Subgroups — C₄₁×#17776 #23793977

Order 10496.

988 conjugacy classes of subgroups.

The subgroup lattice has 542 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×#2991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1561	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1557	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1553	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1547	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1544	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1540	C₂	1	2	yes	maximal, complemented
12	C₄₁×#559	—	2	1	no	—
13	C₄₁×#559	—	2	1	no	—
14	C₄₁×#550	—	2	1	no	—
15	C₄₁×#550	—	2	1	no	—
16	C₄₁×#546	—	2	2	no	—
17	C₄₁×#546	—	2	2	no	—
18	C₄₁×#540	C₄	1	4	yes	—
19	C₄₁×#535	C₄	1	2	yes	—
20	C₄₁×#533	C₂²	1	1	yes	characteristic
21	C₄₁×#532	C₂²	1	2	yes	—
22	C₄₁×#532	C₂²	1	2	yes	—
23	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#518	C₄	1	2	yes	—
25	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#446	—	2	1	no	—
27	C₈×Q₈×C₄₁	—	2	1	no	—
28	C₄₁×#436	—	2	1	no	—
29	C₄₁×#436	—	2	2	no	—
30	C₄₁×#435	—	2	2	no	—
31	C₈×D₄×C₄₁	—	2	1	no	—
32	C₄₁×#432	C₂²	1	2	yes	complemented
33	C₄₁×#431	C₂²	1	1	yes	characteristic
34	C₄₁×#431	C₂²	1	1	yes	characteristic
35	C₄₁×#431	C₂²	1	2	yes	complemented
36	C₄₁×#423	C₂²	1	1	yes	characteristic
37	C₄₁×#423	C₂²	1	1	yes	characteristic
38	C₄₁×#423	C₂²	1	2	yes	complemented
39	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	complemented
40	C₄₁×#332	C₂²	1	1	yes	characteristic, complemented
41	C₄₁×#332	C₂²	1	2	yes	complemented
42	C₄₁×#332	C₂²	1	1	yes	characteristic, complemented
43	C₄₁×#332	C₂²	1	1	yes	characteristic, complemented
44	C₄₁×#332	C₂²	1	1	yes	characteristic, complemented
45	C₄₁×#332	C₂²	1	2	yes	complemented
46	C₄₁×#331	C₂²	1	2	yes	—
47	C₄₁×#331	C₂²	1	2	yes	—
48	C₄₁×#331	C₂²	1	2	yes	—
49	C₄₁×#331	C₂²	1	2	yes	—
50	C₂×#41×C₄₁	—	2	2	no	—
51	C₂×#41×C₄₁	—	2	2	no	—
52	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×D₄×C₄₁	—	2	1	no	—
54	C₂²×D₄×C₄₁	—	2	1	no	—
55	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₄₁	—	2	1	no	—
57	C₂×M₁₆×C₄₁	—	2	1	no	—
58	C₂×M₁₆×C₄₁	—	2	2	no	—
59	C₂²×C₈×C₄₁	—	2	2	no	—
60	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
62	#127×C₄₁	—	2	2	no	—
63	#127×C₄₁	—	2	1	no	—
64	#127×C₄₁	—	2	1	no	—
65	#127×C₄₁	C₂×C₄	1	2	yes	—
66	#127×C₄₁	C₂³	1	2	yes	—
67	#125×C₄₁	C₂×C₄	1	2	yes	—
68	#124×C₄₁	—	2	2	no	—
69	#124×C₄₁	—	2	2	no	—
70	#124×C₄₁	—	2	2	no	—
71	#123×C₄₁	—	2	4	no	—
72	#123×C₄₁	—	2	4	no	—
73	C₂²⋊Q₈×C₄₁	—	2	1	no	—
74	C₂²⋊Q₈×C₄₁	—	2	1	no	—
75	#121×C₄₁	—	2	2	no	—
76	#121×C₄₁	—	2	2	no	—
77	#121×C₄₁	—	2	4	no	—
78	#121×C₄₁	—	2	4	no	—
79	#121×C₄₁	C₂×C₄	1	4	yes	—
80	#121×C₄₁	C₂×C₄	1	4	yes	—
81	#120×C₄₁	—	2	4	no	—
82	C₄×Q₈×C₄₁	—	2	1	no	—
