Subgroups — C₄×C₆₁×#453 #29524902

Order 15616.

1578 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#453	C₂	1	1	yes	maximal, characteristic
3	C₄×AGL(1,ℤ/8)×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₄×#127×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1883	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1859	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1856	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1792	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#104×C₆₁	C₂	1	2	yes	maximal, complemented
11	C₄×#100×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₆₁	C₂²	1	2	yes	—
14	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#520	—	2	2	no	—
16	C₆₁×#520	C₂²	1	2	yes	—
17	C₆₁×#518	—	2	2	no	—
18	C₆₁×#518	—	2	2	no	—
19	C₆₁×#518	C₂²	1	1	yes	characteristic
20	C₄×#41×C₆₁	—	2	2	no	—
21	C₄×#41×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₁	—	2	2	no	—
23	C₂×C₄×D₄×C₆₁	—	2	2	no	—
24	C₂×C₄×D₄×C₆₁	—	2	2	no	—
25	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic, complemented
27	C₆₁×#453	C₄	1	8	yes	direct product factor
28	C₆₁×#442	—	2	2	no	—
29	C₆₁×#440	—	2	2	no	—
30	C₄×SD₁₆×C₆₁	—	2	2	no	—
31	C₄×D₈×C₆₁	—	2	2	no	—
32	C₆₁×#428	C₂²	1	2	yes	complemented
33	C₂×#104×C₆₁	C₂²	1	2	yes	—
34	C₂×#104×C₆₁	C₂²	1	2	yes	—
35	C₆₁×#418	C₂²	1	2	yes	complemented
36	C₆₁×#417	C₂²	1	2	yes	—
37	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#100×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×M₁₆×C₆₁	C₂²	1	2	yes	complemented
40	C₆₁×#390	—	2	2	no	—
41	C₆₁×#390	C₂²	1	1	yes	characteristic
42	C₆₁×#390	C₂²	1	1	yes	characteristic, complemented
43	C₆₁×#384	C₂²	1	1	yes	characteristic, complemented
44	C₄³×C₆₁	C₂²	1	1	yes	characteristic, complemented
45	C₆₁×#343	C₂²	1	2	yes	complemented
46	C₆₁×#339	C₂²	1	2	yes	complemented
47	C₆₁×#339	C₂²	1	2	yes	complemented
48	H(5,𝔽₂)×C₆₁	—	2	4	no	—
49	H(5,𝔽₂)×C₆₁	—	2	2	no	—
50	H(5,𝔽₂)×C₆₁	C₂×C₄	1	4	yes	complemented
51	C₂×#41×C₆₁	—	2	2	no	—
52	C₂×#41×C₆₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₁	—	2	2	no	—
54	C₂²×D₄×C₆₁	—	2	2	no	—
55	C₂²×D₄×C₆₁	—	2	2	no	—
56	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₁	—	2	2	no	—
59	C₂³×C₄×C₆₁	—	2	1	no	—
60	C₂³×C₄×C₆₁	D₄	1	2	yes	—
61	AGL(1,ℤ/8)×C₆₁	—	2	4	no	—
62	AGL(1,ℤ/8)×C₆₁	C₂×C₄	1	8	yes	complemented
63	C₂×SD₁₆×C₆₁	—	2	2	no	—
64	C₂×D₈×C₆₁	—	2	2	no	—
65	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
67	#127×C₆₁	—	2	2	no	—
68	#127×C₆₁	C₂×C₄	1	4	yes	complemented
69	C₄×Q₈×C₆₁	—	2	2	no	—
70	C₄×Q₈×C₆₁	D₄	1	2	yes	complemented
71	C₄×Q₈×C₆₁	D₄	1	2	yes	complemented
72	C₄×D₄×C₆₁	—	2	4	no	—
73	C₄×D₄×C₆₁	—	2	4	no	—
74	C₄×D₄×C₆₁	—	2	4	no	—
75	C₄×D₄×C₆₁	—	2	4	no	—
