Subgroups — C₄₁×#2445 #19106225

Order 5248.

696 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#123×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#394	C₂	1	2	yes	maximal, complemented
5	C₄₁×#394	C₂	1	2	yes	maximal, complemented
6	C₄₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#386	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
15	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₄₁	C₂²	1	2	yes	—
17	#127×C₄₁	—	2	4	no	—
18	#123×C₄₁	—	2	8	no	—
19	C₄×D₄×C₄₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
35	#95×C₄₁	C₂²	1	1	yes	characteristic
36	#95×C₄₁	C₂²	1	1	yes	characteristic
37	#95×C₄₁	C₂²	1	2	yes	—
38	C₂⁴×C₄₁	C₂³	1	2	yes	—
39	C₂⁴×C₄₁	C₂³	1	2	yes	—
40	C₂×D₄×C₄₁	—	2	4	no	—
41	C₂×D₄×C₄₁	—	2	4	no	—
42	C₂×D₄×C₄₁	—	2	4	no	—
43	C₂×D₄×C₄₁	—	2	2	no	—
44	C₂×D₄×C₄₁	—	2	2	no	—
45	C₂×D₄×C₄₁	—	2	2	no	—
46	C₂×D₄×C₄₁	—	2	4	no	—
47	C₂×D₄×C₄₁	—	2	2	no	—
48	C₂×D₄×C₄₁	—	2	2	no	—
49	C₂×D₄×C₄₁	—	4	2	no	—
50	C₂×D₄×C₄₁	—	4	2	no	—
51	C₂×D₄×C₄₁	D₄	1	4	yes	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	4	no	—
54	C₂²×C₄×C₄₁	—	4	2	no	—
55	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
59	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₄₁	—	2	2	no	—
65	C₄⋊C₄×C₄₁	—	2	4	no	—
66	C₄⋊C₄×C₄₁	—	2	2	no	—
67	C₄⋊C₄×C₄₁	—	2	2	no	—
68	C₄⋊C₄×C₄₁	D₄	1	4	yes	complemented
69	C₂²⋊C₄×C₄₁	—	2	4	no	—
70	C₂²⋊C₄×C₄₁	—	2	4	no	—
71	C₂²⋊C₄×C₄₁	—	2	4	no	—
72	C₂²⋊C₄×C₄₁	—	2	4	no	—
73	C₂²⋊C₄×C₄₁	—	4	2	no	—
74	C₂²⋊C₄×C₄₁	—	4	2	no	—
75	C₂²⋊C₄×C₄₁	—	4	2	no	—
76	C₄²×C₄₁	—	2	2	no	—
77	C₄²×C₄₁	—	2	2	no	—
78	C₄²×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	—	2	2	no	—
80	C₂³×C₄₁	—	2	2	no	—
81	C₂³×C₄₁	—	2	2	no	—
82	C₂³×C₄₁	—	2	2	no	—
83	C₂³×C₄₁	—	2	2	no	—
84	C₂³×C₄₁	—	2	2	no	—
85	C₂³×C₄₁	—	2	2	no	—
86	C₂³×C₄₁	—	2	2	no	—
87	C₂³×C₄₁	—	2	2	no	—
88	C₂³×C₄₁	—	2	2	no	—
89	C₂³×C₄₁	—	2	2	no	—
90	C₂³×C₄₁	—	2	2	no	—
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	C₂×D₄	1	4	yes	—
93	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
94	D₄×C₄₁	—	2	8	no	—
95	D₄×C₄₁	—	4	8	no	—
96	D₄×C₄₁	—	4	8	no	—
97	D₄×C₄₁	—	4	4	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	1	no	—
104	C₂×C₄×C₄₁	—	2	1	no	—
105	C₂×C₄×C₄₁	—	2	1	no	—
106	C₂×C₄×C₄₁	—	2	1	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	2	no	—
110	C₂×C₄×C₄₁	—	2	1	no	—
111	C₂×C₄×C₄₁	—	2	1	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	1	no	—
114	C₂×C₄×C₄₁	—	2	1	no	—
115	C₂×C₄×C₄₁	—	2	1	no	—
116	C₂×C₄×C₄₁	—	2	2	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	4	8	no	—
119	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₄₁	#41	1	2	yes	—
124	C₂×C₄×C₄₁	#41	1	2	yes	—
125	C₂×C₄×C₄₁	#41	1	2	yes	—
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	2	4	no	—
128	C₂²×C₄₁	—	2	4	no	—
129	C₂²×C₄₁	—	4	2	no	—
130	C₂²×C₄₁	—	4	2	no	—
131	C₂²×C₄₁	—	4	2	no	—
132	C₂²×C₄₁	—	4	2	no	—
133	C₂²×C₄₁	—	4	2	no	—
134	C₂²×C₄₁	—	4	2	no	—
135	C₂²×C₄₁	—	4	2	no	—
136	C₂²×C₄₁	—	4	2	no	—
137	C₂²×C₄₁	—	4	2	no	—
138	C₂²×C₄₁	—	4	2	no	—
139	C₂²×C₄₁	—	4	2	no	—
140	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
143	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
144	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
145	C₂²×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₄×C₄₁	—	2	2	no	—
148	C₄×C₄₁	—	2	2	no	—
149	C₄×C₄₁	—	2	2	no	—
150	C₄×C₄₁	—	2	2	no	—
151	C₄×C₄₁	—	4	1	no	—
152	C₄×C₄₁	—	4	2	no	—
153	C₄×C₄₁	—	4	1	no	—
154	C₄×C₄₁	—	4	2	no	—
155	C₄×C₄₁	#123	1	4	yes	complemented
156	#2445	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₄₁	—	4	4	no	—
158	C₂×C₄₁	—	8	2	no	—
159	C₂×C₄₁	C₂×#123	1	1	yes	characteristic
160	C₂×C₄₁	#532	1	1	yes	characteristic
161	C₂×C₄₁	#540	1	1	yes	characteristic
162	C₂×C₄₁	D₄²	1	1	yes	characteristic
163	C₂×C₄₁	#546	1	1	yes	characteristic
164	C₂×C₄₁	#550	1	1	yes	characteristic
165	C₂×C₄₁	#559	1	1	yes	characteristic
166	C₂×#127	C₂×C₄₁	1	1	yes	characteristic
167	C₂×#123	C₂×C₄₁	1	2	yes	complemented
