Subgroups — C₄₁×#21187 #23797388

Order 10496.

1670 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#408	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1954	C₂	1	8	yes	maximal, complemented
6	C₂²×#41×C₄₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#532	—	2	4	no	—
10	C₄₁×#532	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₄₁	C₄	1	1	yes	characteristic
12	C₂×#127×C₄₁	C₄	1	1	yes	characteristic
13	C₂×#124×C₄₁	C₄	1	2	yes	—
14	C₂×#121×C₄₁	C₄	1	4	yes	—
15	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
16	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#423	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
19	C₄₁×#408	—	2	16	no	—
20	C₂×C₂²⋊C₈×C₄₁	C₂²	1	16	yes	—
21	C₂×C₄×C₈×C₄₁	—	2	4	no	—
22	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
23	C₂×#41×C₄₁	—	2	4	no	—
24	C₂×#41×C₄₁	—	2	4	no	—
25	C₂×#41×C₄₁	—	2	8	no	—
26	C₂×#41×C₄₁	—	2	8	no	—
27	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
28	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
29	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
30	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₄₁	—	2	8	no	—
35	C₂×M₁₆×C₄₁	—	2	4	no	—
36	C₂²×C₈×C₄₁	—	2	4	no	—
37	C₂²×C₈×C₄₁	—	2	4	no	—
38	C₂²×C₈×C₄₁	D₄	1	8	yes	complemented
39	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
40	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₄₁	—	2	4	no	—
42	#127×C₄₁	—	2	4	no	—
43	#124×C₄₁	—	2	8	no	—
44	#121×C₄₁	—	2	16	no	—
45	C₄×D₄×C₄₁	—	2	8	no	—
46	C₄×D₄×C₄₁	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
49	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₁	D₄	1	4	yes	—
52	C₂×C₄²×C₄₁	D₄	1	4	yes	—
53	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
54	C₄⋊C₈×C₄₁	—	2	8	no	—
55	C₂²⋊C₈×C₄₁	—	2	32	no	—
56	C₄×C₈×C₄₁	—	2	8	no	—
57	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
58	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
59	#41×C₄₁	—	4	16	no	—
60	#41×C₄₁	—	4	8	no	—
61	#41×C₄₁	—	4	8	no	—
62	C₂×Q₈×C₄₁	—	2	4	no	—
63	C₂×Q₈×C₄₁	—	2	8	no	—
64	C₂×D₄×C₄₁	—	2	8	no	—
65	C₂×D₄×C₄₁	—	2	16	no	—
66	C₂×D₄×C₄₁	—	2	8	no	—
67	C₂×D₄×C₄₁	—	2	4	no	—
68	C₂×D₄×C₄₁	—	2	8	no	—
69	C₂×D₄×C₄₁	—	2	16	no	—
70	C₂²×C₄×C₄₁	—	2	8	no	—
71	C₂²×C₄×C₄₁	—	2	8	no	—
72	C₂²×C₄×C₄₁	—	2	4	no	—
73	C₂²×C₄×C₄₁	—	2	4	no	—
74	C₂²×C₄×C₄₁	—	2	8	no	—
75	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
76	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₁	#41	1	4	yes	—
84	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₄₁	—	4	8	no	—
86	C₂×C₈×C₄₁	—	2	16	no	—
87	C₂×C₈×C₄₁	—	2	4	no	—
88	C₂×C₈×C₄₁	—	2	4	no	—
89	C₂×C₈×C₄₁	—	2	8	no	—
90	C₂×C₈×C₄₁	—	2	4	no	—
91	C₂×C₈×C₄₁	C₂×D₄	1	8	yes	—
92	C₄⋊C₄×C₄₁	—	2	8	no	—
93	C₂²⋊C₄×C₄₁	—	2	16	no	—
94	C₄²×C₄₁	C₂²⋊C₄	1	4	yes	—
95	C₄²×C₄₁	C₂²⋊C₄	1	4	yes	—
96	C₄²×C₄₁	C₂×D₄	1	4	yes	—
97	C₄²×C₄₁	C₂×D₄	1	4	yes	—
98	C₂³×C₄₁	—	2	4	no	—
99	C₂³×C₄₁	—	2	4	no	—
100	C₂³×C₄₁	—	2	8	no	—
101	C₂³×C₄₁	—	2	16	no	—
102	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₄₁	C₄×D₄	1	4	yes	—
104	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
105	Q₈×C₄₁	—	4	8	no	—
106	D₄×C₄₁	—	4	16	no	—
107	D₄×C₄₁	—	4	8	no	—
108	D₄×C₄₁	—	4	16	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	16	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	8	no	—
113	C₂×C₄×C₄₁	—	4	8	no	—
114	C₂×C₄×C₄₁	—	4	4	no	—
115	C₂×C₄×C₄₁	—	4	4	no	—
116	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
120	C₂×C₄×C₄₁	#121	1	8	yes	—
121	C₂×C₄×C₄₁	#124	1	4	yes	—
122	C₂×C₄×C₄₁	#127	1	4	yes	—
123	C₂×C₄×C₄₁	#131	1	8	yes	—
124	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
128	C₈×C₄₁	—	2	8	no	—
129	C₈×C₄₁	—	4	4	no	—
130	#21187	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₄₁	—	2	4	no	—
132	C₂²×C₄₁	—	2	8	no	—
133	C₂²×C₄₁	—	4	8	no	—
134	C₂²×C₄₁	—	4	16	no	—
135	C₂²×C₄₁	—	4	4	no	—
136	C₂²×C₄₁	#390	1	4	yes	—
137	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₄₁	C₂×C₄×D₄	1	2	yes	—
139	C₂²×C₄₁	C₂×#131	1	4	yes	—
140	C₄×C₄₁	—	2	4	no	—
141	C₄×C₄₁	—	4	4	no	—
142	C₄×C₄₁	#390	1	4	yes	—
143	C₄×C₄₁	#408	1	8	yes	complemented
144	C₄×C₄₁	C₂×#121	1	2	yes	—
145	C₄×C₄₁	C₂×#124	1	1	yes	characteristic
