Subgroups — C₄×C₆₁×#417 #29523847

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#417	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1763	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1762	C₂	1	2	yes	maximal, complemented
7	C₄×#103×C₆₁	C₂	1	2	yes	maximal, complemented
8	C₄×#102×C₆₁	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1736	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
12	C₄×#41×C₆₁	—	2	4	no	—
13	C₄×#41×C₆₁	C₄	1	8	yes	complemented
14	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₁	—	2	1	no	—
17	C₂×C₄×D₄×C₆₁	—	2	1	no	—
18	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₁	—	2	1	no	—
21	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₁	—	2	1	no	—
24	C₂²×C₄²×C₆₁	—	2	1	no	—
25	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#417	C₄	1	16	yes	direct product factor
27	C₂×#103×C₆₁	C₂²	1	4	yes	—
28	C₂×#102×C₆₁	C₂²	1	4	yes	—
29	C₄×M₁₆×C₆₁	—	2	4	no	—
30	C₂×#97×C₆₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
36	C₂×#95×C₆₁	—	2	1	no	—
37	C₆₁×#340	C₂²	1	4	yes	—
38	C₆₁×#340	C₂²	1	4	yes	—
39	C₂×#41×C₆₁	—	2	2	no	—
40	C₂×#41×C₆₁	—	2	4	no	—
41	C₂×#41×C₆₁	C₂×C₄	1	8	yes	—
42	C₂×#41×C₆₁	C₂×C₄	1	4	yes	—
43	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₆₁	—	2	1	no	—
45	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₁	—	2	1	no	—
47	C₂³×C₄×C₆₁	—	2	2	no	—
48	C₂³×C₄×C₆₁	—	2	1	no	—
49	C₂³×C₄×C₆₁	—	2	1	no	—
50	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₆₁	—	2	4	no	—
54	C₂×M₁₆×C₆₁	—	2	4	no	—
55	C₂×M₁₆×C₆₁	C₂×C₄	1	8	yes	complemented
56	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₄×Q₈×C₆₁	—	2	2	no	—
59	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	complemented
60	C₄×Q₈×C₆₁	C₂×C₄	1	4	yes	complemented
61	C₄×D₄×C₆₁	—	2	8	no	—
62	C₄×D₄×C₆₁	—	2	8	no	—
63	C₄×D₄×C₆₁	—	2	2	no	—
64	C₄×D₄×C₆₁	—	2	2	no	—
65	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	complemented
66	C₄×D₄×C₆₁	C₂×C₄	1	4	yes	complemented
67	#117×C₆₁	—	2	2	no	—
68	#117×C₆₁	—	2	8	no	—
69	#117×C₆₁	C₂×C₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	complemented
75	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
81	C₂×C₄²×C₆₁	—	2	1	no	—
82	C₂×C₄²×C₆₁	—	2	1	no	—
83	C₂×C₄²×C₆₁	—	2	1	no	—
84	C₂×C₄²×C₆₁	—	2	8	no	—
85	C₂×C₄²×C₆₁	—	2	2	no	—
86	C₂×C₄²×C₆₁	—	2	2	no	—
87	C₂×C₄²×C₆₁	—	4	4	no	—
88	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
89	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₁	D₄	1	4	yes	—
92	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
94	#103×C₆₁	C₂×C₄	1	16	yes	complemented
95	#102×C₆₁	C₂×C₄	1	16	yes	complemented
96	#97×C₆₁	—	2	4	no	—
97	C₄×C₈×C₆₁	—	2	4	no	—
98	#95×C₆₁	—	2	2	no	—
99	#95×C₆₁	—	2	2	no	—
100	C₂⁴×C₆₁	—	2	1	no	—
101	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
102	#41×C₆₁	—	2	8	no	—
103	#41×C₆₁	—	2	16	no	—
104	#41×C₆₁	C₄²	1	16	yes	complemented
105	C₂×Q₈×C₆₁	—	2	1	no	—
106	C₂×Q₈×C₆₁	—	2	2	no	—
107	C₂×Q₈×C₆₁	C₂²×C₄	1	4	yes	—
108	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
109	C₂×D₄×C₆₁	—	2	4	no	—
110	C₂×D₄×C₆₁	—	2	8	no	—
111	C₂×D₄×C₆₁	—	2	1	no	—
112	C₂×D₄×C₆₁	—	2	2	no	—
113	C₂×D₄×C₆₁	C₂²×C₄	1	4	yes	—
114	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₆₁	—	2	1	no	—
116	C₂²×C₄×C₆₁	—	2	4	no	—
117	C₂²×C₄×C₆₁	—	2	8	no	—
