Subgroups — C₄₁×#24196 #23800397

Order 10496.

1428 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#442	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2580	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2041	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2011	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1879	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1864	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₁	—	2	1	no	—
13	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#550	—	2	4	no	—
16	C₄₁×#550	—	2	4	no	—
17	C₂×#127×C₄₁	C₂²	1	2	yes	—
18	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#121×C₄₁	—	2	2	no	—
20	C₂×#121×C₄₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#459	—	2	4	no	—
26	C₄₁×#442	—	2	4	no	—
27	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#102×C₄₁	C₂²	1	2	yes	—
31	C₂×#102×C₄₁	C₂²	1	2	yes	—
32	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×#102×C₄₁	C₂²	1	2	yes	—
34	C₂×#102×C₄₁	C₂²	1	2	yes	—
35	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#399	—	2	1	no	—
37	C₄₁×#393	—	2	1	no	—
38	C₄₁×#390	C₂²	1	2	yes	—
39	C₄₁×#389	C₂²	1	2	yes	—
40	C₄₁×#385	—	2	2	no	—
41	C₄₁×#382	C₂²	1	1	yes	characteristic
42	C₄₁×#340	C₂²	1	2	yes	—
43	C₄₁×#336	C₂²	1	2	yes	—
44	C₄₁×#336	C₂²	1	2	yes	—
45	C₂×#41×C₄₁	—	2	8	no	—
46	C₂×#41×C₄₁	—	4	8	no	—
47	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₄₁	—	2	1	no	—
49	C₂²×D₄×C₄₁	—	2	2	no	—
50	C₂²×D₄×C₄₁	—	2	2	no	—
51	C₂²×D₄×C₄₁	—	2	1	no	—
52	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₄₁	—	2	2	no	—
56	C₂³×C₄×C₄₁	—	2	1	no	—
57	C₂³×C₄×C₄₁	—	2	1	no	—
58	C₂×D₈×C₄₁	—	2	2	no	—
59	C₂×D₈×C₄₁	—	2	2	no	—
60	C₂×D₈×C₄₁	—	2	4	no	—
61	C₂×D₈×C₄₁	—	2	4	no	—
62	C₂×D₈×C₄₁	—	2	2	no	—
63	C₂×D₈×C₄₁	—	4	2	no	—
64	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
67	#127×C₄₁	—	2	4	no	—
68	#127×C₄₁	—	2	4	no	—
69	#127×C₄₁	—	2	2	no	—
70	#127×C₄₁	D₄	1	4	yes	—
71	#121×C₄₁	—	2	8	no	—
72	#121×C₄₁	—	2	16	no	—
73	C₄×Q₈×C₄₁	—	2	2	no	—
74	C₄×Q₈×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	—	2	4	no	—
76	C₄×D₄×C₄₁	—	2	4	no	—
77	C₄×D₄×C₄₁	—	2	2	no	—
78	C₄×D₄×C₄₁	—	2	2	no	—
79	C₄×D₄×C₄₁	—	2	2	no	—
80	C₄×D₄×C₄₁	D₄	1	4	yes	—
81	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
92	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
93	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
94	#106×C₄₁	—	2	2	no	—
95	C₄⋊C₈×C₄₁	—	2	2	no	—
96	#102×C₄₁	—	2	2	no	—
97	#102×C₄₁	—	2	4	no	—
98	#102×C₄₁	—	2	4	no	—
99	#102×C₄₁	—	2	2	no	—
100	#102×C₄₁	—	4	2	no	—
101	#102×C₄₁	—	4	2	no	—
102	#97×C₄₁	—	2	2	no	—
103	#95×C₄₁	—	2	1	no	—
104	#95×C₄₁	—	2	1	no	—
105	#95×C₄₁	—	2	2	no	—
106	C₂⁴×C₄₁	—	2	1	no	—
107	C₂⁴×C₄₁	—	2	2	no	—
108	C₂⁴×C₄₁	—	2	1	no	—
109	#41×C₄₁	—	4	32	no	—
110	C₂×Q₈×C₄₁	—	2	2	no	—
111	C₂×Q₈×C₄₁	—	2	4	no	—
112	C₂×D₄×C₄₁	—	2	8	no	—
113	C₂×D₄×C₄₁	—	2	8	no	—
114	C₂×D₄×C₄₁	—	2	4	no	—
115	C₂×D₄×C₄₁	—	2	4	no	—
116	C₂×D₄×C₄₁	—	2	1	no	—
117	C₂×D₄×C₄₁	—	2	1	no	—
118	C₂×D₄×C₄₁	—	2	2	no	—
119	C₂×D₄×C₄₁	—	2	1	no	—
120	C₂×D₄×C₄₁	—	2	2	no	—
