Subgroups — C₂×C₄₁×#1631 #23794706

Order 10496.

1322 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#550	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1631	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₁	—	2	1	no	—
10	C₂²×SD₁₆×C₄₁	—	2	2	no	—
11	C₄₁×#550	—	2	2	no	—
12	C₄₁×#550	—	2	2	no	—
13	C₄₁×#550	—	2	2	no	—
14	C₄₁×#550	C₂²	1	4	yes	—
15	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#127×C₄₁	—	2	1	no	—
17	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#121×C₄₁	—	2	2	no	—
19	C₂×#121×C₄₁	—	2	1	no	—
20	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	—	2	1	no	—
22	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#492	—	2	2	no	—
24	C₄₁×#492	C₂²	1	4	yes	complemented
25	C₄₁×#477	—	2	2	no	—
26	C₄₁×#477	C₂²	1	4	yes	complemented
27	C₄₁×#476	—	2	2	no	—
28	C₄₁×#476	C₂²	1	4	yes	complemented
29	C₂×#107×C₄₁	—	2	1	no	—
30	C₂×#106×C₄₁	—	2	1	no	—
31	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×#103×C₄₁	—	2	1	no	—
34	C₂×#102×C₄₁	—	2	1	no	—
35	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#331	C₂²	1	4	yes	complemented
37	C₄₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
38	C₄₁×D₄⋊C₈	C₂²	1	4	yes	complemented
39	C₂×#41×C₄₁	—	2	4	no	—
40	C₂×#41×C₄₁	—	2	4	no	—
41	C₂×#41×C₄₁	—	4	4	no	—
42	C₂×#41×C₄₁	—	4	4	no	—
43	C₂²×Q₈×C₄₁	—	2	1	no	—
44	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	—	2	2	no	—
47	C₂²×D₄×C₄₁	—	2	1	no	—
48	C₂²×D₄×C₄₁	—	2	1	no	—
49	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₄₁	D₄	1	2	yes	—
51	C₂³×C₄×C₄₁	—	2	2	no	—
52	C₂³×C₄×C₄₁	—	2	1	no	—
53	C₂×SD₁₆×C₄₁	—	2	8	no	—
54	C₂×SD₁₆×C₄₁	—	4	8	no	—
55	C₂²×C₈×C₄₁	—	2	1	no	—
56	C₂²×C₈×C₄₁	—	2	1	no	—
57	C₂²×C₈×C₄₁	—	2	1	no	—
58	C₄⋊Q₈×C₄₁	—	2	1	no	—
59	C₄⋊Q₈×C₄₁	—	2	1	no	—
60	C₄⋊Q₈×C₄₁	—	2	1	no	—
61	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
62	#127×C₄₁	—	2	4	no	—
63	#127×C₄₁	—	2	4	no	—
64	#127×C₄₁	—	2	1	no	—
65	#127×C₄₁	—	2	1	no	—
66	#127×C₄₁	D₄	1	2	yes	—
67	#127×C₄₁	C₂³	1	2	yes	—
68	#121×C₄₁	—	2	4	no	—
69	#121×C₄₁	—	2	8	no	—
70	#121×C₄₁	—	2	8	no	—
71	#121×C₄₁	—	2	4	no	—
72	C₄×Q₈×C₄₁	—	2	1	no	—
73	C₄×Q₈×C₄₁	—	2	1	no	—
74	C₄×Q₈×C₄₁	D₄	1	2	yes	—
75	C₄×Q₈×C₄₁	C₂³	1	2	yes	—
76	C₄×D₄×C₄₁	—	2	4	no	—
77	C₄×D₄×C₄₁	—	2	4	no	—
78	C₄×D₄×C₄₁	—	2	1	no	—
79	C₄×D₄×C₄₁	—	2	1	no	—
80	C₄×D₄×C₄₁	D₄	1	2	yes	—
81	C₄×D₄×C₄₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
89	C₂×C₄²×C₄₁	—	2	1	no	—
90	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
91	#107×C₄₁	—	2	4	no	—
92	#106×C₄₁	—	2	4	no	—
93	C₄⋊C₈×C₄₁	—	2	1	no	—
94	C₄⋊C₈×C₄₁	—	2	1	no	—
95	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
96	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
97	#103×C₄₁	—	2	4	no	—
98	#102×C₄₁	—	2	4	no	—
99	C₄×C₈×C₄₁	—	2	1	no	—
100	C₄×C₈×C₄₁	C₂³	1	2	yes	—
101	C₂⁴×C₄₁	—	2	1	no	—
102	C₂⁴×C₄₁	—	2	2	no	—
103	#41×C₄₁	—	4	8	no	—
