Subgroups — C₄₁×#22998 #23799199

Order 10496.

926 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#462	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₄₁	C₂	1	2	yes	maximal
3	C₄₁×#2365	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1956	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1947	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1933	C₂	1	2	yes	maximal
8	C₄₁×#1878	C₂	1	2	yes	maximal
9	C₄₁×#1863	C₂	1	2	yes	maximal
10	C₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
11	C₄×#103×C₄₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#462	C₄	1	8	yes	complemented
18	C₄×Q₁₆×C₄₁	—	2	8	no	—
19	C₂×#107×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#103×C₄₁	C₂²	1	2	yes	—
23	C₂×#103×C₄₁	C₂²	1	2	yes	—
24	C₂×#103×C₄₁	C₂²	1	2	yes	—
25	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
28	C₄₁×#391	C₂²	1	2	yes	—
29	C₄₁×#389	—	2	1	no	—
30	C₄₁×#389	—	2	2	no	—
31	C₄₁×#389	C₂²	1	2	yes	—
32	C₄₁×#387	—	2	2	no	—
33	C₄₁×#384	—	2	1	no	—
34	C₄₁×#384	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
36	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
39	C₄₁×#340	C₂²	1	1	yes	characteristic
40	C₄₁×#340	C₂²	1	1	yes	characteristic
41	C₄₁×#340	C₂²	1	2	yes	—
42	C₄₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
44	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₄₁	—	2	4	no	—
46	C₂×Q₁₆×C₄₁	—	2	4	no	—
47	C₂×Q₁₆×C₄₁	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₁	—	2	2	no	—
52	C₄⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
53	C₄×Q₈×C₄₁	—	2	2	no	—
54	C₄×Q₈×C₄₁	—	2	4	no	—
55	C₄×Q₈×C₄₁	—	2	4	no	—
56	C₄×Q₈×C₄₁	—	4	2	no	—
57	C₄×Q₈×C₄₁	—	4	2	no	—
58	C₄×Q₈×C₄₁	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₄₁	—	2	1	no	—
74	C₂×C₄²×C₄₁	—	2	2	no	—
75	C₂×C₄²×C₄₁	—	2	2	no	—
76	C₂×C₄²×C₄₁	—	2	1	no	—
77	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
78	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
79	#107×C₄₁	—	2	4	no	—
80	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
81	#103×C₄₁	—	2	2	no	—
82	#103×C₄₁	—	2	4	no	—
83	#103×C₄₁	—	4	2	no	—
84	#103×C₄₁	C₂×C₄	1	8	yes	—
85	#103×C₄₁	D₄	1	4	yes	—
86	C₄×C₈×C₄₁	—	2	4	no	—
87	C₄×C₈×C₄₁	—	2	2	no	—
88	#95×C₄₁	—	2	2	no	—
89	#95×C₄₁	—	2	2	no	—
90	#95×C₄₁	—	2	1	no	—
91	#95×C₄₁	—	2	2	no	—
92	#95×C₄₁	D₄	1	2	yes	—
93	C₂×Q₈×C₄₁	—	2	1	no	—
94	C₂×Q₈×C₄₁	—	2	2	no	—
95	C₂×Q₈×C₄₁	—	2	1	no	—
96	C₂×Q₈×C₄₁	—	2	1	no	—
97	C₂×Q₈×C₄₁	—	2	2	no	—
98	C₂×Q₈×C₄₁	—	2	2	no	—
99	C₂×Q₈×C₄₁	—	2	2	no	—
100	C₂×Q₈×C₄₁	—	2	2	no	—
101	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
102	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
103	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
104	C₂×Q₈×C₄₁	#41	1	2	yes	—
105	C₂²×C₄×C₄₁	—	2	1	no	—
106	C₂²×C₄×C₄₁	—	2	1	no	—
107	C₂²×C₄×C₄₁	—	2	2	no	—
108	C₂²×C₄×C₄₁	—	2	1	no	—
109	C₂²×C₄×C₄₁	—	2	1	no	—
110	C₂²×C₄×C₄₁	—	2	2	no	—
111	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
116	Q₁₆×C₄₁	—	2	16	no	—
117	C₂×C₈×C₄₁	—	2	2	no	—
118	C₂×C₈×C₄₁	—	2	1	no	—
119	C₂×C₈×C₄₁	—	2	2	no	—