76	C₄×D₄×C₆₁	—	2	2	no	—
77	C₄×D₄×C₆₁	—	2	2	no	—
78	C₄×D₄×C₆₁	—	2	4	no	—
79	C₄×D₄×C₆₁	—	2	2	no	—
80	C₄×D₄×C₆₁	—	2	2	no	—
81	C₄×D₄×C₆₁	—	2	2	no	—
82	C₄×D₄×C₆₁	—	2	2	no	—
83	C₄×D₄×C₆₁	—	2	2	no	—
84	C₄×D₄×C₆₁	—	2	2	no	—
85	C₄×D₄×C₆₁	—	2	2	no	—
86	C₄×D₄×C₆₁	—	4	8	no	—
87	C₄×D₄×C₆₁	D₄	1	2	yes	complemented
88	C₄×D₄×C₆₁	D₄	1	2	yes	complemented
89	#117×C₆₁	—	2	4	no	—
90	#117×C₆₁	—	2	2	no	—
91	#117×C₆₁	—	2	2	no	—
92	#117×C₆₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
102	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
105	C₂×C₄²×C₆₁	—	2	2	no	—
106	C₂×C₄²×C₆₁	—	2	1	no	—
107	C₂×C₄²×C₆₁	—	2	2	no	—
108	C₂×C₄²×C₆₁	—	2	1	no	—
109	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
112	#107×C₆₁	—	2	2	no	—
113	#106×C₆₁	—	2	2	no	—
114	#104×C₆₁	—	2	4	no	—
115	#104×C₆₁	C₂×C₄	1	8	yes	complemented
116	#103×C₆₁	—	2	2	no	—
117	#102×C₆₁	—	2	2	no	—
118	#102×C₆₁	—	2	2	no	—
119	#102×C₆₁	—	2	2	no	—
120	#100×C₆₁	—	2	2	no	—
121	#100×C₆₁	C₂×C₄	1	4	yes	complemented
122	#97×C₆₁	—	2	2	no	—
123	C₄×C₈×C₆₁	—	2	2	no	—
124	C₂⁴×C₆₁	—	2	2	no	—
125	C₂⁴×C₆₁	—	2	1	no	—
126	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
127	#41×C₆₁	—	2	8	no	—
128	#41×C₆₁	—	2	8	no	—
129	#41×C₆₁	—	2	2	no	—
130	#41×C₆₁	—	2	2	no	—
131	#41×C₆₁	C₂²×C₄	1	4	yes	—
132	#41×C₆₁	#41	1	4	yes	—
133	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₆₁	—	2	4	no	—
135	C₂×D₄×C₆₁	—	2	8	no	—
136	C₂×D₄×C₆₁	—	2	8	no	—
137	C₂×D₄×C₆₁	—	2	8	no	—
138	C₂×D₄×C₆₁	—	2	2	no	—
139	C₂×D₄×C₆₁	—	2	8	no	—
140	C₂×D₄×C₆₁	—	2	8	no	—
141	C₂×D₄×C₆₁	—	2	2	no	—
142	C₂×D₄×C₆₁	—	2	2	no	—
143	C₂×D₄×C₆₁	—	2	2	no	—
144	C₂×D₄×C₆₁	—	2	2	no	—
145	C₂×D₄×C₆₁	—	2	1	no	—
146	C₂×D₄×C₆₁	—	4	4	no	—
147	C₂×D₄×C₆₁	—	4	4	no	—
148	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₆₁	C₂²×C₄	1	2	yes	complemented
150	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
151	C₂×D₄×C₆₁	#41	1	2	yes	—
152	C₂²×C₄×C₆₁	—	2	2	no	—
153	C₂²×C₄×C₆₁	—	2	1	no	—
154	C₂²×C₄×C₆₁	—	2	2	no	—
155	C₂²×C₄×C₆₁	—	2	1	no	—
156	C₂²×C₄×C₆₁	—	2	2	no	—
157	C₂²×C₄×C₆₁	—	2	4	no	—
158	C₂²×C₄×C₆₁	—	2	2	no	—
159	C₂²×C₄×C₆₁	—	2	2	no	—
160	C₂²×C₄×C₆₁	—	2	2	no	—
161	C₂²×C₄×C₆₁	—	2	2	no	—
162	C₂²×C₄×C₆₁	—	2	2	no	—
163	C₂²×C₄×C₆₁	—	2	1	no	—
164	C₂²×C₄×C₆₁	—	4	4	no	—
165	C₂²×C₄×C₆₁	—	4	4	no	—
166	C₂²×C₄×C₆₁	—	4	1	no	—
167	C₂²×C₄×C₆₁	—	4	1	no	—
168	C₂²×C₄×C₆₁	—	8	4	no	—
169	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
173	SD₁₆×C₆₁	—	2	8	no	—
174	D₈×C₆₁	—	2	8	no	—
175	M₁₆×C₆₁	—	2	2	no	—
176	M₁₆×C₆₁	—	2	2	no	—
177	M₁₆×C₆₁	—	2	2	no	—
178	M₁₆×C₆₁	C₂²×C₄	1	4	yes	complemented
179	C₂×C₈×C₆₁	—	2	2	no	—
180	C₂×C₈×C₆₁	—	2	2	no	—
181	C₄⋊C₄×C₆₁	—	2	4	no	—
182	C₄⋊C₄×C₆₁	—	2	2	no	—
183	C₄⋊C₄×C₆₁	—	2	2	no	—
