Subgroups — C₄₁×#23998 #23800199

Order 10496.

1266 conjugacy classes of subgroups.

The subgroup lattice has 586 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, complemented
2	C₄₁×#3019	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#453	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2883	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2031	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2008	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1906	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1904	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1887	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1829	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#575	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#559	—	2	2	no	—
16	C₄₁×#550	—	2	2	no	—
17	C₄₁×#546	—	2	4	no	—
18	C₄₁×#540	—	2	2	no	—
19	C₄₁×#535	C₂²	1	1	yes	characteristic
20	C₄₁×#535	C₂²	1	1	yes	characteristic, complemented
21	C₄₁×#533	C₂²	1	1	yes	characteristic, complemented
22	C₄₁×#532	—	2	2	no	—
23	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#518	C₂²	1	1	yes	characteristic
25	C₄₁×#518	C₂²	1	1	yes	characteristic
26	C₄₁×#518	C₂²	1	1	yes	characteristic, complemented
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic, complemented
28	C₄₁×#455	—	2	2	no	—
29	C₄₁×#453	—	2	2	no	—
30	C₄₁×#453	—	2	2	no	—
31	C₄₁×#453	—	2	2	no	—
32	C₄₁×#421	C₂²	1	2	yes	—
33	C₄₁×#421	C₂²	1	2	yes	complemented
34	C₂×#104×C₄₁	C₂²	1	2	yes	—
35	C₄₁×#418	C₂²	1	2	yes	complemented
36	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
37	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#410	C₂²	1	2	yes	—
39	C₄₁×#410	C₂²	1	2	yes	—
40	C₂×#99×C₄₁	—	2	4	no	—
41	C₄₁×#344	C₂²	1	2	yes	complemented
42	C₄₁×#344	C₂²	1	2	yes	complemented
43	C₄₁×#339	C₂²	1	2	yes	complemented
44	C₄₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
45	H(5,𝔽₂)×C₄₁	—	2	2	no	—
46	H(5,𝔽₂)×C₄₁	—	2	2	no	—
47	H(5,𝔽₂)×C₄₁	—	4	2	no	—
48	C₂×#41×C₄₁	—	2	4	no	—
49	C₂×#41×C₄₁	—	2	2	no	—
50	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×#41×C₄₁	C₂³	1	2	yes	—
52	C₂²×D₄×C₄₁	—	2	2	no	—
53	C₂²×D₄×C₄₁	—	2	2	no	—
54	C₂²×D₄×C₄₁	—	2	2	no	—
55	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
57	#137×C₄₁	—	2	2	no	—
58	#137×C₄₁	—	2	2	no	—
59	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
60	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
61	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
62	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
63	#135×C₄₁	—	4	2	no	—
64	C₂×SD₁₆×C₄₁	—	2	2	no	—
65	C₂×D₈×C₄₁	—	2	2	no	—
66	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₄₁	—	2	1	no	—
69	#127×C₄₁	—	2	2	no	—
70	#127×C₄₁	—	2	1	no	—
71	#127×C₄₁	D₄	1	2	yes	—
72	#127×C₄₁	D₄	1	2	yes	—
73	#127×C₄₁	D₄	1	2	yes	—
74	#125×C₄₁	—	2	1	no	—
75	#124×C₄₁	—	2	4	no	—
76	#124×C₄₁	—	2	1	no	—
77	#124×C₄₁	D₄	1	2	yes	—
