Subgroups — C₄₁×#21934 #23798135

Order 10496.

1218 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#407	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1851	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1797	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₄₁	C₂²	1	4	yes	—
7	C₂²×C₂²⋊C₄×C₄₁	C₄	1	2	yes	—
8	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₄₁	C₂²	1	8	yes	complemented
10	C₄₁×#407	—	2	8	no	—
11	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
13	C₄×C₂²⋊C₄×C₄₁	C₄	1	4	yes	—
14	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
15	C₂×#95×C₄₁	C₄	1	1	yes	characteristic
16	C₂×#95×C₄₁	C₄	1	1	yes	characteristic
17	C₄₁×#336	C₂²	1	8	yes	complemented
18	C₂⁵×C₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₁	—	4	4	no	—
20	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₁	D₄	1	4	yes	—
25	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
26	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
28	C₂×M₁₆×C₄₁	—	2	8	no	—
29	C₂×M₁₆×C₄₁	—	4	8	no	—
30	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₄₁	—	4	4	no	—
34	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
36	C₄⋊C₈×C₄₁	—	4	8	no	—
37	C₂²⋊C₈×C₄₁	—	2	8	no	—
38	C₂²⋊C₈×C₄₁	—	2	8	no	—
39	#95×C₄₁	C₂×C₄	1	4	yes	—
40	#95×C₄₁	C₂×C₄	1	4	yes	—
41	C₂⁴×C₄₁	—	2	4	no	—
42	C₂⁴×C₄₁	—	4	2	no	—
43	C₂⁴×C₄₁	—	4	1	no	—
44	C₂⁴×C₄₁	—	4	1	no	—
45	C₂⁴×C₄₁	C₄⋊C₄	1	4	yes	—
46	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
47	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	—	2	4	no	—
49	C₂²×C₄×C₄₁	—	2	8	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	4	8	no	—
52	C₂²×C₄×C₄₁	—	4	8	no	—
53	C₂²×C₄×C₄₁	—	4	2	no	—
54	C₂²×C₄×C₄₁	—	4	2	no	—
55	C₂²×C₄×C₄₁	—	4	4	no	—
56	C₂²×C₄×C₄₁	—	4	2	no	—
57	C₂²×C₄×C₄₁	—	4	2	no	—
58	C₂²×C₄×C₄₁	—	4	2	no	—
59	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₁	M₁₆	1	8	yes	complemented
62	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₄₁	—	4	16	no	—
72	C₂×C₈×C₄₁	—	2	8	no	—
73	C₂×C₈×C₄₁	—	2	16	no	—
74	C₂²⋊C₄×C₄₁	—	2	16	no	—
75	C₄²×C₄₁	—	2	8	no	—
76	C₄²×C₄₁	—	4	4	no	—
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	1	no	—
79	C₂³×C₄₁	—	2	1	no	—
80	C₂³×C₄₁	—	2	4	no	—
81	C₂³×C₄₁	—	2	4	no	—
82	C₂³×C₄₁	—	2	2	no	—
83	C₂³×C₄₁	—	2	2	no	—
84	C₂³×C₄₁	—	2	8	no	—
85	C₂³×C₄₁	—	4	4	no	—
86	C₂³×C₄₁	—	4	4	no	—
87	C₂³×C₄₁	—	4	8	no	—
88	C₂³×C₄₁	—	4	8	no	—
89	C₂³×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
90	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₁	C₂×M₁₆	1	4	yes	—
92	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₁	—	2	4	no	—
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	4	no	—
97	C₂×C₄×C₄₁	—	2	8	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	4	no	—
100	C₂×C₄×C₄₁	—	2	8	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	4	4	no	—
105	C₂×C₄×C₄₁	—	4	8	no	—
106	C₂×C₄×C₄₁	—	4	8	no	—
107	C₂×C₄×C₄₁	—	4	8	no	—
108	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₁	C₂×M₁₆	1	8	yes	—
111	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
114	C₈×C₄₁	—	4	8	no	—
115	#21934	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₄₁	—	2	2	no	—
117	C₂²×C₄₁	—	2	2	no	—
118	C₂²×C₄₁	—	2	4	no	—
119	C₂²×C₄₁	—	2	4	no	—
120	C₂²×C₄₁	—	2	4	no	—
121	C₂²×C₄₁	—	2	4	no	—
122	C₂²×C₄₁	—	2	4	no	—
123	C₂²×C₄₁	—	4	16	no	—
124	C₂²×C₄₁	—	4	4	no	—
125	C₂²×C₄₁	—	4	4	no	—
126	C₂²×C₄₁	#423	1	4	yes	—
127	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₄₁	#518	1	2	yes	—
129	C₂²×C₄₁	C₂²×M₁₆	1	2	yes	—
130	C₂²×C₄₁	#568	1	1	yes	characteristic
131	C₂²×C₄₁	#568	1	1	yes	characteristic
132	C₄×C₄₁	—	2	2	no	—
133	C₄×C₄₁	—	2	4	no	—
134	C₄×C₄₁	—	2	8	no	—
135	C₄×C₄₁	—	4	4	no	—
136	C₄×C₄₁	#534	1	2	yes	—
137	C₄×C₄₁	#543	1	2	yes	—
138	C₂×#407	C₂×C₄₁	1	2	yes	—
139	C₂×C₄×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic
140	#1851	C₂×C₄₁	1	8	yes	complemented
141	#1797	C₂×C₄₁	1	4	yes	complemented
142	C₂×C₄₁	—	2	4	no	—
143	C₂×C₄₁	—	4	4	no	—
144	C₂×C₄₁	#2315	1	1	yes	characteristic
145	C₂×C₄₁	C₂×#423	1	1	yes	characteristic
146	C₂×C₄₁	#2905	1	1	yes	characteristic
147	C₂×C₄₁	#2973	1	2	yes	—
148	C₂×C₄₁	#2975	1	2	yes	—
149	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
150	C₂²×M₁₆	C₂²×C₄₁	1	4	yes	—