184	C₄⋊C₄×C₆₁	—	2	2	no	—
185	C₄⋊C₄×C₆₁	—	2	2	no	—
186	C₄⋊C₄×C₆₁	—	4	4	no	—
187	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
188	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
189	C₂²⋊C₄×C₆₁	—	2	4	no	—
190	C₂²⋊C₄×C₆₁	—	2	4	no	—
191	C₂²⋊C₄×C₆₁	—	2	2	no	—
192	C₂²⋊C₄×C₆₁	—	2	2	no	—
193	C₂²⋊C₄×C₆₁	—	2	2	no	—
194	C₂²⋊C₄×C₆₁	—	2	4	no	—
195	C₂²⋊C₄×C₆₁	—	4	4	no	—
196	C₂²⋊C₄×C₆₁	—	4	4	no	—
197	C₂²⋊C₄×C₆₁	—	4	2	no	—
198	C₄²×C₆₁	—	2	4	no	—
199	C₄²×C₆₁	—	2	4	no	—
200	C₄²×C₆₁	—	2	2	no	—
201	C₄²×C₆₁	—	2	2	no	—
202	C₄²×C₆₁	—	2	2	no	—
203	C₄²×C₆₁	—	2	1	no	—
204	C₄²×C₆₁	—	2	2	no	—
205	C₄²×C₆₁	—	4	2	no	—
206	C₄²×C₆₁	C₂²×C₄	1	2	yes	complemented
207	C₄²×C₆₁	C₂×D₄	1	2	yes	—
208	C₄²×C₆₁	C₂×D₄	1	2	yes	—
209	C₂³×C₆₁	—	2	8	no	—
210	C₂³×C₆₁	—	2	2	no	—
211	C₂³×C₆₁	—	2	2	no	—
212	C₂³×C₆₁	—	2	2	no	—
213	C₂³×C₆₁	—	2	2	no	—
214	C₂³×C₆₁	—	4	2	no	—
215	C₂³×C₆₁	—	4	2	no	—
216	C₂³×C₆₁	—	4	2	no	—
217	C₂³×C₆₁	—	4	2	no	—
218	C₂³×C₆₁	—	4	1	no	—
219	C₂³×C₆₁	—	4	1	no	—
220	C₂³×C₆₁	—	8	2	no	—
221	C₂³×C₆₁	—	8	2	no	—
222	C₂³×C₆₁	C₄×D₄	1	4	yes	—
223	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₁	—	2	2	no	—
225	Q₈×C₆₁	C₄×D₄	1	4	yes	complemented
226	D₄×C₆₁	—	2	16	no	—
227	D₄×C₆₁	—	2	4	no	—
228	D₄×C₆₁	—	2	4	no	—
229	D₄×C₆₁	—	2	4	no	—
230	D₄×C₆₁	—	2	4	no	—
231	D₄×C₆₁	—	2	2	no	—
232	D₄×C₆₁	—	2	4	no	—
233	D₄×C₆₁	—	4	16	no	—
234	D₄×C₆₁	—	4	2	no	—
235	D₄×C₆₁	C₄×D₄	1	4	yes	complemented
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	8	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	4	no	—
242	C₂×C₄×C₆₁	—	2	4	no	—
243	C₂×C₄×C₆₁	—	2	2	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	2	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	2	no	—
248	C₂×C₄×C₆₁	—	4	1	no	—
249	C₂×C₄×C₆₁	—	4	1	no	—
250	C₂×C₄×C₆₁	—	4	1	no	—
251	C₂×C₄×C₆₁	—	4	4	no	—
252	C₂×C₄×C₆₁	—	4	4	no	—
253	C₂×C₄×C₆₁	—	4	2	no	—
254	C₂×C₄×C₆₁	—	4	2	no	—
255	C₂×C₄×C₆₁	—	4	1	no	—
256	C₂×C₄×C₆₁	—	8	2	no	—
257	C₂×C₄×C₆₁	—	8	2	no	—
258	C₂×C₄×C₆₁	—	8	2	no	—
259	C₂×C₄×C₆₁	—	8	2	no	—
260	C₂×C₄×C₆₁	—	8	4	no	—
261	C₂×C₄×C₆₁	—	8	2	no	—
262	C₂×C₄×C₆₁	—	8	2	no	—
263	C₂×C₄×C₆₁	#120	1	2	yes	—
264	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
265	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
266	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
267	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
269	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
270	C₈×C₆₁	—	2	4	no	—
271	C₈×C₆₁	—	4	2	no	—
272	C₄×#453	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂²×C₆₁	—	2	1	no	—
274	C₂²×C₆₁	—	2	1	no	—
275	C₂²×C₆₁	—	2	4	no	—
276	C₂²×C₆₁	—	2	4	no	—
277	C₂²×C₆₁	—	4	4	no	—
278	C₂²×C₆₁	—	4	4	no	—
279	C₂²×C₆₁	—	4	2	no	—
280	C₂²×C₆₁	—	4	2	no	—