78	#123×C₄₁	—	2	4	no	—
79	#123×C₄₁	—	2	4	no	—
80	C₂²⋊Q₈×C₄₁	—	2	2	no	—
81	#121×C₄₁	—	2	4	no	—
82	#121×C₄₁	—	2	4	no	—
83	#121×C₄₁	—	2	4	no	—
84	#121×C₄₁	—	2	2	no	—
85	#121×C₄₁	—	2	2	no	—
86	#120×C₄₁	—	2	2	no	—
87	#120×C₄₁	—	2	2	no	—
88	C₄×Q₈×C₄₁	—	2	1	no	—
89	C₄×Q₈×C₄₁	—	2	2	no	—
90	C₄×D₄×C₄₁	—	2	4	no	—
91	C₄×D₄×C₄₁	—	2	1	no	—
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	—	2	2	no	—
94	C₄×D₄×C₄₁	—	2	2	no	—
95	C₄×D₄×C₄₁	—	2	2	no	—
96	C₄×D₄×C₄₁	—	2	2	no	—
97	C₄×D₄×C₄₁	—	4	2	no	—
98	#117×C₄₁	—	2	1	no	—
99	#117×C₄₁	C₂³	1	2	yes	—
100	#117×C₄₁	C₂³	1	1	yes	characteristic
101	#117×C₄₁	C₂³	1	1	yes	characteristic
102	#117×C₄₁	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₄₁	D₄	1	4	yes	—
106	C₂×C₄²×C₄₁	—	2	1	no	—
107	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
108	#104×C₄₁	—	2	2	no	—
109	#104×C₄₁	—	2	2	no	—
110	#104×C₄₁	—	2	4	no	—
111	#104×C₄₁	—	4	2	no	—
112	#102×C₄₁	—	4	2	no	—
113	#100×C₄₁	—	2	2	no	—
114	#100×C₄₁	—	2	2	no	—
115	#99×C₄₁	—	2	4	no	—
116	#99×C₄₁	—	2	4	no	—
117	C₂⁴×C₄₁	—	2	2	no	—
118	C₂⁴×C₄₁	—	2	1	no	—
119	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
120	#41×C₄₁	—	2	2	no	—
121	#41×C₄₁	—	2	2	no	—
122	#41×C₄₁	—	2	2	no	—
123	#41×C₄₁	—	2	2	no	—
124	#41×C₄₁	—	2	2	no	—
125	#41×C₄₁	—	2	2	no	—
126	#41×C₄₁	—	4	4	no	—
127	#41×C₄₁	—	4	4	no	—
128	#41×C₄₁	—	8	8	no	—
129	C₂×Q₈×C₄₁	—	2	2	no	—
130	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₄₁	#41	1	2	yes	complemented
132	C₂×D₄×C₄₁	—	2	4	no	—
133	C₂×D₄×C₄₁	—	2	4	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	1	no	—
136	C₂×D₄×C₄₁	—	2	4	no	—
137	C₂×D₄×C₄₁	—	2	4	no	—
138	C₂×D₄×C₄₁	—	2	4	no	—
139	C₂×D₄×C₄₁	—	2	1	no	—
140	C₂×D₄×C₄₁	—	2	2	no	—
141	C₂×D₄×C₄₁	—	2	2	no	—
142	C₂×D₄×C₄₁	—	2	2	no	—
143	C₂×D₄×C₄₁	—	2	2	no	—
144	C₂×D₄×C₄₁	—	4	4	no	—
145	C₂×D₄×C₄₁	—	4	4	no	—
146	C₂×D₄×C₄₁	—	4	4	no	—
147	C₂×D₄×C₄₁	—	4	4	no	—
148	C₂×D₄×C₄₁	—	8	4	no	—
149	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
151	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
153	C₂×D₄×C₄₁	#41	1	2	yes	complemented
154	C₂²×C₄×C₄₁	—	2	2	no	—
155	C₂²×C₄×C₄₁	—	2	2	no	—
156	C₂²×C₄×C₄₁	—	2	1	no	—
157	C₂²×C₄×C₄₁	—	2	4	no	—
158	C₂²×C₄×C₄₁	—	2	4	no	—
159	C₂²×C₄×C₄₁	—	2	1	no	—
160	C₂²×C₄×C₄₁	—	2	2	no	—
161	C₂²×C₄×C₄₁	—	4	2	no	—
162	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
163	Q₁₆×C₄₁	—	4	2	no	—
164	SD₁₆×C₄₁	—	4	4	no	—
165	SD₁₆×C₄₁	—	4	2	no	—
166	SD₁₆×C₄₁	—	4	2	no	—
167	D₈×C₄₁	—	4	4	no	—
168	D₈×C₄₁	—	4	2	no	—
169	M₁₆×C₄₁	—	2	2	no	—
170	M₁₆×C₄₁	—	2	2	no	—
171	M₁₆×C₄₁	—	2	2	no	—
172	M₁₆×C₄₁	—	2	2	no	—
173	C₂×C₈×C₄₁	—	2	2	no	—
174	C₂×C₈×C₄₁	—	2	2	no	—
175	C₄⋊C₄×C₄₁	—	2	2	no	—
176	C₄⋊C₄×C₄₁	—	2	2	no	—
177	C₄⋊C₄×C₄₁	—	2	2	no	—
178	C₄⋊C₄×C₄₁	—	2	2	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	1	no	—
182	C₄⋊C₄×C₄₁	—	2	2	no	—
183	C₂²⋊C₄×C₄₁	—	2	4	no	—
184	C₂²⋊C₄×C₄₁	—	2	2	no	—
185	C₂²⋊C₄×C₄₁	—	2	4	no	—
186	C₂²⋊C₄×C₄₁	—	2	4	no	—