83	C₄×Q₈×C₄₁	—	2	1	no	—
84	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
86	C₄×D₄×C₄₁	—	2	2	no	—
87	C₄×D₄×C₄₁	—	2	2	no	—
88	C₄×D₄×C₄₁	—	2	2	no	—
89	C₄×D₄×C₄₁	—	2	2	no	—
90	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
91	C₄×D₄×C₄₁	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₄₁	D₄	1	2	yes	—
93	C₄×D₄×C₄₁	D₄	1	2	yes	—
94	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
95	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
96	#117×C₄₁	C₂×C₄	1	2	yes	—
97	#117×C₄₁	C₂×C₄	1	1	yes	characteristic
98	#117×C₄₁	C₂×C₄	1	2	yes	—
99	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
100	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
102	C₄⋊C₈×C₄₁	—	2	2	no	—
103	C₄⋊C₈×C₄₁	—	2	1	no	—
104	C₄⋊C₈×C₄₁	—	2	1	no	—
105	C₄⋊C₈×C₄₁	—	2	2	no	—
106	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
108	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
109	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
112	C₂²⋊C₈×C₄₁	—	2	1	no	—
113	C₂²⋊C₈×C₄₁	—	2	1	no	—
114	C₂²⋊C₈×C₄₁	—	2	2	no	—
115	C₂²⋊C₈×C₄₁	—	2	2	no	—
116	#97×C₄₁	—	2	2	no	—
117	C₄×C₈×C₄₁	—	2	1	no	—
118	C₄×C₈×C₄₁	—	2	1	no	—
119	C₄×C₈×C₄₁	—	2	2	no	—
120	C₂⁴×C₄₁	—	2	2	no	—
121	#41×C₄₁	—	4	2	no	—
122	#41×C₄₁	—	8	2	no	—
123	#41×C₄₁	—	8	2	no	—
124	#41×C₄₁	—	8	2	no	—
125	#41×C₄₁	—	8	2	no	—
126	C₂×Q₈×C₄₁	—	2	1	no	—
127	C₂×Q₈×C₄₁	—	2	1	no	—
128	C₂×Q₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×D₄×C₄₁	—	2	4	no	—
130	C₂×D₄×C₄₁	—	2	4	no	—
131	C₂×D₄×C₄₁	—	2	2	no	—
132	C₂×D₄×C₄₁	—	2	2	no	—
133	C₂×D₄×C₄₁	—	2	2	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	2	no	—
136	C₂×D₄×C₄₁	—	4	4	no	—
137	C₂×D₄×C₄₁	—	8	1	no	—
138	C₂×D₄×C₄₁	—	8	1	no	—
139	C₂×D₄×C₄₁	—	8	1	no	—
140	C₂×D₄×C₄₁	—	8	1	no	—
141	C₂×D₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
142	C₂×D₄×C₄₁	C₂²⋊C₄	1	2	yes	—
143	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
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	2	yes	—
148	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
152	M₁₆×C₄₁	—	8	1	no	—
153	M₁₆×C₄₁	—	8	1	no	—
154	M₁₆×C₄₁	—	8	2	no	—
155	C₂×C₈×C₄₁	—	2	1	no	—
156	C₂×C₈×C₄₁	—	2	1	no	—
157	C₂×C₈×C₄₁	—	2	1	no	—
158	C₂×C₈×C₄₁	—	2	1	no	—
159	C₂×C₈×C₄₁	—	2	2	no	—
160	C₂×C₈×C₄₁	—	2	2	no	—
161	C₂×C₈×C₄₁	—	8	2	no	—
162	C₄⋊C₄×C₄₁	—	2	2	no	—
163	C₄⋊C₄×C₄₁	—	2	2	no	—
164	C₄⋊C₄×C₄₁	—	2	1	no	—
165	C₄⋊C₄×C₄₁	—	2	1	no	—
166	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
169	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
170	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
171	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
172	C₂²⋊C₄×C₄₁	—	2	2	no	—
173	C₂²⋊C₄×C₄₁	—	2	4	no	—
174	C₂²⋊C₄×C₄₁	—	2	4	no	—
175	C₂²⋊C₄×C₄₁	—	2	4	no	—
176	C₂²⋊C₄×C₄₁	—	2	4	no	—
