Subgroups — C₄×C₄₁×#454 #23797672

Order 10496.

1390 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×2₋¹⁺⁴×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#454	C₂	1	1	yes	maximal, characteristic
3	C₄×AGL(1,ℤ/8)×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₄×#124×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1888	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1883	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1860	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1857	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1793	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#104×C₄₁	C₂	1	2	yes	maximal, complemented
11	C₄×#101×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
14	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#520	—	2	2	no	—
16	C₄₁×#520	—	2	2	no	—
17	C₄₁×#520	C₂²	1	2	yes	—
18	C₄₁×#519	—	2	2	no	—
19	C₄₁×#519	C₂²	1	1	yes	characteristic
20	C₄×#41×C₄₁	—	2	2	no	—
21	C₄×#41×C₄₁	—	2	2	no	—
22	C₄×#41×C₄₁	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₄₁	—	2	2	no	—
24	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#454	C₄	1	8	yes	direct product factor
27	C₄₁×#442	—	2	2	no	—
28	C₄₁×#440	—	2	2	no	—
29	C₄×SD₁₆×C₄₁	—	2	2	no	—
30	C₄×D₈×C₄₁	—	2	2	no	—
31	C₄₁×#428	C₂²	1	2	yes	—
32	C₂×#104×C₄₁	C₂²	1	2	yes	—
33	C₂×#104×C₄₁	C₂²	1	2	yes	—
34	C₄₁×#418	C₂²	1	2	yes	—
35	C₄₁×#417	C₂²	1	2	yes	—
36	C₂×#101×C₄₁	C₂²	1	1	yes	characteristic
37	C₂×#101×C₄₁	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₄₁	C₂²	1	2	yes	—
39	C₄₁×#391	C₂²	1	1	yes	characteristic
40	C₄₁×#390	C₂²	1	1	yes	characteristic
41	C₄₁×#388	—	2	2	no	—
42	C₄₁×#382	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
44	C₄³×C₄₁	C₂²	1	1	yes	characteristic
45	C₄₁×#343	C₂²	1	2	yes	—
46	C₄₁×#339	C₂²	1	2	yes	—
47	C₄₁×#339	C₂²	1	2	yes	—
48	2₋¹⁺⁴×C₄₁	—	2	4	no	—
49	2₋¹⁺⁴×C₄₁	—	2	2	no	—
50	2₋¹⁺⁴×C₄₁	C₂×C₄	1	4	yes	complemented
51	C₂×#41×C₄₁	—	2	2	no	—
52	C₂×#41×C₄₁	—	2	2	no	—
53	C₂×#41×C₄₁	C₂³	1	2	yes	—
54	C₂²×Q₈×C₄₁	—	2	2	no	—
55	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₄₁	—	2	1	no	—
58	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
59	AGL(1,ℤ/8)×C₄₁	C₂×C₄	1	8	yes	complemented
60	C₂×SD₁₆×C₄₁	—	2	2	no	—
61	C₂×D₈×C₄₁	—	2	2	no	—
62	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
64	#124×C₄₁	—	2	2	no	—
65	#124×C₄₁	C₂×C₄	1	4	yes	complemented
66	C₄×Q₈×C₄₁	—	2	2	no	—
67	C₄×Q₈×C₄₁	—	2	4	no	—
68	C₄×Q₈×C₄₁	—	2	2	no	—
69	C₄×Q₈×C₄₁	—	2	2	no	—
70	C₄×Q₈×C₄₁	—	2	2	no	—
71	C₄×Q₈×C₄₁	—	2	2	no	—
72	C₄×Q₈×C₄₁	—	2	2	no	—
73	C₄×Q₈×C₄₁	—	2	2	no	—
74	C₄×Q₈×C₄₁	D₄	1	2	yes	—
75	C₄×Q₈×C₄₁	D₄	1	2	yes	—
76	C₄×D₄×C₄₁	—	2	2	no	—
77	C₄×D₄×C₄₁	—	2	4	no	—
78	C₄×D₄×C₄₁	—	2	2	no	—
79	C₄×D₄×C₄₁	—	2	2	no	—
80	C₄×D₄×C₄₁	—	2	2	no	—
