Subgroups — C₄₁×#22571 #23798772

Order 10496.

1002 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#103×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2344	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1933	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1927	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1865	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1823	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#1807	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#1787	C₂	1	1	yes	maximal, characteristic
10	C₄×#103×C₄₁	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
15	C₂×#117×C₄₁	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#106×C₄₁	C₂²	1	2	yes	—
18	C₂×#103×C₄₁	C₄	1	8	yes	complemented
19	C₂×#103×C₄₁	C₂²	1	2	yes	—
20	C₂×#103×C₄₁	C₂²	1	2	yes	—
21	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
27	C₄₁×#391	—	2	2	no	—
28	C₄₁×#391	C₂²	1	2	yes	—
29	C₄₁×#389	C₂²	1	2	yes	—
30	C₄₁×#387	—	2	2	no	—
31	C₄₁×#381	—	2	1	no	—
32	C₄₁×#380	—	2	1	no	—
33	C₄₁×#380	C₂²	1	1	yes	characteristic
34	C₄₁×#380	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
37	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#340	C₂²	1	2	yes	—
39	C₄₁×#340	C₂²	1	2	yes	—
40	C₄₁×#340	C₂²	1	2	yes	—
41	C₄₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₄₁	—	2	1	no	—
43	C₂²×Q₈×C₄₁	—	2	2	no	—
44	C₂²×Q₈×C₄₁	—	2	4	no	—
45	C₂²×Q₈×C₄₁	—	4	2	no	—
46	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
47	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₁	—	2	1	no	—
50	C₂³×C₄×C₄₁	—	2	1	no	—
51	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₁	—	4	2	no	—
53	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₄₁	—	2	4	no	—
58	C₄×Q₈×C₄₁	—	4	2	no	—
59	#117×C₄₁	—	4	4	no	—
60	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₄₁	—	2	2	no	—
74	C₂×C₄²×C₄₁	—	2	2	no	—
75	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
76	#106×C₄₁	—	4	2	no	—
77	#103×C₄₁	—	2	4	no	—
78	#103×C₄₁	—	4	2	no	—
79	#103×C₄₁	C₂×C₄	1	4	yes	—
80	#103×C₄₁	C₂×C₄	1	4	yes	—
81	#103×C₄₁	C₂×C₄	1	4	yes	—
82	#103×C₄₁	C₂×C₄	1	4	yes	—
83	C₂²⋊C₈×C₄₁	—	2	2	no	—
84	C₄×C₈×C₄₁	—	4	2	no	—
85	#95×C₄₁	—	2	2	no	—
86	#95×C₄₁	—	2	2	no	—
87	#95×C₄₁	—	2	2	no	—
88	#95×C₄₁	—	2	1	no	—
89	#95×C₄₁	—	2	1	no	—
90	#95×C₄₁	—	2	2	no	—
91	#95×C₄₁	D₄	1	2	yes	—
92	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
93	C₂×Q₈×C₄₁	—	2	4	no	—
94	C₂×Q₈×C₄₁	—	2	1	no	—
95	C₂×Q₈×C₄₁	—	2	1	no	—
96	C₂×Q₈×C₄₁	—	2	2	no	—
97	C₂×Q₈×C₄₁	—	2	8	no	—
98	C₂×Q₈×C₄₁	—	2	2	no	—
99	C₂×Q₈×C₄₁	—	2	2	no	—
100	C₂×Q₈×C₄₁	—	4	4	no	—
101	C₂×Q₈×C₄₁	—	4	4	no	—
102	C₂×Q₈×C₄₁	—	4	8	no	—
103	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
104	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
105	C₂×Q₈×C₄₁	#41	1	4	yes	—
106	C₂²×C₄×C₄₁	—	2	1	no	—
107	C₂²×C₄×C₄₁	—	2	2	no	—
108	C₂²×C₄×C₄₁	—	2	2	no	—
109	C₂²×C₄×C₄₁	—	2	4	no	—
110	C₂²×C₄×C₄₁	—	2	4	no	—
111	C₂²×C₄×C₄₁	—	2	1	no	—
112	C₂²×C₄×C₄₁	—	2	1	no	—
113	C₂²×C₄×C₄₁	—	2	1	no	—
114	C₂²×C₄×C₄₁	—	2	2	no	—
115	C₂²×C₄×C₄₁	—	2	2	no	—
116	C₂²×C₄×C₄₁	—	4	4	no	—
117	C₂²×C₄×C₄₁	—	4	2	no	—
