	—	2	8	no	—
228	C₄²×C₆₁	C₂×D₄	1	4	yes	—
229	C₄²×C₆₁	C₂⁴	1	4	yes	—
230	C₂³×C₆₁	—	2	2	no	—
231	C₂³×C₆₁	—	2	2	no	—
232	C₂³×C₆₁	—	2	4	no	—
233	C₂³×C₆₁	—	2	2	no	—
234	C₂³×C₆₁	—	2	2	no	—
235	C₂³×C₆₁	—	4	2	no	—
236	C₂³×C₆₁	—	4	2	no	—
237	C₂³×C₆₁	—	4	2	no	—
238	C₂³×C₆₁	—	4	2	no	—
239	C₂³×C₆₁	—	4	2	no	—
240	C₂³×C₆₁	#121	1	4	yes	—
241	C₂³×C₆₁	C₂²×D₄	1	2	yes	—
242	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
243	Q₈×C₆₁	—	2	4	no	—
244	Q₈×C₆₁	—	2	2	no	—
245	Q₈×C₆₁	—	2	4	no	—
246	Q₈×C₆₁	—	2	2	no	—
247	Q₈×C₆₁	—	2	4	no	—
248	Q₈×C₆₁	—	2	2	no	—
249	Q₈×C₆₁	—	4	8	no	—
250	Q₈×C₆₁	—	4	4	no	—
251	Q₈×C₆₁	—	4	2	no	—
252	Q₈×C₆₁	C₂²×D₄	1	4	yes	—
253	D₄×C₆₁	—	2	16	no	—
254	D₄×C₆₁	—	2	4	no	—
255	D₄×C₆₁	—	2	2	no	—
256	D₄×C₆₁	—	2	2	no	—
257	D₄×C₆₁	—	2	2	no	—
258	D₄×C₆₁	—	4	8	no	—
259	D₄×C₆₁	—	4	2	no	—
260	D₄×C₆₁	C₂²×D₄	1	4	yes	—
261	C₂×C₄×C₆₁	—	2	4	no	—
262	C₂×C₄×C₆₁	—	2	4	no	—
263	C₂×C₄×C₆₁	—	2	4	no	—
264	C₂×C₄×C₆₁	—	2	2	no	—
265	C₂×C₄×C₆₁	—	2	16	no	—
266	C₂×C₄×C₆₁	—	2	4	no	—
267	C₂×C₄×C₆₁	—	2	4	no	—
268	C₂×C₄×C₆₁	—	2	2	no	—
269	C₂×C₄×C₆₁	—	4	2	no	—
270	C₂×C₄×C₆₁	—	4	8	no	—
271	C₂×C₄×C₆₁	—	4	4	no	—
272	C₂×C₄×C₆₁	—	4	4	no	—
273	C₂×C₄×C₆₁	—	4	8	no	—
274	C₂×C₄×C₆₁	—	8	8	no	—
275	C₂×C₄×C₆₁	—	8	4	no	—
276	C₂×C₄×C₆₁	#121	1	4	yes	—
277	C₂×C₄×C₆₁	#121	1	8	yes	—
278	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
279	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
280	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
284	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
285	C₂×C₄×C₆₁	C₂×#41	1	4	yes	—
286	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
287	C₈×C₆₁	—	2	2	no	—
288	C₈×C₆₁	—	2	2	no	—
289	C₈×C₆₁	—	2	4	no	—
290	C₂×#3046	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂²×C₆₁	—	2	8	no	—
292	C₂²×C₆₁	—	2	2	no	—
293	C₂²×C₆₁	—	2	2	no	—
294	C₂²×C₆₁	—	2	4	no	—
295	C₂²×C₆₁	—	4	8	no	—
296	C₂²×C₆₁	—	4	2	no	—
297	C₂²×C₆₁	—	4	2	no	—
298	C₂²×C₆₁	—	4	2	no	—
299	C₂²×C₆₁	—	4	2	no	—
300	C₂²×C₆₁	—	4	2	no	—
301	C₂²×C₆₁	—	8	4	no	—
302	C₂²×C₆₁	C₂×#121	1	4	yes	—
303	C₂²×C₆₁	C₂×#121	1	2	yes	—
304	C₂²×C₆₁	#577	1	2	yes	—
305	C₂²×C₆₁	#578	1	2	yes	—
306	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
307	C₄×C₆₁	—	2	4	no	—
308	C₄×C₆₁	—	2	4	no	—
309	C₄×C₆₁	—	2	4	no	—
310	C₄×C₆₁	—	4	2	no	—
311	C₄×C₆₁	—	4	4	no	—
312	C₄×C₆₁	—	4	4	no	—
313	C₄×C₆₁	C₂×#121	1	4	yes	—
314	C₄×C₆₁	C₂×#121	1	2	yes	—
315	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
316	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
317	C₂²×#137	C₂×C₆₁	1	1	yes	characteristic, complemented
318	C₂²×#135	C₂×C₆₁	1	1	yes	characteristic
319	C₂×#536	C₂×C₆₁	1	1	yes	characteristic, complemented
320	C₂×#520	C₂×C₆₁	1	1	yes	characteristic, complemented
321	#3046	C₂×C₆₁	1	16	yes	direct product factor
322	C₂×#464	C₂×C₆₁	1	2	yes	complemented
323	C₂×#462	C₂×C₆₁	1	2	yes	complemented
324	C₂×#463	C₂×C₆₁	1	2	yes	complemented
325	C₂×#461	C₂×C₆₁	1	2	yes	complemented
