Subgroups — C₄₁×#36727 #23812928

Order 10496.

2214 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2847	C₂	1	2	yes	maximal, complemented
2	C₄₁×#2748	C₂	1	4	yes	maximal, complemented
3	C₄₁×#2741	C₂	1	4	yes	maximal, complemented
4	C₄₁×#2703	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2671	C₂	1	4	yes	maximal, complemented
6	C₄₁×#2656	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#2646	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#2443	C₂	1	4	yes	maximal, complemented
9	C₄₁×#2424	C₂	1	4	yes	maximal, complemented
10	C₄₁×#2325	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#123×C₄₁	C₂²	1	4	yes	complemented
14	C₂×#121×C₄₁	C₂²	1	2	yes	—
15	C₂×#121×C₄₁	C₂²	1	2	yes	—
16	C₂×#121×C₄₁	C₂²	1	2	yes	—
17	C₂×#121×C₄₁	C₂²	1	4	yes	—
18	C₂×#121×C₄₁	C₂²	1	4	yes	—
19	C₂×#120×C₄₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₄₁×#400	C₂²	1	4	yes	complemented
24	C₄₁×#399	C₂²	1	4	yes	complemented
25	C₄₁×#399	C₂²	1	4	yes	complemented
26	C₄₁×#398	C₂²	1	4	yes	complemented
27	C₄₁×#398	C₂²	1	4	yes	complemented
28	C₄₁×#397	C₂²	1	4	yes	complemented
29	C₄₁×#396	C₂²	1	4	yes	—
30	C₄₁×#394	C₂²	1	4	yes	—
31	C₄₁×#394	C₂²	1	4	yes	—
32	C₄₁×#394	C₂²	1	4	yes	complemented
33	C₄₁×#392	C₂²	1	4	yes	—
34	C₄₁×#392	C₂²	1	4	yes	complemented
35	C₄₁×#390	C₂²	1	4	yes	—
36	C₄₁×#388	C₂²	1	4	yes	—
37	C₄₁×#388	C₂²	1	4	yes	complemented
38	C₄₁×#388	C₂²	1	4	yes	complemented
39	C₄₁×#387	C₂²	1	4	yes	complemented
40	C₄₁×#387	C₂²	1	4	yes	complemented
41	C₄₁×#386	C₂²	1	4	yes	—
42	C₄₁×#386	C₂²	1	4	yes	—
43	C₄₁×#386	C₂²	1	4	yes	—
44	C₄₁×#386	C₂²	1	4	yes	—
45	C₄₁×#386	C₂²	1	4	yes	—
46	C₄₁×#386	C₂²	1	4	yes	complemented
47	C₄₁×#385	C₂²	1	4	yes	—
48	C₄₁×#385	C₂²	1	4	yes	complemented
49	C₄₁×#385	C₂²	1	4	yes	complemented
50	C₄₁×#380	C₂²	1	4	yes	—
51	C₄₁×#380	C₂²	1	4	yes	—
52	C₄₁×#380	C₂²	1	2	yes	—
53	C₄₁×#380	C₂²	1	2	yes	—
54	C₄₁×#380	C₂²	1	2	yes	—
55	C₂×#95×C₄₁	C₂²	1	1	yes	characteristic
56	C₂×#95×C₄₁	C₂²	1	1	yes	characteristic
57	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
59	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
60	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
61	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
62	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
63	C₂³×C₄×C₄₁	—	2	4	no	—
64	C₂³×C₄×C₄₁	—	4	4	no	—
65	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
72	#123×C₄₁	—	4	8	no	—
73	#121×C₄₁	—	2	8	no	—
74	#121×C₄₁	—	4	4	no	—
75	#121×C₄₁	—	4	4	no	—
76	#121×C₄₁	—	4	8	no	—
77	#121×C₄₁	—	4	8	no	—
78	#120×C₄₁	—	4	8	no	—
79	C₄×D₄×C₄₁	—	4	8	no	—
80	C₂×C₄⋊C₄×C₄₁	—	4	4	no	—
81	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₄₁	—	4	4	no	—
91	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