118	C₂²×C₄×C₄₁	—	4	2	no	—
119	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₄₁	#41	1	2	yes	—
124	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
125	C₂×C₈×C₄₁	—	2	2	no	—
126	C₂×C₈×C₄₁	—	2	2	no	—
127	C₂×C₈×C₄₁	—	2	2	no	—
128	C₂×C₈×C₄₁	—	2	2	no	—
129	C₂×C₈×C₄₁	—	2	2	no	—
130	C₂×C₈×C₄₁	—	4	4	no	—
131	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₄₁	—	2	4	no	—
134	C₄⋊C₄×C₄₁	—	2	1	no	—
135	C₄⋊C₄×C₄₁	—	2	1	no	—
136	C₄⋊C₄×C₄₁	—	2	2	no	—
137	C₄⋊C₄×C₄₁	—	2	2	no	—
138	C₄⋊C₄×C₄₁	—	2	2	no	—
139	C₄⋊C₄×C₄₁	—	2	2	no	—
140	C₄⋊C₄×C₄₁	—	4	4	no	—
141	C₄⋊C₄×C₄₁	—	4	4	no	—
142	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₄₁	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₄₁	—	2	2	no	—
145	C₂²⋊C₄×C₄₁	—	4	2	no	—
146	C₂²⋊C₄×C₄₁	—	4	2	no	—
147	C₂²⋊C₄×C₄₁	SD₁₆	1	4	yes	—
148	C₄²×C₄₁	—	2	2	no	—
149	C₄²×C₄₁	—	2	2	no	—
150	C₄²×C₄₁	—	4	4	no	—
151	C₄²×C₄₁	—	4	4	no	—
152	C₂³×C₄₁	—	2	2	no	—
153	C₂³×C₄₁	—	2	2	no	—
154	C₂³×C₄₁	C₄×D₄	1	2	yes	—
155	C₂³×C₄₁	C₂×SD₁₆	1	2	yes	—
156	C₂³×C₄₁	#135	1	2	yes	—
157	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₄₁	—	2	2	no	—
159	Q₈×C₄₁	—	2	4	no	—
160	Q₈×C₄₁	—	2	2	no	—
161	Q₈×C₄₁	—	2	2	no	—
162	Q₈×C₄₁	—	4	8	no	—
163	Q₈×C₄₁	—	4	2	no	—
164	Q₈×C₄₁	#117	1	4	yes	complemented
165	C₂×C₄×C₄₁	—	2	2	no	—
166	C₂×C₄×C₄₁	—	2	2	no	—
167	C₂×C₄×C₄₁	—	2	4	no	—
168	C₂×C₄×C₄₁	—	2	2	no	—
169	C₂×C₄×C₄₁	—	2	2	no	—
170	C₂×C₄×C₄₁	—	2	2	no	—
171	C₂×C₄×C₄₁	—	2	2	no	—
172	C₂×C₄×C₄₁	—	2	4	no	—
173	C₂×C₄×C₄₁	—	2	2	no	—
174	C₂×C₄×C₄₁	—	2	2	no	—
175	C₂×C₄×C₄₁	—	2	2	no	—
176	C₂×C₄×C₄₁	—	2	2	no	—
177	C₂×C₄×C₄₁	—	2	2	no	—
178	C₂×C₄×C₄₁	—	2	2	no	—
179	C₂×C₄×C₄₁	—	4	8	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	1	no	—
184	C₂×C₄×C₄₁	—	4	1	no	—
185	C₂×C₄×C₄₁	—	4	8	no	—
186	C₂×C₄×C₄₁	—	4	2	no	—
187	C₂×C₄×C₄₁	—	4	2	no	—
188	C₂×C₄×C₄₁	—	4	2	no	—
189	C₂×C₄×C₄₁	—	4	2	no	—
190	C₂×C₄×C₄₁	—	4	2	no	—
191	C₂×C₄×C₄₁	—	4	2	no	—
192	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
194	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
196	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
198	C₈×C₄₁	—	2	2	no	—
199	C₈×C₄₁	—	2	2	no	—
200	C₈×C₄₁	—	4	2	no	—
201	#22571	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂²×C₄₁	—	2	2	no	—
203	C₂²×C₄₁	—	2	2	no	—
204	C₂²×C₄₁	—	2	4	no	—
205	C₂²×C₄₁	—	2	4	no	—
206	C₂²×C₄₁	C₄×SD₁₆	1	4	yes	—
207	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₄₁	#546	1	2	yes	—
209	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
210	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
211	C₂²×C₄₁	#578	1	2	yes	—
212	C₄×C₄₁	—	2	2	no	—
213	C₄×C₄₁	—	2	4	no	—
214	C₄×C₄₁	—	2	4	no	—
215	C₄×C₄₁	—	4	2	no	—
216	C₄×C₄₁	—	4	2	no	—
217	C₄×C₄₁	—	4	2	no	—
218	C₄×C₄₁	—	4	2	no	—
219	C₄×C₄₁	—	8	2	no	—
220	C₄×C₄₁	—	8	2	no	—
221	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
222	C₄×C₄₁	#518	1	1	yes	characteristic
223	C₄×C₄₁	#551	1	2	yes	—
224	C₂²×#103	C₂×C₄₁	1	1	yes	characteristic
225	#2344	C₂×C₄₁	1	1	yes	characteristic
226	Q₈×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic
227	#1933	C₂×C₄₁	1	2	yes	complemented