120	C₂×C₈×C₄₁	—	2	1	no	—
121	C₂×C₈×C₄₁	—	2	1	no	—
122	C₂×C₈×C₄₁	—	2	1	no	—
123	C₂×C₈×C₄₁	—	2	1	no	—
124	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
125	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₄₁	—	2	4	no	—
127	C₄⋊C₄×C₄₁	—	2	4	no	—
128	C₄⋊C₄×C₄₁	—	2	4	no	—
129	C₄⋊C₄×C₄₁	—	2	2	no	—
130	C₄⋊C₄×C₄₁	—	2	1	no	—
131	C₄⋊C₄×C₄₁	—	2	1	no	—
132	C₄⋊C₄×C₄₁	—	2	2	no	—
133	C₄⋊C₄×C₄₁	—	2	1	no	—
134	C₄⋊C₄×C₄₁	—	2	2	no	—
135	C₄⋊C₄×C₄₁	—	2	1	no	—
136	C₄⋊C₄×C₄₁	—	4	4	no	—
137	C₄⋊C₄×C₄₁	—	4	4	no	—
138	C₄⋊C₄×C₄₁	—	4	2	no	—
139	C₄⋊C₄×C₄₁	—	8	1	no	—
140	C₄⋊C₄×C₄₁	—	8	2	no	—
141	C₄⋊C₄×C₄₁	Q₁₆	1	4	yes	complemented
142	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
144	C₄²×C₄₁	—	2	4	no	—
145	C₄²×C₄₁	—	2	2	no	—
146	C₄²×C₄₁	—	2	2	no	—
147	C₄²×C₄₁	—	4	4	no	—
148	C₄²×C₄₁	—	4	4	no	—
149	C₄²×C₄₁	—	4	2	no	—
150	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
151	C₄²×C₄₁	#41	1	2	yes	—
152	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₄₁	—	2	2	no	—
154	Q₈×C₄₁	—	2	2	no	—
155	Q₈×C₄₁	—	2	2	no	—
156	Q₈×C₄₁	—	2	4	no	—
157	Q₈×C₄₁	—	2	4	no	—
158	Q₈×C₄₁	—	4	2	no	—
159	Q₈×C₄₁	—	4	2	no	—
160	Q₈×C₄₁	C₄×D₄	1	4	yes	—
161	C₂×C₄×C₄₁	—	2	2	no	—
162	C₂×C₄×C₄₁	—	2	2	no	—
163	C₂×C₄×C₄₁	—	2	2	no	—
164	C₂×C₄×C₄₁	—	2	2	no	—
165	C₂×C₄×C₄₁	—	2	1	no	—
166	C₂×C₄×C₄₁	—	2	1	no	—
167	C₂×C₄×C₄₁	—	2	2	no	—
168	C₂×C₄×C₄₁	—	2	2	no	—
169	C₂×C₄×C₄₁	—	2	4	no	—
170	C₂×C₄×C₄₁	—	2	2	no	—
171	C₂×C₄×C₄₁	—	2	2	no	—
172	C₂×C₄×C₄₁	—	2	2	no	—
173	C₂×C₄×C₄₁	—	2	2	no	—
174	C₂×C₄×C₄₁	—	4	2	no	—
175	C₂×C₄×C₄₁	—	4	2	no	—
176	C₂×C₄×C₄₁	—	4	1	no	—
177	C₂×C₄×C₄₁	—	4	1	no	—
178	C₂×C₄×C₄₁	—	4	1	no	—
179	C₂×C₄×C₄₁	—	4	1	no	—
180	C₂×C₄×C₄₁	—	4	2	no	—
181	C₂×C₄×C₄₁	—	4	1	no	—
182	C₂×C₄×C₄₁	—	4	1	no	—
183	C₂×C₄×C₄₁	—	4	2	no	—
184	C₂×C₄×C₄₁	—	4	2	no	—
185	C₂×C₄×C₄₁	—	4	1	no	—
186	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₄₁	C₂×Q₁₆	1	4	yes	—
189	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
190	C₂×C₄×C₄₁	#135	1	2	yes	—
191	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
197	C₈×C₄₁	—	2	4	no	—
198	C₈×C₄₁	—	2	2	no	—
199	C₈×C₄₁	—	4	1	no	—
200	#22998	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₄₁	D₄²	1	1	yes	characteristic
203	C₂²×C₄₁	#548	1	1	yes	characteristic
204	C₂²×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
205	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
206	C₂²×C₄₁	#577	1	1	yes	characteristic
207	C₂²×C₄₁	#578	1	1	yes	characteristic
208	C₄×C₄₁	—	2	4	no	—
209	C₄×C₄₁	—	2	2	no	—
210	C₄×C₄₁	—	2	4	no	—
211	C₄×C₄₁	—	4	4	no	—
212	C₄×C₄₁	—	4	2	no	—
213	C₄×C₄₁	—	4	2	no	—
214	C₄×C₄₁	—	4	2	no	—
215	C₄×C₄₁	—	8	1	no	—
216	C₄×C₄₁	—	8	2	no	—
217	C₄×C₄₁	C₄×Q₁₆	1	4	yes	complemented
218	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
219	C₄×C₄₁	#520	1	1	yes	characteristic
220	C₄×C₄₁	#547	1	1	yes	characteristic
221	C₄×C₄₁	#553	1	1	yes	characteristic
222	C₂×#462	C₂×C₄₁	1	1	yes	characteristic
223	C₂×C₄×Q₁₆	C₂×C₄₁	1	2	yes	—
224	#2365	C₂×C₄₁	1	1	yes	characteristic
225	Q₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