121	C₂×D₄×C₄₁	—	2	1	no	—
122	C₂×D₄×C₄₁	—	2	2	no	—
123	C₂×D₄×C₄₁	—	2	1	no	—
124	C₂×D₄×C₄₁	—	2	1	no	—
125	C₂×D₄×C₄₁	—	2	1	no	—
126	C₂×D₄×C₄₁	—	2	2	no	—
127	C₂×D₄×C₄₁	—	2	1	no	—
128	C₂×D₄×C₄₁	—	2	1	no	—
129	C₂×D₄×C₄₁	—	2	2	no	—
130	C₂×D₄×C₄₁	—	2	1	no	—
131	C₂×D₄×C₄₁	—	4	4	no	—
132	C₂×D₄×C₄₁	—	4	4	no	—
133	C₂×D₄×C₄₁	—	4	4	no	—
134	C₂×D₄×C₄₁	—	8	2	no	—
135	C₂×D₄×C₄₁	—	8	2	no	—
136	C₂×D₄×C₄₁	—	8	2	no	—
137	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
138	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
139	C₂×D₄×C₄₁	#41	1	2	yes	complemented
140	C₂²×C₄×C₄₁	—	2	8	no	—
141	C₂²×C₄×C₄₁	—	2	1	no	—
142	C₂²×C₄×C₄₁	—	2	4	no	—
143	C₂²×C₄×C₄₁	—	2	2	no	—
144	C₂²×C₄×C₄₁	—	2	1	no	—
145	C₂²×C₄×C₄₁	—	4	4	no	—
146	C₂²×C₄×C₄₁	—	4	4	no	—
147	C₂²×C₄×C₄₁	—	4	4	no	—
148	C₂²×C₄×C₄₁	—	4	2	no	—
149	C₂²×C₄×C₄₁	—	8	2	no	—
150	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
155	D₈×C₄₁	—	2	8	no	—
156	D₈×C₄₁	—	4	8	no	—
157	C₂×C₈×C₄₁	—	2	1	no	—
158	C₂×C₈×C₄₁	—	2	1	no	—
159	C₂×C₈×C₄₁	—	2	1	no	—
160	C₂×C₈×C₄₁	—	2	2	no	—
161	C₂×C₈×C₄₁	—	2	2	no	—
162	C₂×C₈×C₄₁	—	2	1	no	—
163	C₂×C₈×C₄₁	—	2	2	no	—
164	C₂×C₈×C₄₁	—	2	1	no	—
165	C₂×C₈×C₄₁	#41	1	2	yes	complemented
166	C₄⋊C₄×C₄₁	—	2	4	no	—
167	C₄⋊C₄×C₄₁	—	2	2	no	—
168	C₄⋊C₄×C₄₁	—	2	1	no	—
169	C₄⋊C₄×C₄₁	—	2	1	no	—
170	C₄⋊C₄×C₄₁	—	2	1	no	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	8	2	no	—
174	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
175	C₂²⋊C₄×C₄₁	—	2	8	no	—
176	C₂²⋊C₄×C₄₁	—	2	4	no	—
177	C₂²⋊C₄×C₄₁	—	4	2	no	—
178	C₂²⋊C₄×C₄₁	—	4	4	no	—
179	C₂²⋊C₄×C₄₁	—	8	2	no	—
180	C₄²×C₄₁	—	2	2	no	—
181	C₄²×C₄₁	—	2	2	no	—
182	C₄²×C₄₁	C₂×D₄	1	2	yes	—
183	C₄²×C₄₁	C₂×D₄	1	2	yes	—
184	C₂³×C₄₁	—	2	4	no	—
185	C₂³×C₄₁	—	2	2	no	—
186	C₂³×C₄₁	—	2	2	no	—
187	C₂³×C₄₁	—	4	1	no	—
188	C₂³×C₄₁	—	4	1	no	—
189	C₂³×C₄₁	—	4	2	no	—
190	C₂³×C₄₁	—	4	2	no	—
191	C₂³×C₄₁	—	4	1	no	—
192	C₂³×C₄₁	—	4	1	no	—
193	C₂³×C₄₁	—	4	2	no	—
194	C₂³×C₄₁	—	4	2	no	—
195	C₂³×C₄₁	—	4	2	no	—
196	C₂³×C₄₁	—	4	2	no	—
197	C₂³×C₄₁	—	4	2	no	—
198	C₂³×C₄₁	—	4	2	no	—
199	C₂³×C₄₁	—	4	2	no	—
200	C₂³×C₄₁	—	4	2	no	—
201	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₄₁	—	2	8	no	—
203	D₄×C₄₁	—	2	2	no	—
204	D₄×C₄₁	—	2	2	no	—
205	D₄×C₄₁	—	2	2	no	—
206	D₄×C₄₁	—	2	2	no	—
207	D₄×C₄₁	—	2	4	no	—
208	D₄×C₄₁	—	2	4	no	—
209	D₄×C₄₁	—	4	8	no	—
210	D₄×C₄₁	—	4	16	no	—
211	D₄×C₄₁	—	4	2	no	—
212	D₄×C₄₁	—	4	1	no	—
213	D₄×C₄₁	—	4	2	no	—
214	D₄×C₄₁	—	4	1	no	—
215	D₄×C₄₁	—	4	1	no	—
216	D₄×C₄₁	—	4	1	no	—
217	D₄×C₄₁	—	8	4	no	—
218	D₄×C₄₁	—	8	8	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	—	2	2	no	—
223	C₂×C₄×C₄₁	—	2	2	no	—
224	C₂×C₄×C₄₁	—	2	2	no	—
225	C₂×C₄×C₄₁	—	2	2	no	—
226	C₂×C₄×C₄₁	—	2	2	no	—
227	C₂×C₄×C₄₁	—	4	8	no	—
228	C₂×C₄×C₄₁	—	4	16	no	—
229	C₂×C₄×C₄₁	—	4	2	no	—
230	C₂×C₄×C₄₁	—	4	1	no	—
231	C₂×C₄×C₄₁	—	4	2	no	—
232	C₂×C₄×C₄₁	—	4	2	no	—
233	C₂×C₄×C₄₁	—	4	2	no	—
234	C₂×C₄×C₄₁	—	4	2	no	—
235	C₂×C₄×C₄₁	—	8	4	no	—
236	C₂×C₄×C₄₁	#120	1	2	yes	—
237	C₂×C₄×C₄₁	#120	1	2	yes	—
238	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
