Subgroups — C₄₁×#21165 #23797366

Order 10496.

914 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#430	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2911	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2905	C₂	1	2	yes	maximal
4	C₂×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1946	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1945	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₁×#339	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×#117×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#430	—	2	4	no	—
14	C₂×#108×C₄₁	C₂²	1	4	yes	—
15	C₄₁×#428	C₄	1	8	yes	complemented
16	C₄₁×#427	C₂²	1	4	yes	—
17	C₄₁×#424	—	2	4	no	—
18	C₄₁×#424	C₂²	1	8	yes	—
19	C₄₁×#423	—	2	4	no	—
20	C₄×M₁₆×C₄₁	—	2	4	no	—
21	C₂×#97×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
23	C₄₁×#339	C₂²	1	4	yes	—
24	C₄₁×#339	C₂²	1	4	yes	—
25	C₂³×C₄×C₄₁	—	2	1	no	—
26	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₄₁	—	2	2	no	—
28	C₂×M₁₆×C₄₁	—	2	2	no	—
29	C₂×M₁₆×C₄₁	—	2	2	no	—
30	C₂×M₁₆×C₄₁	—	2	4	no	—
31	C₂×M₁₆×C₄₁	—	2	4	no	—
32	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₄₁	D₄	1	4	yes	—
34	C₂×M₁₆×C₄₁	D₄	1	4	yes	—
35	C₂×M₁₆×C₄₁	Q₈	1	4	yes	—
36	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
37	C₂²×C₈×C₄₁	—	2	2	no	—
38	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
40	#117×C₄₁	—	2	2	no	—
41	#117×C₄₁	—	2	2	no	—
42	#117×C₄₁	C₂×C₄	1	4	yes	—
43	#117×C₄₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	—	2	2	no	—
47	C₂×C₄²×C₄₁	—	2	2	no	—
48	C₂×C₄²×C₄₁	—	2	1	no	—
49	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	#108×C₄₁	—	2	8	no	—
53	#107×C₄₁	C₂×C₄	1	8	yes	complemented
54	#106×C₄₁	C₂×C₄	1	8	yes	complemented
55	C₄⋊C₈×C₄₁	—	2	16	no	—
56	#97×C₄₁	—	2	4	no	—
57	C₄×C₈×C₄₁	—	2	4	no	—
58	C₂⁴×C₄₁	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	—	2	1	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	1	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	4	4	no	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₁	#41	1	2	yes	—
71	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₄₁	—	2	8	no	—
73	M₁₆×C₄₁	—	2	4	no	—
74	M₁₆×C₄₁	—	2	4	no	—
75	M₁₆×C₄₁	—	4	4	no	—
76	M₁₆×C₄₁	C₄⋊C₄	1	16	yes	—
77	C₂×C₈×C₄₁	—	2	2	no	—
78	C₂×C₈×C₄₁	—	2	2	no	—
79	C₂×C₈×C₄₁	—	2	4	no	—
80	C₂×C₈×C₄₁	—	4	4	no	—
81	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₄₁	C₂×Q₈	1	4	yes	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
87	C₂²⋊C₄×C₄₁	—	2	4	no	—
88	C₂²⋊C₄×C₄₁	—	2	4	no	—
89	C₄²×C₄₁	—	2	4	no	—
90	C₄²×C₄₁	—	2	4	no	—
91	C₄²×C₄₁	—	2	1	no	—
92	C₄²×C₄₁	—	2	1	no	—
93	C₄²×C₄₁	—	2	4	no	—
94	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
95	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
96	C₂³×C₄₁	—	4	2	no	—
97	C₂³×C₄₁	C₄×D₄	1	2	yes	—
98	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₄₁	#125	1	2	yes	—
100	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	8	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	4	8	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	—	8	4	no	—
108	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₁	C₄×Q₈	1	4	yes	—
111	C₂×C₄×C₄₁	#121	1	4	yes	—
112	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₁	#125	1	2	yes	—
114	C₂×C₄×C₄₁	C₄⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
121	C₈×C₄₁	—	2	4	no	—
122	C₈×C₄₁	—	4	4	no	—
123	C₈×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
124	#21165	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₁	—	2	2	no	—
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	4	2	no	—
128	C₂²×C₄₁	—	4	2	no	—
129	C₂²×C₄₁	#389	1	4	yes	—
130	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄₁	C₂×#125	1	1	yes	characteristic
133	C₄×C₄₁	—	2	4	no	—
134	C₄×C₄₁	—	4	2	no	—
135	C₄×C₄₁	—	4	8	no	—
136	C₄×C₄₁	#389	1	4	yes	—
137	C₄×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₄₁	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
140	C₄×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×#430	C₂×C₄₁	1	1	yes	characteristic
142	#2911	C₂×C₄₁	1	1	yes	characteristic
143	#2905	C₂×C₄₁	1	2	yes	—