226	#1956	C₂×C₄₁	1	1	yes	characteristic
227	#1947	C₂×C₄₁	1	1	yes	characteristic
228	#1933	C₂×C₄₁	1	2	yes	—
229	#1878	C₂×C₄₁	1	2	yes	—
230	#1863	C₂×C₄₁	1	2	yes	—
231	C₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
232	C₄×#103	C₂×C₄₁	1	1	yes	characteristic
233	C₂×C₄₁	#2361	1	1	yes	characteristic
234	C₂×C₄₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
235	C₂×C₄₁	#2947	1	1	yes	characteristic
236	C₂×C₄₁	#3284	1	1	yes	characteristic
237	C₂×C₄₁	D₄×Q₁₆	1	1	yes	characteristic
238	C₂×C₄₁	#3363	1	1	yes	characteristic
239	C₂×C₄₁	#3364	1	1	yes	characteristic
240	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
241	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
242	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
243	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
244	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
245	#462	C₄×C₄₁	1	8	yes	complemented
246	C₄×Q₁₆	—	2	8	no	—
247	C₂×#107	C₂²×C₄₁	1	2	yes	—
248	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
249	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
250	C₂×#103	C₂²×C₄₁	1	2	yes	—
251	C₂×#103	C₂²×C₄₁	1	2	yes	—
252	C₂×#103	C₂²×C₄₁	1	2	yes	—
253	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
256	#391	C₂²×C₄₁	1	2	yes	—
257	#389	—	2	1	no	—
258	#389	—	2	2	no	—
259	#389	C₂²×C₄₁	1	2	yes	—
260	#387	—	2	2	no	—
261	#384	—	2	1	no	—
262	#384	C₂²×C₄₁	1	1	yes	characteristic
263	C₄×C₄⋊C₄	—	2	1	no	—
264	C₄×C₄⋊C₄	—	2	1	no	—
265	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
266	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
267	#340	C₂²×C₄₁	1	1	yes	characteristic
268	#340	C₂²×C₄₁	1	2	yes	—
269	#340	C₂²×C₄₁	1	1	yes	characteristic
270	#336	C₂²×C₄₁	1	2	yes	—
271	C₄₁	#22998	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
273	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
274	C₂×Q₁₆	—	2	4	no	—
275	C₂×Q₁₆	—	2	4	no	—
276	C₂×Q₁₆	C₂×C₄×C₄₁	1	8	yes	—
277	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
279	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
280	C₄⋊Q₈	—	2	2	no	—
281	C₄⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
282	C₄×Q₈	—	2	2	no	—
283	C₄×Q₈	—	2	4	no	—
284	C₄×Q₈	—	2	4	no	—
285	C₄×Q₈	—	4	2	no	—
286	C₄×Q₈	—	4	2	no	—
287	C₄×Q₈	C₂×C₄×C₄₁	1	4	yes	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
307	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
308	#107	—	2	4	no	—
309	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
310	#103	—	2	2	no	—
311	#103	—	2	4	no	—
312	#103	—	4	2	no	—
313	#103	C₂×C₄×C₄₁	1	8	yes	—
314	#103	D₄×C₄₁	1	4	yes	—
315	C₄×C₈	—	2	4	no	—
316	C₄×C₈	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	—	2	2	no	—
321	#95	D₄×C₄₁	1	2	yes	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
331	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
332	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
333	C₂×Q₈	#41×C₄₁	1	2	yes	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
343	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