187	C₂²⋊C₄×C₄₁	—	2	4	no	—
188	C₂²⋊C₄×C₄₁	—	2	1	no	—
189	C₂²⋊C₄×C₄₁	—	2	2	no	—
190	C₂²⋊C₄×C₄₁	—	4	2	no	—
191	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
192	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
193	C₄²×C₄₁	—	2	2	no	—
194	C₄²×C₄₁	—	2	1	no	—
195	C₄²×C₄₁	—	2	1	no	—
196	C₄²×C₄₁	—	2	1	no	—
197	C₄²×C₄₁	—	2	2	no	—
198	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₄₁	C₂×D₄	1	2	yes	—
201	C₂³×C₄₁	—	2	2	no	—
202	C₂³×C₄₁	—	2	2	no	—
203	C₂³×C₄₁	—	2	2	no	—
204	C₂³×C₄₁	—	2	1	no	—
205	C₂³×C₄₁	—	2	2	no	—
206	C₂³×C₄₁	—	4	4	no	—
207	C₂³×C₄₁	—	4	1	no	—
208	C₂³×C₄₁	—	4	1	no	—
209	C₂³×C₄₁	—	8	2	no	—
210	C₂³×C₄₁	—	8	2	no	—
211	C₂³×C₄₁	—	16	2	no	—
212	C₂³×C₄₁	#120	1	4	yes	—
213	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
214	Q₈×C₄₁	—	2	2	no	—
215	Q₈×C₄₁	—	2	2	no	—
216	Q₈×C₄₁	—	2	2	no	—
217	Q₈×C₄₁	—	4	4	no	—
218	D₄×C₄₁	—	2	4	no	—
219	D₄×C₄₁	—	2	4	no	—
220	D₄×C₄₁	—	2	2	no	—
221	D₄×C₄₁	—	2	4	no	—
222	D₄×C₄₁	—	2	2	no	—
223	D₄×C₄₁	—	2	2	no	—
224	D₄×C₄₁	—	4	8	no	—
225	D₄×C₄₁	—	4	8	no	—
226	D₄×C₄₁	—	4	2	no	—
227	D₄×C₄₁	—	4	1	no	—
228	D₄×C₄₁	—	4	2	no	—
229	D₄×C₄₁	—	4	1	no	—
230	D₄×C₄₁	—	4	2	no	—
231	D₄×C₄₁	—	8	4	no	—
232	D₄×C₄₁	—	8	4	no	—
233	D₄×C₄₁	—	8	4	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	2	2	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	2	2	no	—
240	C₂×C₄×C₄₁	—	2	2	no	—
241	C₂×C₄×C₄₁	—	4	2	no	—
242	C₂×C₄×C₄₁	—	4	4	no	—
243	C₂×C₄×C₄₁	—	4	4	no	—
244	C₂×C₄×C₄₁	—	4	2	no	—
245	C₂×C₄×C₄₁	—	4	2	no	—
246	C₂×C₄×C₄₁	—	4	2	no	—
247	C₂×C₄×C₄₁	—	4	2	no	—
248	C₂×C₄×C₄₁	—	8	4	no	—
249	C₂×C₄×C₄₁	—	8	4	no	—
250	C₂×C₄×C₄₁	—	16	2	no	—
251	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
254	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
255	C₈×C₄₁	—	4	2	no	—
256	C₈×C₄₁	—	4	2	no	—
257	#23998	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₄₁	—	2	4	no	—
259	C₂²×C₄₁	—	4	1	no	—
260	C₂²×C₄₁	—	4	4	no	—
261	C₂²×C₄₁	—	4	2	no	—
262	C₂²×C₄₁	—	4	4	no	—
263	C₂²×C₄₁	—	4	2	no	—
264	C₂²×C₄₁	—	4	1	no	—
265	C₂²×C₄₁	—	4	2	no	—
266	C₂²×C₄₁	—	8	2	no	—
267	C₂²×C₄₁	—	8	2	no	—
268	C₂²×C₄₁	—	8	2	no	—
269	C₂²×C₄₁	—	16	1	no	—
270	C₂²×C₄₁	—	16	1	no	—
271	C₂²×C₄₁	—	16	4	no	—
272	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
273	C₂²×C₄₁	#525	1	1	yes	characteristic
274	C₂²×C₄₁	#535	1	1	yes	characteristic
275	C₄×C₄₁	—	2	2	no	—
276	C₄×C₄₁	—	4	2	no	—
277	C₄×C₄₁	—	4	1	no	—
278	C₄×C₄₁	—	4	2	no	—
279	C₄×C₄₁	—	4	2	no	—
280	C₄×C₄₁	—	4	2	no	—
281	C₄×C₄₁	—	4	1	no	—
282	C₄×C₄₁	—	8	2	no	—
283	C₄×C₄₁	#546	1	2	yes	—
284	C₄×C₄₁	#546	1	2	yes	—
285	#3508	C₂×C₄₁	1	1	yes	characteristic, complemented
286	#3019	C₂×C₄₁	1	1	yes	characteristic, complemented
287	C₂×#453	C₂×C₄₁	1	1	yes	characteristic, complemented
288	#2883	C₂×C₄₁	1	1	yes	characteristic, complemented
289	#2031	C₂×C₄₁	1	2	yes	complemented
290	#2008	C₂×C₄₁	1	2	yes	complemented
291	#1911	C₂×C₄₁	1	1	yes	characteristic, complemented