281	C₂²×C₆₁	—	4	2	no	—
282	C₂²×C₆₁	—	4	2	no	—
283	C₂²×C₆₁	—	4	2	no	—
284	C₂²×C₆₁	—	8	2	no	—
285	C₂²×C₆₁	—	8	2	no	—
286	C₂²×C₆₁	—	8	8	no	—
287	C₂²×C₆₁	—	8	1	no	—
288	C₂²×C₆₁	—	8	1	no	—
289	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
290	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
291	C₂²×C₆₁	#525	1	1	yes	characteristic
292	C₄×C₆₁	—	2	4	no	—
293	C₄×C₆₁	—	2	1	no	—
294	C₄×C₆₁	—	2	2	no	—
295	C₄×C₆₁	—	2	4	no	—
296	C₄×C₆₁	—	4	2	no	—
297	C₄×C₆₁	—	4	2	no	—
298	C₄×C₆₁	—	4	2	no	—
299	C₄×C₆₁	—	4	2	no	—
300	C₄×C₆₁	—	8	1	no	—
301	C₄×C₆₁	#453	1	2	yes	direct product factor
302	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
303	C₄×C₆₁	#525	1	2	yes	—
304	C₄×H(5,𝔽₂)	C₂×C₆₁	1	1	yes	characteristic, complemented
305	C₂×#453	C₂×C₆₁	1	1	yes	characteristic
306	C₄×AGL(1,ℤ/8)	C₂×C₆₁	1	2	yes	complemented
307	C₄×#127	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#1887	C₂×C₆₁	1	1	yes	characteristic, complemented
309	#1883	C₂×C₆₁	1	2	yes	complemented
310	#1859	C₂×C₆₁	1	2	yes	complemented
311	#1856	C₂×C₆₁	1	1	yes	characteristic, complemented
312	#1792	C₂×C₆₁	1	1	yes	characteristic, complemented
313	C₄×#104	C₂×C₆₁	1	2	yes	complemented
314	C₄×#100	C₂×C₆₁	1	1	yes	characteristic, complemented
315	C₂×C₆₁	—	2	1	no	—
316	C₂×C₆₁	—	2	1	no	—
317	C₂×C₆₁	—	4	4	no	—
318	C₂×C₆₁	—	4	4	no	—
319	C₂×C₆₁	—	8	2	no	—
320	C₂×C₆₁	C₄×#120	1	1	yes	characteristic
321	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
322	C₂×C₆₁	#3018	1	1	yes	characteristic
323	C₂×H(5,𝔽₂)	C₂²×C₆₁	1	1	yes	characteristic
324	C₂×AGL(1,ℤ/8)	C₂²×C₆₁	1	2	yes	—
325	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
326	#520	—	2	2	no	—
327	#520	C₂²×C₆₁	1	2	yes	—
328	#518	—	2	2	no	—
329	#518	—	2	2	no	—
330	#518	C₂²×C₆₁	1	1	yes	characteristic
331	C₄×#41	—	2	2	no	—
332	C₄×#41	C₂²×C₆₁	1	2	yes	—
333	C₂×C₄×D₄	—	2	2	no	—
334	C₂×C₄×D₄	—	2	2	no	—
335	C₂×C₄×D₄	—	2	2	no	—
336	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
337	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic, complemented
338	#453	C₄×C₆₁	1	8	yes	direct product factor
339	#442	—	2	2	no	—
340	#440	—	2	2	no	—
341	C₄×SD₁₆	—	2	2	no	—
342	C₄×D₈	—	2	2	no	—
343	#428	C₂²×C₆₁	1	2	yes	complemented
344	C₂×#104	C₂²×C₆₁	1	2	yes	—
345	C₂×#104	C₂²×C₆₁	1	2	yes	—
346	#418	C₂²×C₆₁	1	2	yes	complemented
347	#417	C₂²×C₆₁	1	2	yes	—
348	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×#100	C₂²×C₆₁	1	1	yes	characteristic
350	C₄×M₁₆	C₂²×C₆₁	1	2	yes	complemented
351	#390	—	2	2	no	—
352	#390	C₂²×C₆₁	1	1	yes	characteristic
353	#390	C₂²×C₆₁	1	1	yes	characteristic, complemented
354	#384	C₂²×C₆₁	1	1	yes	characteristic, complemented
355	C₄³	C₂²×C₆₁	1	1	yes	characteristic, complemented