104	#41×C₄₁	—	4	16	no	—
105	#41×C₄₁	—	4	8	no	—
106	C₂×Q₈×C₄₁	—	2	1	no	—
107	C₂×Q₈×C₄₁	—	2	2	no	—
108	C₂×Q₈×C₄₁	—	2	2	no	—
109	C₂×Q₈×C₄₁	—	4	4	no	—
110	C₂×Q₈×C₄₁	—	4	1	no	—
111	C₂×Q₈×C₄₁	SD₁₆	1	4	yes	complemented
112	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
113	C₂×D₄×C₄₁	—	2	8	no	—
114	C₂×D₄×C₄₁	—	2	8	no	—
115	C₂×D₄×C₄₁	—	2	1	no	—
116	C₂×D₄×C₄₁	—	2	2	no	—
117	C₂×D₄×C₄₁	—	2	2	no	—
118	C₂×D₄×C₄₁	—	2	2	no	—
119	C₂×D₄×C₄₁	—	2	2	no	—
120	C₂×D₄×C₄₁	—	2	2	no	—
121	C₂×D₄×C₄₁	—	2	2	no	—
122	C₂×D₄×C₄₁	—	2	2	no	—
123	C₂×D₄×C₄₁	—	4	8	no	—
124	C₂×D₄×C₄₁	—	4	4	no	—
125	C₂×D₄×C₄₁	—	4	2	no	—
126	C₂×D₄×C₄₁	—	4	2	no	—
127	C₂×D₄×C₄₁	—	4	2	no	—
128	C₂×D₄×C₄₁	—	4	2	no	—
129	C₂×D₄×C₄₁	—	8	4	no	—
130	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₄₁	—	2	1	no	—
133	C₂²×C₄×C₄₁	—	2	8	no	—
134	C₂²×C₄×C₄₁	—	2	1	no	—
135	C₂²×C₄×C₄₁	—	2	1	no	—
136	C₂²×C₄×C₄₁	—	2	2	no	—
137	C₂²×C₄×C₄₁	—	2	2	no	—
138	C₂²×C₄×C₄₁	—	4	8	no	—
139	C₂²×C₄×C₄₁	—	4	2	no	—
140	C₂²×C₄×C₄₁	—	4	2	no	—
141	C₂²×C₄×C₄₁	—	4	1	no	—
142	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
145	SD₁₆×C₄₁	—	4	16	no	—
146	C₂×C₈×C₄₁	—	2	4	no	—
147	C₂×C₈×C₄₁	—	2	2	no	—
148	C₂×C₈×C₄₁	—	2	2	no	—
149	C₂×C₈×C₄₁	—	2	2	no	—
150	C₂×C₈×C₄₁	—	4	2	no	—
151	C₂×C₈×C₄₁	—	4	2	no	—
152	C₄⋊C₄×C₄₁	—	2	2	no	—
153	C₄⋊C₄×C₄₁	—	2	1	no	—
154	C₄⋊C₄×C₄₁	—	2	1	no	—
155	C₄⋊C₄×C₄₁	—	2	2	no	—
156	C₄⋊C₄×C₄₁	—	2	2	no	—
157	C₄⋊C₄×C₄₁	—	2	2	no	—
158	C₄⋊C₄×C₄₁	—	4	1	no	—
159	C₄⋊C₄×C₄₁	—	4	1	no	—
160	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₄₁	—	2	8	no	—
163	C₂²⋊C₄×C₄₁	—	2	2	no	—
164	C₂²⋊C₄×C₄₁	—	2	2	no	—
165	C₄²×C₄₁	—	2	2	no	—
166	C₄²×C₄₁	—	2	2	no	—
167	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
171	C₂³×C₄₁	—	2	2	no	—
172	C₂³×C₄₁	—	2	4	no	—
173	C₂³×C₄₁	—	4	4	no	—
174	C₂³×C₄₁	—	4	2	no	—
175	C₂³×C₄₁	—	4	2	no	—
176	C₂³×C₄₁	—	4	4	no	—
177	C₂³×C₄₁	—	4	4	no	—
178	C₂³×C₄₁	—	4	2	no	—
179	C₂³×C₄₁	#120	1	1	yes	characteristic
180	Q₈×C₄₁	—	2	2	no	—
181	Q₈×C₄₁	—	2	4	no	—
182	Q₈×C₄₁	—	4	4	no	—
183	Q₈×C₄₁	—	8	2	no	—
184	Q₈×C₄₁	C₂×SD₁₆	1	4	yes	complemented
185	D₄×C₄₁	—	2	2	no	—
186	D₄×C₄₁	—	2	2	no	—
187	D₄×C₄₁	—	4	4	no	—
188	D₄×C₄₁	—	4	16	no	—
189	D₄×C₄₁	—	4	2	no	—
190	D₄×C₄₁	—	4	2	no	—
191	D₄×C₄₁	—	4	2	no	—
192	D₄×C₄₁	—	4	2	no	—
193	D₄×C₄₁	—	4	4	no	—
194	D₄×C₄₁	—	4	2	no	—
195	D₄×C₄₁	—	4	4	no	—
196	D₄×C₄₁	—	8	2	no	—
197	D₄×C₄₁	—	8	8	no	—
198	C₂×C₄×C₄₁	—	2	2	no	—
199	C₂×C₄×C₄₁	—	2	4	no	—
200	C₂×C₄×C₄₁	—	2	2	no	—
201	C₂×C₄×C₄₁	—	2	2	no	—
202	C₂×C₄×C₄₁	—	2	2	no	—
203	C₂×C₄×C₄₁	—	4	16	no	—
204	C₂×C₄×C₄₁	—	4	2	no	—
205	C₂×C₄×C₄₁	—	4	2	no	—
206	C₂×C₄×C₄₁	—	4	2	no	—
207	C₂×C₄×C₄₁	—	4	2	no	—
208	C₂×C₄×C₄₁	—	4	4	no	—
209	C₂×C₄×C₄₁	—	4	2	no	—
210	C₂×C₄×C₄₁	—	8	2	no	—
211	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
212	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
213	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