81	C₄×D₄×C₄₁	—	2	4	no	—
82	C₄×D₄×C₄₁	—	2	2	no	—
83	C₄×D₄×C₄₁	—	2	2	no	—
84	C₄×D₄×C₄₁	—	2	2	no	—
85	C₄×D₄×C₄₁	D₄	1	2	yes	—
86	C₄×D₄×C₄₁	D₄	1	2	yes	—
87	#117×C₄₁	—	2	4	no	—
88	#117×C₄₁	—	2	4	no	—
89	#117×C₄₁	—	2	2	no	—
90	#117×C₄₁	—	2	2	no	—
91	#117×C₄₁	—	2	2	no	—
92	#117×C₄₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
94	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
96	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
101	C₂×C₄²×C₄₁	—	2	4	no	—
102	C₂×C₄²×C₄₁	—	2	2	no	—
103	C₂×C₄²×C₄₁	—	2	1	no	—
104	C₂×C₄²×C₄₁	—	2	1	no	—
105	C₂×C₄²×C₄₁	D₄	1	2	yes	—
106	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
109	#107×C₄₁	—	2	2	no	—
110	#106×C₄₁	—	2	2	no	—
111	#104×C₄₁	—	2	4	no	—
112	#104×C₄₁	C₂×C₄	1	8	yes	complemented
113	#103×C₄₁	—	2	2	no	—
114	#102×C₄₁	—	2	2	no	—
115	#102×C₄₁	—	2	2	no	—
116	#102×C₄₁	—	2	2	no	—
117	#101×C₄₁	—	2	2	no	—
118	#101×C₄₁	C₂×C₄	1	4	yes	complemented
119	#97×C₄₁	—	2	2	no	—
120	C₄×C₈×C₄₁	—	2	2	no	—
121	#95×C₄₁	—	2	2	no	—
122	#95×C₄₁	—	2	2	no	—
123	C₂⁴×C₄₁	—	2	1	no	—
124	#41×C₄₁	—	2	8	no	—
125	#41×C₄₁	—	2	8	no	—
126	#41×C₄₁	—	2	8	no	—
127	#41×C₄₁	—	2	8	no	—
128	#41×C₄₁	—	2	2	no	—
129	#41×C₄₁	—	2	2	no	—
130	#41×C₄₁	C₂²×C₄	1	4	yes	—
131	#41×C₄₁	#41	1	4	yes	—
132	C₂×Q₈×C₄₁	—	2	2	no	—
133	C₂×Q₈×C₄₁	—	2	8	no	—
134	C₂×Q₈×C₄₁	—	2	8	no	—
135	C₂×Q₈×C₄₁	—	2	2	no	—
136	C₂×Q₈×C₄₁	—	2	2	no	—
137	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
138	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
139	C₂×Q₈×C₄₁	#41	1	2	yes	—
140	C₂×D₄×C₄₁	—	2	2	no	—
141	C₂×D₄×C₄₁	—	2	2	no	—
142	C₂×D₄×C₄₁	—	2	2	no	—
143	C₂×D₄×C₄₁	—	2	2	no	—
144	C₂×D₄×C₄₁	—	2	1	no	—
145	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
146	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₄₁	—	2	4	no	—
148	C₂²×C₄×C₄₁	—	2	2	no	—
149	C₂²×C₄×C₄₁	—	2	1	no	—
150	C₂²×C₄×C₄₁	—	2	1	no	—
151	C₂²×C₄×C₄₁	—	2	4	no	—
152	C₂²×C₄×C₄₁	—	2	2	no	—
153	C₂²×C₄×C₄₁	—	2	2	no	—
154	C₂²×C₄×C₄₁	—	2	1	no	—
155	C₂²×C₄×C₄₁	—	4	4	no	—
156	C₂²×C₄×C₄₁	—	4	1	no	—
157	C₂²×C₄×C₄₁	—	4	1	no	—
158	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
163	SD₁₆×C₄₁	—	2	8	no	—
164	D₈×C₄₁	—	2	8	no	—
165	M₁₆×C₄₁	—	2	2	no	—
166	M₁₆×C₄₁	—	2	2	no	—
167	M₁₆×C₄₁	—	2	2	no	—
168	M₁₆×C₄₁	C₂²×C₄	1	4	yes	—
169	C₂×C₈×C₄₁	—	2	2	no	—
170	C₂×C₈×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	2	4	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	2	2	no	—
174	C₄⋊C₄×C₄₁	—	2	2	no	—
175	C₄⋊C₄×C₄₁	—	2	4	no	—
176	C₄⋊C₄×C₄₁	—	2	2	no	—
177	C₄⋊C₄×C₄₁	—	2	2	no	—
178	C₄⋊C₄×C₄₁	—	2	2	no	—
179	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₄₁	—	2	4	no	—
182	C₂²⋊C₄×C₄₁	—	2	2	no	—
183	C₂²⋊C₄×C₄₁	—	2	8	no	—
184	C₂²⋊C₄×C₄₁	—	2	2	no	—
185	C₂²⋊C₄×C₄₁	—	2	2	no	—
186	C₂²⋊C₄×C₄₁	—	4	4	no	—