356	#343	C₂²×C₆₁	1	2	yes	complemented
357	#339	C₂²×C₆₁	1	2	yes	complemented
358	#339	C₂²×C₆₁	1	2	yes	complemented
359	C₆₁	C₄×#453	1	1	yes	characteristic, direct product factor, Sylow
360	H(5,𝔽₂)	—	2	4	no	—
361	H(5,𝔽₂)	—	2	2	no	—
362	H(5,𝔽₂)	C₂×C₄×C₆₁	1	4	yes	complemented
363	C₂×#41	—	2	2	no	—
364	C₂×#41	C₂³×C₆₁	1	2	yes	—
365	C₂²×D₄	—	2	2	no	—
366	C₂²×D₄	—	2	2	no	—
367	C₂²×D₄	—	2	2	no	—
368	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
370	C₂³×C₄	—	2	2	no	—
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	D₄×C₆₁	1	2	yes	—
373	AGL(1,ℤ/8)	—	2	4	no	—
374	AGL(1,ℤ/8)	C₂×C₄×C₆₁	1	8	yes	complemented
375	C₂×SD₁₆	—	2	2	no	—
376	C₂×D₈	—	2	2	no	—
377	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
378	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
379	#127	—	2	2	no	—
380	#127	C₂×C₄×C₆₁	1	4	yes	complemented
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	D₄×C₆₁	1	2	yes	complemented
383	C₄×Q₈	D₄×C₆₁	1	2	yes	complemented
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	4	no	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	4	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	4	8	no	—
399	C₄×D₄	D₄×C₆₁	1	2	yes	complemented
400	C₄×D₄	D₄×C₆₁	1	2	yes	complemented
401	#117	—	2	4	no	—
402	#117	—	2	2	no	—
403	#117	—	2	2	no	—
404	#117	C₂³×C₆₁	1	2	yes	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	—	2	2	no	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	4	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	—	2	1	no	—
419	C₂×C₄²	—	2	2	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
424	#107	—	2	2	no	—
425	#106	—	2	2	no	—
426	#104	—	2	4	no	—
427	#104	C₂×C₄×C₆₁	1	8	yes	complemented
428	#103	—	2	2	no	—
429	#102	—	2	2	no	—
430	#102	—	2	2	no	—
431	#102	—	2	2	no	—
432	#100	—	2	2	no	—
433	#100	C₂×C₄×C₆₁	1	4	yes	complemented
434	#97	—	2	2	no	—
435	C₄×C₈	—	2	2	no	—
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	2	1	no	—
438	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
439	#41	—	2	8	no	—
440	#41	—	2	8	no	—
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	#41	C₂²×C₄×C₆₁	1	4	yes	—
444	#41	#41×C₆₁	1	4	yes	—
445	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	8	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	—
461	C₂×D₄	C₂²×C₄×C₆₁	1	2	yes	complemented
462	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
463	C₂×D₄	#41×C₆₁	1	2	yes	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	—	4	4	no	—
478	C₂²×C₄	—	4	1	no	—
479	C₂²×C₄	—	4	1	no	—
480	C₂²×C₄	—	8	4	no	—
481	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
483	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
485	SD₁₆	—	2	8	no	—
