Subgroups — C₄₁×#22899 #23799100

Order 10496.

1016 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3018	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2956	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2901	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1932	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1906	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1884	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1861	C₂	1	2	yes	maximal, complemented
9	C₄×#104×C₄₁	C₂	1	2	yes	maximal, complemented
10	C₄₁×#585	C₂²	1	1	yes	characteristic
11	C₄₁×#575	C₂²	1	2	yes	—
12	C₄₁×#568	C₂²	1	2	yes	—
13	C₄₁×#568	C₂²	1	1	yes	characteristic, complemented
14	C₄₁×#532	C₂²	1	1	yes	characteristic
15	C₄×#41×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#456	C₄	1	2	yes	complemented
17	C₄₁×#455	C₄	1	2	yes	complemented
18	C₄₁×#454	C₄	1	2	yes	complemented
19	C₄₁×#453	C₄	1	2	yes	complemented
20	C₄₁×#444	—	2	4	no	—
21	C₄₁×#443	—	2	4	no	—
22	C₄₁×#430	C₂²	1	2	yes	complemented
23	C₄₁×#423	C₂²	1	1	yes	characteristic, complemented
24	C₄₁×#421	—	2	4	no	—
25	C₄₁×#421	C₂²	1	2	yes	—
26	C₄₁×#421	C₂²	1	2	yes	—
27	C₂×#104×C₄₁	—	2	4	no	—
28	C₂×#104×C₄₁	C₂²	1	2	yes	—
29	C₂×#104×C₄₁	C₂²	1	2	yes	—
30	C₂×#104×C₄₁	C₂²	1	2	yes	—
31	C₂×#104×C₄₁	C₂²	1	2	yes	complemented
32	C₄₁×#413	C₂²	1	1	yes	characteristic
33	C₄₁×#410	C₂²	1	1	yes	characteristic, complemented
34	C₄×M₁₆×C₄₁	C₂²	1	2	yes	complemented
35	C₄³×C₄₁	C₂²	1	1	yes	characteristic, complemented
36	C₄₁×#344	C₂²	1	2	yes	complemented
37	C₄₁×#339	C₂²	1	2	yes	complemented
38	C₄₁×#339	C₂²	1	2	yes	complemented
39	2₋¹⁺⁴×C₄₁	—	2	2	no	—
40	2₋¹⁺⁴×C₄₁	C₂×C₄	1	2	yes	—
41	H(5,𝔽₂)×C₄₁	—	2	2	no	—
42	H(5,𝔽₂)×C₄₁	—	4	1	no	—
43	H(5,𝔽₂)×C₄₁	C₂×C₄	1	2	yes	—
44	C₂×#41×C₄₁	—	2	4	no	—
45	C₂×#41×C₄₁	—	4	1	no	—
46	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
47	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₄₁	C₂³	1	2	yes	—
49	#137×C₄₁	—	2	2	no	—
50	#137×C₄₁	C₂×C₄	1	4	yes	complemented
51	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
52	AGL(1,ℤ/8)×C₄₁	C₂×C₄	1	4	yes	complemented
53	#135×C₄₁	—	2	4	no	—
54	#131×C₄₁	—	2	4	no	—
55	#131×C₄₁	—	2	4	no	—
56	#131×C₄₁	—	4	2	no	—
57	C₂×M₁₆×C₄₁	—	2	4	no	—
58	C₂×M₁₆×C₄₁	—	4	1	no	—
59	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
63	#127×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
64	#124×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
65	#124×C₄₁	C₂×C₄	1	1	yes	characteristic, complemented
66	#121×C₄₁	—	4	1	no	—
67	C₄×Q₈×C₄₁	—	2	2	no	—
68	C₄×D₄×C₄₁	—	2	2	no	—
69	C₄×D₄×C₄₁	—	4	1	no	—
70	C₄×D₄×C₄₁	—	4	2	no	—
71	#117×C₄₁	—	2	2	no	—