104	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
105	#95×C₄₁	C₂³	1	4	yes	—
106	#95×C₄₁	C₂³	1	4	yes	—
107	#95×C₄₁	C₂³	1	4	yes	—
108	#95×C₄₁	C₂³	1	4	yes	—
109	#95×C₄₁	C₂³	1	4	yes	—
110	#95×C₄₁	C₂³	1	4	yes	—
111	#95×C₄₁	C₂³	1	4	yes	—
112	#95×C₄₁	C₂³	1	4	yes	—
113	#95×C₄₁	C₂³	1	4	yes	—
114	C₂⁴×C₄₁	—	2	2	no	—
115	C₂⁴×C₄₁	—	2	2	no	—
116	C₂⁴×C₄₁	—	2	2	no	—
117	C₂⁴×C₄₁	—	2	4	no	—
118	C₂⁴×C₄₁	—	4	2	no	—
119	C₂⁴×C₄₁	C₂⁴	1	4	yes	—
120	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
121	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
122	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
123	C₂×D₄×C₄₁	—	2	8	no	—
124	C₂×D₄×C₄₁	—	2	8	no	—
125	C₂×D₄×C₄₁	—	2	8	no	—
126	C₂×D₄×C₄₁	—	2	8	no	—
127	C₂×D₄×C₄₁	—	2	8	no	—
128	C₂×D₄×C₄₁	—	2	4	no	—
129	C₂×D₄×C₄₁	—	4	4	no	—
130	C₂×D₄×C₄₁	—	4	4	no	—
131	C₂×D₄×C₄₁	—	4	2	no	—
132	C₂×D₄×C₄₁	—	4	2	no	—
133	C₂×D₄×C₄₁	—	4	4	no	—
134	C₂×D₄×C₄₁	—	4	4	no	—
135	C₂×D₄×C₄₁	—	4	4	no	—
136	C₂×D₄×C₄₁	—	4	4	no	—
137	C₂×D₄×C₄₁	—	4	4	no	—
138	C₂²×C₄×C₄₁	—	2	2	no	—
139	C₂²×C₄×C₄₁	—	2	4	no	—
140	C₂²×C₄×C₄₁	—	2	4	no	—
141	C₂²×C₄×C₄₁	—	2	4	no	—
142	C₂²×C₄×C₄₁	—	2	4	no	—
143	C₂²×C₄×C₄₁	—	2	4	no	—
144	C₂²×C₄×C₄₁	—	2	4	no	—
145	C₂²×C₄×C₄₁	—	2	4	no	—
146	C₂²×C₄×C₄₁	—	4	2	no	—
147	C₂²×C₄×C₄₁	—	4	2	no	—
148	C₂²×C₄×C₄₁	—	4	2	no	—
149	C₂²×C₄×C₄₁	—	4	2	no	—
150	C₂²×C₄×C₄₁	—	4	2	no	—
151	C₂²×C₄×C₄₁	—	4	2	no	—
152	C₂²×C₄×C₄₁	—	4	2	no	—
153	C₂²×C₄×C₄₁	—	4	4	no	—
154	C₂²×C₄×C₄₁	—	4	4	no	—
155	C₂²×C₄×C₄₁	—	4	4	no	—
156	C₂²×C₄×C₄₁	—	4	16	no	—
157	C₂²×C₄×C₄₁	#41	1	4	yes	—
158	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
165	C₄⋊C₄×C₄₁	—	2	4	no	—
166	C₄⋊C₄×C₄₁	—	2	8	no	—
167	C₄⋊C₄×C₄₁	—	4	4	no	—
168	C₄⋊C₄×C₄₁	—	4	4	no	—
169	C₄⋊C₄×C₄₁	—	4	2	no	—
170	C₄⋊C₄×C₄₁	—	4	2	no	—
171	C₄⋊C₄×C₄₁	—	4	4	no	—
172	C₄⋊C₄×C₄₁	—	4	4	no	—
173	C₄⋊C₄×C₄₁	—	4	4	no	—
174	C₂²⋊C₄×C₄₁	—	2	8	no	—
175	C₂²⋊C₄×C₄₁	—	2	8	no	—
176	C₂²⋊C₄×C₄₁	—	2	8	no	—
177	C₂²⋊C₄×C₄₁	—	2	8	no	—
178	C₂²⋊C₄×C₄₁	—	2	8	no	—
179	C₂²⋊C₄×C₄₁	—	4	4	no	—
180	C₂²⋊C₄×C₄₁	—	4	4	no	—
181	C₂²⋊C₄×C₄₁	—	4	4	no	—
182	C₂²⋊C₄×C₄₁	—	4	4	no	—
183	C₂²⋊C₄×C₄₁	—	4	4	no	—
184	C₂²⋊C₄×C₄₁	—	4	4	no	—
185	C₂²⋊C₄×C₄₁	—	4	4	no	—
186	C₂²⋊C₄×C₄₁	—	4	4	no	—
187	C₄²×C₄₁	—	4	4	no	—
188	C₂³×C₄₁	—	2	2	no	—
189	C₂³×C₄₁	—	2	1	no	—
190	C₂³×C₄₁	—	2	1	no	—
191	C₂³×C₄₁	—	2	8	no	—
192	C₂³×C₄₁	—	2	1	no	—
193	C₂³×C₄₁	—	2	1	no	—
194	C₂³×C₄₁	—	2	1	no	—
195	C₂³×C₄₁	—	2	2	no	—
196	C₂³×C₄₁	—	2	2	no	—
197	C₂³×C₄₁	—	2	2	no	—
198	C₂³×C₄₁	—	2	2	no	—
199	C₂³×C₄₁	—	2	1	no	—
200	C₂³×C₄₁	—	2	4	no	—
201	C₂³×C₄₁	—	2	4	no	—
202	C₂³×C₄₁	—	2	4	no	—
203	C₂³×C₄₁	—	2	4	no	—
204	C₂³×C₄₁	—	2	4	no	—
205	C₂³×C₄₁	—	2	4	no	—
206	C₂³×C₄₁	—	2	4	no	—
207	C₂³×C₄₁	—	4	8	no	—
208	C₂³×C₄₁	—	4	8	no	—