486	D₈	—	2	8	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	C₂²×C₄×C₆₁	1	4	yes	complemented
491	C₂×C₈	—	2	2	no	—
492	C₂×C₈	—	2	2	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	4	4	no	—
499	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
500	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	4	4	no	—
508	C₂²⋊C₄	—	4	4	no	—
509	C₂²⋊C₄	—	4	2	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	4	2	no	—
518	C₄²	C₂²×C₄×C₆₁	1	2	yes	complemented
519	C₄²	C₂×D₄×C₆₁	1	2	yes	—
520	C₄²	C₂×D₄×C₆₁	1	2	yes	—
521	C₂³	—	2	8	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	4	2	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	4	2	no	—
529	C₂³	—	4	2	no	—
530	C₂³	—	4	1	no	—
531	C₂³	—	4	1	no	—
532	C₂³	—	8	2	no	—
533	C₂³	—	8	2	no	—
534	C₂³	C₄×D₄×C₆₁	1	4	yes	—
535	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
536	Q₈	—	2	2	no	—
537	Q₈	C₄×D₄×C₆₁	1	4	yes	complemented
538	D₄	—	2	16	no	—
539	D₄	—	2	4	no	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	2	no	—
544	D₄	—	2	4	no	—
545	D₄	—	4	16	no	—
546	D₄	—	4	2	no	—
547	D₄	C₄×D₄×C₆₁	1	4	yes	complemented
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	8	2	no	—
569	C₂×C₄	—	8	2	no	—
570	C₂×C₄	—	8	2	no	—
571	C₂×C₄	—	8	2	no	—
572	C₂×C₄	—	8	4	no	—
573	C₂×C₄	—	8	2	no	—
574	C₂×C₄	—	8	2	no	—
575	C₂×C₄	#120×C₆₁	1	2	yes	—
576	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
577	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
578	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
579	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
580	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
581	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
582	C₈	—	2	4	no	—
583	C₈	—	4	2	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	4	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	4	4	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	8	2	no	—
596	C₂²	—	8	2	no	—
597	C₂²	—	8	8	no	—
598	C₂²	—	8	1	no	—
599	C₂²	—	8	1	no	—
600	C₂²	C₂×C₄×D₄×C₆₁	1	1	yes	characteristic
601	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
602	C₂²	C₆₁×#525	1	1	yes	characteristic
603	C₄	—	2	4	no	—
604	C₄	—	2	4	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	2	no	—
607	C₄	—	4	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	8	1	no	—
612	C₄	C₆₁×#453	1	2	yes	direct product factor
613	C₄	C₂×C₄×D₄×C₆₁	1	2	yes	—
614	C₄	C₆₁×#525	1	2	yes	—
615	C₂	—	2	1	no	—
616	C₂	—	2	1	no	—
617	C₂	—	4	4	no	—
618	C₂	—	4	4	no	—
619	C₂	—	8	2	no	—
620	C₂	C₄×#120×C₆₁	1	1	yes	characteristic
621	C₂	C₂×C₆₁×#453	1	1	yes	characteristic
622	C₂	C₆₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#120
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂×#120