118	C₂²×C₄×C₆₁	—	2	2	no	—
119	C₂²×C₄×C₆₁	—	2	2	no	—
120	C₂²×C₄×C₆₁	—	2	8	no	—
121	C₂²×C₄×C₆₁	—	2	4	no	—
122	C₂²×C₄×C₆₁	—	2	1	no	—
123	C₂²×C₄×C₆₁	—	2	1	no	—
124	C₂²×C₄×C₆₁	—	2	4	no	—
125	C₂²×C₄×C₆₁	—	2	4	no	—
126	C₂²×C₄×C₆₁	—	4	2	no	—
127	C₂²×C₄×C₆₁	—	4	4	no	—
128	C₂²×C₄×C₆₁	—	4	4	no	—
129	C₂²×C₄×C₆₁	—	4	4	no	—
130	C₂²×C₄×C₆₁	—	4	2	no	—
131	C₂²×C₄×C₆₁	—	4	2	no	—
132	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
133	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
134	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₁	#41	1	4	yes	—
143	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
144	M₁₆×C₆₁	—	2	16	no	—
145	C₂×C₈×C₆₁	—	2	4	no	—
146	C₂×C₈×C₆₁	—	2	4	no	—
147	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
148	C₄⋊C₄×C₆₁	—	2	4	no	—
149	C₄⋊C₄×C₆₁	—	2	4	no	—
150	C₄⋊C₄×C₆₁	—	2	8	no	—
151	C₄⋊C₄×C₆₁	—	2	1	no	—
152	C₄⋊C₄×C₆₁	—	2	1	no	—
153	C₄⋊C₄×C₆₁	—	2	4	no	—
154	C₄⋊C₄×C₆₁	C₂²×C₄	1	8	yes	—
155	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
156	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
157	C₂²⋊C₄×C₆₁	—	2	4	no	—
158	C₂²⋊C₄×C₆₁	—	2	4	no	—
159	C₂²⋊C₄×C₆₁	—	2	4	no	—
160	C₄²×C₆₁	—	2	4	no	—
161	C₄²×C₆₁	—	2	4	no	—
162	C₄²×C₆₁	—	2	4	no	—
163	C₄²×C₆₁	—	4	4	no	—
164	C₄²×C₆₁	—	4	4	no	—
165	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
166	C₄²×C₆₁	C₂²⋊C₄	1	4	yes	—
167	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
168	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
169	C₄²×C₆₁	C₂×D₄	1	2	yes	—
170	C₄²×C₆₁	C₂×D₄	1	2	yes	—
171	C₂³×C₆₁	—	2	1	no	—
172	C₂³×C₆₁	—	2	1	no	—
173	C₂³×C₆₁	—	2	2	no	—
174	C₂³×C₆₁	—	2	2	no	—
175	C₂³×C₆₁	—	2	4	no	—
176	C₂³×C₆₁	—	4	1	no	—
177	C₂³×C₆₁	—	4	1	no	—
178	C₂³×C₆₁	—	4	2	no	—
179	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₂³×C₆₁	C₄×D₄	1	4	yes	—
181	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₆₁	—	2	2	no	—
183	Q₈×C₆₁	—	2	4	no	—
184	Q₈×C₆₁	C₂×C₄²	1	4	yes	complemented
185	D₄×C₆₁	—	2	16	no	—
186	D₄×C₆₁	—	2	2	no	—
187	D₄×C₆₁	—	2	4	no	—
188	D₄×C₆₁	C₂×C₄²	1	4	yes	complemented
189	C₂×C₄×C₆₁	—	2	2	no	—
190	C₂×C₄×C₆₁	—	2	4	no	—
191	C₂×C₄×C₆₁	—	2	1	no	—
192	C₂×C₄×C₆₁	—	2	1	no	—
193	C₂×C₄×C₆₁	—	2	2	no	—
194	C₂×C₄×C₆₁	—	2	2	no	—
195	C₂×C₄×C₆₁	—	2	16	no	—
196	C₂×C₄×C₆₁	—	2	8	no	—
197	C₂×C₄×C₆₁	—	2	2	no	—
198	C₂×C₄×C₆₁	—	2	2	no	—
199	C₂×C₄×C₆₁	—	2	8	no	—
200	C₂×C₄×C₆₁	—	4	4	no	—
201	C₂×C₄×C₆₁	—	4	8	no	—
202	C₂×C₄×C₆₁	—	4	16	no	—
203	C₂×C₄×C₆₁	—	4	4	no	—
204	C₂×C₄×C₆₁	—	4	1	no	—
205	C₂×C₄×C₆₁	—	4	1	no	—
206	C₂×C₄×C₆₁	—	4	2	no	—
207	C₂×C₄×C₆₁	—	4	4	no	—
208	C₂×C₄×C₆₁	—	4	8	no	—
209	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
210	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
211	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
212	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
213	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
214	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
215	C₂×C₄×C₆₁	#117	1	4	yes	—
216	C₂×C₄×C₆₁	C₄×D₄	1	8	yes	—
217	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
218	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₆₁	#137	1	2	yes	—
220	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
224	C₈×C₆₁	—	2	8	no	—
225	C₄×#417	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂²×C₆₁	—	2	2	no	—