177	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
178	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
179	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
180	C₄²×C₄₁	—	2	2	no	—
181	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
182	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
183	C₄²×C₄₁	C₂×D₄	1	2	yes	—
184	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₄₁	C₂×D₄	1	2	yes	—
187	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
188	C₂³×C₄₁	—	2	4	no	—
189	C₂³×C₄₁	—	4	4	no	—
190	C₂³×C₄₁	—	8	2	no	—
191	C₂³×C₄₁	—	8	2	no	—
192	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
194	Q₈×C₄₁	—	2	2	no	—
195	Q₈×C₄₁	—	4	1	no	—
196	Q₈×C₄₁	—	4	1	no	—
197	Q₈×C₄₁	—	4	1	no	—
198	Q₈×C₄₁	—	4	1	no	—
199	D₄×C₄₁	—	2	2	no	—
200	D₄×C₄₁	—	4	2	no	—
201	D₄×C₄₁	—	4	4	no	—
202	D₄×C₄₁	—	4	2	no	—
203	D₄×C₄₁	—	4	2	no	—
204	D₄×C₄₁	—	4	2	no	—
205	D₄×C₄₁	—	4	2	no	—
206	D₄×C₄₁	—	4	4	no	—
207	D₄×C₄₁	—	8	2	no	—
208	D₄×C₄₁	—	8	2	no	—
209	D₄×C₄₁	—	8	4	no	—
210	D₄×C₄₁	—	8	4	no	—
211	C₂×C₄×C₄₁	—	2	2	no	—
212	C₂×C₄×C₄₁	—	2	1	no	—
213	C₂×C₄×C₄₁	—	2	1	no	—
214	C₂×C₄×C₄₁	—	2	2	no	—
215	C₂×C₄×C₄₁	—	2	1	no	—
216	C₂×C₄×C₄₁	—	2	1	no	—
217	C₂×C₄×C₄₁	—	2	2	no	—
218	C₂×C₄×C₄₁	—	2	2	no	—
219	C₂×C₄×C₄₁	—	2	2	no	—
220	C₂×C₄×C₄₁	—	2	2	no	—
221	C₂×C₄×C₄₁	—	2	2	no	—
222	C₂×C₄×C₄₁	—	4	2	no	—
223	C₂×C₄×C₄₁	—	8	4	no	—
224	C₂×C₄×C₄₁	—	8	4	no	—
225	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
227	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
232	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
234	C₈×C₄₁	—	4	1	no	—
235	C₈×C₄₁	—	4	2	no	—
236	C₈×C₄₁	—	4	1	no	—
237	C₈×C₄₁	—	4	2	no	—
238	C₈×C₄₁	—	4	1	no	—
239	C₈×C₄₁	—	4	1	no	—
240	#17776	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂²×C₄₁	—	2	1	no	—
242	C₂²×C₄₁	—	2	2	no	—
243	C₂²×C₄₁	—	2	2	no	—
244	C₂²×C₄₁	—	2	2	no	—
245	C₂²×C₄₁	—	2	1	no	—
246	C₂²×C₄₁	—	2	1	no	—
247	C₂²×C₄₁	—	4	4	no	—
248	C₂²×C₄₁	—	4	4	no	—
249	C₂²×C₄₁	—	4	4	no	—
250	C₂²×C₄₁	—	16	4	no	—
251	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
252	C₄×C₄₁	—	2	1	no	—
253	C₄×C₄₁	—	2	2	no	—
254	C₄×C₄₁	—	2	2	no	—
255	C₄×C₄₁	—	2	2	no	—
256	C₄×C₄₁	—	2	1	no	—
257	C₄×C₄₁	—	2	1	no	—
258	C₄×C₄₁	—	4	2	no	—
259	C₄×C₄₁	—	4	2	no	—
260	C₄×C₄₁	#413	1	2	yes	—
261	C₄×C₄₁	#576	1	1	yes	characteristic
262	C₄×C₄₁	#576	1	1	yes	characteristic
263	C₄×C₄₁	#577	1	1	yes	characteristic
264	C₄×C₄₁	#577	1	1	yes	characteristic
265	#3508	C₂×C₄₁	1	1	yes	characteristic
266	#2991	C₂×C₄₁	1	1	yes	characteristic, complemented
267	#2990	C₂×C₄₁	1	1	yes	characteristic, complemented
268	#1561	C₂×C₄₁	1	1	yes	characteristic, complemented
269	#1557	C₂×C₄₁	1	2	yes	complemented
270	#1553	C₂×C₄₁	1	2	yes	complemented
271	#1547	C₂×C₄₁	1	2	yes	complemented
272	#1545	C₂×C₄₁	1	1	yes	characteristic, complemented