228	#1927	C₂×C₄₁	1	2	yes	complemented
229	#1865	C₂×C₄₁	1	2	yes	complemented
230	#1823	C₂×C₄₁	1	1	yes	characteristic
231	#1807	C₂×C₄₁	1	1	yes	characteristic
232	#1787	C₂×C₄₁	1	1	yes	characteristic
233	C₄×#103	C₂×C₄₁	1	2	yes	complemented
234	C₈×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic
235	C₂×C₄₁	—	2	4	no	—
236	C₂×C₄₁	#2342	1	1	yes	characteristic
237	C₂×C₄₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
238	C₂×C₄₁	#2950	1	1	yes	characteristic
239	C₂×C₄₁	#3298	1	2	yes	—
240	C₂×C₄₁	#3300	1	2	yes	—
241	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
242	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
243	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
244	C₂×#117	—	2	1	no	—
245	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
246	C₂×#106	C₂²×C₄₁	1	2	yes	—
247	C₂×#103	C₄×C₄₁	1	8	yes	complemented
248	C₂×#103	C₂²×C₄₁	1	2	yes	—
249	C₂×#103	C₂²×C₄₁	1	2	yes	—
250	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
251	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
252	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
253	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
254	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
255	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
256	#391	—	2	2	no	—
257	#391	C₂²×C₄₁	1	2	yes	—
258	#389	C₂²×C₄₁	1	2	yes	—
259	#387	—	2	2	no	—
260	#381	—	2	1	no	—
261	#380	—	2	1	no	—
262	#380	C₂²×C₄₁	1	1	yes	characteristic
263	#380	C₂²×C₄₁	1	1	yes	characteristic
264	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
265	C₄×C₂²⋊C₄	—	2	1	no	—
266	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
267	#340	C₂²×C₄₁	1	2	yes	—
268	#340	C₂²×C₄₁	1	2	yes	—
269	#340	C₂²×C₄₁	1	2	yes	—
270	#336	C₂²×C₄₁	1	2	yes	—
271	C₄₁	#22571	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×Q₈	—	2	1	no	—
273	C₂²×Q₈	—	2	2	no	—
274	C₂²×Q₈	—	2	4	no	—
275	C₂²×Q₈	—	4	2	no	—
276	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
277	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
278	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
282	C₂²×C₈	—	4	2	no	—
283	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
284	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
285	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
287	C₄×Q₈	—	2	4	no	—
288	C₄×Q₈	—	4	2	no	—
289	#117	—	4	4	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	2	no	—
294	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
295	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
302	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
306	#106	—	4	2	no	—
307	#103	—	2	4	no	—
308	#103	—	4	2	no	—
309	#103	C₂×C₄×C₄₁	1	4	yes	—
310	#103	C₂×C₄×C₄₁	1	4	yes	—
311	#103	C₂×C₄×C₄₁	1	4	yes	—
312	#103	C₂×C₄×C₄₁	1	4	yes	—
313	C₂²⋊C₈	—	2	2	no	—
314	C₄×C₈	—	4	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	1	no	—
320	#95	—	2	2	no	—
321	#95	D₄×C₄₁	1	2	yes	—
322	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
323	C₂×Q₈	—	2	4	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	8	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	4	4	no	—