209	C₂³×C₄₁	—	4	4	no	—
210	C₂³×C₄₁	—	4	4	no	—
211	C₂³×C₄₁	C₂×#41	1	2	yes	—
212	C₂³×C₄₁	H(5,𝔽₂)	1	2	yes	—
213	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
214	C₂³×C₄₁	2₋¹⁺⁴	1	2	yes	—
215	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
216	D₄×C₄₁	—	4	16	no	—
217	D₄×C₄₁	—	8	4	no	—
218	D₄×C₄₁	—	8	8	no	—
219	D₄×C₄₁	—	8	8	no	—
220	D₄×C₄₁	—	8	8	no	—
221	C₂×C₄×C₄₁	—	2	8	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	4	no	—
227	C₂×C₄×C₄₁	—	2	4	no	—
228	C₂×C₄×C₄₁	—	2	4	no	—
229	C₂×C₄×C₄₁	—	2	4	no	—
230	C₂×C₄×C₄₁	—	2	4	no	—
231	C₂×C₄×C₄₁	—	2	4	no	—
232	C₂×C₄×C₄₁	—	2	4	no	—
233	C₂×C₄×C₄₁	—	2	4	no	—
234	C₂×C₄×C₄₁	—	2	4	no	—
235	C₂×C₄×C₄₁	—	2	4	no	—
236	C₂×C₄×C₄₁	—	2	4	no	—
237	C₂×C₄×C₄₁	—	2	4	no	—
238	C₂×C₄×C₄₁	—	2	4	no	—
239	C₂×C₄×C₄₁	—	2	4	no	—
240	C₂×C₄×C₄₁	—	2	4	no	—
241	C₂×C₄×C₄₁	—	4	4	no	—
242	C₂×C₄×C₄₁	—	4	8	no	—
243	C₂×C₄×C₄₁	—	4	8	no	—
244	C₂×C₄×C₄₁	—	8	4	no	—
245	C₂×C₄×C₄₁	—	8	4	no	—
246	C₂×C₄×C₄₁	—	8	8	no	—
247	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
248	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
249	#36727	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂²×C₄₁	—	2	2	no	—
251	C₂²×C₄₁	—	2	2	no	—
252	C₂²×C₄₁	—	2	4	no	—
253	C₂²×C₄₁	—	2	4	no	—
254	C₂²×C₄₁	—	2	2	no	—
255	C₂²×C₄₁	—	2	2	no	—
256	C₂²×C₄₁	—	2	4	no	—
257	C₂²×C₄₁	—	2	4	no	—
258	C₂²×C₄₁	—	4	16	no	—
259	C₂²×C₄₁	—	4	1	no	—
260	C₂²×C₄₁	—	4	1	no	—
261	C₂²×C₄₁	—	4	1	no	—
262	C₂²×C₄₁	—	4	1	no	—
263	C₂²×C₄₁	—	4	2	no	—
264	C₂²×C₄₁	—	4	2	no	—
265	C₂²×C₄₁	—	4	4	no	—
266	C₂²×C₄₁	—	4	4	no	—
267	C₂²×C₄₁	—	4	4	no	—
268	C₂²×C₄₁	—	4	4	no	—
269	C₂²×C₄₁	—	4	4	no	—
270	C₂²×C₄₁	—	4	4	no	—
271	C₂²×C₄₁	—	4	4	no	—
272	C₂²×C₄₁	#540	1	4	yes	—
273	C₂²×C₄₁	C₂²×#41	1	1	yes	characteristic
274	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
275	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	2	yes	—
276	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
277	C₂²×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
278	C₄×C₄₁	—	2	4	no	—
279	C₄×C₄₁	—	2	4	no	—
280	C₄×C₄₁	—	4	4	no	—
281	C₄×C₄₁	—	4	4	no	—
282	C₄×C₄₁	—	4	4	no	—
283	C₄×C₄₁	—	4	4	no	—
284	C₄×C₄₁	—	4	4	no	—
285	#2847	C₂×C₄₁	1	2	yes	complemented
286	#2748	C₂×C₄₁	1	4	yes	complemented
287	#2741	C₂×C₄₁	1	4	yes	complemented
288	#2703	C₂×C₄₁	1	4	yes	complemented
289	#2671	C₂×C₄₁	1	4	yes	complemented
290	#2656	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#2646	C₂×C₄₁	1	1	yes	characteristic, complemented
292	#2443	C₂×C₄₁	1	4	yes	complemented
293	#2424	C₂×C₄₁	1	4	yes	complemented
294	#2325	C₂×C₄₁	1	2	yes	complemented
295	C₂×#380	C₂×C₄₁	1	1	yes	characteristic, complemented
296	C₂×C₄₁	—	2	4	no	—
297	C₂×C₄₁	—	4	2	no	—
298	C₂×C₄₁	—	4	2	no	—
