Subgroups — C₄₁×#22814 #23799015

Order 10496.

770 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2355	C₂	1	2	yes	maximal
3	C₄₁×#1971	C₂	1	2	yes	maximal
4	C₄₁×#1927	C₂	1	4	yes	maximal
5	C₄₁×#1844	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1803	C₂	1	2	yes	maximal
7	C₄₁×#1776	C₂	1	2	yes	maximal
8	C₄₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₄₁	C₄	1	8	yes	complemented
12	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#106×C₄₁	C₂²	1	2	yes	—
14	C₂×#106×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#103×C₄₁	C₂²	1	4	yes	—
17	C₂×#103×C₄₁	C₂²	1	2	yes	—
18	C₂×#97×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#391	C₂²	1	4	yes	—
21	C₄₁×#387	—	2	4	no	—
22	C₄₁×#384	C₂²	1	2	yes	—
23	C₄₁×#382	—	2	2	no	—
24	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
25	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#340	C₂²	1	4	yes	—
28	C₄₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₄₁	—	2	2	no	—
31	C₂×Q₁₆×C₄₁	—	2	2	no	—
32	C₂×Q₁₆×C₄₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₄₁	—	2	4	no	—
37	C₄×Q₈×C₄₁	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	—	2	4	no	—
46	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	#107×C₄₁	C₂×C₄	1	4	yes	—
49	#106×C₄₁	—	2	4	no	—
50	#106×C₄₁	—	4	2	no	—
51	C₄⋊C₈×C₄₁	—	2	2	no	—
52	#103×C₄₁	—	4	4	no	—
53	#103×C₄₁	C₂×C₄	1	8	yes	—
54	#97×C₄₁	—	2	4	no	—
55	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
56	#95×C₄₁	—	2	4	no	—
57	#95×C₄₁	—	2	4	no	—
58	#95×C₄₁	—	2	4	no	—
59	#95×C₄₁	D₄	1	2	yes	—
60	C₂×Q₈×C₄₁	—	2	2	no	—
61	C₂×Q₈×C₄₁	—	2	2	no	—
62	C₂×Q₈×C₄₁	—	2	2	no	—
63	C₂×Q₈×C₄₁	—	2	2	no	—
64	C₂×Q₈×C₄₁	—	2	2	no	—
65	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₁	—	2	2	no	—
67	C₂²×C₄×C₄₁	—	2	2	no	—
68	C₂²×C₄×C₄₁	—	2	4	no	—
69	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₄₁	—	2	8	no	—
74	C₂×C₈×C₄₁	—	2	1	no	—
75	C₂×C₈×C₄₁	—	2	1	no	—
76	C₂×C₈×C₄₁	—	2	2	no	—
77	C₂×C₈×C₄₁	—	2	2	no	—
78	C₂×C₈×C₄₁	—	2	2	no	—
79	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₁	#41	1	2	yes	—
82	C₂×C₈×C₄₁	#41	1	2	yes	—
83	C₄⋊C₄×C₄₁	—	2	8	no	—
84	C₄⋊C₄×C₄₁	—	2	4	no	—
85	C₄⋊C₄×C₄₁	—	2	2	no	—
86	C₄⋊C₄×C₄₁	—	2	4	no	—
87	C₄⋊C₄×C₄₁	—	2	4	no	—
88	C₄⋊C₄×C₄₁	—	4	4	no	—
89	C₄⋊C₄×C₄₁	—	8	2	no	—
90	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
91	C₄²×C₄₁	—	2	8	no	—
92	C₄²×C₄₁	—	2	2	no	—
93	C₄²×C₄₁	—	2	2	no	—
94	C₄²×C₄₁	—	8	4	no	—
95	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₁	#41	1	2	yes	—
97	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₄₁	—	2	4	no	—
99	Q₈×C₄₁	—	2	4	no	—
100	Q₈×C₄₁	—	4	2	no	—
101	Q₈×C₄₁	—	4	2	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	4	no	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	4	no	—
107	C₂×C₄×C₄₁	—	2	4	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	4	2	no	—
110	C₂×C₄×C₄₁	—	4	2	no	—
111	C₂×C₄×C₄₁	—	4	2	no	—
112	C₂×C₄×C₄₁	—	4	4	no	—
113	C₂×C₄×C₄₁	—	4	4	no	—
114	C₂×C₄×C₄₁	—	4	4	no	—
115	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₁	#137	1	4	yes	—
118	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₈×C₄₁	—	2	2	no	—
125	C₈×C₄₁	—	4	2	no	—
126	C₈×C₄₁	#117	1	4	yes	—
127	#22814	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₁	#547	1	1	yes	characteristic
130	C₂²×C₄₁	#550	1	1	yes	characteristic
131	C₂²×C₄₁	C₂×#137	1	2	yes	—
132	C₂²×C₄₁	#577	1	2	yes	—
133	C₄×C₄₁	—	2	2	no	—
134	C₄×C₄₁	—	4	4	no	—
135	C₄×C₄₁	—	4	4	no	—
136	C₄×C₄₁	—	4	4	no	—
137	C₄×C₄₁	—	8	4	no	—
138	C₄×C₄₁	#441	1	4	yes	—
139	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
140	C₄×C₄₁	#520	1	1	yes	characteristic
141	C₄×C₄₁	#555	1	1	yes	characteristic
142	C₄×C₄₁	#556	1	1	yes	characteristic
143	C₂×C₄×Q₁₆	C₂×C₄₁	1	1	yes	characteristic
144	#2355	C₂×C₄₁	1	2	yes	—
145	#1971	C₂×C₄₁	1	2	yes	—