326	C₂×#424	C₂×C₆₁	1	1	yes	characteristic, complemented
327	C₂×#417	C₂×C₆₁	1	1	yes	characteristic, complemented
328	C₂²×#103	C₂×C₆₁	1	1	yes	characteristic, complemented
329	C₂×C₆₁	—	2	4	no	—
330	C₂×C₆₁	—	4	4	no	—
331	C₂×C₆₁	—	8	2	no	—
332	C₂×C₆₁	#3046	1	4	yes	direct product factor
333	C₂×C₆₁	C₂²×#121	1	1	yes	characteristic
334	C₂×C₆₁	C₂×#577	1	1	yes	characteristic
335	C₂×C₆₁	C₂×#578	1	1	yes	characteristic
336	C₂²×#41	C₂²×C₆₁	1	1	yes	characteristic
337	C₂²×#41	C₂²×C₆₁	1	1	yes	characteristic
338	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
339	C₂×#137	—	2	8	no	—
340	C₂×#137	C₂²×C₆₁	1	8	yes	complemented
341	C₂×#135	—	2	8	no	—
342	C₂×#135	C₂²×C₆₁	1	8	yes	—
343	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
344	C₂²×Q₁₆	C₂²×C₆₁	1	2	yes	—
345	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
346	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
347	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
348	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
349	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
350	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
351	#536	—	2	4	no	—
352	#536	C₂²×C₆₁	1	8	yes	complemented
353	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
354	C₂×#124	C₂²×C₆₁	1	2	yes	—
355	C₂×#123	—	2	2	no	—
356	C₂×C₂²⋊Q₈	—	2	1	no	—
357	C₂×C₂²⋊Q₈	—	2	1	no	—
358	#520	—	2	4	no	—
359	#520	C₂²×C₆₁	1	8	yes	complemented
360	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
361	C₂×C₄×D₄	—	2	2	no	—
362	C₂×C₄×D₄	C₂²×C₆₁	1	2	yes	—
363	C₂×#117	—	2	1	no	—
364	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
365	C₂²×C₄⋊C₄	—	2	1	no	—
366	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
367	#464	C₂²×C₆₁	1	16	yes	complemented
368	#463	C₂²×C₆₁	1	16	yes	complemented
369	#462	C₂²×C₆₁	1	16	yes	complemented
370	#461	C₂²×C₆₁	1	16	yes	complemented
371	#424	C₂²×C₆₁	1	8	yes	complemented
372	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
373	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
374	#417	C₂²×C₆₁	1	8	yes	complemented
375	C₂×#103	C₂²×C₆₁	1	2	yes	—
376	C₂×#103	C₂²×C₆₁	1	2	yes	—
377	C₂×#103	C₂²×C₆₁	1	2	yes	—
378	C₂×#103	C₂²×C₆₁	1	8	yes	complemented
379	C₂×#102	C₂²×C₆₁	1	2	yes	—
380	C₆₁	C₂×#3046	1	1	yes	characteristic, direct product factor, Sylow
381	C₂×#41	—	2	2	no	—
382	C₂×#41	—	2	2	no	—
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	4	no	—
386	C₂×#41	D₄×C₆₁	1	8	yes	—
387	C₂×#41	C₂³×C₆₁	1	4	yes	—
388	C₂×#41	C₂³×C₆₁	1	4	yes	—
389	C₂²×Q₈	—	2	1	no	—
390	C₂²×Q₈	—	2	2	no	—
391	C₂²×Q₈	—	2	4	no	—
392	C₂²×Q₈	—	2	4	no	—
393	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
394	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
395	C₂²×Q₈	C₂³×C₆₁	1	4	yes	—
396	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
397	C₂²×D₄	—	2	1	no	—
398	C₂²×D₄	—	2	1	no	—
399	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