243	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
245	C₈×C₄₁	—	2	2	no	—
246	C₈×C₄₁	—	4	1	no	—
247	C₈×C₄₁	—	4	2	no	—
248	#24196	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂²×C₄₁	—	4	4	no	—
250	C₂²×C₄₁	—	4	2	no	—
251	C₂²×C₄₁	—	4	2	no	—
252	C₂²×C₄₁	—	4	4	no	—
253	C₂²×C₄₁	—	4	2	no	—
254	C₂²×C₄₁	—	4	4	no	—
255	C₂²×C₄₁	—	4	4	no	—
256	C₂²×C₄₁	—	4	2	no	—
257	C₂²×C₄₁	—	8	2	no	—
258	C₂²×C₄₁	—	8	2	no	—
259	C₂²×C₄₁	—	8	2	no	—
260	C₂²×C₄₁	—	8	2	no	—
261	C₂²×C₄₁	—	8	2	no	—
262	C₂²×C₄₁	—	8	2	no	—
263	C₂²×C₄₁	—	8	2	no	—
264	C₂²×C₄₁	—	8	1	no	—
265	C₂²×C₄₁	—	8	1	no	—
266	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
267	C₂²×C₄₁	#548	1	1	yes	characteristic
268	C₂²×C₄₁	#550	1	1	yes	characteristic
269	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
270	C₂²×C₄₁	#576	1	2	yes	—
271	C₄×C₄₁	—	2	2	no	—
272	C₄×C₄₁	—	2	4	no	—
273	C₄×C₄₁	—	2	2	no	—
274	C₄×C₄₁	—	4	2	no	—
275	C₄×C₄₁	—	4	2	no	—
276	C₄×C₄₁	—	8	2	no	—
277	C₄×C₄₁	—	8	1	no	—
278	C₄×C₄₁	#525	1	1	yes	characteristic
279	C₄×C₄₁	#537	1	1	yes	characteristic
280	C₄×C₄₁	#546	1	1	yes	characteristic
281	C₄×C₄₁	#547	1	1	yes	characteristic
282	C₂×#550	C₂×C₄₁	1	1	yes	characteristic, complemented
283	C₂×#459	C₂×C₄₁	1	1	yes	characteristic, complemented
284	C₂×#442	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#2580	C₂×C₄₁	1	1	yes	characteristic, complemented
286	#2041	C₂×C₄₁	1	2	yes	complemented
287	#2011	C₂×C₄₁	1	2	yes	complemented
288	#1985	C₂×C₄₁	1	1	yes	characteristic, complemented
289	#1961	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#1879	C₂×C₄₁	1	2	yes	complemented
291	#1864	C₂×C₄₁	1	2	yes	complemented
292	#1804	C₂×C₄₁	1	1	yes	characteristic, complemented
293	C₂×C₄₁	—	8	2	no	—
294	C₂×C₄₁	—	8	2	no	—
295	C₂×C₄₁	—	8	4	no	—
296	C₂×C₄₁	#2407	1	1	yes	characteristic
297	C₂×C₄₁	#3003	1	1	yes	characteristic
298	C₂×C₄₁	#3008	1	1	yes	characteristic
299	C₂×C₄₁	#3366	1	2	yes	—
300	C₂×C₄₁	#3399	1	2	yes	—
301	C₂²×#41	—	2	1	no	—
302	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
303	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
304	#550	—	2	4	no	—
305	#550	—	2	4	no	—
306	C₂×#127	C₂²×C₄₁	1	2	yes	—
307	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
308	C₂×#121	—	2	2	no	—
309	C₂×#121	—	2	1	no	—
310	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
312	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
313	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
314	#459	—	2	4	no	—
315	#442	—	2	4	no	—
316	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
317	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
318	C₂×#102	C₂²×C₄₁	1	2	yes	—
319	C₂×#102	C₂²×C₄₁	1	2	yes	—
320	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
321	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
322	C₂×#102	C₂²×C₄₁	1	2	yes	—
323	C₂×#102	C₂²×C₄₁	1	2	yes	—
324	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
325	#399	—	2	1	no	—
326	#393	—	2	1	no	—
327	#390	C₂²×C₄₁	1	2	yes	—
328	#389	C₂²×C₄₁	1	2	yes	—
329	#385	—	2	2	no	—
330	#382	C₂²×C₄₁	1	1	yes	characteristic
