Subgroups — C₄₁×#21006 #23797207

Order 10496.

1654 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#568	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#101×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1908	C₂	1	8	yes	maximal, complemented
7	C₄₁×#1742	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×#137×C₄₁	—	2	4	no	—
12	C₂×#137×C₄₁	—	2	2	no	—
13	C₂×#137×C₄₁	C₄	1	8	yes	complemented
14	C₂×#137×C₄₁	C₂²	1	4	yes	—
15	C₂²×Q₁₆×C₄₁	—	2	1	no	—
16	C₂²×SD₁₆×C₄₁	—	2	1	no	—
17	C₄₁×#568	—	2	4	no	—
18	C₄₁×#568	—	2	4	no	—
19	C₂×#131×C₄₁	—	2	2	no	—
20	C₂×#131×C₄₁	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₄₁	—	2	2	no	—
22	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#419	—	2	1	no	—
28	C₄₁×#419	—	2	2	no	—
29	C₄₁×#419	—	2	2	no	—
30	C₄₁×#418	—	2	1	no	—
31	C₄₁×#417	C₂²	1	4	yes	—
32	C₄₁×#416	—	2	2	no	—
33	C₂×#103×C₄₁	C₂²	1	4	yes	complemented
34	C₂×#101×C₄₁	—	2	4	no	—
35	C₂×#101×C₄₁	C₂²	1	4	yes	complemented
36	C₄₁×#343	C₂²	1	4	yes	complemented
37	C₄₁×#341	C₂²	1	4	yes	complemented
38	C₂×#41×C₄₁	—	2	1	no	—
39	C₂×#41×C₄₁	—	2	2	no	—
40	C₂×#41×C₄₁	—	2	2	no	—
41	C₂×#41×C₄₁	—	2	1	no	—
42	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
43	C₂×#41×C₄₁	D₄	1	4	yes	—
44	C₂×#41×C₄₁	D₄	1	2	yes	—
45	C₂×#41×C₄₁	C₂³	1	2	yes	—
46	C₂²×Q₈×C₄₁	—	2	2	no	—
47	C₂²×Q₈×C₄₁	—	2	1	no	—
48	C₂²×Q₈×C₄₁	—	2	1	no	—
49	C₂²×Q₈×C₄₁	—	2	1	no	—
50	C₂²×Q₈×C₄₁	—	2	2	no	—
51	C₂²×Q₈×C₄₁	—	2	1	no	—
52	C₂²×Q₈×C₄₁	—	2	1	no	—
53	C₂²×Q₈×C₄₁	—	2	2	no	—
54	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
55	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
57	C₂²×D₄×C₄₁	—	2	1	no	—
58	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₄₁	—	2	1	no	—
60	C₂³×C₄×C₄₁	—	2	1	no	—
61	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
62	#137×C₄₁	—	2	16	no	—
63	#137×C₄₁	—	2	8	no	—
64	#137×C₄₁	C₂×C₄	1	16	yes	—
65	C₂×Q₁₆×C₄₁	—	2	8	no	—
66	C₂×Q₁₆×C₄₁	—	2	4	no	—
67	C₂×SD₁₆×C₄₁	—	2	8	no	—
68	C₂×SD₁₆×C₄₁	—	2	4	no	—
69	#131×C₄₁	—	2	16	no	—
70	#131×C₄₁	—	2	8	no	—
71	#131×C₄₁	—	2	8	no	—
72	C₂×M₁₆×C₄₁	—	2	16	no	—
73	C₂×M₁₆×C₄₁	—	2	1	no	—
74	C₂×M₁₆×C₄₁	—	2	2	no	—
75	C₂×M₁₆×C₄₁	—	2	2	no	—
76	C₂×M₁₆×C₄₁	—	2	2	no	—
77	C₂×M₁₆×C₄₁	—	2	2	no	—
78	C₂×M₁₆×C₄₁	—	2	1	no	—
79	C₂×M₁₆×C₄₁	—	2	1	no	—
80	C₂×M₁₆×C₄₁	—	2	4	no	—
81	C₂×M₁₆×C₄₁	—	2	2	no	—
82	C₂×M₁₆×C₄₁	—	2	2	no	—
83	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
84	C₂×M₁₆×C₄₁	D₄	1	4	yes	—
85	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
86	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
87	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
88	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
89	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
90	C₂²×C₈×C₄₁	—	2	1	no	—
91	C₂²×C₈×C₄₁	—	2	1	no	—
92	C₂²×C₈×C₄₁	—	2	4	no	—