187	C₂²⋊C₄×C₄₁	—	4	2	no	—
188	C₄²×C₄₁	—	2	4	no	—
189	C₄²×C₄₁	—	2	4	no	—
190	C₄²×C₄₁	—	2	2	no	—
191	C₄²×C₄₁	—	2	2	no	—
192	C₄²×C₄₁	—	2	2	no	—
193	C₄²×C₄₁	—	2	2	no	—
194	C₄²×C₄₁	—	2	2	no	—
195	C₄²×C₄₁	—	2	1	no	—
196	C₄²×C₄₁	—	2	2	no	—
197	C₄²×C₄₁	—	4	2	no	—
198	C₄²×C₄₁	—	4	2	no	—
199	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
200	C₄²×C₄₁	C₂×D₄	1	2	yes	—
201	C₄²×C₄₁	C₂×D₄	1	2	yes	—
202	C₂³×C₄₁	—	2	2	no	—
203	C₂³×C₄₁	—	2	2	no	—
204	C₂³×C₄₁	—	4	2	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	4	1	no	—
207	C₂³×C₄₁	—	4	1	no	—
208	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₄₁	—	2	8	no	—
210	Q₈×C₄₁	—	2	4	no	—
211	Q₈×C₄₁	—	2	2	no	—
212	Q₈×C₄₁	—	2	4	no	—
213	Q₈×C₄₁	C₄×D₄	1	4	yes	—
214	D₄×C₄₁	—	2	8	no	—
215	D₄×C₄₁	—	2	4	no	—
216	D₄×C₄₁	—	2	4	no	—
217	D₄×C₄₁	—	2	4	no	—
218	D₄×C₄₁	—	2	2	no	—
219	D₄×C₄₁	—	4	2	no	—
220	D₄×C₄₁	C₄×D₄	1	4	yes	—
221	C₂×C₄×C₄₁	—	2	16	no	—
222	C₂×C₄×C₄₁	—	2	2	no	—
223	C₂×C₄×C₄₁	—	2	2	no	—
224	C₂×C₄×C₄₁	—	2	2	no	—
225	C₂×C₄×C₄₁	—	2	4	no	—
226	C₂×C₄×C₄₁	—	2	4	no	—
227	C₂×C₄×C₄₁	—	2	2	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	2	2	no	—
231	C₂×C₄×C₄₁	—	2	2	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	4	1	no	—
234	C₂×C₄×C₄₁	—	4	1	no	—
235	C₂×C₄×C₄₁	—	4	1	no	—
236	C₂×C₄×C₄₁	—	4	1	no	—
237	C₂×C₄×C₄₁	—	4	1	no	—
238	C₂×C₄×C₄₁	—	4	4	no	—
239	C₂×C₄×C₄₁	—	4	2	no	—
240	C₂×C₄×C₄₁	—	4	2	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	8	2	no	—
243	C₂×C₄×C₄₁	—	8	2	no	—
244	C₂×C₄×C₄₁	—	8	4	no	—
245	C₂×C₄×C₄₁	—	8	2	no	—
246	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
247	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
248	C₂×C₄×C₄₁	#120	1	2	yes	—
249	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
250	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
253	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
254	C₈×C₄₁	—	2	4	no	—
255	C₈×C₄₁	—	4	2	no	—
256	C₄×#454	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂²×C₄₁	—	2	1	no	—
258	C₂²×C₄₁	—	2	1	no	—
259	C₂²×C₄₁	—	2	4	no	—
260	C₂²×C₄₁	—	4	4	no	—
261	C₂²×C₄₁	—	4	2	no	—
262	C₂²×C₄₁	—	4	2	no	—
263	C₂²×C₄₁	—	4	2	no	—
264	C₂²×C₄₁	—	8	2	no	—
265	C₂²×C₄₁	—	8	1	no	—
266	C₂²×C₄₁	—	8	1	no	—
267	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
269	C₂²×C₄₁	#525	1	1	yes	characteristic
270	C₄×C₄₁	—	2	1	no	—
271	C₄×C₄₁	—	2	2	no	—
272	C₄×C₄₁	—	2	4	no	—
273	C₄×C₄₁	—	2	4	no	—
274	C₄×C₄₁	—	4	2	no	—
275	C₄×C₄₁	—	4	2	no	—
276	C₄×C₄₁	—	4	2	no	—
277	C₄×C₄₁	—	4	2	no	—
278	C₄×C₄₁	—	4	2	no	—
279	C₄×C₄₁	—	8	1	no	—
280	C₄×C₄₁	#454	1	2	yes	direct product factor
281	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
282	C₄×C₄₁	#525	1	2	yes	—
283	C₄×2₋¹⁺⁴	C₂×C₄₁	1	1	yes	characteristic, complemented
284	C₂×#454	C₂×C₄₁	1	1	yes	characteristic