72	#117×C₄₁	C₂³	1	2	yes	—
73	C₂×C₄²×C₄₁	—	2	2	no	—
74	C₂×C₄²×C₄₁	—	4	1	no	—
75	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
77	#108×C₄₁	—	2	4	no	—
78	C₄⋊C₈×C₄₁	—	4	1	no	—
79	#104×C₄₁	—	2	4	no	—
80	#104×C₄₁	—	2	4	no	—
81	#104×C₄₁	—	2	2	no	—
82	#104×C₄₁	—	2	2	no	—
83	#104×C₄₁	—	2	2	no	—
84	#104×C₄₁	—	2	2	no	—
85	#104×C₄₁	C₂×C₄	1	4	yes	complemented
86	#104×C₄₁	C₂×C₄	1	4	yes	complemented
87	#104×C₄₁	D₄	1	4	yes	complemented
88	#104×C₄₁	D₄	1	4	yes	complemented
89	#101×C₄₁	C₂×C₄	1	2	yes	complemented
90	#100×C₄₁	C₂×C₄	1	2	yes	complemented
91	#99×C₄₁	—	4	1	no	—
92	#97×C₄₁	—	2	2	no	—
93	C₄×C₈×C₄₁	—	2	2	no	—
94	#41×C₄₁	—	2	2	no	—
95	#41×C₄₁	—	2	2	no	—
96	#41×C₄₁	—	2	4	no	—
97	#41×C₄₁	—	2	4	no	—
98	#41×C₄₁	—	2	2	no	—
99	#41×C₄₁	—	2	2	no	—
100	#41×C₄₁	—	2	2	no	—
101	#41×C₄₁	—	2	2	no	—
102	#41×C₄₁	—	4	2	no	—
103	#41×C₄₁	—	4	2	no	—
104	#41×C₄₁	—	4	4	no	—
105	#41×C₄₁	—	4	2	no	—
106	#41×C₄₁	C₂²×C₄	1	4	yes	—
107	#41×C₄₁	C₂×D₄	1	4	yes	—
108	C₂×Q₈×C₄₁	—	2	4	no	—
109	C₂×Q₈×C₄₁	—	4	1	no	—
110	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×Q₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×Q₈×C₄₁	#41	1	2	yes	—
113	C₂×D₄×C₄₁	—	2	2	no	—
114	C₂×D₄×C₄₁	—	2	4	no	—
115	C₂×D₄×C₄₁	—	2	1	no	—
116	C₂×D₄×C₄₁	—	2	1	no	—
117	C₂×D₄×C₄₁	—	4	1	no	—
118	C₂×D₄×C₄₁	—	4	2	no	—
119	C₂×D₄×C₄₁	—	4	1	no	—
120	C₂×D₄×C₄₁	—	4	4	no	—
121	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
123	C₂×D₄×C₄₁	#41	1	2	yes	—
124	C₂²×C₄×C₄₁	—	2	2	no	—
125	C₂²×C₄×C₄₁	—	2	2	no	—
126	C₂²×C₄×C₄₁	—	2	1	no	—
127	C₂²×C₄×C₄₁	—	2	1	no	—
128	C₂²×C₄×C₄₁	—	2	1	no	—
129	C₂²×C₄×C₄₁	—	4	2	no	—
130	C₂²×C₄×C₄₁	—	4	2	no	—
131	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₄₁	—	2	4	no	—
133	SD₁₆×C₄₁	—	2	4	no	—
134	SD₁₆×C₄₁	—	2	4	no	—
135	D₈×C₄₁	—	2	4	no	—
136	M₁₆×C₄₁	—	2	2	no	—
137	M₁₆×C₄₁	—	2	2	no	—
138	M₁₆×C₄₁	—	2	2	no	—
139	M₁₆×C₄₁	—	2	2	no	—
140	M₁₆×C₄₁	—	4	2	no	—
141	M₁₆×C₄₁	—	4	4	no	—
142	M₁₆×C₄₁	C₂²×C₄	1	4	yes	—
143	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₄₁	—	2	2	no	—
145	C₂×C₈×C₄₁	—	2	2	no	—
146	C₂×C₈×C₄₁	—	2	2	no	—
147	C₂×C₈×C₄₁	—	4	2	no	—
148	C₂×C₈×C₄₁	—	4	4	no	—
149	C₄⋊C₄×C₄₁	—	2	2	no	—
150	C₄⋊C₄×C₄₁	—	4	1	no	—
151	C₄⋊C₄×C₄₁	—	4	2	no	—
152	C₂²⋊C₄×C₄₁	—	2	2	no	—
153	C₂²⋊C₄×C₄₁	—	4	1	no	—
154	C₂²⋊C₄×C₄₁	—	4	1	no	—
155	C₂²⋊C₄×C₄₁	—	4	2	no	—
156	C₄²×C₄₁	—	2	2	no	—
157	C₄²×C₄₁	—	2	1	no	—
158	C₄²×C₄₁	—	2	2	no	—
159	C₄²×C₄₁	—	4	2	no	—
160	C₄²×C₄₁	—	4	4	no	—
161	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₄₁	C₂×D₄	1	2	yes	—
164	C₄²×C₄₁	C₂×D₄	1	2	yes	—
165	C₂³×C₄₁	—	2	2	no	—
166	C₂³×C₄₁	—	2	1	no	—
167	C₂³×C₄₁	—	2	1	no	—
168	C₂³×C₄₁	—	4	2	no	—