331	C₂×Q₈	—	4	4	no	—
332	C₂×Q₈	—	4	8	no	—
333	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
334	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
335	C₂×Q₈	#41×C₄₁	1	4	yes	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	4	4	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
353	C₂²×C₄	#41×C₄₁	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
355	C₂×C₈	—	2	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	—	4	4	no	—
361	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
373	C₄⋊C₄	C₂²×C₄×C₄₁	1	2	yes	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	SD₁₆×C₄₁	1	4	yes	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	4	4	no	—
381	C₄²	—	4	4	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	C₄×D₄×C₄₁	1	2	yes	—
385	C₂³	C₂×SD₁₆×C₄₁	1	2	yes	—
386	C₂³	#135×C₄₁	1	2	yes	—
387	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
388	Q₈	—	2	2	no	—
389	Q₈	—	2	4	no	—
390	Q₈	—	2	2	no	—
391	Q₈	—	2	2	no	—
392	Q₈	—	4	8	no	—
393	Q₈	—	4	2	no	—
394	Q₈	#117×C₄₁	1	4	yes	complemented
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	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	—	4	8	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
423	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
424	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
426	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
427	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
428	C₈	—	2	2	no	—
429	C₈	—	2	2	no	—
430	C₈	—	4	2	no	—
431	C₂²	—	2	4	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	C₄×SD₁₆×C₄₁	1	4	yes	—
436	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
437	C₂²	C₄₁×#546	1	2	yes	—
438	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
439	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
440	C₂²	C₄₁×#578	1	2	yes	—
441	C₄	—	2	4	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	4	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	8	2	no	—
449	C₄	—	8	2	no	—
450	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
451	C₄	C₄₁×#518	1	1	yes	characteristic
452	C₄	C₄₁×#551	1	2	yes	—
453	C₂	—	2	4	no	—
454	C₂	C₄₁×#2342	1	1	yes	characteristic
455	C₂	C₂×C₄×SD₁₆×C₄₁	1	1	yes	characteristic
456	C₂	C₄₁×#2950	1	1	yes	characteristic
457	C₂	C₄₁×#3298	1	2	yes	—
458	C₂	C₄₁×#3300	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1933	C₂	semidirect
5. C₄₁×#1927	C₂	semidirect
6. C₄₁×#1865	C₂	semidirect
10. C₄×#103×C₄₁	C₂	semidirect
18. C₂×#103×C₄₁	C₄	semidirect
164. Q₈×C₄₁	#117	semidirect
201. #22571	C₄₁	direct (N × H)
227. #1933	C₂×C₄₁	semidirect
228. #1927	C₂×C₄₁	semidirect
229. #1865	C₂×C₄₁	semidirect
233. C₄×#103	C₂×C₄₁	semidirect
247. C₂×#103	C₄×C₄₁	semidirect
271. C₄₁	#22571	direct (N × H)
394. Q₈	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#103×C₄₁	2
C₄₁×#2344	2
Q₈×C₂²⋊C₄×C₄₁	2
C₄₁×#1933	2
C₄₁×#1927	2
C₄₁×#1865	2
C₄₁×#1823	2
C₄₁×#1807	2
C₄₁×#1787	2
C₄×#103×C₄₁	2
C₈×C₂²⋊C₄×C₄₁	2
#22571	41

Subgroups — C41×#22571 #23798772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22571 #23798772