331	#340	C₂²×C₄₁	1	2	yes	—
332	#336	C₂²×C₄₁	1	2	yes	—
333	#336	C₂²×C₄₁	1	2	yes	—
334	C₄₁	#24196	1	1	yes	characteristic, direct product factor, Sylow
335	C₂×#41	—	2	8	no	—
336	C₂×#41	—	4	8	no	—
337	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
338	C₂²×D₄	—	2	1	no	—
339	C₂²×D₄	—	2	2	no	—
340	C₂²×D₄	—	2	2	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
343	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
344	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
345	C₂³×C₄	—	2	2	no	—
346	C₂³×C₄	—	2	1	no	—
347	C₂³×C₄	—	2	1	no	—
348	C₂×D₈	—	2	2	no	—
349	C₂×D₈	—	2	2	no	—
350	C₂×D₈	—	2	4	no	—
351	C₂×D₈	—	2	4	no	—
352	C₂×D₈	—	2	2	no	—
353	C₂×D₈	—	4	2	no	—
354	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
355	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
356	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
357	#127	—	2	4	no	—
358	#127	—	2	4	no	—
359	#127	—	2	2	no	—
360	#127	D₄×C₄₁	1	4	yes	—
361	#121	—	2	8	no	—
362	#121	—	2	16	no	—
363	C₄×Q₈	—	2	2	no	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×D₄	—	2	4	no	—
366	C₄×D₄	—	2	4	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	D₄×C₄₁	1	4	yes	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
373	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	—	2	2	no	—
378	C₂×C₂²⋊C₄	—	2	1	no	—
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	2	no	—
381	C₂×C₂²⋊C₄	—	2	2	no	—
382	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
383	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
384	#106	—	2	2	no	—
385	C₄⋊C₈	—	2	2	no	—
386	#102	—	2	2	no	—
387	#102	—	2	4	no	—
388	#102	—	2	4	no	—
389	#102	—	2	2	no	—
390	#102	—	4	2	no	—
391	#102	—	4	2	no	—
392	#97	—	2	2	no	—
393	#95	—	2	1	no	—
394	#95	—	2	1	no	—
395	#95	—	2	2	no	—
396	C₂⁴	—	2	1	no	—
397	C₂⁴	—	2	2	no	—
398	C₂⁴	—	2	1	no	—
399	#41	—	4	32	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	4	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	—	8	2	no	—
425	C₂×D₄	—	8	2	no	—
426	C₂×D₄	—	8	2	no	—
427	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
428	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
429	C₂×D₄	#41×C₄₁	1	2	yes	complemented
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	4	no	—
437	C₂²×C₄	—	4	4	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	8	2	no	—
440	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
445	D₈	—	2	8	no	—
446	D₈	—	4	8	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	#41×C₄₁	1	2	yes	complemented
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	8	2	no	—
464	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	4	2	no	—
468	C₂²⋊C₄	—	4	4	no	—
469	C₂²⋊C₄	—	8	2	no	—
470	C₄²	—	2	2	no	—
471	C₄²	—	2	2	no	—
472	C₄²	C₂×D₄×C₄₁	1	2	yes	—
473	C₄²	C₂×D₄×C₄₁	1	2	yes	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	1	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	4	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
492	Q₈	—	2	8	no	—
493	D₄	—	2	2	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	2	no	—