169	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
170	Q₈×C₄₁	—	2	2	no	—
171	Q₈×C₄₁	—	2	2	no	—
172	Q₈×C₄₁	—	2	2	no	—
173	Q₈×C₄₁	—	4	2	no	—
174	Q₈×C₄₁	C₄×D₄	1	4	yes	—
175	D₄×C₄₁	—	2	4	no	—
176	D₄×C₄₁	—	2	4	no	—
177	D₄×C₄₁	—	2	2	no	—
178	D₄×C₄₁	—	2	2	no	—
179	D₄×C₄₁	—	2	2	no	—
180	D₄×C₄₁	—	4	4	no	—
181	D₄×C₄₁	—	4	2	no	—
182	D₄×C₄₁	—	4	2	no	—
183	D₄×C₄₁	—	4	2	no	—
184	D₄×C₄₁	—	8	2	no	—
185	D₄×C₄₁	C₄×D₄	1	4	yes	—
186	C₂×C₄×C₄₁	—	2	1	no	—
187	C₂×C₄×C₄₁	—	2	2	no	—
188	C₂×C₄×C₄₁	—	2	2	no	—
189	C₂×C₄×C₄₁	—	2	4	no	—
190	C₂×C₄×C₄₁	—	2	4	no	—
191	C₂×C₄×C₄₁	—	2	1	no	—
192	C₂×C₄×C₄₁	—	2	1	no	—
193	C₂×C₄×C₄₁	—	2	1	no	—
194	C₂×C₄×C₄₁	—	2	2	no	—
195	C₂×C₄×C₄₁	—	4	2	no	—
196	C₂×C₄×C₄₁	—	4	4	no	—
197	C₂×C₄×C₄₁	—	4	4	no	—
198	C₂×C₄×C₄₁	—	4	2	no	—
199	C₂×C₄×C₄₁	—	4	1	no	—
200	C₂×C₄×C₄₁	—	4	1	no	—
201	C₂×C₄×C₄₁	—	4	1	no	—
202	C₂×C₄×C₄₁	—	4	1	no	—
203	C₂×C₄×C₄₁	—	4	2	no	—
204	C₂×C₄×C₄₁	—	8	4	no	—
205	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
208	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
209	C₈×C₄₁	—	2	4	no	—
210	C₈×C₄₁	—	2	4	no	—
211	C₈×C₄₁	—	4	2	no	—
212	#22899	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂²×C₄₁	—	2	4	no	—
214	C₂²×C₄₁	—	4	1	no	—
215	C₂²×C₄₁	—	4	1	no	—
216	C₂²×C₄₁	—	4	1	no	—
217	C₂²×C₄₁	—	8	2	no	—
218	C₂²×C₄₁	—	8	1	no	—
219	C₂²×C₄₁	—	8	1	no	—
220	C₂²×C₄₁	—	16	2	no	—
221	C₂²×C₄₁	#518	1	1	yes	characteristic
222	C₂²×C₄₁	#546	1	2	yes	—
223	C₄×C₄₁	—	2	4	no	—
224	C₄×C₄₁	—	2	2	no	—
225	C₄×C₄₁	—	2	4	no	—
226	C₄×C₄₁	—	4	2	no	—
227	C₄×C₄₁	—	4	1	no	—
228	C₄×C₄₁	—	4	1	no	—
229	C₄×C₄₁	—	4	1	no	—
230	C₄×C₄₁	—	8	2	no	—
231	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
232	C₄×C₄₁	D₄²	1	1	yes	characteristic
233	C₄×C₄₁	#551	1	1	yes	characteristic
234	#3018	C₂×C₄₁	1	1	yes	characteristic
235	#2956	C₂×C₄₁	1	2	yes	complemented
236	#2901	C₂×C₄₁	1	2	yes	complemented
237	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
238	#1932	C₂×C₄₁	1	2	yes	complemented
239	#1906	C₂×C₄₁	1	2	yes	complemented
240	#1884	C₂×C₄₁	1	1	yes	characteristic, complemented
241	#1861	C₂×C₄₁	1	2	yes	complemented
242	C₄×#104	C₂×C₄₁	1	2	yes	complemented
243	C₂×C₄₁	—	2	2	no	—
244	C₂×C₄₁	—	2	1	no	—
245	C₂×C₄₁	—	4	4	no	—
246	C₂×C₄₁	—	8	1	no	—
247	C₂×C₄₁	—	8	1	no	—
248	C₂×C₄₁	#2358	1	1	yes	characteristic
249	#585	C₂²×C₄₁	1	1	yes	characteristic
250	#575	C₂²×C₄₁	1	2	yes	—
251	#568	C₂²×C₄₁	1	2	yes	—
252	#568	C₂²×C₄₁	1	1	yes	characteristic, complemented
253	#532	C₂²×C₄₁	1	1	yes	characteristic
254	C₄×#41	C₂²×C₄₁	1	2	yes	—
255	#456	C₄×C₄₁	1	2	yes	complemented
256	#455	C₄×C₄₁	1	2	yes	complemented
257	#454	C₄×C₄₁	1	2	yes	complemented
258	#453	C₄×C₄₁	1	2	yes	complemented