299	C₂×C₄₁	—	8	4	no	—
300	C₂×C₄₁	C₂×#540	1	1	yes	characteristic
301	C₂×C₄₁	#3490	1	1	yes	characteristic
302	C₂×C₄₁	#3492	1	1	yes	characteristic
303	C₂×C₄₁	#3536	1	2	yes	—
304	C₂×C₄₁	#3537	1	2	yes	—
305	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
306	C₂×#123	C₂²×C₄₁	1	4	yes	complemented
307	C₂×#121	C₂²×C₄₁	1	2	yes	—
308	C₂×#121	C₂²×C₄₁	1	2	yes	—
309	C₂×#121	C₂²×C₄₁	1	2	yes	—
310	C₂×#121	C₂²×C₄₁	1	4	yes	—
311	C₂×#121	C₂²×C₄₁	1	4	yes	—
312	C₂×#120	C₂²×C₄₁	1	4	yes	—
313	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
314	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
315	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
316	#400	C₂²×C₄₁	1	4	yes	complemented
317	#399	C₂²×C₄₁	1	4	yes	complemented
318	#399	C₂²×C₄₁	1	4	yes	complemented
319	#398	C₂²×C₄₁	1	4	yes	complemented
320	#398	C₂²×C₄₁	1	4	yes	complemented
321	#397	C₂²×C₄₁	1	4	yes	complemented
322	#396	C₂²×C₄₁	1	4	yes	—
323	#394	C₂²×C₄₁	1	4	yes	—
324	#394	C₂²×C₄₁	1	4	yes	—
325	#394	C₂²×C₄₁	1	4	yes	complemented
326	#392	C₂²×C₄₁	1	4	yes	—
327	#392	C₂²×C₄₁	1	4	yes	complemented
328	#390	C₂²×C₄₁	1	4	yes	—
329	#388	C₂²×C₄₁	1	4	yes	—
330	#388	C₂²×C₄₁	1	4	yes	complemented
331	#388	C₂²×C₄₁	1	4	yes	complemented
332	#387	C₂²×C₄₁	1	4	yes	complemented
333	#387	C₂²×C₄₁	1	4	yes	complemented
334	#386	C₂²×C₄₁	1	4	yes	—
335	#386	C₂²×C₄₁	1	4	yes	—
336	#386	C₂²×C₄₁	1	4	yes	—
337	#386	C₂²×C₄₁	1	4	yes	—
338	#386	C₂²×C₄₁	1	4	yes	—
339	#386	C₂²×C₄₁	1	4	yes	complemented
340	#385	C₂²×C₄₁	1	4	yes	—
341	#385	C₂²×C₄₁	1	4	yes	complemented
342	#385	C₂²×C₄₁	1	4	yes	complemented
343	#380	C₂²×C₄₁	1	2	yes	—
344	#380	C₂²×C₄₁	1	2	yes	—
345	#380	C₂²×C₄₁	1	2	yes	—
346	#380	C₂²×C₄₁	1	4	yes	—
347	#380	C₂²×C₄₁	1	4	yes	—
348	C₂×#95	C₂²×C₄₁	1	1	yes	characteristic
349	C₂×#95	C₂²×C₄₁	1	1	yes	characteristic
350	C₄₁	#36727	1	1	yes	characteristic, direct product factor, Sylow
351	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
352	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
353	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
354	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
355	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
356	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
357	C₂³×C₄	—	2	4	no	—
358	C₂³×C₄	—	4	4	no	—
359	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
361	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
362	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
363	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
364	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
365	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
366	#123	—	4	8	no	—
367	#121	—	2	8	no	—
368	#121	—	4	4	no	—
369	#121	—	4	4	no	—