93	C₂²×C₈×C₄₁	—	2	1	no	—
94	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
95	#117×C₄₁	—	2	1	no	—
96	#117×C₄₁	—	2	1	no	—
97	#117×C₄₁	—	2	1	no	—
98	#117×C₄₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄²×C₄₁	—	2	1	no	—
102	#103×C₄₁	—	2	4	no	—
103	#103×C₄₁	—	2	8	no	—
104	#102×C₄₁	—	2	4	no	—
105	#101×C₄₁	—	2	8	no	—
106	C₂⁴×C₄₁	—	2	1	no	—
107	C₂⁴×C₄₁	#41	1	1	yes	characteristic
108	#41×C₄₁	—	2	4	no	—
109	#41×C₄₁	—	2	4	no	—
110	#41×C₄₁	—	2	4	no	—
111	#41×C₄₁	—	2	4	no	—
112	#41×C₄₁	—	2	4	no	—
113	#41×C₄₁	—	2	2	no	—
114	#41×C₄₁	—	2	2	no	—
115	#41×C₄₁	C₂²⋊C₄	1	8	yes	—
116	#41×C₄₁	C₂²×C₄	1	4	yes	—
117	#41×C₄₁	C₂×D₄	1	4	yes	—
118	C₂×Q₈×C₄₁	—	2	4	no	—
119	C₂×Q₈×C₄₁	—	2	1	no	—
120	C₂×Q₈×C₄₁	—	2	2	no	—
121	C₂×Q₈×C₄₁	—	2	2	no	—
122	C₂×Q₈×C₄₁	—	2	4	no	—
123	C₂×Q₈×C₄₁	—	2	4	no	—
124	C₂×Q₈×C₄₁	—	2	4	no	—
125	C₂×Q₈×C₄₁	—	2	4	no	—
126	C₂×Q₈×C₄₁	—	2	2	no	—
127	C₂×Q₈×C₄₁	—	2	2	no	—
128	C₂×Q₈×C₄₁	—	2	2	no	—
129	C₂×Q₈×C₄₁	—	2	2	no	—
130	C₂×Q₈×C₄₁	—	4	2	no	—
131	C₂×Q₈×C₄₁	—	4	2	no	—
132	C₂×Q₈×C₄₁	—	4	1	no	—
133	C₂×Q₈×C₄₁	—	4	1	no	—
134	C₂×Q₈×C₄₁	—	4	2	no	—
135	C₂×Q₈×C₄₁	—	4	2	no	—
136	C₂×Q₈×C₄₁	—	4	2	no	—
137	C₂×Q₈×C₄₁	C₂²⋊C₄	1	4	yes	—
138	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
139	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₄₁	—	2	4	no	—
141	C₂×D₄×C₄₁	—	2	2	no	—
142	C₂×D₄×C₄₁	—	2	4	no	—
143	C₂×D₄×C₄₁	—	2	2	no	—
144	C₂×D₄×C₄₁	—	2	2	no	—
145	C₂×D₄×C₄₁	—	2	1	no	—
146	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	—
147	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₄₁	—	2	1	no	—
149	C₂²×C₄×C₄₁	—	2	4	no	—
150	C₂²×C₄×C₄₁	—	2	4	no	—
151	C₂²×C₄×C₄₁	—	2	2	no	—
152	C₂²×C₄×C₄₁	—	2	2	no	—
153	C₂²×C₄×C₄₁	—	2	4	no	—
154	C₂²×C₄×C₄₁	—	2	2	no	—
155	C₂²×C₄×C₄₁	—	2	2	no	—
156	C₂²×C₄×C₄₁	—	4	1	no	—
157	C₂²×C₄×C₄₁	—	4	1	no	—
158	C₂²×C₄×C₄₁	—	4	2	no	—
159	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
160	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
167	C₂²×C₄×C₄₁	#41	1	2	yes	—
168	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
169	Q₁₆×C₄₁	—	2	16	no	—
170	SD₁₆×C₄₁	—	2	16	no	—
171	M₁₆×C₄₁	—	2	16	no	—
172	M₁₆×C₄₁	—	2	4	no	—
173	M₁₆×C₄₁	—	2	4	no	—
174	M₁₆×C₄₁	—	2	4	no	—
175	M₁₆×C₄₁	—	2	4	no	—
176	M₁₆×C₄₁	—	2	2	no	—
177	M₁₆×C₄₁	—	2	2	no	—
178	M₁₆×C₄₁	—	2	2	no	—
179	M₁₆×C₄₁	—	2	2	no	—
180	M₁₆×C₄₁	—	4	2	no	—
181	M₁₆×C₄₁	C₂²×C₄	1	4	yes	—
182	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
183	C₂×C₈×C₄₁	—	2	16	no	—
184	C₂×C₈×C₄₁	—	2	4	no	—
185	C₂×C₈×C₄₁	—	2	4	no	—
186	C₂×C₈×C₄₁	—	2	2	no	—
187	C₂×C₈×C₄₁	—	2	2	no	—
188	C₂×C₈×C₄₁	—	2	2	no	—
189	C₂×C₈×C₄₁	—	2	2	no	—
190	C₂×C₈×C₄₁	—	2	2	no	—
191	C₂×C₈×C₄₁	—	4	2	no	—
192	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₄₁	—	2	2	no	—
194	C₄⋊C₄×C₄₁	—	2	2	no	—
195	C₂²⋊C₄×C₄₁	—	4	1	no	—
196	C₂²⋊C₄×C₄₁	—	4	1	no	—