146	#1927	C₂×C₄₁	1	4	yes	—
147	#1844	C₂×C₄₁	1	1	yes	characteristic
148	#1803	C₂×C₄₁	1	2	yes	—
149	#1776	C₂×C₄₁	1	2	yes	—
150	#1770	C₂×C₄₁	1	1	yes	characteristic
151	C₂×C₄₁	#2354	1	1	yes	characteristic
152	C₂×C₄₁	C₂×#441	1	1	yes	characteristic
153	C₂×C₄₁	#2952	1	1	yes	characteristic
154	C₂×C₄₁	#3356	1	2	yes	—
155	C₂×C₄₁	#3400	1	2	yes	—
156	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
158	C₄×Q₁₆	C₄×C₄₁	1	8	yes	complemented
159	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
160	C₂×#106	C₂²×C₄₁	1	2	yes	—
161	C₂×#106	C₂²×C₄₁	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×#103	C₂²×C₄₁	1	4	yes	—
164	C₂×#103	C₂²×C₄₁	1	2	yes	—
165	C₂×#97	C₂²×C₄₁	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
167	#391	C₂²×C₄₁	1	4	yes	—
168	#387	—	2	4	no	—
169	#384	C₂²×C₄₁	1	2	yes	—
170	#382	—	2	2	no	—
171	C₄×C₄⋊C₄	—	2	2	no	—
172	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
173	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
174	#340	C₂²×C₄₁	1	4	yes	—
175	#336	C₂²×C₄₁	1	2	yes	—
176	C₄₁	#22814	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	C₂×C₄×C₄₁	1	4	yes	—
181	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
182	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	C₂×C₄×C₄₁	1	8	yes	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
190	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
191	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	4	no	—
194	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
195	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
196	#107	C₂×C₄×C₄₁	1	4	yes	—
197	#106	—	2	4	no	—
198	#106	—	4	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	#103	—	4	4	no	—
201	#103	C₂×C₄×C₄₁	1	8	yes	—
202	#97	—	2	4	no	—
203	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
204	#95	—	2	4	no	—
205	#95	—	2	4	no	—
206	#95	—	2	4	no	—
207	#95	D₄×C₄₁	1	2	yes	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
221	Q₁₆	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
229	C₂×C₈	#41×C₄₁	1	2	yes	—
230	C₂×C₈	#41×C₄₁	1	2	yes	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	8	2	no	—
238	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	8	4	no	—
243	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
244	C₄²	#41×C₄₁	1	2	yes	—
245	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	Q₈	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
265	C₂×C₄	#137×C₄₁	1	4	yes	—
266	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
270	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
271	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	4	2	no	—
274	C₈	#117×C₄₁	1	4	yes	—
275	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
276	C₂²	C₄₁×#547	1	1	yes	characteristic
277	C₂²	C₄₁×#550	1	1	yes	characteristic
278	C₂²	C₂×#137×C₄₁	1	2	yes	—
279	C₂²	C₄₁×#577	1	2	yes	—
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	—	8	4	no	—
285	C₄	C₄₁×#441	1	4	yes	—
286	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
287	C₄	C₄₁×#520	1	1	yes	characteristic
288	C₄	C₄₁×#555	1	1	yes	characteristic
289	C₄	C₄₁×#556	1	1	yes	characteristic
290	C₂	C₄₁×#2354	1	1	yes	characteristic
291	C₂	C₂×C₄₁×#441	1	1	yes	characteristic
292	C₂	C₄₁×#2952	1	1	yes	characteristic
293	C₂	C₄₁×#3356	1	2	yes	—
294	C₂	C₄₁×#3400	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄×C₄⋊C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₄₁	C₄	semidirect
127. #22814	C₄₁	direct (N × H)
158. C₄×Q₁₆	C₄×C₄₁	semidirect
176. C₄₁	#22814	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₄₁	2
C₄₁×#2355	2
C₄₁×#1971	2
C₄₁×#1927	2
C₄₁×#1844	2
C₄₁×#1803	2
C₄₁×#1776	2
C₄₁×#1770	2
#22814	41

Subgroups — C41×#22814 #23799015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22814 #23799015