227	C₂²×C₆₁	—	2	2	no	—
228	C₂²×C₆₁	—	2	4	no	—
229	C₂²×C₆₁	—	2	4	no	—
230	C₂²×C₆₁	—	2	4	no	—
231	C₂²×C₆₁	—	4	2	no	—
232	C₂²×C₆₁	—	4	2	no	—
233	C₂²×C₆₁	—	4	4	no	—
234	C₂²×C₆₁	—	4	4	no	—
235	C₂²×C₆₁	C₄×C₂²⋊C₄	1	4	yes	—
236	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₆₁	C₂×C₄×D₄	1	2	yes	—
238	C₂²×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂²×C₆₁	C₂×#137	1	1	yes	characteristic
240	C₂²×C₆₁	#575	1	2	yes	—
241	C₄×C₆₁	—	2	2	no	—
242	C₄×C₆₁	—	2	2	no	—
243	C₄×C₆₁	—	2	2	no	—
244	C₄×C₆₁	—	2	4	no	—
245	C₄×C₆₁	—	4	2	no	—
246	C₄×C₆₁	—	4	2	no	—
247	C₄×C₆₁	—	4	8	no	—
248	C₄×C₆₁	C₄×C₂²⋊C₄	1	4	yes	—
249	C₄×C₆₁	#417	1	4	yes	direct product factor
250	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
251	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
252	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
253	C₂×C₄×#41	C₂×C₆₁	1	1	yes	characteristic
254	C₂×#417	C₂×C₆₁	1	2	yes	—
255	C₂×C₄×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
256	C₂×C₄×C₄⋊C₄	C₂×C₆₁	1	1	yes	characteristic, complemented
257	#1763	C₂×C₆₁	1	2	yes	complemented
258	#1762	C₂×C₆₁	1	2	yes	complemented
259	C₄×#103	C₂×C₆₁	1	2	yes	complemented
260	C₄×#102	C₂×C₆₁	1	2	yes	complemented
261	#1736	C₂×C₆₁	1	2	yes	complemented
262	C₂×C₆₁	—	2	4	no	—
263	C₂×C₆₁	—	4	4	no	—
264	C₂×C₆₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₆₁	C₂×#417	1	1	yes	characteristic
266	C₂×C₆₁	#2896	1	1	yes	characteristic
267	C₂×C₆₁	C₄×AGL(1,ℤ/8)	1	2	yes	—
268	C₂×C₆₁	C₄×#137	1	2	yes	—
269	C₂²×#41	C₂²×C₆₁	1	1	yes	characteristic
270	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
271	C₄×#41	—	2	4	no	—
272	C₄×#41	C₄×C₆₁	1	8	yes	complemented
273	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
274	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
275	C₂×C₄×D₄	—	2	1	no	—
276	C₂×C₄×D₄	—	2	1	no	—
277	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
279	C₂×#117	—	2	1	no	—
280	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
282	C₂²×C₄²	—	2	1	no	—
283	C₂²×C₄²	—	2	1	no	—
284	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
285	#417	C₄×C₆₁	1	16	yes	direct product factor
286	C₂×#103	C₂²×C₆₁	1	4	yes	—
287	C₂×#102	C₂²×C₆₁	1	4	yes	—
288	C₄×M₁₆	—	2	4	no	—
289	C₂×#97	C₂²×C₆₁	1	2	yes	—
290	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
291	C₄×C₄⋊C₄	—	2	1	no	—
292	C₄×C₄⋊C₄	—	2	1	no	—
293	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
294	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
295	C₂×#95	—	2	1	no	—
296	#340	C₂²×C₆₁	1	4	yes	—
297	#340	C₂²×C₆₁	1	4	yes	—
298	C₆₁	C₄×#417	1	1	yes	characteristic, direct product factor, Sylow
299	C₂×#41	—	2	2	no	—
300	C₂×#41	—	2	4	no	—
301	C₂×#41	C₂×C₄×C₆₁	1	4	yes	—
302	C₂×#41	C₂×C₄×C₆₁	1	8	yes	—
303	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
304	C₂²×D₄	—	2	1	no	—
305	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
306	C₂³×C₄	—	2	1	no	—
307	C₂³×C₄	—	2	2	no	—
308	C₂³×C₄	—	2	1	no	—
309	C₂³×C₄	—	2	1	no	—
310	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
311	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
313	C₂×M₁₆	—	2	4	no	—
314	C₂×M₁₆	—	2	4	no	—
315	C₂×M₁₆	C₂×C₄×C₆₁	1	8	yes	complemented
316	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
317	C₄×Q₈	—	2	2	no	—
318	C₄×Q₈	—	2	2	no	—