273	#1545	C₂×C₄₁	1	1	yes	characteristic, complemented
274	#1544	C₂×C₄₁	1	1	yes	characteristic, complemented
275	#1540	C₂×C₄₁	1	2	yes	complemented
276	C₂×C₄₁	—	4	2	no	—
277	C₂×C₄₁	—	4	1	no	—
278	C₂×C₄₁	—	8	4	no	—
279	C₂×C₄₁	C₂×#413	1	1	yes	characteristic
280	C₂×C₄₁	#2897	1	1	yes	characteristic
281	C₂×C₄₁	#2897	1	1	yes	characteristic
282	#559	—	2	1	no	—
283	#559	—	2	1	no	—
284	#550	—	2	1	no	—
285	#550	—	2	1	no	—
286	#546	—	2	2	no	—
287	#546	—	2	2	no	—
288	#540	C₄×C₄₁	1	4	yes	—
289	#535	C₄×C₄₁	1	2	yes	—
290	#533	C₂²×C₄₁	1	1	yes	characteristic
291	#532	C₂²×C₄₁	1	2	yes	—
292	#532	C₂²×C₄₁	1	2	yes	—
293	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
294	#518	C₄×C₄₁	1	2	yes	—
295	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
296	#446	—	2	1	no	—
297	C₈×Q₈	—	2	1	no	—
298	#436	—	2	1	no	—
299	#436	—	2	2	no	—
300	#435	—	2	2	no	—
301	C₈×D₄	—	2	1	no	—
302	#432	C₂²×C₄₁	1	2	yes	complemented
303	#431	C₂²×C₄₁	1	1	yes	characteristic
304	#431	C₂²×C₄₁	1	1	yes	characteristic
305	#431	C₂²×C₄₁	1	2	yes	complemented
306	#423	C₂²×C₄₁	1	1	yes	characteristic
307	#423	C₂²×C₄₁	1	1	yes	characteristic
308	#423	C₂²×C₄₁	1	2	yes	complemented
309	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
310	#332	C₂²×C₄₁	1	1	yes	characteristic, complemented
311	#332	C₂²×C₄₁	1	2	yes	complemented
312	#332	C₂²×C₄₁	1	1	yes	characteristic, complemented
313	#332	C₂²×C₄₁	1	1	yes	characteristic, complemented
314	#332	C₂²×C₄₁	1	1	yes	characteristic, complemented
315	#332	C₂²×C₄₁	1	2	yes	complemented
316	#331	C₂²×C₄₁	1	2	yes	—
317	#331	C₂²×C₄₁	1	2	yes	—
318	#331	C₂²×C₄₁	1	2	yes	—
319	#331	C₂²×C₄₁	1	2	yes	—
320	C₄₁	#17776	1	1	yes	characteristic, direct product factor, Sylow
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
327	C₂×M₁₆	—	2	1	no	—
328	C₂×M₁₆	—	2	1	no	—
329	C₂×M₁₆	—	2	2	no	—
330	C₂²×C₈	—	2	2	no	—
331	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
332	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
333	#127	—	2	2	no	—
334	#127	—	2	1	no	—
335	#127	—	2	1	no	—
336	#127	C₂×C₄×C₄₁	1	2	yes	—
337	#127	C₂³×C₄₁	1	2	yes	—
338	#125	C₂×C₄×C₄₁	1	2	yes	—
339	#124	—	2	2	no	—
340	#124	—	2	2	no	—
341	#124	—	2	2	no	—
342	#123	—	2	4	no	—
343	#123	—	2	4	no	—
344	C₂²⋊Q₈	—	2	1	no	—
345	C₂²⋊Q₈	—	2	1	no	—
346	#121	—	2	2	no	—
347	#121	—	2	2	no	—
348	#121	—	2	4	no	—
349	#121	—	2	4	no	—
350	#121	C₂×C₄×C₄₁	1	4	yes	—
351	#121	C₂×C₄×C₄₁	1	4	yes	—
352	#120	—	2	4	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
356	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
362	C₄×D₄	C₂×C₄×C₄₁	1	4	yes	—
363	C₄×D₄	D₄×C₄₁	1	2	yes	—
364	C₄×D₄	D₄×C₄₁	1	2	yes	—
365	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
366	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