197	C₄²×C₄₁	—	4	1	no	—
198	C₄²×C₄₁	—	4	1	no	—
199	C₂³×C₄₁	—	2	2	no	—
200	C₂³×C₄₁	—	2	4	no	—
201	C₂³×C₄₁	—	4	2	no	—
202	C₂³×C₄₁	—	8	2	no	—
203	C₂³×C₄₁	C₄×D₄	1	2	yes	—
204	C₂³×C₄₁	#121	1	1	yes	characteristic
205	C₂³×C₄₁	#121	1	1	yes	characteristic
206	C₂³×C₄₁	#123	1	2	yes	—
207	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
208	Q₈×C₄₁	—	2	2	no	—
209	Q₈×C₄₁	—	2	4	no	—
210	Q₈×C₄₁	—	2	4	no	—
211	Q₈×C₄₁	—	2	4	no	—
212	Q₈×C₄₁	—	4	2	no	—
213	Q₈×C₄₁	—	4	2	no	—
214	Q₈×C₄₁	—	4	4	no	—
215	Q₈×C₄₁	—	4	4	no	—
216	Q₈×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
217	D₄×C₄₁	—	2	4	no	—
218	D₄×C₄₁	—	2	4	no	—
219	D₄×C₄₁	—	2	8	no	—
220	D₄×C₄₁	—	2	2	no	—
221	D₄×C₄₁	—	2	4	no	—
222	D₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
223	C₂×C₄×C₄₁	—	2	2	no	—
224	C₂×C₄×C₄₁	—	2	4	no	—
225	C₂×C₄×C₄₁	—	2	4	no	—
226	C₂×C₄×C₄₁	—	2	4	no	—
227	C₂×C₄×C₄₁	—	2	8	no	—
228	C₂×C₄×C₄₁	—	2	4	no	—
229	C₂×C₄×C₄₁	—	4	2	no	—
230	C₂×C₄×C₄₁	—	4	2	no	—
231	C₂×C₄×C₄₁	—	4	4	no	—
232	C₂×C₄×C₄₁	—	4	4	no	—
233	C₂×C₄×C₄₁	—	4	2	no	—
234	C₂×C₄×C₄₁	—	8	2	no	—
235	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
236	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
237	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
238	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
239	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
240	C₂×C₄×C₄₁	#120	1	2	yes	—
241	C₂×C₄×C₄₁	#120	1	2	yes	—
242	C₂×C₄×C₄₁	#121	1	2	yes	—
243	C₂×C₄×C₄₁	#121	1	2	yes	—
244	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
245	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
246	C₂×C₄×C₄₁	#123	1	2	yes	—
247	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
248	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
253	C₈×C₄₁	—	2	4	no	—
254	C₈×C₄₁	—	2	4	no	—
255	C₈×C₄₁	—	2	4	no	—
256	C₈×C₄₁	—	4	2	no	—
257	#21006	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₄₁	—	2	2	no	—
259	C₂²×C₄₁	—	2	2	no	—
260	C₂²×C₄₁	—	2	2	no	—
261	C₂²×C₄₁	—	2	4	no	—
262	C₂²×C₄₁	—	4	2	no	—
263	C₂²×C₄₁	—	4	2	no	—
264	C₂²×C₄₁	—	4	4	no	—
265	C₂²×C₄₁	—	8	4	no	—
266	C₂²×C₄₁	#386	1	2	yes	—
267	C₂²×C₄₁	#386	1	2	yes	—
268	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
269	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
270	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
271	C₄×C₄₁	—	2	4	no	—
272	C₄×C₄₁	—	4	4	no	—
273	C₄×C₄₁	—	8	2	no	—
274	C₄×C₄₁	#386	1	2	yes	—
275	C₄×C₄₁	#386	1	2	yes	—
276	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
277	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
278	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
279	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
280	C₂²×#137	C₂×C₄₁	1	1	yes	characteristic
281	C₂×#568	C₂×C₄₁	1	1	yes	characteristic
282	#2896	C₂×C₄₁	1	1	yes	characteristic
