Subgroups — C₄₁×#17771 #23793972

Order 10496.

990 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2990	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1557	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1555	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1553	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1552	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1547	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1540	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#559	—	2	2	no	—
11	C₄₁×#550	—	2	2	no	—
12	C₄₁×#546	—	2	2	no	—
13	C₄₁×#546	—	2	2	no	—
14	C₄₁×#540	C₄	1	4	yes	—
15	C₄₁×#535	C₄	1	2	yes	—
16	C₄₁×#533	C₂²	1	1	yes	characteristic
17	C₄₁×#532	C₂²	1	2	yes	—
18	C₄₁×#532	C₂²	1	2	yes	—
19	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#518	C₄	1	2	yes	—
21	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#446	—	2	2	no	—
23	C₄₁×#435	—	2	2	no	—
24	C₄₁×#435	—	2	2	no	—
25	C₈×D₄×C₄₁	—	2	2	no	—
26	C₄₁×#432	C₂²	1	2	yes	—
27	C₄₁×#432	C₂²	1	2	yes	complemented
28	C₄₁×#431	C₂²	1	2	yes	complemented
29	C₄₁×#423	C₂²	1	2	yes	complemented
30	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
31	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	complemented
32	C₄₁×#332	C₂²	1	2	yes	complemented
33	C₄₁×#332	C₂²	1	2	yes	complemented
34	C₄₁×#332	C₂²	1	2	yes	complemented
35	C₄₁×#332	C₂²	1	2	yes	complemented
36	C₄₁×#331	C₂²	1	1	yes	characteristic
37	C₄₁×#331	C₂²	1	2	yes	—
38	C₄₁×#331	C₂²	1	1	yes	characteristic
39	C₄₁×#331	C₂²	1	2	yes	—
40	C₄₁×#331	C₂²	1	1	yes	characteristic
41	C₄₁×#331	C₂²	1	1	yes	characteristic
42	C₂×#41×C₄₁	—	2	2	no	—
43	C₂×#41×C₄₁	—	2	2	no	—
44	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×M₁₆×C₄₁	—	2	2	no	—
48	C₂²×C₈×C₄₁	—	2	2	no	—
49	C₂²×C₈×C₄₁	—	2	2	no	—
50	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
51	#127×C₄₁	—	2	2	no	—
52	#127×C₄₁	—	2	2	no	—
53	#127×C₄₁	C₂×C₄	1	2	yes	—
54	#127×C₄₁	C₂³	1	1	yes	characteristic
55	#127×C₄₁	C₂³	1	1	yes	characteristic
56	#125×C₄₁	C₂×C₄	1	2	yes	—
57	#124×C₄₁	—	2	2	no	—
58	#124×C₄₁	—	2	2	no	—
59	#124×C₄₁	—	2	2	no	—
60	#123×C₄₁	—	2	4	no	—
61	#123×C₄₁	—	2	4	no	—
62	C₂²⋊Q₈×C₄₁	—	2	2	no	—
63	#121×C₄₁	—	2	2	no	—
64	#121×C₄₁	—	2	2	no	—
65	#121×C₄₁	—	2	4	no	—
66	#121×C₄₁	—	2	4	no	—
67	#121×C₄₁	C₂×C₄	1	4	yes	—
68	#121×C₄₁	C₂×C₄	1	4	yes	—
69	#120×C₄₁	—	2	4	no	—
70	C₄×Q₈×C₄₁	—	2	2	no	—
71	C₄×Q₈×C₄₁	D₄	1	2	yes	—
72	C₄×D₄×C₄₁	—	2	4	no	—
73	C₄×D₄×C₄₁	—	2	2	no	—
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
77	C₄×D₄×C₄₁	D₄	1	2	yes	—
78	C₄×D₄×C₄₁	D₄	1	2	yes	—
79	C₄×D₄×C₄₁	D₄	1	2	yes	—
80	#117×C₄₁	C₂×C₄	1	2	yes	—
81	#117×C₄₁	C₂×C₄	1	1	yes	characteristic
82	#117×C₄₁	C₂×C₄	1	2	yes	—
83	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
85	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