367	#117	C₂×C₄×C₄₁	1	2	yes	—
368	#117	C₂×C₄×C₄₁	1	1	yes	characteristic
369	#117	C₂×C₄×C₄₁	1	2	yes	—
370	C₂×C₂²⋊C₄	—	2	4	no	—
371	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
372	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
373	C₄⋊C₈	—	2	2	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	2	no	—
377	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
378	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
379	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
380	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
382	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
383	C₂²⋊C₈	—	2	1	no	—
384	C₂²⋊C₈	—	2	1	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	#97	—	2	2	no	—
388	C₄×C₈	—	2	1	no	—
389	C₄×C₈	—	2	1	no	—
390	C₄×C₈	—	2	2	no	—
391	C₂⁴	—	2	2	no	—
392	#41	—	4	2	no	—
393	#41	—	8	2	no	—
394	#41	—	8	2	no	—
395	#41	—	8	2	no	—
396	#41	—	8	2	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	8	1	no	—
409	C₂×D₄	—	8	1	no	—
410	C₂×D₄	—	8	1	no	—
411	C₂×D₄	—	8	1	no	—
412	C₂×D₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
413	C₂×D₄	C₂²⋊C₄×C₄₁	1	2	yes	—
414	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
418	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
423	M₁₆	—	8	1	no	—
424	M₁₆	—	8	1	no	—
425	M₁₆	—	8	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	8	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
438	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
439	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
440	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
441	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
442	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
449	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
450	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
451	C₄²	—	2	2	no	—
452	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
453	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
454	C₄²	C₂×D₄×C₄₁	1	2	yes	—
455	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
457	C₄²	C₂×D₄×C₄₁	1	2	yes	—
458	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
459	C₂³	—	2	4	no	—
460	C₂³	—	4	4	no	—
461	C₂³	—	8	2	no	—
462	C₂³	—	8	2	no	—
463	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
464	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
465	Q₈	—	2	2	no	—
466	Q₈	—	4	1	no	—
467	Q₈	—	4	1	no	—
468	Q₈	—	4	1	no	—
469	Q₈	—	4	1	no	—
470	D₄	—	2	2	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	4	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	4	no	—
478	D₄	—	8	2	no	—
479	D₄	—	8	2	no	—
480	D₄	—	8	4	no	—
481	D₄	—	8	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	8	4	no	—
495	C₂×C₄	—	8	4	no	—
496	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