283	C₂×#419	C₂×C₄₁	1	1	yes	characteristic, complemented
284	C₂²×#101	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#1908	C₂×C₄₁	1	8	yes	complemented
286	#1742	C₂×C₄₁	1	1	yes	characteristic, complemented
287	#1738	C₂×C₄₁	1	1	yes	characteristic, complemented
288	C₂×C₄₁	—	2	2	no	—
289	C₂×C₄₁	—	2	1	no	—
290	C₂×C₄₁	—	2	1	no	—
291	C₂×C₄₁	—	2	2	no	—
292	C₂×C₄₁	—	4	4	no	—
293	C₂×C₄₁	C₂×#386	1	1	yes	characteristic
294	C₂×C₄₁	#3069	1	2	yes	—
295	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
296	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
297	C₂×#137	—	2	4	no	—
298	C₂×#137	—	2	2	no	—
299	C₂×#137	C₄×C₄₁	1	8	yes	complemented
300	C₂×#137	C₂²×C₄₁	1	4	yes	—
301	C₂²×Q₁₆	—	2	1	no	—
302	C₂²×SD₁₆	—	2	1	no	—
303	#568	—	2	4	no	—
304	#568	—	2	4	no	—
305	C₂×#131	—	2	2	no	—
306	C₂×#131	C₂²×C₄₁	1	4	yes	—
307	C₂²×M₁₆	—	2	2	no	—
308	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
309	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
310	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
311	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
312	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
313	#419	—	2	1	no	—
314	#419	—	2	2	no	—
315	#419	—	2	2	no	—
316	#418	—	2	1	no	—
317	#417	C₂²×C₄₁	1	4	yes	—
318	#416	—	2	2	no	—
319	C₂×#103	C₂²×C₄₁	1	4	yes	complemented
320	C₂×#101	—	2	4	no	—
321	C₂×#101	C₂²×C₄₁	1	4	yes	complemented
322	#343	C₂²×C₄₁	1	4	yes	complemented
323	#341	C₂²×C₄₁	1	4	yes	complemented
324	C₄₁	#21006	1	1	yes	characteristic, direct product factor, Sylow
325	C₂×#41	—	2	1	no	—
326	C₂×#41	—	2	2	no	—
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	2	1	no	—
329	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
330	C₂×#41	D₄×C₄₁	1	4	yes	—
331	C₂×#41	D₄×C₄₁	1	2	yes	—
332	C₂×#41	C₂³×C₄₁	1	2	yes	—
333	C₂²×Q₈	—	2	2	no	—
334	C₂²×Q₈	—	2	1	no	—
335	C₂²×Q₈	—	2	1	no	—
336	C₂²×Q₈	—	2	1	no	—
337	C₂²×Q₈	—	2	2	no	—
338	C₂²×Q₈	—	2	1	no	—
339	C₂²×Q₈	—	2	1	no	—
340	C₂²×Q₈	—	2	2	no	—
341	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
342	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
343	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
346	C₂³×C₄	—	2	1	no	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
349	#137	—	2	16	no	—
350	#137	—	2	8	no	—
351	#137	C₂×C₄×C₄₁	1	16	yes	—
352	C₂×Q₁₆	—	2	8	no	—
353	C₂×Q₁₆	—	2	4	no	—
354	C₂×SD₁₆	—	2	8	no	—
355	C₂×SD₁₆	—	2	4	no	—
356	#131	—	2	16	no	—
357	#131	—	2	8	no	—
358	#131	—	2	8	no	—
359	C₂×M₁₆	—	2	16	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₂×M₁₆	—	2	2	no	—
362	C₂×M₁₆	—	2	2	no	—
363	C₂×M₁₆	—	2	2	no	—
364	C₂×M₁₆	—	2	2	no	—
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	—	2	1	no	—
367	C₂×M₁₆	—	2	4	no	—
368	C₂×M₁₆	—	2	2	no	—
369	C₂×M₁₆	—	2	2	no	—