144	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic
145	#1946	C₂×C₄₁	1	4	yes	complemented
146	#1945	C₂×C₄₁	1	4	yes	complemented
147	C₂×#339	C₂×C₄₁	1	2	yes	—
148	C₂×C₄₁	—	2	1	no	—
149	C₂×C₄₁	—	2	1	no	—
150	C₂×C₄₁	—	2	4	no	—
151	C₂×C₄₁	C₂×#389	1	1	yes	characteristic
152	C₂×C₄₁	#2956	1	2	yes	—
153	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×#117	C₂²×C₄₁	1	2	yes	—
157	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
158	#430	—	2	4	no	—
159	C₂×#108	C₂²×C₄₁	1	4	yes	—
160	#428	C₄×C₄₁	1	8	yes	complemented
161	#427	C₂²×C₄₁	1	4	yes	—
162	#424	—	2	4	no	—
163	#424	C₂²×C₄₁	1	8	yes	—
164	#423	—	2	4	no	—
165	C₄×M₁₆	—	2	4	no	—
166	C₂×#97	C₂²×C₄₁	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
168	#339	C₂²×C₄₁	1	4	yes	—
169	#339	C₂²×C₄₁	1	4	yes	—
170	C₄₁	#21165	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	2	4	no	—
178	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
179	C₂×M₁₆	D₄×C₄₁	1	4	yes	—
180	C₂×M₁₆	D₄×C₄₁	1	4	yes	—
181	C₂×M₁₆	Q₈×C₄₁	1	4	yes	—
182	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
185	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×C₄×C₄₁	1	4	yes	—
189	#117	C₂³×C₄₁	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
196	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
198	#108	—	2	8	no	—
199	#107	C₂×C₄×C₄₁	1	8	yes	complemented
200	#106	C₂×C₄×C₄₁	1	8	yes	complemented
201	C₄⋊C₈	—	2	16	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	C₂⁴	#41×C₄₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
213	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
214	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
215	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	#41×C₄₁	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
218	M₁₆	—	2	8	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	4	no	—
222	M₁₆	C₄⋊C₄×C₄₁	1	16	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
229	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
230	C₂×C₈	C₂×Q₈×C₄₁	1	4	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	4	no	—
240	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
241	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×D₄×C₄₁	1	2	yes	—
244	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
245	C₂³	#125×C₄₁	1	2	yes	—
246	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
255	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
256	C₂×C₄	C₄×Q₈×C₄₁	1	4	yes	—
257	C₂×C₄	#121×C₄₁	1	4	yes	—
258	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
259	C₂×C₄	#125×C₄₁	1	2	yes	—
260	C₂×C₄	C₄⋊Q₈×C₄₁	1	4	yes	—
261	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
265	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	4	no	—
269	C₈	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	C₄₁×#389	1	4	yes	—
275	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
276	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
277	C₂²	C₂×#125×C₄₁	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	8	no	—
281	C₄	C₄₁×#389	1	4	yes	—
282	C₄	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
283	C₄	C₂×C₄×Q₈×C₄₁	1	1	yes	characteristic
284	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
285	C₄	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄₁×#389	1	1	yes	characteristic
290	C₂	C₄₁×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×C₄×D₄
prime radical	C₂⁴×C₄₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂²×M₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄₁×#1946	C₂	semidirect
6. C₄₁×#1945	C₂	semidirect
15. C₄₁×#428	C₄	semidirect
53. #107×C₄₁	C₂×C₄	semidirect
54. #106×C₄₁	C₂×C₄	semidirect
124. #21165	C₄₁	direct (N × H)
145. #1946	C₂×C₄₁	semidirect
146. #1945	C₂×C₄₁	semidirect
160. #428	C₄×C₄₁	semidirect
170. C₄₁	#21165	direct (N × H)
199. #107	C₂×C₄×C₄₁	semidirect
200. #106	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
C₂×C₄₁×#430	2
C₄₁×#2911	2
C₄₁×#2905	2
C₂×C₄×M₁₆×C₄₁	2
C₄₁×#1946	2
C₄₁×#1945	2
C₂×C₄₁×#339	2
#21165	41

Subgroups — C41×#21165 #23797366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21165 #23797366