168	C₂×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic, complemented
169	#394	C₂×C₄₁	1	2	yes	complemented
170	#394	C₂×C₄₁	1	2	yes	complemented
171	#390	C₂×C₄₁	1	1	yes	characteristic, complemented
172	#390	C₂×C₄₁	1	1	yes	characteristic, complemented
173	#390	C₂×C₄₁	1	1	yes	characteristic, complemented
174	#389	C₂×C₄₁	1	1	yes	characteristic, complemented
175	#386	C₂×C₄₁	1	2	yes	complemented
176	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
177	C₄₁	#2445	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
179	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
180	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
181	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
182	C₂³×C₄	C₂²×C₄₁	1	2	yes	—
183	#127	—	2	4	no	—
184	#123	—	2	8	no	—
185	C₄×D₄	—	2	4	no	—
186	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
188	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
195	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
198	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
199	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
200	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
201	#95	C₂²×C₄₁	1	1	yes	characteristic
202	#95	C₂²×C₄₁	1	1	yes	characteristic
203	#95	C₂²×C₄₁	1	2	yes	—
204	C₂⁴	C₂³×C₄₁	1	2	yes	—
205	C₂⁴	C₂³×C₄₁	1	2	yes	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	D₄×C₄₁	1	4	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
225	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	D₄×C₄₁	1	4	yes	complemented
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×D₄×C₄₁	1	4	yes	—
259	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	D₄	—	4	8	no	—
262	D₄	—	4	8	no	—
263	D₄	—	4	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
286	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
287	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
288	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
289	C₂×C₄	#41×C₄₁	1	2	yes	—
290	C₂×C₄	#41×C₄₁	1	2	yes	—
291	C₂×C₄	#41×C₄₁	1	2	yes	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
307	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
308	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
309	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
310	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
311	C₂²	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
312	C₂²	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	2	no	—
321	C₄	#123×C₄₁	1	4	yes	complemented
322	C₂	—	4	4	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#123×C₄₁	1	1	yes	characteristic
325	C₂	C₄₁×#532	1	1	yes	characteristic
326	C₂	C₄₁×#540	1	1	yes	characteristic
327	C₂	D₄²×C₄₁	1	1	yes	characteristic
328	C₂	C₄₁×#546	1	1	yes	characteristic
329	C₂	C₄₁×#550	1	1	yes	characteristic
330	C₂	C₄₁×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₄₁
prime radical	C₂×#127×C₄₁	C₂
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
2. C₂×#123×C₄₁	C₂	semidirect
3. C₂×C₄×D₄×C₄₁	C₂	semidirect
4. C₄₁×#394	C₂	semidirect
5. C₄₁×#394	C₂	semidirect
6. C₄₁×#390	C₂	semidirect
7. C₄₁×#390	C₂	semidirect
8. C₄₁×#390	C₂	semidirect
9. C₄₁×#389	C₂	semidirect
10. C₄₁×#386	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
68. C₄⋊C₄×C₄₁	D₄	semidirect
155. C₄×C₄₁	#123	semidirect
156. #2445	C₄₁	direct (N × H)
167. C₂×#123	C₂×C₄₁	semidirect
168. C₂×C₄×D₄	C₂×C₄₁	semidirect
169. #394	C₂×C₄₁	semidirect
170. #394	C₂×C₄₁	semidirect
171. #390	C₂×C₄₁	semidirect
172. #390	C₂×C₄₁	semidirect
173. #390	C₂×C₄₁	semidirect
174. #389	C₂×C₄₁	semidirect
175. #386	C₂×C₄₁	semidirect
176. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
177. C₄₁	#2445	direct (N × H)
234. C₄⋊C₄	D₄×C₄₁	semidirect
321. C₄	#123×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₄₁	2
C₂×#123×C₄₁	2
C₂×C₄×D₄×C₄₁	2
C₄₁×#394	2
C₄₁×#390	2
C₄₁×#389	2
C₄₁×#386	2
C₄×C₄⋊C₄×C₄₁	2
#2445	41

Subgroups — C41×#2445 #19106225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2445 #19106225