370	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
371	C₂×M₁₆	D₄×C₄₁	1	4	yes	—
372	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
373	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
374	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
375	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
376	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
377	C₂²×C₈	—	2	1	no	—
378	C₂²×C₈	—	2	1	no	—
379	C₂²×C₈	—	2	4	no	—
380	C₂²×C₈	—	2	1	no	—
381	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	—	2	1	no	—
385	#117	—	2	1	no	—
386	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₄²	—	2	1	no	—
389	#103	—	2	4	no	—
390	#103	—	2	8	no	—
391	#102	—	2	4	no	—
392	#101	—	2	8	no	—
393	C₂⁴	—	2	1	no	—
394	C₂⁴	#41×C₄₁	1	1	yes	characteristic
395	#41	—	2	4	no	—
396	#41	—	2	4	no	—
397	#41	—	2	4	no	—
398	#41	—	2	4	no	—
399	#41	—	2	4	no	—
400	#41	—	2	2	no	—
401	#41	—	2	2	no	—
402	#41	C₂²⋊C₄×C₄₁	1	8	yes	—
403	#41	C₂²×C₄×C₄₁	1	4	yes	—
404	#41	C₂×D₄×C₄₁	1	4	yes	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	4	2	no	—
418	C₂×Q₈	—	4	2	no	—
419	C₂×Q₈	—	4	1	no	—
420	C₂×Q₈	—	4	1	no	—
421	C₂×Q₈	—	4	2	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	C₂²⋊C₄×C₄₁	1	4	yes	—
425	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
426	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	—
434	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	4	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
447	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
448	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
453	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
454	C₂²×C₄	#41×C₄₁	1	2	yes	—
455	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
456	Q₁₆	—	2	16	no	—
457	SD₁₆	—	2	16	no	—
458	M₁₆	—	2	16	no	—
459	M₁₆	—	2	4	no	—
460	M₁₆	—	2	4	no	—
461	M₁₆	—	2	4	no	—
462	M₁₆	—	2	4	no	—
463	M₁₆	—	2	2	no	—
464	M₁₆	—	2	2	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	2	2	no	—
467	M₁₆	—	4	2	no	—
468	M₁₆	C₂²×C₄×C₄₁	1	4	yes	—
469	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
470	C₂×C₈	—	2	16	no	—
471	C₂×C₈	—	2	4	no	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	4	2	no	—
479	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₄²	—	4	1	no	—
485	C₄²	—	4	1	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	8	2	no	—
490	C₂³	C₄×D₄×C₄₁	1	2	yes	—
491	C₂³	#121×C₄₁	1	1	yes	characteristic
492	C₂³	#121×C₄₁	1	1	yes	characteristic
493	C₂³	#123×C₄₁	1	2	yes	—
494	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
495	Q₈	—	2	2	no	—
496	Q₈	—	2	4	no	—
497	Q₈	—	2	4	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	4	2	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	4	no	—
502	Q₈	—	4	4	no	—
503	Q₈	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