370	#121	—	4	8	no	—
371	#121	—	4	8	no	—
372	#120	—	4	8	no	—
373	C₄×D₄	—	4	8	no	—
374	C₂×C₄⋊C₄	—	4	4	no	—
375	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
376	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
380	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
383	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
384	C₂×C₂²⋊C₄	—	4	4	no	—
385	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
398	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
399	#95	C₂³×C₄₁	1	4	yes	—
400	#95	C₂³×C₄₁	1	4	yes	—
401	#95	C₂³×C₄₁	1	4	yes	—
402	#95	C₂³×C₄₁	1	4	yes	—
403	#95	C₂³×C₄₁	1	4	yes	—
404	#95	C₂³×C₄₁	1	4	yes	—
405	#95	C₂³×C₄₁	1	4	yes	—
406	#95	C₂³×C₄₁	1	4	yes	—
407	#95	C₂³×C₄₁	1	4	yes	—
408	C₂⁴	—	2	2	no	—
409	C₂⁴	—	2	2	no	—
410	C₂⁴	—	2	2	no	—
411	C₂⁴	—	2	4	no	—
412	C₂⁴	—	4	2	no	—
413	C₂⁴	C₂⁴×C₄₁	1	4	yes	—
414	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
415	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
416	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	2	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	—	4	4	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	4	no	—
431	C₂×D₄	—	4	4	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	4	no	—
450	C₂²×C₄	—	4	16	no	—
451	C₂²×C₄	#41×C₄₁	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
453	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
454	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	8	no	—
461	C₄⋊C₄	—	4	4	no	—
462	C₄⋊C₄	—	4	4	no	—
463	C₄⋊C₄	—	4	2	no	—
464	C₄⋊C₄	—	4	2	no	—
465	C₄⋊C₄	—	4	4	no	—
466	C₄⋊C₄	—	4	4	no	—
467	C₄⋊C₄	—	4	4	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	4	4	no	—
474	C₂²⋊C₄	—	4	4	no	—
475	C₂²⋊C₄	—	4	4	no	—
476	C₂²⋊C₄	—	4	4	no	—
477	C₂²⋊C₄	—	4	4	no	—
478	C₂²⋊C₄	—	4	4	no	—
479	C₂²⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	4	4	no	—
481	C₄²	—	4	4	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	8	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	4	8	no	—
502	C₂³	—	4	8	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	4	4	no	—
505	C₂³	C₂×#41×C₄₁	1	2	yes	—
506	C₂³	H(5,𝔽₂)×C₄₁	1	2	yes	—
507	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
508	C₂³	2₋¹⁺⁴×C₄₁	1	2	yes	—
509	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
510	D₄	—	4	16	no	—
511	D₄	—	8	4	no	—
512	D₄	—	8	8	no	—
513	D₄	—	8	8	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	8	no	—
537	C₂×C₄	—	4	8	no	—
538	C₂×C₄	—	8	4	no	—
539	C₂×C₄	—	8	4	no	—
540	C₂×C₄	—	8	8	no	—
541	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
542	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	4	no	—
546	C₂²	—	2	4	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	4	16	no	—
