Subgroups — C₄₁×#2813 #19106593

Order 5248.

578 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#396	C₂	1	2	yes	maximal, complemented
6	C₄₁×#394	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#394	C₂	1	2	yes	maximal, complemented
8	C₄₁×#394	C₂	1	2	yes	maximal, complemented
9	C₄₁×#392	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#392	C₂	1	2	yes	maximal, complemented
11	C₄₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
16	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
17	#123×C₄₁	—	2	4	no	—
18	#121×C₄₁	—	2	4	no	—
19	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
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×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
33	#95×C₄₁	C₂²	1	1	yes	characteristic
34	#95×C₄₁	C₂²	1	1	yes	characteristic
35	#95×C₄₁	C₂²	1	2	yes	—
36	#95×C₄₁	C₂²	1	2	yes	—
37	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
38	C₂⁴×C₄₁	C₂³	1	2	yes	—
39	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₄₁	—	2	4	no	—
41	C₂×D₄×C₄₁	—	2	2	no	—
42	C₂×D₄×C₄₁	—	2	2	no	—
43	C₂×D₄×C₄₁	—	4	2	no	—
44	C₂×D₄×C₄₁	—	4	2	no	—
45	C₂×D₄×C₄₁	—	4	2	no	—
46	C₂×D₄×C₄₁	—	4	2	no	—
47	C₂×D₄×C₄₁	—	4	2	no	—
48	C₂×D₄×C₄₁	—	4	2	no	—
49	C₂×D₄×C₄₁	—	4	2	no	—
50	C₂²×C₄×C₄₁	—	4	2	no	—
51	C₂²×C₄×C₄₁	D₄	1	4	yes	complemented
52	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₄₁	—	2	2	no	—
61	C₄⋊C₄×C₄₁	—	2	2	no	—
62	C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₄⋊C₄×C₄₁	—	4	2	no	—
64	C₄⋊C₄×C₄₁	—	4	2	no	—
65	C₂²⋊C₄×C₄₁	—	2	2	no	—
66	C₂²⋊C₄×C₄₁	—	2	4	no	—
67	C₂²⋊C₄×C₄₁	—	2	2	no	—
68	C₂²⋊C₄×C₄₁	—	2	2	no	—
69	C₂²⋊C₄×C₄₁	—	4	2	no	—
70	C₂²⋊C₄×C₄₁	—	4	2	no	—
71	C₂²⋊C₄×C₄₁	—	4	2	no	—
72	C₂²⋊C₄×C₄₁	—	4	2	no	—
73	C₂²⋊C₄×C₄₁	—	4	2	no	—
74	C₂³×C₄₁	—	2	1	no	—
75	C₂³×C₄₁	—	2	2	no	—
76	C₂³×C₄₁	—	2	2	no	—
77	C₂³×C₄₁	—	2	1	no	—
78	C₂³×C₄₁	—	2	1	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	1	no	—
88	C₂³×C₄₁	—	2	1	no	—
89	C₂³×C₄₁	—	2	2	no	—
90	C₂³×C₄₁	C₂×D₄	1	2	yes	—
91	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
92	D₄×C₄₁	—	4	4	no	—
93	D₄×C₄₁	—	4	4	no	—
94	D₄×C₄₁	—	4	8	no	—
95	D₄×C₄₁	—	4	8	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	2	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	2	no	—
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	1	no	—
114	C₂×C₄×C₄₁	—	4	4	no	—
115	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄₁	—	2	2	no	—
118	C₂²×C₄₁	—	2	2	no	—
119	C₂²×C₄₁	—	2	2	no	—
120	C₂²×C₄₁	—	4	2	no	—
121	C₂²×C₄₁	—	4	2	no	—
122	C₂²×C₄₁	—	4	2	no	—
123	C₂²×C₄₁	—	4	1	no	—
124	C₂²×C₄₁	—	4	2	no	—
125	C₂²×C₄₁	—	4	1	no	—
126	C₂²×C₄₁	—	4	1	no	—
127	C₂²×C₄₁	—	4	2	no	—
128	C₂²×C₄₁	—	4	2	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	C₂²×D₄	1	1	yes	characteristic
135	C₂²×C₄₁	H(5,𝔽₂)	1	2	yes	—
136	C₂²×C₄₁	H(5,𝔽₂)	1	2	yes	—
137	C₂²×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂²×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
139	C₄×C₄₁	—	2	2	no	—
140	C₄×C₄₁	—	2	2	no	—
141	C₄×C₄₁	—	4	1	no	—
142	C₄×C₄₁	—	4	1	no	—
143	C₄×C₄₁	—	4	2	no	—
144	C₄×C₄₁	—	4	1	no	—
145	C₄×C₄₁	—	4	2	no	—
146	C₄×C₄₁	—	4	2	no	—
147	#2813	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₄₁	—	4	2	no	—
149	C₂×C₄₁	—	8	2	no	—
150	C₂×C₄₁	—	8	1	no	—
151	C₂×C₄₁	#534	1	1	yes	characteristic