400	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
401	C₂³×C₄	—	2	1	no	—
402	C₂³×C₄	—	2	1	no	—
403	C₂³×C₄	—	2	1	no	—
404	C₂³×C₄	—	2	1	no	—
405	C₂³×C₄	—	2	2	no	—
406	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
407	#137	—	2	32	no	—
408	#135	—	2	32	no	—
409	C₂×Q₁₆	—	2	4	no	—
410	C₂×Q₁₆	—	2	8	no	—
411	C₂×Q₁₆	C₂³×C₆₁	1	4	yes	—
412	C₂×Q₁₆	C₂³×C₆₁	1	8	yes	—
413	C₂×SD₁₆	—	2	4	no	—
414	C₂×SD₁₆	—	2	4	no	—
415	C₂×SD₁₆	—	2	8	no	—
416	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
417	C₂×SD₁₆	C₂³×C₆₁	1	4	yes	—
418	C₂×SD₁₆	C₂³×C₆₁	1	8	yes	—
419	C₂×D₈	—	2	4	no	—
420	C₂×D₈	C₂³×C₆₁	1	4	yes	—
421	C₂×M₁₆	—	2	4	no	—
422	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
423	C₂²×C₈	—	2	4	no	—
424	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
426	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
427	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
428	C₄⋊Q₈	—	2	4	no	—
429	C₄⋊Q₈	C₂³×C₆₁	1	8	yes	—
430	#124	—	2	4	no	—
431	#124	C₂³×C₆₁	1	8	yes	—
432	#123	—	2	16	no	—
433	C₂²⋊Q₈	—	2	8	no	—
434	C₂²⋊Q₈	—	2	8	no	—
435	C₄×Q₈	—	2	4	no	—
436	C₄×Q₈	C₂³×C₆₁	1	8	yes	—
437	C₄×D₄	—	2	16	no	—
438	C₄×D₄	—	2	4	no	—
439	C₄×D₄	C₂³×C₆₁	1	8	yes	—
440	#117	—	2	8	no	—
441	#117	D₄×C₆₁	1	4	yes	—
442	#117	C₂³×C₆₁	1	4	yes	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	—	2	2	no	—
446	C₂×C₄⋊C₄	—	2	2	no	—
447	C₂×C₄⋊C₄	—	2	2	no	—
448	C₂×C₄⋊C₄	—	2	8	no	—
449	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
450	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
453	C₂×C₂²⋊C₄	—	2	2	no	—
454	C₂×C₂²⋊C₄	—	2	2	no	—
455	C₂×C₂²⋊C₄	—	2	2	no	—
456	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
457	C₂×C₄²	—	2	2	no	—
458	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
459	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
460	C₄⋊C₈	C₂³×C₆₁	1	8	yes	—
461	#103	C₂³×C₆₁	1	8	yes	—
462	#103	C₂³×C₆₁	1	8	yes	—
463	#103	C₂³×C₆₁	1	8	yes	—
464	#102	C₂³×C₆₁	1	8	yes	—
465	C₂⁴	—	2	1	no	—
466	C₂⁴	—	2	1	no	—
467	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
468	#41	—	2	16	no	—
469	#41	—	2	8	no	—
470	#41	—	2	8	no	—
471	#41	—	2	8	no	—
472	#41	—	4	8	no	—
473	#41	C₂×D₄×C₆₁	1	16	yes	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	2	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	2	4	no	—
480	C₂×Q₈	—	2	2	no	—
481	C₂×Q₈	—	2	8	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	8	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	—	4	4	no	—
486	C₂×Q₈	—	4	8	no	—
487	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
488	C₂×Q₈	C₂⁴×C₆₁	1	4	yes	—
489	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
490	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
491	C₂×D₄	—	2	8	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	2	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	4	4	no	—
500	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