496	D₄	—	2	2	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	4	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	16	no	—
501	D₄	—	4	2	no	—
502	D₄	—	4	1	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	1	no	—
506	D₄	—	4	1	no	—
507	D₄	—	8	4	no	—
508	D₄	—	8	8	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	16	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	8	4	no	—
526	C₂×C₄	#120×C₄₁	1	2	yes	—
527	C₂×C₄	#120×C₄₁	1	2	yes	—
528	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
529	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
531	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
532	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
533	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
534	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
535	C₈	—	2	2	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	2	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	8	2	no	—
547	C₂²	—	8	2	no	—
548	C₂²	—	8	2	no	—
549	C₂²	—	8	2	no	—
550	C₂²	—	8	2	no	—
551	C₂²	—	8	2	no	—
552	C₂²	—	8	2	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
556	C₂²	C₄₁×#548	1	1	yes	characteristic
557	C₂²	C₄₁×#550	1	1	yes	characteristic
558	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
559	C₂²	C₄₁×#576	1	2	yes	—
560	C₄	—	2	2	no	—
561	C₄	—	2	4	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	2	no	—
565	C₄	—	8	2	no	—
566	C₄	—	8	1	no	—
567	C₄	C₄₁×#525	1	1	yes	characteristic
568	C₄	C₄₁×#537	1	1	yes	characteristic
569	C₄	C₄₁×#546	1	1	yes	characteristic
570	C₄	C₄₁×#547	1	1	yes	characteristic
571	C₂	—	8	2	no	—
572	C₂	—	8	2	no	—
573	C₂	—	8	4	no	—
574	C₂	C₄₁×#2407	1	1	yes	characteristic
575	C₂	C₄₁×#3003	1	1	yes	characteristic
576	C₂	C₄₁×#3008	1	1	yes	characteristic
577	C₂	C₄₁×#3366	1	2	yes	—
578	C₂	C₄₁×#3399	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×Q₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#550	C₂	semidirect
2. C₂×C₄₁×#459	C₂	semidirect
3. C₂×C₄₁×#442	C₂	semidirect
4. C₄₁×#2580	C₂	semidirect
5. C₄₁×#2041	C₂	semidirect
6. C₄₁×#2011	C₂	semidirect
7. C₄₁×#1985	C₂	semidirect
8. C₄₁×#1961	C₂	semidirect
9. C₄₁×#1879	C₂	semidirect
10. C₄₁×#1864	C₂	semidirect
11. C₄₁×#1804	C₂	semidirect
139. C₂×D₄×C₄₁	#41	semidirect
165. C₂×C₈×C₄₁	#41	semidirect
248. #24196	C₄₁	direct (N × H)
282. C₂×#550	C₂×C₄₁	semidirect
283. C₂×#459	C₂×C₄₁	semidirect
284. C₂×#442	C₂×C₄₁	semidirect
285. #2580	C₂×C₄₁	semidirect
286. #2041	C₂×C₄₁	semidirect
287. #2011	C₂×C₄₁	semidirect
288. #1985	C₂×C₄₁	semidirect
289. #1961	C₂×C₄₁	semidirect
290. #1879	C₂×C₄₁	semidirect
291. #1864	C₂×C₄₁	semidirect
292. #1804	C₂×C₄₁	semidirect
334. C₄₁	#24196	direct (N × H)
429. C₂×D₄	#41×C₄₁	semidirect
455. C₂×C₈	#41×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×#550	2
C₂×C₄₁×#459	2
C₂×C₄₁×#442	2
C₄₁×#2580	2
C₄₁×#2041	2
C₄₁×#2011	2
C₄₁×#1985	2
C₄₁×#1961	2
C₄₁×#1879	2
C₄₁×#1864	2
C₄₁×#1804	2
#24196	41

Subgroups — C41×#24196 #23800397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24196 #23800397