285	C₄×AGL(1,ℤ/8)	C₂×C₄₁	1	2	yes	complemented
286	C₄×#124	C₂×C₄₁	1	1	yes	characteristic, complemented
287	#1888	C₂×C₄₁	1	1	yes	characteristic, complemented
288	#1883	C₂×C₄₁	1	2	yes	complemented
289	#1860	C₂×C₄₁	1	2	yes	complemented
290	#1857	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#1793	C₂×C₄₁	1	1	yes	characteristic, complemented
292	C₄×#104	C₂×C₄₁	1	2	yes	complemented
293	C₄×#101	C₂×C₄₁	1	1	yes	characteristic, complemented
294	C₂×C₄₁	—	2	1	no	—
295	C₂×C₄₁	—	2	1	no	—
296	C₂×C₄₁	—	4	4	no	—
297	C₂×C₄₁	—	8	2	no	—
298	C₂×C₄₁	C₄×#120	1	1	yes	characteristic
299	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
300	C₂×C₄₁	#3018	1	1	yes	characteristic
301	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	1	yes	characteristic
302	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
303	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
304	#520	—	2	2	no	—
305	#520	—	2	2	no	—
306	#520	C₂²×C₄₁	1	2	yes	—
307	#519	—	2	2	no	—
308	#519	C₂²×C₄₁	1	1	yes	characteristic
309	C₄×#41	—	2	2	no	—
310	C₄×#41	—	2	2	no	—
311	C₄×#41	C₂²×C₄₁	1	2	yes	—
312	C₂×C₄×Q₈	—	2	2	no	—
313	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
314	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
315	#454	C₄×C₄₁	1	8	yes	direct product factor
316	#442	—	2	2	no	—
317	#440	—	2	2	no	—
318	C₄×SD₁₆	—	2	2	no	—
319	C₄×D₈	—	2	2	no	—
320	#428	C₂²×C₄₁	1	2	yes	—
321	C₂×#104	C₂²×C₄₁	1	2	yes	—
322	C₂×#104	C₂²×C₄₁	1	2	yes	—
323	#418	C₂²×C₄₁	1	2	yes	—
324	#417	C₂²×C₄₁	1	2	yes	—
325	C₂×#101	C₂²×C₄₁	1	1	yes	characteristic
326	C₂×#101	C₂²×C₄₁	1	1	yes	characteristic
327	C₄×M₁₆	C₂²×C₄₁	1	2	yes	—
328	#391	C₂²×C₄₁	1	1	yes	characteristic
329	#390	C₂²×C₄₁	1	1	yes	characteristic
330	#388	—	2	2	no	—
331	#382	C₂²×C₄₁	1	1	yes	characteristic
332	C₄×C₂²⋊C₄	—	2	2	no	—
333	C₄³	C₂²×C₄₁	1	1	yes	characteristic
334	#343	C₂²×C₄₁	1	2	yes	—
335	#339	C₂²×C₄₁	1	2	yes	—
336	#339	C₂²×C₄₁	1	2	yes	—
337	C₄₁	C₄×#454	1	1	yes	characteristic, direct product factor, Sylow
338	2₋¹⁺⁴	—	2	4	no	—
339	2₋¹⁺⁴	—	2	2	no	—
340	2₋¹⁺⁴	C₂×C₄×C₄₁	1	4	yes	complemented
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	C₂³×C₄₁	1	2	yes	—
344	C₂²×Q₈	—	2	2	no	—
345	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
346	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	AGL(1,ℤ/8)	—	2	4	no	—
349	AGL(1,ℤ/8)	C₂×C₄×C₄₁	1	8	yes	complemented
350	C₂×SD₁₆	—	2	2	no	—
351	C₂×D₈	—	2	2	no	—
352	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
353	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
354	#124	—	2	2	no	—
355	#124	C₂×C₄×C₄₁	1	4	yes	complemented
356	C₄×Q₈	—	2	2	no	—
357	C₄×Q₈	—	2	4	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	2	no	—