prime radical	C₂×C₆₁×#453	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×H(5,𝔽₂)×C₆₁	C₂	semidirect
3. C₄×AGL(1,ℤ/8)×C₆₁	C₂	semidirect
4. C₄×#127×C₆₁	C₂	semidirect
5. C₆₁×#1887	C₂	semidirect
6. C₆₁×#1883	C₂	semidirect
7. C₆₁×#1859	C₂	semidirect
8. C₆₁×#1856	C₂	semidirect
9. C₆₁×#1792	C₂	semidirect
10. C₄×#104×C₆₁	C₂	semidirect
11. C₄×#100×C₆₁	C₂	semidirect
26. C₂×C₄×D₄×C₆₁	C₂²	semidirect
27. C₆₁×#453	C₄	direct (N × H)
32. C₆₁×#428	C₂²	semidirect
35. C₆₁×#418	C₂²	semidirect
39. C₄×M₁₆×C₆₁	C₂²	semidirect
42. C₆₁×#390	C₂²	semidirect
43. C₆₁×#384	C₂²	semidirect
44. C₄³×C₆₁	C₂²	semidirect
45. C₆₁×#343	C₂²	semidirect
46. C₆₁×#339	C₂²	semidirect
47. C₆₁×#339	C₂²	semidirect
50. H(5,𝔽₂)×C₆₁	C₂×C₄	semidirect
62. AGL(1,ℤ/8)×C₆₁	C₂×C₄	semidirect
68. #127×C₆₁	C₂×C₄	semidirect
70. C₄×Q₈×C₆₁	D₄	semidirect
71. C₄×Q₈×C₆₁	D₄	semidirect
87. C₄×D₄×C₆₁	D₄	semidirect
88. C₄×D₄×C₆₁	D₄	semidirect
115. #104×C₆₁	C₂×C₄	semidirect
121. #100×C₆₁	C₂×C₄	semidirect
149. C₂×D₄×C₆₁	C₂²×C₄	semidirect
178. M₁₆×C₆₁	C₂²×C₄	semidirect
206. C₄²×C₆₁	C₂²×C₄	semidirect
225. Q₈×C₆₁	C₄×D₄	semidirect
235. D₄×C₆₁	C₄×D₄	semidirect
272. C₄×#453	C₆₁	direct (N × H)
301. C₄×C₆₁	#453	direct (N × H)
304. C₄×H(5,𝔽₂)	C₂×C₆₁	semidirect
306. C₄×AGL(1,ℤ/8)	C₂×C₆₁	semidirect
307. C₄×#127	C₂×C₆₁	semidirect
308. #1887	C₂×C₆₁	semidirect
309. #1883	C₂×C₆₁	semidirect
310. #1859	C₂×C₆₁	semidirect
311. #1856	C₂×C₆₁	semidirect
312. #1792	C₂×C₆₁	semidirect
313. C₄×#104	C₂×C₆₁	semidirect
314. C₄×#100	C₂×C₆₁	semidirect
337. C₂×C₄×D₄	C₂²×C₆₁	semidirect
338. #453	C₄×C₆₁	direct (N × H)
343. #428	C₂²×C₆₁	semidirect
346. #418	C₂²×C₆₁	semidirect
350. C₄×M₁₆	C₂²×C₆₁	semidirect
353. #390	C₂²×C₆₁	semidirect
354. #384	C₂²×C₆₁	semidirect
355. C₄³	C₂²×C₆₁	semidirect
356. #343	C₂²×C₆₁	semidirect
357. #339	C₂²×C₆₁	semidirect
358. #339	C₂²×C₆₁	semidirect
359. C₆₁	C₄×#453	direct (N × H)
362. H(5,𝔽₂)	C₂×C₄×C₆₁	semidirect
374. AGL(1,ℤ/8)	C₂×C₄×C₆₁	semidirect
380. #127	C₂×C₄×C₆₁	semidirect
382. C₄×Q₈	D₄×C₆₁	semidirect
383. C₄×Q₈	D₄×C₆₁	semidirect
399. C₄×D₄	D₄×C₆₁	semidirect
400. C₄×D₄	D₄×C₆₁	semidirect
427. #104	C₂×C₄×C₆₁	semidirect
433. #100	C₂×C₄×C₆₁	semidirect
461. C₂×D₄	C₂²×C₄×C₆₁	semidirect
490. M₁₆	C₂²×C₄×C₆₁	semidirect
518. C₄²	C₂²×C₄×C₆₁	semidirect
537. Q₈	C₄×D₄×C₆₁	semidirect
547. D₄	C₄×D₄×C₆₁	semidirect
612. C₄	C₆₁×#453	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×H(5,𝔽₂)×C₆₁	2
C₂×C₆₁×#453	2
C₄×AGL(1,ℤ/8)×C₆₁	2
C₄×#127×C₆₁	2
C₆₁×#1887	2
C₆₁×#1883	2
C₆₁×#1859	2
C₆₁×#1856	2
C₆₁×#1792	2
C₄×#104×C₆₁	2
C₄×#100×C₆₁	2
C₄×#453	61

Subgroups — C4×C61×#453 #29524902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#453 #29524902