319	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	complemented
320	C₄×Q₈	C₂×C₄×C₆₁	1	4	yes	complemented
321	C₄×D₄	—	2	8	no	—
322	C₄×D₄	—	2	8	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	complemented
326	C₄×D₄	C₂×C₄×C₆₁	1	4	yes	complemented
327	#117	—	2	2	no	—
328	#117	—	2	8	no	—
329	#117	C₂×C₄×C₆₁	1	4	yes	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	4	no	—
334	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	complemented
335	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₄²	—	2	1	no	—
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	—	2	8	no	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	—	2	2	no	—
347	C₂×C₄²	—	4	4	no	—
348	C₂×C₄²	C₂×C₄×C₆₁	1	4	yes	—
349	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
350	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₆₁	1	4	yes	—
352	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
354	#103	C₂×C₄×C₆₁	1	16	yes	complemented
355	#102	C₂×C₄×C₆₁	1	16	yes	complemented
356	#97	—	2	4	no	—
357	C₄×C₈	—	2	4	no	—
358	#95	—	2	2	no	—
359	#95	—	2	2	no	—
360	C₂⁴	—	2	1	no	—
361	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
362	#41	—	2	8	no	—
363	#41	—	2	16	no	—
364	#41	C₄²×C₆₁	1	16	yes	complemented
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	C₂²×C₄×C₆₁	1	4	yes	—
368	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	8	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	C₂²×C₄×C₆₁	1	4	yes	—
374	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	—	4	4	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
393	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
394	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
395	C₂²×C₄	C₂²×C₄×C₆₁	1	4	yes	—
396	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
402	C₂²×C₄	#41×C₆₁	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
404	M₁₆	—	2	16	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	C₂²×C₄×C₆₁	1	8	yes	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	C₂²×C₄×C₆₁	1	8	yes	—
415	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
416	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₄²	—	2	4	no	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	4	4	no	—
424	C₄²	—	4	4	no	—
425	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
426	C₄²	C₂²⋊C₄×C₆₁	1	4	yes	—
427	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
428	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
429	C₄²	C₂×D₄×C₆₁	1	2	yes	—
430	C₄²	C₂×D₄×C₆₁	1	2	yes	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	2	no	—
439	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
440	C₂³	C₄×D₄×C₆₁	1	4	yes	—
441	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
442	Q₈	—	2	2	no	—
443	Q₈	—	2	4	no	—
444	Q₈	C₂×C₄²×C₆₁	1	4	yes	complemented
445	D₄	—	2	16	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	4	no	—
448	D₄	C₂×C₄²×C₆₁	1	4	yes	complemented
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	16	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	8	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	8	no	—
462	C₂×C₄	—	4	16	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	C₂×C₄²×C₆₁	1	4	yes	—
470	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