360	C₄×Q₈	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	—	2	2	no	—
364	C₄×Q₈	D₄×C₄₁	1	2	yes	—
365	C₄×Q₈	D₄×C₄₁	1	2	yes	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	D₄×C₄₁	1	2	yes	—
376	C₄×D₄	D₄×C₄₁	1	2	yes	—
377	#117	—	2	4	no	—
378	#117	—	2	4	no	—
379	#117	—	2	2	no	—
380	#117	—	2	2	no	—
381	#117	—	2	2	no	—
382	#117	C₂³×C₄₁	1	2	yes	—
383	C₂×C₄⋊C₄	—	2	4	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₄²	—	2	4	no	—
392	C₂×C₄²	—	2	2	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	D₄×C₄₁	1	2	yes	—
396	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
399	#107	—	2	2	no	—
400	#106	—	2	2	no	—
401	#104	—	2	4	no	—
402	#104	C₂×C₄×C₄₁	1	8	yes	complemented
403	#103	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#101	—	2	2	no	—
408	#101	C₂×C₄×C₄₁	1	4	yes	complemented
409	#97	—	2	2	no	—
410	C₄×C₈	—	2	2	no	—
411	#95	—	2	2	no	—
412	#95	—	2	2	no	—
413	C₂⁴	—	2	1	no	—
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	#41	—	2	8	no	—
417	#41	—	2	8	no	—
418	#41	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	C₂²×C₄×C₄₁	1	4	yes	—
421	#41	#41×C₄₁	1	4	yes	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	8	no	—
424	C₂×Q₈	—	2	8	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
428	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
429	C₂×Q₈	#41×C₄₁	1	2	yes	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
436	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
449	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
453	SD₁₆	—	2	8	no	—
454	D₈	—	2	8	no	—
455	M₁₆	—	2	2	no	—
456	M₁₆	—	2	2	no	—
457	M₁₆	—	2	2	no	—
458	M₁₆	C₂²×C₄×C₄₁	1	4	yes	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
470	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	4	4	no	—
477	C₂²⋊C₄	—	4	2	no	—
478	C₄²	—	2	4	no	—
479	C₄²	—	2	4	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	2	no	—
484	C₄²	—	2	2	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	2	no	—
487	C₄²	—	4	2	no	—
488	C₄²	—	4	2	no	—
489	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
490	C₄²	C₂×D₄×C₄₁	1	2	yes	—
491	C₄²	C₂×D₄×C₄₁	1	2	yes	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	1	no	—
497	C₂³	—	4	1	no	—
498	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
499	Q₈	—	2	8	no	—
500	Q₈	—	2	4	no	—
501	Q₈	—	2	2	no	—
502	Q₈	—	2	4	no	—
503	Q₈	C₄×D₄×C₄₁	1	4	yes	—
504	D₄	—	2	8	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	4	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	2	no	—
509	D₄	—	4	2	no	—
510	D₄	C₄×D₄×C₄₁	1	4	yes	—
511	C₂×C₄	—	2	16	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	8	2	no	—
533	C₂×C₄	—	8	2	no	—
534	C₂×C₄	—	8	4	no	—
535	C₂×C₄	—	8	2	no	—
536	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
537	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