259	#444	—	2	4	no	—
260	#443	—	2	4	no	—
261	#430	C₂²×C₄₁	1	2	yes	complemented
262	#423	C₂²×C₄₁	1	1	yes	characteristic, complemented
263	#421	—	2	4	no	—
264	#421	C₂²×C₄₁	1	2	yes	—
265	#421	C₂²×C₄₁	1	2	yes	—
266	C₂×#104	—	2	4	no	—
267	C₂×#104	C₂²×C₄₁	1	2	yes	—
268	C₂×#104	C₂²×C₄₁	1	2	yes	—
269	C₂×#104	C₂²×C₄₁	1	2	yes	—
270	C₂×#104	C₂²×C₄₁	1	2	yes	complemented
271	#413	C₂²×C₄₁	1	1	yes	characteristic
272	#410	C₂²×C₄₁	1	1	yes	characteristic, complemented
273	C₄×M₁₆	C₂²×C₄₁	1	2	yes	complemented
274	C₄³	C₂²×C₄₁	1	1	yes	characteristic, complemented
275	#344	C₂²×C₄₁	1	2	yes	complemented
276	#339	C₂²×C₄₁	1	2	yes	complemented
277	#339	C₂²×C₄₁	1	2	yes	complemented
278	C₄₁	#22899	1	1	yes	characteristic, direct product factor, Sylow
279	2₋¹⁺⁴	—	2	2	no	—
280	2₋¹⁺⁴	C₂×C₄×C₄₁	1	2	yes	—
281	H(5,𝔽₂)	—	2	2	no	—
282	H(5,𝔽₂)	—	4	1	no	—
283	H(5,𝔽₂)	C₂×C₄×C₄₁	1	2	yes	—
284	C₂×#41	—	2	4	no	—
285	C₂×#41	—	4	1	no	—
286	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
287	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
288	C₂×#41	C₂³×C₄₁	1	2	yes	—
289	#137	—	2	2	no	—
290	#137	C₂×C₄×C₄₁	1	4	yes	complemented
291	AGL(1,ℤ/8)	—	2	2	no	—
292	AGL(1,ℤ/8)	C₂×C₄×C₄₁	1	4	yes	complemented
293	#135	—	2	4	no	—
294	#131	—	2	4	no	—
295	#131	—	2	4	no	—
296	#131	—	4	2	no	—
297	C₂×M₁₆	—	2	4	no	—
298	C₂×M₁₆	—	4	1	no	—
299	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
300	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
301	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
302	C₄⋊Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
303	#127	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
304	#124	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
305	#124	C₂×C₄×C₄₁	1	1	yes	characteristic, complemented
306	#121	—	4	1	no	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	4	1	no	—
310	C₄×D₄	—	4	2	no	—
311	#117	—	2	2	no	—
312	#117	C₂³×C₄₁	1	2	yes	—
313	C₂×C₄²	—	2	2	no	—
314	C₂×C₄²	—	4	1	no	—
315	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
317	#108	—	2	4	no	—
318	C₄⋊C₈	—	4	1	no	—
319	#104	—	2	4	no	—
320	#104	—	2	4	no	—
321	#104	—	2	2	no	—
322	#104	—	2	2	no	—
323	#104	—	2	2	no	—
324	#104	—	2	2	no	—
325	#104	C₂×C₄×C₄₁	1	4	yes	complemented
326	#104	C₂×C₄×C₄₁	1	4	yes	complemented
327	#104	D₄×C₄₁	1	4	yes	complemented
328	#104	D₄×C₄₁	1	4	yes	complemented
329	#101	C₂×C₄×C₄₁	1	2	yes	complemented
330	#100	C₂×C₄×C₄₁	1	2	yes	complemented
331	#99	—	4	1	no	—
332	#97	—	2	2	no	—
333	C₄×C₈	—	2	2	no	—
334	#41	—	2	2	no	—
335	#41	—	2	2	no	—
336	#41	—	2	4	no	—
337	#41	—	2	4	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	#41	—	4	2	no	—
343	#41	—	4	2	no	—