345	Q₁₆	—	2	16	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
354	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	—	2	4	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	4	4	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	8	1	no	—
369	C₄⋊C₄	—	8	2	no	—
370	C₄⋊C₄	Q₁₆×C₄₁	1	4	yes	complemented
371	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
372	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
373	C₄²	—	2	4	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	4	4	no	—
377	C₄²	—	4	4	no	—
378	C₄²	—	4	2	no	—
379	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
380	C₄²	#41×C₄₁	1	2	yes	—
381	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	2	no	—
384	Q₈	—	2	2	no	—
385	Q₈	—	2	4	no	—
386	Q₈	—	2	4	no	—
387	Q₈	—	4	2	no	—
388	Q₈	—	4	2	no	—
389	Q₈	C₄×D₄×C₄₁	1	4	yes	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
416	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
417	C₂×C₄	C₂×Q₁₆×C₄₁	1	4	yes	—
418	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
419	C₂×C₄	#135×C₄₁	1	2	yes	—
420	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
422	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
424	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
425	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
426	C₈	—	2	4	no	—
427	C₈	—	2	2	no	—
428	C₈	—	4	1	no	—
429	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
430	C₂²	D₄²×C₄₁	1	1	yes	characteristic
431	C₂²	C₄₁×#548	1	1	yes	characteristic
432	C₂²	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
433	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
434	C₂²	C₄₁×#577	1	1	yes	characteristic
435	C₂²	C₄₁×#578	1	1	yes	characteristic
436	C₄	—	2	4	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	4	no	—
439	C₄	—	4	4	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	4	2	no	—
443	C₄	—	8	1	no	—
444	C₄	—	8	2	no	—
445	C₄	C₄×Q₁₆×C₄₁	1	4	yes	complemented
446	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
447	C₄	C₄₁×#520	1	1	yes	characteristic
448	C₄	C₄₁×#547	1	1	yes	characteristic
449	C₄	C₄₁×#553	1	1	yes	characteristic
450	C₂	C₄₁×#2361	1	1	yes	characteristic
451	C₂	C₂×C₄×Q₁₆×C₄₁	1	1	yes	characteristic
452	C₂	C₄₁×#2947	1	1	yes	characteristic
453	C₂	C₄₁×#3284	1	1	yes	characteristic
454	C₂	D₄×Q₁₆×C₄₁	1	1	yes	characteristic
455	C₂	C₄₁×#3363	1	1	yes	characteristic
456	C₂	C₄₁×#3364	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄×C₄⋊C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₄₁×#462	C₄	semidirect
141. C₄⋊C₄×C₄₁	Q₁₆	semidirect
200. #22998	C₄₁	direct (N × H)
217. C₄×C₄₁	C₄×Q₁₆	semidirect
245. #462	C₄×C₄₁	semidirect
271. C₄₁	#22998	direct (N × H)
370. C₄⋊C₄	Q₁₆×C₄₁	semidirect
445. C₄	C₄×Q₁₆×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#462	2
C₂×C₄×Q₁₆×C₄₁	2
C₄₁×#2365	2
Q₈×C₄⋊C₄×C₄₁	2
C₄₁×#1956	2
C₄₁×#1947	2
C₄₁×#1933	2
C₄₁×#1878	2
C₄₁×#1863	2
C₈×C₄⋊C₄×C₄₁	2
C₄×#103×C₄₁	2
#22998	41

Subgroups — C41×#22998 #23799199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22998 #23799199