87	C₄⋊C₈×C₄₁	—	2	2	no	—
88	C₄⋊C₈×C₄₁	—	2	2	no	—
89	C₄⋊C₈×C₄₁	—	2	2	no	—
90	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
91	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
92	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
93	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
94	C₂²⋊C₈×C₄₁	—	2	2	no	—
95	C₂²⋊C₈×C₄₁	—	2	2	no	—
96	C₂²⋊C₈×C₄₁	—	2	2	no	—
97	#97×C₄₁	—	2	2	no	—
98	#97×C₄₁	—	2	2	no	—
99	C₄×C₈×C₄₁	—	2	2	no	—
100	C₂⁴×C₄₁	—	2	2	no	—
101	#41×C₄₁	—	4	2	no	—
102	#41×C₄₁	—	8	2	no	—
103	#41×C₄₁	—	8	2	no	—
104	#41×C₄₁	—	8	2	no	—
105	#41×C₄₁	—	8	2	no	—
106	C₂×Q₈×C₄₁	—	2	2	no	—
107	C₂×Q₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×D₄×C₄₁	—	2	4	no	—
109	C₂×D₄×C₄₁	—	2	4	no	—
110	C₂×D₄×C₄₁	—	2	2	no	—
111	C₂×D₄×C₄₁	—	2	4	no	—
112	C₂×D₄×C₄₁	—	2	2	no	—
113	C₂×D₄×C₄₁	—	2	2	no	—
114	C₂×D₄×C₄₁	—	4	4	no	—
115	C₂×D₄×C₄₁	—	8	2	no	—
116	C₂×D₄×C₄₁	—	8	2	no	—
117	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
118	C₂×D₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
119	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₄₁	—	2	4	no	—
122	C₂²×C₄×C₄₁	—	2	4	no	—
123	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
128	M₁₆×C₄₁	—	8	2	no	—
129	C₂×C₈×C₄₁	—	2	2	no	—
130	C₂×C₈×C₄₁	—	2	2	no	—
131	C₂×C₈×C₄₁	—	2	2	no	—
132	C₂×C₈×C₄₁	—	2	2	no	—
133	C₂×C₈×C₄₁	—	8	2	no	—
134	C₂×C₈×C₄₁	—	8	2	no	—
135	C₄⋊C₄×C₄₁	—	2	4	no	—
136	C₄⋊C₄×C₄₁	—	2	2	no	—
137	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
138	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
139	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
140	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
141	C₂²⋊C₄×C₄₁	—	2	2	no	—
142	C₂²⋊C₄×C₄₁	—	2	4	no	—
143	C₂²⋊C₄×C₄₁	—	2	4	no	—
144	C₂²⋊C₄×C₄₁	—	2	4	no	—
145	C₂²⋊C₄×C₄₁	—	2	4	no	—
146	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
149	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
150	C₄²×C₄₁	—	2	2	no	—
151	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
152	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
153	C₄²×C₄₁	C₂×D₄	1	2	yes	—
154	C₄²×C₄₁	C₂×D₄	1	2	yes	—
155	C₄²×C₄₁	C₂×D₄	1	2	yes	—
156	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
157	C₂³×C₄₁	—	2	2	no	—
158	C₂³×C₄₁	—	2	2	no	—
159	C₂³×C₄₁	—	4	4	no	—
160	C₂³×C₄₁	—	8	4	no	—
161	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	Q₈×C₄₁	—	4	1	no	—
164	Q₈×C₄₁	—	4	2	no	—
165	Q₈×C₄₁	—	4	2	no	—
166	D₄×C₄₁	—	2	4	no	—
167	D₄×C₄₁	—	4	1	no	—
168	D₄×C₄₁	—	4	2	no	—
169	D₄×C₄₁	—	4	2	no	—
170	D₄×C₄₁	—	4	4	no	—
171	D₄×C₄₁	—	4	2	no	—
172	D₄×C₄₁	—	4	2	no	—
173	D₄×C₄₁	—	4	2	no	—
174	D₄×C₄₁	—	4	2	no	—
175	D₄×C₄₁	—	8	4	no	—
176	D₄×C₄₁	—	8	4	no	—
177	D₄×C₄₁	—	8	4	no	—
178	C₂×C₄×C₄₁	—	2	2	no	—
179	C₂×C₄×C₄₁	—	2	2	no	—
180	C₂×C₄×C₄₁	—	2	2	no	—