344	#41	—	4	4	no	—
345	#41	—	4	2	no	—
346	#41	C₂²×C₄×C₄₁	1	4	yes	—
347	#41	C₂×D₄×C₄₁	1	4	yes	—
348	C₂×Q₈	—	2	4	no	—
349	C₂×Q₈	—	4	1	no	—
350	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
351	C₂×Q₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
352	C₂×Q₈	#41×C₄₁	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	4	no	—
361	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
362	C₂×D₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
363	C₂×D₄	#41×C₄₁	1	2	yes	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	—	4	2	no	—
371	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
372	Q₁₆	—	2	4	no	—
373	SD₁₆	—	2	4	no	—
374	SD₁₆	—	2	4	no	—
375	D₈	—	2	4	no	—
376	M₁₆	—	2	2	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	2	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	4	2	no	—
381	M₁₆	—	4	4	no	—
382	M₁₆	C₂²×C₄×C₄₁	1	4	yes	—
383	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	4	2	no	—
388	C₂×C₈	—	4	4	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	4	1	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₂²⋊C₄	—	2	2	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	1	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	4	2	no	—
400	C₄²	—	4	4	no	—
401	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
402	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
403	C₄²	C₂×D₄×C₄₁	1	2	yes	—
404	C₄²	C₂×D₄×C₄₁	1	2	yes	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	4	2	no	—
409	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
410	Q₈	—	2	2	no	—
411	Q₈	—	2	2	no	—
412	Q₈	—	2	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	C₄×D₄×C₄₁	1	4	yes	—
415	D₄	—	2	4	no	—
416	D₄	—	2	4	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	4	no	—
421	D₄	—	4	2	no	—
422	D₄	—	4	2	no	—
423	D₄	—	4	2	no	—
424	D₄	—	8	2	no	—
425	D₄	C₄×D₄×C₄₁	1	4	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	4	4	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	8	4	no	—
445	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
447	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
448	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
449	C₈	—	2	4	no	—
450	C₈	—	2	4	no	—
451	C₈	—	4	2	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	8	2	no	—
457	C₂²	—	8	1	no	—
458	C₂²	—	8	1	no	—
459	C₂²	—	16	2	no	—
460	C₂²	C₄₁×#518	1	1	yes	characteristic
461	C₂²	C₄₁×#546	1	2	yes	—
462	C₄	—	2	4	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	4	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	1	no	—
468	C₄	—	4	1	no	—
469	C₄	—	8	2	no	—
470	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
471	C₄	D₄²×C₄₁	1	1	yes	characteristic