214	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
215	C₂×C₄×C₄₁	#135	1	2	yes	—
216	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
217	C₂×C₄×C₄₁	#137	1	2	yes	—
218	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
221	C₈×C₄₁	—	2	4	no	—
222	C₈×C₄₁	—	4	2	no	—
223	C₈×C₄₁	—	4	2	no	—
224	C₂×#1631	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂²×C₄₁	—	4	4	no	—
226	C₂²×C₄₁	—	4	2	no	—
227	C₂²×C₄₁	—	4	2	no	—
228	C₂²×C₄₁	—	4	4	no	—
229	C₂²×C₄₁	—	4	4	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	8	4	no	—
232	C₂²×C₄₁	—	8	8	no	—
233	C₂²×C₄₁	#450	1	2	yes	—
234	C₂²×C₄₁	#452	1	2	yes	—
235	C₂²×C₄₁	#453	1	2	yes	—
236	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
237	C₄×C₄₁	—	2	4	no	—
238	C₄×C₄₁	—	2	1	no	—
239	C₄×C₄₁	—	2	1	no	—
240	C₄×C₄₁	—	4	2	no	—
241	C₄×C₄₁	—	4	2	no	—
242	C₄×C₄₁	—	8	2	no	—
243	C₄×C₄₁	#450	1	2	yes	—
244	C₄×C₄₁	#452	1	2	yes	—
245	C₄×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
246	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
247	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
249	C₂×#550	C₂×C₄₁	1	1	yes	characteristic
250	C₂×#492	C₂×C₄₁	1	1	yes	characteristic, complemented
251	C₂×#477	C₂×C₄₁	1	1	yes	characteristic, complemented
252	C₂×#476	C₂×C₄₁	1	1	yes	characteristic, complemented
253	#1631	C₂×C₄₁	1	8	yes	direct product factor
254	C₂×#331	C₂×C₄₁	1	1	yes	characteristic, complemented
255	C₂×Q₈⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
256	C₂×D₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
257	C₂×C₄₁	—	8	2	no	—
258	C₂×C₄₁	—	8	4	no	—
259	C₂×C₄₁	#1631	1	4	yes	direct product factor
260	C₂×C₄₁	C₂×#450	1	1	yes	characteristic
261	C₂×C₄₁	C₂×#452	1	1	yes	characteristic
262	C₂×C₄₁	C₂×#453	1	1	yes	characteristic
263	C₂²×#41	—	2	1	no	—
264	C₂²×SD₁₆	—	2	2	no	—
265	#550	—	2	2	no	—
266	#550	—	2	2	no	—
267	#550	—	2	2	no	—
268	#550	C₂²×C₄₁	1	4	yes	—
269	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
270	C₂×#127	—	2	1	no	—
271	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
272	C₂×#121	—	2	2	no	—
273	C₂×#121	—	2	1	no	—
274	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
275	C₂×C₄×D₄	—	2	1	no	—
276	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
277	#492	—	2	2	no	—
278	#492	C₂²×C₄₁	1	4	yes	complemented
279	#477	—	2	2	no	—
280	#477	C₂²×C₄₁	1	4	yes	complemented
281	#476	—	2	2	no	—
282	#476	C₂²×C₄₁	1	4	yes	complemented
283	C₂×#107	—	2	1	no	—
284	C₂×#106	—	2	1	no	—
285	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
286	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
287	C₂×#103	—	2	1	no	—
288	C₂×#102	—	2	1	no	—
289	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
290	#331	C₂²×C₄₁	1	4	yes	complemented
291	Q₈⋊C₈	C₂²×C₄₁	1	4	yes	complemented
292	D₄⋊C₈	C₂²×C₄₁	1	4	yes	complemented
293	C₄₁	C₂×#1631	1	1	yes	characteristic, direct product factor, Sylow