538	C₂×C₄	#120×C₄₁	1	2	yes	—
539	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
540	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
542	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
543	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
544	C₈	—	2	4	no	—
545	C₈	—	4	2	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	4	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	8	2	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
557	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
558	C₂²	C₄₁×#525	1	1	yes	characteristic
559	C₄	—	2	4	no	—
560	C₄	—	2	4	no	—
561	C₄	—	2	1	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	2	no	—
565	C₄	—	4	2	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	2	no	—
568	C₄	—	8	1	no	—
569	C₄	C₄₁×#454	1	2	yes	direct product factor
570	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
571	C₄	C₄₁×#525	1	2	yes	—
572	C₂	—	2	1	no	—
573	C₂	—	2	1	no	—
574	C₂	—	4	4	no	—
575	C₂	—	8	2	no	—
576	C₂	C₄×#120×C₄₁	1	1	yes	characteristic
577	C₂	C₂×C₄₁×#454	1	1	yes	characteristic
578	C₂	C₄₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	#120
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×#120
prime radical	C₂×C₄₁×#454	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×2₋¹⁺⁴×C₄₁	C₂	semidirect
3. C₄×AGL(1,ℤ/8)×C₄₁	C₂	semidirect
4. C₄×#124×C₄₁	C₂	semidirect
5. C₄₁×#1888	C₂	semidirect
6. C₄₁×#1883	C₂	semidirect
7. C₄₁×#1860	C₂	semidirect
8. C₄₁×#1857	C₂	semidirect
9. C₄₁×#1793	C₂	semidirect
10. C₄×#104×C₄₁	C₂	semidirect
11. C₄×#101×C₄₁	C₂	semidirect
26. C₄₁×#454	C₄	direct (N × H)
50. 2₋¹⁺⁴×C₄₁	C₂×C₄	semidirect
59. AGL(1,ℤ/8)×C₄₁	C₂×C₄	semidirect
65. #124×C₄₁	C₂×C₄	semidirect
112. #104×C₄₁	C₂×C₄	semidirect
118. #101×C₄₁	C₂×C₄	semidirect
256. C₄×#454	C₄₁	direct (N × H)
280. C₄×C₄₁	#454	direct (N × H)
283. C₄×2₋¹⁺⁴	C₂×C₄₁	semidirect
285. C₄×AGL(1,ℤ/8)	C₂×C₄₁	semidirect
286. C₄×#124	C₂×C₄₁	semidirect
287. #1888	C₂×C₄₁	semidirect
288. #1883	C₂×C₄₁	semidirect
289. #1860	C₂×C₄₁	semidirect
290. #1857	C₂×C₄₁	semidirect
291. #1793	C₂×C₄₁	semidirect
292. C₄×#104	C₂×C₄₁	semidirect
293. C₄×#101	C₂×C₄₁	semidirect
315. #454	C₄×C₄₁	direct (N × H)
337. C₄₁	C₄×#454	direct (N × H)
340. 2₋¹⁺⁴	C₂×C₄×C₄₁	semidirect
349. AGL(1,ℤ/8)	C₂×C₄×C₄₁	semidirect
355. #124	C₂×C₄×C₄₁	semidirect
402. #104	C₂×C₄×C₄₁	semidirect
408. #101	C₂×C₄×C₄₁	semidirect
569. C₄	C₄₁×#454	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×2₋¹⁺⁴×C₄₁	2
C₂×C₄₁×#454	2
C₄×AGL(1,ℤ/8)×C₄₁	2
C₄×#124×C₄₁	2
C₄₁×#1888	2
C₄₁×#1883	2
C₄₁×#1860	2
C₄₁×#1857	2
C₄₁×#1793	2
C₄×#104×C₄₁	2
C₄×#101×C₄₁	2
C₄×#454	41

Subgroups — C4×C41×#454 #23797672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×#454 #23797672