504	D₄	—	2	4	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	8	no	—
507	D₄	—	2	2	no	—
508	D₄	—	2	4	no	—
509	D₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	8	2	no	—
522	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
523	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
524	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
525	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
526	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
527	C₂×C₄	#120×C₄₁	1	2	yes	—
528	C₂×C₄	#120×C₄₁	1	2	yes	—
529	C₂×C₄	#121×C₄₁	1	2	yes	—
530	C₂×C₄	#121×C₄₁	1	2	yes	—
531	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
532	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
533	C₂×C₄	#123×C₄₁	1	2	yes	—
534	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
539	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
540	C₈	—	2	4	no	—
541	C₈	—	2	4	no	—
542	C₈	—	2	4	no	—
543	C₈	—	4	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	4	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	8	4	no	—
552	C₂²	C₄₁×#386	1	2	yes	—
553	C₂²	C₄₁×#386	1	2	yes	—
554	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
555	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
556	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
557	C₄	—	2	4	no	—
558	C₄	—	4	4	no	—
559	C₄	—	8	2	no	—
560	C₄	C₄₁×#386	1	2	yes	—
561	C₄	C₄₁×#386	1	2	yes	—
562	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
563	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
564	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
565	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
566	C₂	—	2	2	no	—
567	C₂	—	2	1	no	—
568	C₂	—	2	1	no	—
569	C₂	—	2	2	no	—
570	C₂	—	4	4	no	—
571	C₂	C₂×C₄₁×#386	1	1	yes	characteristic
572	C₂	C₄₁×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×#41×C₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₄₁×#419	C₂	semidirect
5. C₂²×#101×C₄₁	C₂	semidirect
6. C₄₁×#1908	C₂	semidirect
7. C₄₁×#1742	C₂	semidirect
8. C₄₁×#1738	C₂	semidirect
13. C₂×#137×C₄₁	C₄	semidirect
33. C₂×#103×C₄₁	C₂²	semidirect
35. C₂×#101×C₄₁	C₂²	semidirect
36. C₄₁×#343	C₂²	semidirect
37. C₄₁×#341	C₂²	semidirect
257. #21006	C₄₁	direct (N × H)
283. C₂×#419	C₂×C₄₁	semidirect
284. C₂²×#101	C₂×C₄₁	semidirect
285. #1908	C₂×C₄₁	semidirect
286. #1742	C₂×C₄₁	semidirect
287. #1738	C₂×C₄₁	semidirect
299. C₂×#137	C₄×C₄₁	semidirect
319. C₂×#103	C₂²×C₄₁	semidirect
321. C₂×#101	C₂²×C₄₁	semidirect
322. #343	C₂²×C₄₁	semidirect
323. #341	C₂²×C₄₁	semidirect
324. C₄₁	#21006	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#137×C₄₁	2
C₂×C₄₁×#568	2
C₄₁×#2896	2
C₂×C₄₁×#419	2
C₂²×#101×C₄₁	2
C₄₁×#1908	2
C₄₁×#1742	2
C₄₁×#1738	2
#21006	41

Subgroups — C41×#21006 #23797207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21006 #23797207