181	C₂×C₄×C₄₁	—	2	2	no	—
182	C₂×C₄×C₄₁	—	2	2	no	—
183	C₂×C₄×C₄₁	—	2	2	no	—
184	C₂×C₄×C₄₁	—	2	2	no	—
185	C₂×C₄×C₄₁	—	2	4	no	—
186	C₂×C₄×C₄₁	—	4	2	no	—
187	C₂×C₄×C₄₁	—	8	4	no	—
188	C₂×C₄×C₄₁	—	8	4	no	—
189	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
192	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
194	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
196	C₈×C₄₁	—	4	2	no	—
197	C₈×C₄₁	—	4	2	no	—
198	C₈×C₄₁	—	4	2	no	—
199	C₈×C₄₁	—	4	2	no	—
200	#17771	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂²×C₄₁	—	2	2	no	—
202	C₂²×C₄₁	—	2	2	no	—
203	C₂²×C₄₁	—	2	2	no	—
204	C₂²×C₄₁	—	2	2	no	—
205	C₂²×C₄₁	—	4	2	no	—
206	C₂²×C₄₁	—	4	2	no	—
207	C₂²×C₄₁	—	4	4	no	—
208	C₂²×C₄₁	—	4	4	no	—
209	C₂²×C₄₁	—	16	4	no	—
210	C₂²×C₄₁	#413	1	2	yes	—
211	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
212	C₄×C₄₁	—	2	1	no	—
213	C₄×C₄₁	—	2	1	no	—
214	C₄×C₄₁	—	2	2	no	—
215	C₄×C₄₁	—	2	2	no	—
216	C₄×C₄₁	—	2	2	no	—
217	C₄×C₄₁	—	2	2	no	—
218	C₄×C₄₁	—	4	4	no	—
219	C₄×C₄₁	#576	1	2	yes	—
220	C₄×C₄₁	#577	1	2	yes	—
221	#3508	C₂×C₄₁	1	1	yes	characteristic
222	#2990	C₂×C₄₁	1	2	yes	complemented
223	#1557	C₂×C₄₁	1	2	yes	complemented
224	#1555	C₂×C₄₁	1	1	yes	characteristic, complemented
225	#1553	C₂×C₄₁	1	2	yes	complemented
226	#1552	C₂×C₄₁	1	2	yes	complemented
227	#1547	C₂×C₄₁	1	2	yes	complemented
228	#1540	C₂×C₄₁	1	2	yes	complemented
229	C₂×#331	C₂×C₄₁	1	1	yes	characteristic, complemented
230	C₂×C₄₁	—	2	2	no	—
231	C₂×C₄₁	—	4	2	no	—
232	C₂×C₄₁	—	8	2	no	—
233	C₂×C₄₁	—	8	2	no	—
234	C₂×C₄₁	C₂×#413	1	1	yes	characteristic
235	C₂×C₄₁	#2897	1	2	yes	—
236	#559	—	2	2	no	—
237	#550	—	2	2	no	—
238	#546	—	2	2	no	—
239	#546	—	2	2	no	—
240	#540	C₄×C₄₁	1	4	yes	—
241	#535	C₄×C₄₁	1	2	yes	—
242	#533	C₂²×C₄₁	1	1	yes	characteristic
243	#532	C₂²×C₄₁	1	2	yes	—
244	#532	C₂²×C₄₁	1	2	yes	—
245	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
246	#518	C₄×C₄₁	1	2	yes	—
247	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
248	#446	—	2	2	no	—
249	#435	—	2	2	no	—
250	#435	—	2	2	no	—
251	C₈×D₄	—	2	2	no	—
252	#432	C₂²×C₄₁	1	2	yes	—
253	#432	C₂²×C₄₁	1	2	yes	complemented
254	#431	C₂²×C₄₁	1	2	yes	complemented
255	#423	C₂²×C₄₁	1	2	yes	complemented
256	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
258	#332	C₂²×C₄₁	1	2	yes	complemented
259	#332	C₂²×C₄₁	1	2	yes	complemented
260	#332	C₂²×C₄₁	1	2	yes	complemented
261	#332	C₂²×C₄₁	1	2	yes	complemented
262	#331	C₂²×C₄₁	1	1	yes	characteristic
263	#331	C₂²×C₄₁	1	2	yes	—
264	#331	C₂²×C₄₁	1	1	yes	characteristic
265	#331	C₂²×C₄₁	1	2	yes	—
266	#331	C₂²×C₄₁	1	1	yes	characteristic
267	#331	C₂²×C₄₁	1	1	yes	characteristic
268	C₄₁	#17771	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×#41	—	2	2	no	—
270	C₂×#41	—	2	2	no	—
271	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