151	C₂²×C₂²⋊C₄	C₄×C₄₁	1	2	yes	—
152	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₄₁	1	8	yes	complemented
154	#407	—	2	8	no	—
155	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
156	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
157	C₄×C₂²⋊C₄	C₄×C₄₁	1	4	yes	—
158	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
159	C₂×#95	C₄×C₄₁	1	1	yes	characteristic
160	C₂×#95	C₄×C₄₁	1	1	yes	characteristic
161	#336	C₂²×C₄₁	1	8	yes	complemented
162	C₄₁	#21934	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁵	C₂×C₄×C₄₁	1	1	yes	characteristic
164	C₂³×C₄	—	4	4	no	—
165	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
166	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
167	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
168	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
169	C₂³×C₄	D₄×C₄₁	1	4	yes	—
170	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
171	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
172	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
173	C₂×M₁₆	—	2	8	no	—
174	C₂×M₁₆	—	4	8	no	—
175	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
176	C₂×C₂²⋊C₄	—	2	8	no	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
178	C₂×C₄²	—	4	4	no	—
179	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
180	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
181	C₄⋊C₈	—	4	8	no	—
182	C₂²⋊C₈	—	2	8	no	—
183	C₂²⋊C₈	—	2	8	no	—
184	#95	C₂×C₄×C₄₁	1	4	yes	—
185	#95	C₂×C₄×C₄₁	1	4	yes	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	4	2	no	—
188	C₂⁴	—	4	1	no	—
189	C₂⁴	—	4	1	no	—
190	C₂⁴	C₄⋊C₄×C₄₁	1	4	yes	—
191	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
192	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	8	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
205	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
206	C₂²×C₄	M₁₆×C₄₁	1	8	yes	complemented
207	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
208	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
211	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
213	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
214	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
216	M₁₆	—	4	16	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	16	no	—
219	C₂²⋊C₄	—	2	16	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	4	4	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	4	8	no	—
233	C₂³	—	4	8	no	—
234	C₂³	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
235	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
236	C₂³	C₂×M₁₆×C₄₁	1	4	yes	—
237	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
255	C₂×C₄	C₂×M₁₆×C₄₁	1	8	yes	—
256	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
258	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
259	C₈	—	4	8	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	16	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₄₁×#423	1	4	yes	—
271	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
272	C₂²	C₄₁×#518	1	2	yes	—
273	C₂²	C₂²×M₁₆×C₄₁	1	2	yes	—
274	C₂²	C₄₁×#568	1	1	yes	characteristic
275	C₂²	C₄₁×#568	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	8	no	—
279	C₄	—	4	4	no	—
280	C₄	C₄₁×#534	1	2	yes	—
281	C₄	C₄₁×#543	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	4	4	no	—
284	C₂	C₄₁×#2315	1	1	yes	characteristic
285	C₂	C₂×C₄₁×#423	1	1	yes	characteristic
286	C₂	C₄₁×#2905	1	1	yes	characteristic
287	C₂	C₄₁×#2973	1	2	yes	—
288	C₂	C₄₁×#2975	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂⁵×C₄₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1851	C₂	semidirect
4. C₄₁×#1797	C₂	semidirect
9. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
17. C₄₁×#336	C₂²	semidirect
61. C₂²×C₄×C₄₁	M₁₆	semidirect
115. #21934	C₄₁	direct (N × H)
140. #1851	C₂×C₄₁	semidirect
141. #1797	C₂×C₄₁	semidirect
153. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
161. #336	C₂²×C₄₁	semidirect
162. C₄₁	#21934	direct (N × H)
206. C₂²×C₄	M₁₆×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×#407	2
C₂×C₄×C₂²⋊C₄×C₄₁	2
C₄₁×#1851	2
C₄₁×#1797	2
#21934	41

Subgroups — C41×#21934 #23798135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21934 #23798135