552	C₂²	—	4	1	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	C₄₁×#540	1	4	yes	—
566	C₂²	C₂²×#41×C₄₁	1	1	yes	characteristic
567	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
568	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	2	yes	—
569	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
570	C₂²	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
571	C₄	—	2	4	no	—
572	C₄	—	2	4	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	4	no	—
577	C₄	—	4	4	no	—
578	C₂	—	2	4	no	—
579	C₂	—	4	2	no	—
580	C₂	—	4	2	no	—
581	C₂	—	8	4	no	—
582	C₂	C₂×C₄₁×#540	1	1	yes	characteristic
583	C₂	C₄₁×#3490	1	1	yes	characteristic
584	C₂	C₄₁×#3492	1	1	yes	characteristic
585	C₂	C₄₁×#3536	1	2	yes	—
586	C₂	C₄₁×#3537	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#2847	C₂	semidirect
2. C₄₁×#2748	C₂	semidirect
3. C₄₁×#2741	C₂	semidirect
4. C₄₁×#2703	C₂	semidirect
5. C₄₁×#2671	C₂	semidirect
6. C₄₁×#2656	C₂	semidirect
7. C₄₁×#2646	C₂	semidirect
8. C₄₁×#2443	C₂	semidirect
9. C₄₁×#2424	C₂	semidirect
10. C₄₁×#2325	C₂	semidirect
11. C₂×C₄₁×#380	C₂	semidirect
13. C₂×#123×C₄₁	C₂²	semidirect
23. C₄₁×#400	C₂²	semidirect
24. C₄₁×#399	C₂²	semidirect
25. C₄₁×#399	C₂²	semidirect
26. C₄₁×#398	C₂²	semidirect
27. C₄₁×#398	C₂²	semidirect
28. C₄₁×#397	C₂²	semidirect
32. C₄₁×#394	C₂²	semidirect
34. C₄₁×#392	C₂²	semidirect
37. C₄₁×#388	C₂²	semidirect
38. C₄₁×#388	C₂²	semidirect
39. C₄₁×#387	C₂²	semidirect
40. C₄₁×#387	C₂²	semidirect
46. C₄₁×#386	C₂²	semidirect
48. C₄₁×#385	C₂²	semidirect
49. C₄₁×#385	C₂²	semidirect
249. #36727	C₄₁	direct (N × H)
285. #2847	C₂×C₄₁	semidirect
286. #2748	C₂×C₄₁	semidirect
287. #2741	C₂×C₄₁	semidirect
288. #2703	C₂×C₄₁	semidirect
289. #2671	C₂×C₄₁	semidirect
290. #2656	C₂×C₄₁	semidirect
291. #2646	C₂×C₄₁	semidirect
292. #2443	C₂×C₄₁	semidirect
293. #2424	C₂×C₄₁	semidirect
294. #2325	C₂×C₄₁	semidirect
295. C₂×#380	C₂×C₄₁	semidirect
306. C₂×#123	C₂²×C₄₁	semidirect
316. #400	C₂²×C₄₁	semidirect
317. #399	C₂²×C₄₁	semidirect
318. #399	C₂²×C₄₁	semidirect
319. #398	C₂²×C₄₁	semidirect
320. #398	C₂²×C₄₁	semidirect
321. #397	C₂²×C₄₁	semidirect
325. #394	C₂²×C₄₁	semidirect
327. #392	C₂²×C₄₁	semidirect
330. #388	C₂²×C₄₁	semidirect
331. #388	C₂²×C₄₁	semidirect
332. #387	C₂²×C₄₁	semidirect
333. #387	C₂²×C₄₁	semidirect
339. #386	C₂²×C₄₁	semidirect
341. #385	C₂²×C₄₁	semidirect
342. #385	C₂²×C₄₁	semidirect
350. C₄₁	#36727	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2847	2
C₄₁×#2748	2
C₄₁×#2741	2
C₄₁×#2703	2
C₄₁×#2671	2
C₄₁×#2656	2
C₄₁×#2646	2
C₄₁×#2443	2
C₄₁×#2424	2
C₄₁×#2325	2
C₂×C₄₁×#380	2
#36727	41

Subgroups — C41×#36727 #23812928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#36727 #23812928