497	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
498	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
499	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
500	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
501	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
502	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
505	C₈	—	4	1	no	—
506	C₈	—	4	2	no	—
507	C₈	—	4	1	no	—
508	C₈	—	4	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	1	no	—
511	C₂²	—	2	1	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	2	no	—
515	C₂²	—	2	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	4	no	—
520	C₂²	—	16	4	no	—
521	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
522	C₄	—	2	1	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	C₄₁×#413	1	2	yes	—
531	C₄	C₄₁×#576	1	1	yes	characteristic
532	C₄	C₄₁×#576	1	1	yes	characteristic
533	C₄	C₄₁×#577	1	1	yes	characteristic
534	C₄	C₄₁×#577	1	1	yes	characteristic
535	C₂	—	4	1	no	—
536	C₂	—	4	2	no	—
537	C₂	—	8	4	no	—
538	C₂	C₂×C₄₁×#413	1	1	yes	characteristic
539	C₂	C₄₁×#2897	1	1	yes	characteristic
540	C₂	C₄₁×#2897	1	1	yes	characteristic

	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×#2991	C₂	semidirect
3. C₄₁×#2990	C₂	semidirect
4. C₄₁×#1561	C₂	semidirect
5. C₄₁×#1557	C₂	semidirect
6. C₄₁×#1553	C₂	semidirect
7. C₄₁×#1547	C₂	semidirect
8. C₄₁×#1545	C₂	semidirect
9. C₄₁×#1545	C₂	semidirect
10. C₄₁×#1544	C₂	semidirect
11. C₄₁×#1540	C₂	semidirect
32. C₄₁×#432	C₂²	semidirect
35. C₄₁×#431	C₂²	semidirect
38. C₄₁×#423	C₂²	semidirect
39. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
40. C₄₁×#332	C₂²	semidirect
41. C₄₁×#332	C₂²	semidirect
42. C₄₁×#332	C₂²	semidirect
43. C₄₁×#332	C₂²	semidirect
44. C₄₁×#332	C₂²	semidirect
45. C₄₁×#332	C₂²	semidirect
240. #17776	C₄₁	direct (N × H)
266. #2991	C₂×C₄₁	semidirect
267. #2990	C₂×C₄₁	semidirect
268. #1561	C₂×C₄₁	semidirect
269. #1557	C₂×C₄₁	semidirect
270. #1553	C₂×C₄₁	semidirect
271. #1547	C₂×C₄₁	semidirect
272. #1545	C₂×C₄₁	semidirect
273. #1545	C₂×C₄₁	semidirect
274. #1544	C₂×C₄₁	semidirect
275. #1540	C₂×C₄₁	semidirect
302. #432	C₂²×C₄₁	semidirect
305. #431	C₂²×C₄₁	semidirect
308. #423	C₂²×C₄₁	semidirect
309. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
310. #332	C₂²×C₄₁	semidirect
311. #332	C₂²×C₄₁	semidirect
312. #332	C₂²×C₄₁	semidirect
313. #332	C₂²×C₄₁	semidirect
314. #332	C₂²×C₄₁	semidirect
315. #332	C₂²×C₄₁	semidirect
320. C₄₁	#17776	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×#2991	2
C₄₁×#2990	2
C₄₁×#1561	2
C₄₁×#1557	2
C₄₁×#1553	2
C₄₁×#1547	2
C₄₁×#1545	2
C₄₁×#1544	2
C₄₁×#1540	2
#17776	41

Subgroups — C41×#17776 #23793977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17776 #23793977