274	C₂×M₁₆	—	2	2	no	—
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	—	2	2	no	—
277	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
278	#127	—	2	2	no	—
279	#127	—	2	2	no	—
280	#127	C₂×C₄×C₄₁	1	2	yes	—
281	#127	C₂³×C₄₁	1	1	yes	characteristic
282	#127	C₂³×C₄₁	1	1	yes	characteristic
283	#125	C₂×C₄×C₄₁	1	2	yes	—
284	#124	—	2	2	no	—
285	#124	—	2	2	no	—
286	#124	—	2	2	no	—
287	#123	—	2	4	no	—
288	#123	—	2	4	no	—
289	C₂²⋊Q₈	—	2	2	no	—
290	#121	—	2	2	no	—
291	#121	—	2	2	no	—
292	#121	—	2	4	no	—
293	#121	—	2	4	no	—
294	#121	C₂×C₄×C₄₁	1	4	yes	—
295	#121	C₂×C₄×C₄₁	1	4	yes	—
296	#120	—	2	4	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₄×Q₈	D₄×C₄₁	1	2	yes	—
299	C₄×D₄	—	2	4	no	—
300	C₄×D₄	—	2	2	no	—
301	C₄×D₄	—	2	2	no	—
302	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
303	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
304	C₄×D₄	D₄×C₄₁	1	2	yes	—
305	C₄×D₄	D₄×C₄₁	1	2	yes	—
306	C₄×D₄	D₄×C₄₁	1	2	yes	—
307	#117	C₂×C₄×C₄₁	1	2	yes	—
308	#117	C₂×C₄×C₄₁	1	1	yes	characteristic
309	#117	C₂×C₄×C₄₁	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
314	C₄⋊C₈	—	2	2	no	—
315	C₄⋊C₈	—	2	2	no	—
316	C₄⋊C₈	—	2	2	no	—
317	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
318	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
319	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
320	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₂²⋊C₈	—	2	2	no	—
323	C₂²⋊C₈	—	2	2	no	—
324	#97	—	2	2	no	—
325	#97	—	2	2	no	—
326	C₄×C₈	—	2	2	no	—
327	C₂⁴	—	2	2	no	—
328	#41	—	4	2	no	—
329	#41	—	8	2	no	—
330	#41	—	8	2	no	—
331	#41	—	8	2	no	—
332	#41	—	8	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	4	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	4	4	no	—
342	C₂×D₄	—	8	2	no	—
343	C₂×D₄	—	8	2	no	—
344	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
345	C₂×D₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
346	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
347	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
355	M₁₆	—	8	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	8	2	no	—
361	C₂×C₈	—	8	2	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
365	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
366	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
367	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
368	C₂²⋊C₄	—	2	2	no	—
369	C₂²⋊C₄	—	2	4	no	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
374	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
375	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
376	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
377	C₄²	—	2	2	no	—
378	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
379	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