294	C₂×#41	—	2	4	no	—
295	C₂×#41	—	2	4	no	—
296	C₂×#41	—	4	4	no	—
297	C₂×#41	—	4	4	no	—
298	C₂²×Q₈	—	2	1	no	—
299	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
300	C₂²×D₄	—	2	2	no	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
305	C₂²×D₄	D₄×C₄₁	1	2	yes	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	—	2	1	no	—
308	C₂×SD₁₆	—	2	8	no	—
309	C₂×SD₁₆	—	4	8	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	—	2	1	no	—
312	C₂²×C₈	—	2	1	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
317	#127	—	2	4	no	—
318	#127	—	2	4	no	—
319	#127	—	2	1	no	—
320	#127	—	2	1	no	—
321	#127	D₄×C₄₁	1	2	yes	—
322	#127	C₂³×C₄₁	1	2	yes	—
323	#121	—	2	4	no	—
324	#121	—	2	8	no	—
325	#121	—	2	8	no	—
326	#121	—	2	4	no	—
327	C₄×Q₈	—	2	1	no	—
328	C₄×Q₈	—	2	1	no	—
329	C₄×Q₈	D₄×C₄₁	1	2	yes	—
330	C₄×Q₈	C₂³×C₄₁	1	2	yes	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	—	2	4	no	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	D₄×C₄₁	1	2	yes	—
336	C₄×D₄	C₂³×C₄₁	1	2	yes	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	1	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
346	#107	—	2	4	no	—
347	#106	—	2	4	no	—
348	C₄⋊C₈	—	2	1	no	—
349	C₄⋊C₈	—	2	1	no	—
350	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
351	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
352	#103	—	2	4	no	—
353	#102	—	2	4	no	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	C₂³×C₄₁	1	2	yes	—
356	C₂⁴	—	2	1	no	—
357	C₂⁴	—	2	2	no	—
358	#41	—	4	8	no	—
359	#41	—	4	16	no	—
360	#41	—	4	8	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	—	4	1	no	—
366	C₂×Q₈	SD₁₆×C₄₁	1	4	yes	complemented
367	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	8	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	4	8	no	—
379	C₂×D₄	—	4	4	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	8	4	no	—
385	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
386	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	4	8	no	—
394	C₂²×C₄	—	4	2	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
400	SD₁₆	—	4	16	no	—
401	C₂×C₈	—	2	4	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	4	2	no	—
406	C₂×C₈	—	4	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	4	1	no	—
414	C₄⋊C₄	—	4	1	no	—
415	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
416	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
423	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
425	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
426	C₂³	—	2	2	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	4	4	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	4	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	4	2	no	—
434	C₂³	#120×C₄₁	1	1	yes	characteristic
435	Q₈	—	2	2	no	—
436	Q₈	—	2	4	no	—
437	Q₈	—	4	4	no	—
438	Q₈	—	8	2	no	—
439	Q₈	C₂×SD₁₆×C₄₁	1	4	yes	complemented