292	#1906	C₂×C₄₁	1	2	yes	complemented
293	#1904	C₂×C₄₁	1	2	yes	complemented
294	#1887	C₂×C₄₁	1	1	yes	characteristic, complemented
295	#1829	C₂×C₄₁	1	1	yes	characteristic, complemented
296	C₂×C₄₁	—	2	1	no	—
297	C₂×C₄₁	—	2	1	no	—
298	C₂×C₄₁	—	2	1	no	—
299	C₂×C₄₁	—	4	4	no	—
300	C₂×C₄₁	—	8	1	no	—
301	C₂×C₄₁	—	8	2	no	—
302	C₂×C₄₁	—	8	2	no	—
303	C₂×C₄₁	#2404	1	1	yes	characteristic
304	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	1	yes	characteristic
305	#575	C₂²×C₄₁	1	2	yes	—
306	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
307	#559	—	2	2	no	—
308	#550	—	2	2	no	—
309	#546	—	2	4	no	—
310	#540	—	2	2	no	—
311	#535	C₂²×C₄₁	1	1	yes	characteristic
312	#535	C₂²×C₄₁	1	1	yes	characteristic, complemented
313	#533	C₂²×C₄₁	1	1	yes	characteristic, complemented
314	#532	—	2	2	no	—
315	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
316	#518	C₂²×C₄₁	1	1	yes	characteristic
317	#518	C₂²×C₄₁	1	1	yes	characteristic
318	#518	C₂²×C₄₁	1	1	yes	characteristic, complemented
319	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic, complemented
320	#455	—	2	2	no	—
321	#453	—	2	2	no	—
322	#453	—	2	2	no	—
323	#453	—	2	2	no	—
324	#421	C₂²×C₄₁	1	2	yes	—
325	#421	C₂²×C₄₁	1	2	yes	complemented
326	C₂×#104	C₂²×C₄₁	1	2	yes	—
327	#418	C₂²×C₄₁	1	2	yes	complemented
328	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
329	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
330	#410	C₂²×C₄₁	1	2	yes	—
331	#410	C₂²×C₄₁	1	2	yes	—
332	C₂×#99	—	2	4	no	—
333	#344	C₂²×C₄₁	1	2	yes	complemented
334	#344	C₂²×C₄₁	1	2	yes	complemented
335	#339	C₂²×C₄₁	1	2	yes	complemented
336	H(3,ℤ/4)	C₂²×C₄₁	1	2	yes	complemented
337	C₄₁	#23998	1	1	yes	characteristic, direct product factor, Sylow
338	H(5,𝔽₂)	—	2	2	no	—
339	H(5,𝔽₂)	—	2	2	no	—
340	H(5,𝔽₂)	—	4	2	no	—
341	C₂×#41	—	2	4	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
344	C₂×#41	C₂³×C₄₁	1	2	yes	—
345	C₂²×D₄	—	2	2	no	—
346	C₂²×D₄	—	2	2	no	—
347	C₂²×D₄	—	2	2	no	—
348	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
349	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
350	#137	—	2	2	no	—
351	#137	—	2	2	no	—
352	AGL(1,ℤ/8)	—	2	4	no	—
353	AGL(1,ℤ/8)	—	2	2	no	—
354	AGL(1,ℤ/8)	—	2	2	no	—
355	AGL(1,ℤ/8)	—	4	2	no	—
356	#135	—	4	2	no	—
357	C₂×SD₁₆	—	2	2	no	—
358	C₂×D₈	—	2	2	no	—
359	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
360	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
361	C₄⋊Q₈	—	2	1	no	—
362	#127	—	2	2	no	—
363	#127	—	2	1	no	—
364	#127	D₄×C₄₁	1	2	yes	—
365	#127	D₄×C₄₁	1	2	yes	—
366	#127	D₄×C₄₁	1	2	yes	—
367	#125	—	2	1	no	—
368	#124	—	2	4	no	—
369	#124	—	2	1	no	—
370	#124	D₄×C₄₁	1	2	yes	—
371	#123	—	2	4	no	—
372	#123	—	2	4	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	#121	—	2	4	no	—
375	#121	—	2	4	no	—
376	#121	—	2	4	no	—
377	#121	—	2	2	no	—
378	#121	—	2	2	no	—
379	#120	—	2	2	no	—