471	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
472	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
473	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
474	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
475	C₂×C₄	#117×C₆₁	1	4	yes	—
476	C₂×C₄	C₄×D₄×C₆₁	1	4	yes	—
477	C₂×C₄	C₄×D₄×C₆₁	1	8	yes	—
478	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
479	C₂×C₄	#137×C₆₁	1	2	yes	—
480	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
481	C₂×C₄	C₂³×C₄×C₆₁	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
484	C₈	—	2	8	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	C₄×C₂²⋊C₄×C₆₁	1	4	yes	—
495	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
496	C₂²	C₂×C₄×D₄×C₆₁	1	2	yes	—
497	C₂²	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
498	C₂²	C₂×#137×C₆₁	1	1	yes	characteristic
499	C₂²	C₆₁×#575	1	2	yes	—
500	C₄	—	2	4	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	8	no	—
507	C₄	C₄×C₂²⋊C₄×C₆₁	1	4	yes	—
508	C₄	C₆₁×#417	1	4	yes	direct product factor
509	C₄	C₂²×C₄²×C₆₁	1	1	yes	characteristic
510	C₄	C₂×#117×C₆₁	1	1	yes	characteristic
511	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
512	C₂	—	2	4	no	—
513	C₂	—	4	4	no	—
514	C₂	C₂×C₄×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
515	C₂	C₂×C₆₁×#417	1	1	yes	characteristic
516	C₂	C₆₁×#2896	1	1	yes	characteristic
517	C₂	C₄×AGL(1,ℤ/8)×C₆₁	1	2	yes	—
518	C₂	C₄×#137×C₆₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₁	C₂×D₄
commutator	C₄	C₂²×C₄²×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂³×C₆₁	C₂²×D₄
prime radical	C₂²×#41×C₆₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₆₁	C₂	semidirect
4. C₂×C₄×C₄⋊C₄×C₆₁	C₂	semidirect
5. C₆₁×#1763	C₂	semidirect
6. C₆₁×#1762	C₂	semidirect
7. C₄×#103×C₆₁	C₂	semidirect
8. C₄×#102×C₆₁	C₂	semidirect
9. C₆₁×#1736	C₂	semidirect
13. C₄×#41×C₆₁	C₄	semidirect
26. C₆₁×#417	C₄	direct (N × H)
55. C₂×M₁₆×C₆₁	C₂×C₄	semidirect
59. C₄×Q₈×C₆₁	C₂×C₄	semidirect
60. C₄×Q₈×C₆₁	C₂×C₄	semidirect
65. C₄×D₄×C₆₁	C₂×C₄	semidirect
66. C₄×D₄×C₆₁	C₂×C₄	semidirect
74. C₂×C₄⋊C₄×C₆₁	C₂×C₄	semidirect
94. #103×C₆₁	C₂×C₄	semidirect
95. #102×C₆₁	C₂×C₄	semidirect
104. #41×C₆₁	C₄²	semidirect
184. Q₈×C₆₁	C₂×C₄²	semidirect
188. D₄×C₆₁	C₂×C₄²	semidirect
225. C₄×#417	C₆₁	direct (N × H)
249. C₄×C₆₁	#417	direct (N × H)
255. C₂×C₄×M₁₆	C₂×C₆₁	semidirect
256. C₂×C₄×C₄⋊C₄	C₂×C₆₁	semidirect
257. #1763	C₂×C₆₁	semidirect
258. #1762	C₂×C₆₁	semidirect
259. C₄×#103	C₂×C₆₁	semidirect
260. C₄×#102	C₂×C₆₁	semidirect
261. #1736	C₂×C₆₁	semidirect
272. C₄×#41	C₄×C₆₁	semidirect
285. #417	C₄×C₆₁	direct (N × H)
298. C₆₁	C₄×#417	direct (N × H)
315. C₂×M₁₆	C₂×C₄×C₆₁	semidirect
319. C₄×Q₈	C₂×C₄×C₆₁	semidirect
320. C₄×Q₈	C₂×C₄×C₆₁	semidirect
325. C₄×D₄	C₂×C₄×C₆₁	semidirect
326. C₄×D₄	C₂×C₄×C₆₁	semidirect
334. C₂×C₄⋊C₄	C₂×C₄×C₆₁	semidirect
354. #103	C₂×C₄×C₆₁	semidirect
355. #102	C₂×C₄×C₆₁	semidirect
364. #41	C₄²×C₆₁	semidirect
444. Q₈	C₂×C₄²×C₆₁	semidirect
448. D₄	C₂×C₄²×C₆₁	semidirect
508. C₄	C₆₁×#417	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₁	2
C₂×C₆₁×#417	2
C₂×C₄×M₁₆×C₆₁	2
C₂×C₄×C₄⋊C₄×C₆₁	2
C₆₁×#1763	2
C₆₁×#1762	2
C₄×#103×C₆₁	2
C₄×#102×C₆₁	2
C₆₁×#1736	2
C₄×#417	61

Subgroups — C4×C61×#417 #29523847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#417 #29523847