501	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
502	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	8	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	4	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	8	no	—
513	C₂²×C₄	—	4	4	no	—
514	C₂²×C₄	—	4	4	no	—
515	C₂²×C₄	—	8	4	no	—
516	C₂²×C₄	—	8	2	no	—
517	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
518	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
519	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
520	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
521	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
522	C₂²×C₄	#41×C₆₁	1	4	yes	—
523	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
524	C₂²×C₄	C₂⁴×C₆₁	1	2	yes	—
525	Q₁₆	—	2	8	no	—
526	Q₁₆	—	2	16	no	—
527	SD₁₆	—	2	8	no	—
528	SD₁₆	—	2	8	no	—
529	SD₁₆	—	2	16	no	—
530	D₈	—	2	8	no	—
531	M₁₆	—	2	8	no	—
532	C₂×C₈	—	2	2	no	—
533	C₂×C₈	—	2	2	no	—
534	C₂×C₈	—	2	8	no	—
535	C₂×C₈	—	2	4	no	—
536	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
537	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
538	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	8	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	8	no	—
544	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
545	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
546	C₄⋊C₄	C₂⁴×C₆₁	1	4	yes	—
547	C₂²⋊C₄	—	2	8	no	—
548	C₂²⋊C₄	—	2	8	no	—
549	C₂²⋊C₄	—	2	8	no	—
550	C₂²⋊C₄	C₂×D₄×C₆₁	1	8	yes	—
551	C₄²	—	2	8	no	—
552	C₄²	C₂×D₄×C₆₁	1	4	yes	—
553	C₄²	C₂⁴×C₆₁	1	4	yes	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	4	2	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	—	4	2	no	—
563	C₂³	—	4	2	no	—
564	C₂³	#121×C₆₁	1	4	yes	—
565	C₂³	C₂²×D₄×C₆₁	1	2	yes	—
566	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
567	Q₈	—	2	2	no	—
568	Q₈	—	2	4	no	—
569	Q₈	—	2	4	no	—
570	Q₈	—	2	2	no	—
571	Q₈	—	2	4	no	—
572	Q₈	—	2	2	no	—
573	Q₈	—	4	8	no	—
574	Q₈	—	4	4	no	—
575	Q₈	—	4	2	no	—
576	Q₈	C₂²×D₄×C₆₁	1	4	yes	—
577	D₄	—	2	16	no	—
578	D₄	—	2	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	2	2	no	—
581	D₄	—	2	4	no	—
582	D₄	—	4	8	no	—
583	D₄	—	4	2	no	—
584	D₄	C₂²×D₄×C₆₁	1	4	yes	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	16	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	8	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	—	4	8	no	—
598	C₂×C₄	—	8	8	no	—
599	C₂×C₄	—	8	4	no	—
600	C₂×C₄	#121×C₆₁	1	4	yes	—
601	C₂×C₄	#121×C₆₁	1	8	yes	—
602	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
603	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
604	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
605	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
606	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
607	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
608	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
609	C₂×C₄	C₂×#41×C₆₁	1	4	yes	—