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	4	4	no	—
443	D₄	—	4	16	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	2	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	4	no	—
451	D₄	—	8	2	no	—
452	D₄	—	8	8	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	4	16	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
467	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
468	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
469	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
470	C₂×C₄	#135×C₄₁	1	2	yes	—
471	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
472	C₂×C₄	#137×C₄₁	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
476	C₈	—	2	4	no	—
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	8	4	no	—
486	C₂²	—	8	8	no	—
487	C₂²	C₄₁×#450	1	2	yes	—
488	C₂²	C₄₁×#452	1	2	yes	—
489	C₂²	C₄₁×#453	1	2	yes	—
490	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
491	C₄	—	2	4	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	8	2	no	—
497	C₄	C₄₁×#450	1	2	yes	—
498	C₄	C₄₁×#452	1	2	yes	—
499	C₄	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
500	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
501	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
502	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
503	C₂	—	8	2	no	—
504	C₂	—	8	4	no	—
505	C₂	C₄₁×#1631	1	4	yes	direct product factor
506	C₂	C₂×C₄₁×#450	1	1	yes	characteristic
507	C₂	C₂×C₄₁×#452	1	1	yes	characteristic
508	C₂	C₂×C₄₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#492	C₂	semidirect
3. C₂×C₄₁×#477	C₂	semidirect
4. C₂×C₄₁×#476	C₂	semidirect
5. C₄₁×#1631	C₂	direct (N × H)
6. C₂×C₄₁×#331	C₂	semidirect
7. C₂×C₄₁×Q₈⋊C₈	C₂	semidirect
8. C₂×C₄₁×D₄⋊C₈	C₂	semidirect
24. C₄₁×#492	C₂²	semidirect
26. C₄₁×#477	C₂²	semidirect
28. C₄₁×#476	C₂²	semidirect
36. C₄₁×#331	C₂²	semidirect
37. C₄₁×Q₈⋊C₈	C₂²	semidirect
38. C₄₁×D₄⋊C₈	C₂²	semidirect
111. C₂×Q₈×C₄₁	SD₁₆	semidirect
184. Q₈×C₄₁	C₂×SD₁₆	semidirect
224. C₂×#1631	C₄₁	direct (N × H)
250. C₂×#492	C₂×C₄₁	semidirect
251. C₂×#477	C₂×C₄₁	semidirect
252. C₂×#476	C₂×C₄₁	semidirect
253. #1631	C₂×C₄₁	direct (N × H)
254. C₂×#331	C₂×C₄₁	semidirect
255. C₂×Q₈⋊C₈	C₂×C₄₁	semidirect
256. C₂×D₄⋊C₈	C₂×C₄₁	semidirect
259. C₂×C₄₁	#1631	direct (N × H)
278. #492	C₂²×C₄₁	semidirect
280. #477	C₂²×C₄₁	semidirect
282. #476	C₂²×C₄₁	semidirect
290. #331	C₂²×C₄₁	semidirect
291. Q₈⋊C₈	C₂²×C₄₁	semidirect
292. D₄⋊C₈	C₂²×C₄₁	semidirect
293. C₄₁	C₂×#1631	direct (N × H)
366. C₂×Q₈	SD₁₆×C₄₁	semidirect
439. Q₈	C₂×SD₁₆×C₄₁	semidirect
505. C₂	C₄₁×#1631	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×C₄₁×#550	2
C₂×C₄₁×#492	2
C₂×C₄₁×#477	2
C₂×C₄₁×#476	2
C₄₁×#1631	2
C₂×C₄₁×#331	2
C₂×C₄₁×Q₈⋊C₈	2
C₂×C₄₁×D₄⋊C₈	2
C₂×#1631	41

Subgroups — C2×C41×#1631 #23794706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1631 #23794706