152	C₂×C₄₁	#535	1	1	yes	characteristic
153	C₂×C₄₁	#537	1	1	yes	characteristic
154	C₂×C₄₁	#560	1	2	yes	—
155	C₂×C₄₁	#562	1	2	yes	—
156	C₂×#123	C₂×C₄₁	1	1	yes	characteristic, complemented
157	C₂×#121	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#400	C₂×C₄₁	1	1	yes	characteristic, complemented
159	#399	C₂×C₄₁	1	1	yes	characteristic, complemented
160	#396	C₂×C₄₁	1	2	yes	complemented
161	#394	C₂×C₄₁	1	1	yes	characteristic, complemented
162	#394	C₂×C₄₁	1	2	yes	complemented
163	#394	C₂×C₄₁	1	2	yes	complemented
164	#392	C₂×C₄₁	1	1	yes	characteristic, complemented
165	#392	C₂×C₄₁	1	2	yes	complemented
166	#380	C₂×C₄₁	1	1	yes	characteristic, complemented
167	C₄₁	#2813	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
169	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
170	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
171	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
172	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
173	#123	—	2	4	no	—
174	#121	—	2	4	no	—
175	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
189	#95	C₂²×C₄₁	1	1	yes	characteristic
190	#95	C₂²×C₄₁	1	1	yes	characteristic
191	#95	C₂²×C₄₁	1	2	yes	—
192	#95	C₂²×C₄₁	1	2	yes	—
193	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
194	C₂⁴	C₂³×C₄₁	1	2	yes	—
195	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	D₄×C₄₁	1	4	yes	complemented
208	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
209	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
211	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
213	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₂×D₄×C₄₁	1	2	yes	—
247	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	8	no	—
251	D₄	—	4	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
291	C₂²	H(5,𝔽₂)×C₄₁	1	2	yes	—
292	C₂²	H(5,𝔽₂)×C₄₁	1	2	yes	—
293	C₂²	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
294	C₂²	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	8	1	no	—
306	C₂	C₄₁×#534	1	1	yes	characteristic
307	C₂	C₄₁×#535	1	1	yes	characteristic
308	C₂	C₄₁×#537	1	1	yes	characteristic
309	C₂	C₄₁×#560	1	2	yes	—
310	C₂	C₄₁×#562	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×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. C₂×#123×C₄₁	C₂	semidirect
2. C₂×#121×C₄₁	C₂	semidirect
3. C₄₁×#400	C₂	semidirect
4. C₄₁×#399	C₂	semidirect
5. C₄₁×#396	C₂	semidirect
6. C₄₁×#394	C₂	semidirect
7. C₄₁×#394	C₂	semidirect
8. C₄₁×#394	C₂	semidirect
9. C₄₁×#392	C₂	semidirect
10. C₄₁×#392	C₂	semidirect
11. C₄₁×#380	C₂	semidirect
51. C₂²×C₄×C₄₁	D₄	semidirect
147. #2813	C₄₁	direct (N × H)
156. C₂×#123	C₂×C₄₁	semidirect
157. C₂×#121	C₂×C₄₁	semidirect
158. #400	C₂×C₄₁	semidirect
159. #399	C₂×C₄₁	semidirect
160. #396	C₂×C₄₁	semidirect
161. #394	C₂×C₄₁	semidirect
162. #394	C₂×C₄₁	semidirect
163. #394	C₂×C₄₁	semidirect
164. #392	C₂×C₄₁	semidirect
165. #392	C₂×C₄₁	semidirect
166. #380	C₂×C₄₁	semidirect
167. C₄₁	#2813	direct (N × H)
207. C₂²×C₄	D₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₄₁	2
C₂×#121×C₄₁	2
C₄₁×#400	2
C₄₁×#399	2
C₄₁×#396	2
C₄₁×#394	2
C₄₁×#392	2
C₄₁×#380	2
#2813	41

Subgroups — C41×#2813 #19106593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2813 #19106593