472	C₄	C₄₁×#551	1	1	yes	characteristic
473	C₂	—	2	2	no	—
474	C₂	—	2	1	no	—
475	C₂	—	4	4	no	—
476	C₂	—	8	1	no	—
477	C₂	—	8	1	no	—
478	C₂	C₄₁×#2358	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂×C₄₁	#2358
prime radical	C₄₁×#3018	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2956	C₂	semidirect
3. C₄₁×#2901	C₂	semidirect
4. C₄₁×#1962	C₂	semidirect
5. C₄₁×#1932	C₂	semidirect
6. C₄₁×#1906	C₂	semidirect
7. C₄₁×#1884	C₂	semidirect
8. C₄₁×#1861	C₂	semidirect
9. C₄×#104×C₄₁	C₂	semidirect
13. C₄₁×#568	C₂²	semidirect
16. C₄₁×#456	C₄	semidirect
17. C₄₁×#455	C₄	semidirect
18. C₄₁×#454	C₄	semidirect
19. C₄₁×#453	C₄	semidirect
22. C₄₁×#430	C₂²	semidirect
23. C₄₁×#423	C₂²	semidirect
31. C₂×#104×C₄₁	C₂²	semidirect
33. C₄₁×#410	C₂²	semidirect
34. C₄×M₁₆×C₄₁	C₂²	semidirect
35. C₄³×C₄₁	C₂²	semidirect
36. C₄₁×#344	C₂²	semidirect
37. C₄₁×#339	C₂²	semidirect
38. C₄₁×#339	C₂²	semidirect
50. #137×C₄₁	C₂×C₄	semidirect
52. AGL(1,ℤ/8)×C₄₁	C₂×C₄	semidirect
62. C₄⋊Q₈×C₄₁	C₂×C₄	semidirect
63. #127×C₄₁	C₂×C₄	semidirect
64. #124×C₄₁	C₂×C₄	semidirect
65. #124×C₄₁	C₂×C₄	semidirect
85. #104×C₄₁	C₂×C₄	semidirect
86. #104×C₄₁	C₂×C₄	semidirect
87. #104×C₄₁	D₄	semidirect
88. #104×C₄₁	D₄	semidirect
89. #101×C₄₁	C₂×C₄	semidirect
90. #100×C₄₁	C₂×C₄	semidirect
212. #22899	C₄₁	direct (N × H)
235. #2956	C₂×C₄₁	semidirect
236. #2901	C₂×C₄₁	semidirect
237. #1962	C₂×C₄₁	semidirect
238. #1932	C₂×C₄₁	semidirect
239. #1906	C₂×C₄₁	semidirect
240. #1884	C₂×C₄₁	semidirect
241. #1861	C₂×C₄₁	semidirect
242. C₄×#104	C₂×C₄₁	semidirect
252. #568	C₂²×C₄₁	semidirect
255. #456	C₄×C₄₁	semidirect
256. #455	C₄×C₄₁	semidirect
257. #454	C₄×C₄₁	semidirect
258. #453	C₄×C₄₁	semidirect
261. #430	C₂²×C₄₁	semidirect
262. #423	C₂²×C₄₁	semidirect
270. C₂×#104	C₂²×C₄₁	semidirect
272. #410	C₂²×C₄₁	semidirect
273. C₄×M₁₆	C₂²×C₄₁	semidirect
274. C₄³	C₂²×C₄₁	semidirect
275. #344	C₂²×C₄₁	semidirect
276. #339	C₂²×C₄₁	semidirect
277. #339	C₂²×C₄₁	semidirect
278. C₄₁	#22899	direct (N × H)
290. #137	C₂×C₄×C₄₁	semidirect
292. AGL(1,ℤ/8)	C₂×C₄×C₄₁	semidirect
302. C₄⋊Q₈	C₂×C₄×C₄₁	semidirect
303. #127	C₂×C₄×C₄₁	semidirect
304. #124	C₂×C₄×C₄₁	semidirect
305. #124	C₂×C₄×C₄₁	semidirect
325. #104	C₂×C₄×C₄₁	semidirect
326. #104	C₂×C₄×C₄₁	semidirect
327. #104	D₄×C₄₁	semidirect
328. #104	D₄×C₄₁	semidirect
329. #101	C₂×C₄×C₄₁	semidirect
330. #100	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3018	2
C₄₁×#2956	2
C₄₁×#2901	2
C₄₁×#1962	2
C₄₁×#1932	2
C₄₁×#1906	2
C₄₁×#1884	2
C₄₁×#1861	2
C₄×#104×C₄₁	2
#22899	41

Subgroups — C41×#22899 #23799100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22899 #23799100