610	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
611	C₈	—	2	2	no	—
612	C₈	—	2	2	no	—
613	C₈	—	2	4	no	—
614	C₂²	—	2	8	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	4	no	—
618	C₂²	—	4	8	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	2	no	—
622	C₂²	—	4	2	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	8	4	no	—
625	C₂²	C₂×#121×C₆₁	1	4	yes	—
626	C₂²	C₂×#121×C₆₁	1	2	yes	—
627	C₂²	C₆₁×#577	1	2	yes	—
628	C₂²	C₆₁×#578	1	2	yes	—
629	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
630	C₄	—	2	4	no	—
631	C₄	—	2	4	no	—
632	C₄	—	2	4	no	—
633	C₄	—	4	4	no	—
634	C₄	—	4	2	no	—
635	C₄	—	4	4	no	—
636	C₄	C₂×#121×C₆₁	1	4	yes	—
637	C₄	C₂×#121×C₆₁	1	2	yes	—
638	C₄	C₂³×D₄×C₆₁	1	1	yes	characteristic
639	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
640	C₂	—	2	4	no	—
641	C₂	—	4	4	no	—
642	C₂	—	8	2	no	—
643	C₂	C₆₁×#3046	1	4	yes	direct product factor
644	C₂	C₂²×#121×C₆₁	1	1	yes	characteristic
645	C₂	C₂×C₆₁×#577	1	1	yes	characteristic
646	C₂	C₂×C₆₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
prime radical	C₂²×#135×C₆₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×#117×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₆₁	C₂	semidirect
3. C₂×C₆₁×#536	C₂	semidirect
4. C₂×C₆₁×#520	C₂	semidirect
5. C₆₁×#3046	C₂	direct (N × H)
6. C₂×C₆₁×#464	C₂	semidirect
7. C₂×C₆₁×#462	C₂	semidirect
8. C₂×C₆₁×#463	C₂	semidirect
9. C₂×C₆₁×#461	C₂	semidirect
10. C₂×C₆₁×#424	C₂	semidirect
11. C₂×C₆₁×#417	C₂	semidirect
12. C₂²×#103×C₆₁	C₂	semidirect
17. C₂×#137×C₆₁	C₂²	semidirect
29. C₆₁×#536	C₂²	semidirect
36. C₆₁×#520	C₂²	semidirect
44. C₆₁×#464	C₂²	semidirect
45. C₆₁×#463	C₂²	semidirect
46. C₆₁×#462	C₂²	semidirect
47. C₆₁×#461	C₂²	semidirect
48. C₆₁×#424	C₂²	semidirect
51. C₆₁×#417	C₂²	semidirect
55. C₂×#103×C₆₁	C₂²	semidirect
290. C₂×#3046	C₆₁	direct (N × H)
317. C₂²×#137	C₂×C₆₁	semidirect
319. C₂×#536	C₂×C₆₁	semidirect
320. C₂×#520	C₂×C₆₁	semidirect
321. #3046	C₂×C₆₁	direct (N × H)
322. C₂×#464	C₂×C₆₁	semidirect
323. C₂×#462	C₂×C₆₁	semidirect
324. C₂×#463	C₂×C₆₁	semidirect
325. C₂×#461	C₂×C₆₁	semidirect
326. C₂×#424	C₂×C₆₁	semidirect
327. C₂×#417	C₂×C₆₁	semidirect
328. C₂²×#103	C₂×C₆₁	semidirect
332. C₂×C₆₁	#3046	direct (N × H)
340. C₂×#137	C₂²×C₆₁	semidirect
352. #536	C₂²×C₆₁	semidirect
359. #520	C₂²×C₆₁	semidirect
367. #464	C₂²×C₆₁	semidirect
368. #463	C₂²×C₆₁	semidirect
369. #462	C₂²×C₆₁	semidirect
370. #461	C₂²×C₆₁	semidirect
371. #424	C₂²×C₆₁	semidirect
374. #417	C₂²×C₆₁	semidirect
378. C₂×#103	C₂²×C₆₁	semidirect
380. C₆₁	C₂×#3046	direct (N × H)
643. C₂	C₆₁×#3046	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#137×C₆₁	2
C₂²×#135×C₆₁	2
C₂×C₆₁×#536	2
C₂×C₆₁×#520	2
C₆₁×#3046	2
C₂×C₆₁×#464	2
C₂×C₆₁×#462	2
C₂×C₆₁×#463	2
C₂×C₆₁×#461	2
C₂×C₆₁×#424	2
C₂×C₆₁×#417	2
C₂²×#103×C₆₁	2
C₂×#3046	61

Subgroups — C2×C61×#3046 #29563854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#3046 #29563854