380	#120	—	2	2	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×D₄	—	2	4	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	4	2	no	—
391	#117	—	2	1	no	—
392	#117	C₂³×C₄₁	1	2	yes	—
393	#117	C₂³×C₄₁	1	1	yes	characteristic
394	#117	C₂³×C₄₁	1	1	yes	characteristic
395	#117	C₂³×C₄₁	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	D₄×C₄₁	1	4	yes	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
401	#104	—	2	2	no	—
402	#104	—	2	2	no	—
403	#104	—	2	4	no	—
404	#104	—	4	2	no	—
405	#102	—	4	2	no	—
406	#100	—	2	2	no	—
407	#100	—	2	2	no	—
408	#99	—	2	4	no	—
409	#99	—	2	4	no	—
410	C₂⁴	—	2	2	no	—
411	C₂⁴	—	2	1	no	—
412	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
413	#41	—	2	2	no	—
414	#41	—	2	2	no	—
415	#41	—	2	2	no	—
416	#41	—	2	2	no	—
417	#41	—	2	2	no	—
418	#41	—	2	2	no	—
419	#41	—	4	4	no	—
420	#41	—	4	4	no	—
421	#41	—	8	8	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
424	C₂×Q₈	#41×C₄₁	1	2	yes	complemented
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	4	4	no	—
438	C₂×D₄	—	4	4	no	—
439	C₂×D₄	—	4	4	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	8	4	no	—
442	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
443	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
444	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
445	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
446	C₂×D₄	#41×C₄₁	1	2	yes	complemented
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	4	2	no	—
455	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
456	Q₁₆	—	4	2	no	—
457	SD₁₆	—	4	4	no	—
458	SD₁₆	—	4	2	no	—
459	SD₁₆	—	4	2	no	—
460	D₈	—	4	4	no	—
461	D₈	—	4	2	no	—
462	M₁₆	—	2	2	no	—
463	M₁₆	—	2	2	no	—
464	M₁₆	—	2	2	no	—
465	M₁₆	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	2	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	4	2	no	—
484	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
485	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
486	C₄²	—	2	2	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	2	no	—
491	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
492	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
493	C₄²	C₂×D₄×C₄₁	1	2	yes	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	4	4	no	—
500	C₂³	—	4	1	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	8	2	no	—
503	C₂³	—	8	2	no	—
504	C₂³	—	16	2	no	—
505	C₂³	#120×C₄₁	1	4	yes	—
506	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
507	Q₈	—	2	2	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	2	2	no	—
510	Q₈	—	4	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	2	no	—
517	D₄	—	4	8	no	—
518	D₄	—	4	8	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	2	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	2	no	—
524	D₄	—	8	4	no	—
525	D₄	—	8	4	no	—
526	D₄	—	8	4	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	4	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	8	4	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	—	16	2	no	—