380	C₄²	C₂×D₄×C₄₁	1	2	yes	—
381	C₄²	C₂×D₄×C₄₁	1	2	yes	—
382	C₄²	C₂×D₄×C₄₁	1	2	yes	—
383	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	4	4	no	—
387	C₂³	—	8	4	no	—
388	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
389	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
390	Q₈	—	4	1	no	—
391	Q₈	—	4	2	no	—
392	Q₈	—	4	2	no	—
393	D₄	—	2	4	no	—
394	D₄	—	4	1	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	4	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	8	4	no	—
403	D₄	—	8	4	no	—
404	D₄	—	8	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	8	4	no	—
415	C₂×C₄	—	8	4	no	—
416	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
417	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
418	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
419	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
420	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
421	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
423	C₈	—	4	2	no	—
424	C₈	—	4	2	no	—
425	C₈	—	4	2	no	—
426	C₈	—	4	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	4	no	—
435	C₂²	—	16	4	no	—
436	C₂²	C₄₁×#413	1	2	yes	—
437	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	2	no	—
444	C₄	—	4	4	no	—
445	C₄	C₄₁×#576	1	2	yes	—
446	C₄	C₄₁×#577	1	2	yes	—
447	C₂	—	2	2	no	—
448	C₂	—	4	2	no	—
449	C₂	—	8	2	no	—
450	C₂	—	8	2	no	—
451	C₂	C₂×C₄₁×#413	1	1	yes	characteristic
452	C₂	C₄₁×#2897	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2990	C₂	semidirect
3. C₄₁×#1557	C₂	semidirect
4. C₄₁×#1555	C₂	semidirect
5. C₄₁×#1553	C₂	semidirect
6. C₄₁×#1552	C₂	semidirect
7. C₄₁×#1547	C₂	semidirect
8. C₄₁×#1540	C₂	semidirect
9. C₂×C₄₁×#331	C₂	semidirect
27. C₄₁×#432	C₂²	semidirect
28. C₄₁×#431	C₂²	semidirect
29. C₄₁×#423	C₂²	semidirect
31. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
32. C₄₁×#332	C₂²	semidirect
33. C₄₁×#332	C₂²	semidirect
34. C₄₁×#332	C₂²	semidirect
35. C₄₁×#332	C₂²	semidirect
200. #17771	C₄₁	direct (N × H)
222. #2990	C₂×C₄₁	semidirect
223. #1557	C₂×C₄₁	semidirect
224. #1555	C₂×C₄₁	semidirect
225. #1553	C₂×C₄₁	semidirect
226. #1552	C₂×C₄₁	semidirect
227. #1547	C₂×C₄₁	semidirect
228. #1540	C₂×C₄₁	semidirect
229. C₂×#331	C₂×C₄₁	semidirect
253. #432	C₂²×C₄₁	semidirect
254. #431	C₂²×C₄₁	semidirect
255. #423	C₂²×C₄₁	semidirect
257. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
258. #332	C₂²×C₄₁	semidirect
259. #332	C₂²×C₄₁	semidirect
260. #332	C₂²×C₄₁	semidirect
261. #332	C₂²×C₄₁	semidirect
268. C₄₁	#17771	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3508	2
C₄₁×#2990	2
C₄₁×#1557	2
C₄₁×#1555	2
C₄₁×#1553	2
C₄₁×#1552	2
C₄₁×#1547	2
C₄₁×#1540	2
C₂×C₄₁×#331	2
#17771	41

Subgroups — C41×#17771 #23793972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17771 #23793972