544	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
545	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
546	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
547	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
548	C₈	—	4	2	no	—
549	C₈	—	4	2	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	2	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	8	2	no	—
559	C₂²	—	8	2	no	—
560	C₂²	—	8	2	no	—
561	C₂²	—	16	1	no	—
562	C₂²	—	16	1	no	—
563	C₂²	—	16	4	no	—
564	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
565	C₂²	C₄₁×#525	1	1	yes	characteristic
566	C₂²	C₄₁×#535	1	1	yes	characteristic
567	C₄	—	2	2	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	2	no	—
571	C₄	—	4	2	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	1	no	—
574	C₄	—	8	2	no	—
575	C₄	C₄₁×#546	1	2	yes	—
576	C₄	C₄₁×#546	1	2	yes	—
577	C₂	—	2	1	no	—
578	C₂	—	2	1	no	—
579	C₂	—	2	1	no	—
580	C₂	—	4	4	no	—
581	C₂	—	8	2	no	—
582	C₂	—	8	1	no	—
583	C₂	—	8	2	no	—
584	C₂	C₄₁×#2404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#2404
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3508	C₂	semidirect
2. C₄₁×#3019	C₂	semidirect
3. C₂×C₄₁×#453	C₂	semidirect
4. C₄₁×#2883	C₂	semidirect
5. C₄₁×#2031	C₂	semidirect
6. C₄₁×#2008	C₂	semidirect
7. C₄₁×#1911	C₂	semidirect
8. C₄₁×#1906	C₂	semidirect
9. C₄₁×#1904	C₂	semidirect
10. C₄₁×#1887	C₂	semidirect
11. C₄₁×#1829	C₂	semidirect
20. C₄₁×#535	C₂²	semidirect
21. C₄₁×#533	C₂²	semidirect
26. C₄₁×#518	C₂²	semidirect
27. C₄×#41×C₄₁	C₂²	semidirect
33. C₄₁×#421	C₂²	semidirect
35. C₄₁×#418	C₂²	semidirect
41. C₄₁×#344	C₂²	semidirect
42. C₄₁×#344	C₂²	semidirect
43. C₄₁×#339	C₂²	semidirect
44. C₄₁×H(3,ℤ/4)	C₂²	semidirect
131. C₂×Q₈×C₄₁	#41	semidirect
153. C₂×D₄×C₄₁	#41	semidirect
257. #23998	C₄₁	direct (N × H)
285. #3508	C₂×C₄₁	semidirect
286. #3019	C₂×C₄₁	semidirect
287. C₂×#453	C₂×C₄₁	semidirect
288. #2883	C₂×C₄₁	semidirect
289. #2031	C₂×C₄₁	semidirect
290. #2008	C₂×C₄₁	semidirect
291. #1911	C₂×C₄₁	semidirect
292. #1906	C₂×C₄₁	semidirect
293. #1904	C₂×C₄₁	semidirect
294. #1887	C₂×C₄₁	semidirect
295. #1829	C₂×C₄₁	semidirect
312. #535	C₂²×C₄₁	semidirect
313. #533	C₂²×C₄₁	semidirect
318. #518	C₂²×C₄₁	semidirect
319. C₄×#41	C₂²×C₄₁	semidirect
325. #421	C₂²×C₄₁	semidirect
327. #418	C₂²×C₄₁	semidirect
333. #344	C₂²×C₄₁	semidirect
334. #344	C₂²×C₄₁	semidirect
335. #339	C₂²×C₄₁	semidirect
336. H(3,ℤ/4)	C₂²×C₄₁	semidirect
337. C₄₁	#23998	direct (N × H)
424. C₂×Q₈	#41×C₄₁	semidirect
446. C₂×D₄	#41×C₄₁	semidirect

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×#3019	2
C₂×C₄₁×#453	2
C₄₁×#2883	2
C₄₁×#2031	2
C₄₁×#2008	2
C₄₁×#1911	2
C₄₁×#1906	2
C₄₁×#1904	2
C₄₁×#1887	2
C₄₁×#1829	2
#23998	41

Subgroups — C41×#23998 #23800199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23998 #23800199