146	C₄×C₄₁	C₂×#127	1	1	yes	characteristic
147	C₂×#532	C₂×C₄₁	1	1	yes	characteristic
148	C₂×#423	C₂×C₄₁	1	1	yes	characteristic, complemented
149	C₂×#408	C₂×C₄₁	1	4	yes	complemented
150	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
151	#1954	C₂×C₄₁	1	8	yes	complemented
152	C₂×C₄₁	—	2	4	no	—
153	C₂×C₄₁	—	8	4	no	—
154	C₂×C₄₁	C₂×#390	1	1	yes	characteristic
155	C₂×C₄₁	C₂×#408	1	2	yes	—
156	C₂×C₄₁	#2931	1	4	yes	—
157	C₂²×#41	C₂²×C₄₁	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
159	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
160	#532	—	2	4	no	—
161	#532	—	2	4	no	—
162	C₂×C₄⋊Q₈	C₄×C₄₁	1	1	yes	characteristic
163	C₂×#127	C₄×C₄₁	1	1	yes	characteristic
164	C₂×#124	C₄×C₄₁	1	2	yes	—
165	C₂×#121	C₄×C₄₁	1	4	yes	—
166	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
167	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
168	#423	—	2	4	no	—
169	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
170	#408	—	2	16	no	—
171	C₂×C₂²⋊C₈	C₂²×C₄₁	1	16	yes	—
172	C₂×C₄×C₈	—	2	4	no	—
173	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
174	C₄₁	#21187	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×#41	—	2	4	no	—
176	C₂×#41	—	2	4	no	—
177	C₂×#41	—	2	8	no	—
178	C₂×#41	—	2	8	no	—
179	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
180	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
181	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
182	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
184	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
185	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	8	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	D₄×C₄₁	1	8	yes	complemented
191	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
192	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
193	C₄⋊Q₈	—	2	4	no	—
194	#127	—	2	4	no	—
195	#124	—	2	8	no	—
196	#121	—	2	16	no	—
197	C₄×D₄	—	2	8	no	—
198	C₄×D₄	—	2	8	no	—
199	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
201	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₄₁	1	4	yes	—
204	C₂×C₄²	D₄×C₄₁	1	4	yes	—
205	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
206	C₄⋊C₈	—	2	8	no	—
207	C₂²⋊C₈	—	2	32	no	—
208	C₄×C₈	—	2	8	no	—
209	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
210	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
211	#41	—	4	16	no	—
212	#41	—	4	8	no	—
213	#41	—	4	8	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	8	no	—
216	C₂×D₄	—	2	8	no	—
217	C₂×D₄	—	2	16	no	—
218	C₂×D₄	—	2	8	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	8	no	—
221	C₂×D₄	—	2	16	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
228	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
235	C₂²×C₄	#41×C₄₁	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
237	M₁₆	—	4	8	no	—
238	C₂×C₈	—	2	16	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	8	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	C₂×D₄×C₄₁	1	8	yes	—
244	C₄⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	—	2	16	no	—
246	C₄²	C₂²⋊C₄×C₄₁	1	4	yes	—
247	C₄²	C₂²⋊C₄×C₄₁	1	4	yes	—
248	C₄²	C₂×D₄×C₄₁	1	4	yes	—
249	C₄²	C₂×D₄×C₄₁	1	4	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	8	no	—
253	C₂³	—	2	16	no	—
254	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
255	C₂³	C₄×D₄×C₄₁	1	4	yes	—
256	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
257	Q₈	—	4	8	no	—
258	D₄	—	4	16	no	—
259	D₄	—	4	8	no	—
260	D₄	—	4	16	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
271	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
272	C₂×C₄	#121×C₄₁	1	8	yes	—
273	C₂×C₄	#124×C₄₁	1	4	yes	—
274	C₂×C₄	#127×C₄₁	1	4	yes	—
275	C₂×C₄	#131×C₄₁	1	8	yes	—
276	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
278	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
280	C₈	—	2	8	no	—
281	C₈	—	4	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	8	no	—
286	C₂²	—	4	16	no	—
287	C₂²	C₄₁×#390	1	4	yes	—
288	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
289	C₂²	C₂×C₄×D₄×C₄₁	1	2	yes	—
290	C₂²	C₂×#131×C₄₁	1	4	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	4	4	no	—
293	C₄	C₄₁×#390	1	4	yes	—
294	C₄	C₄₁×#408	1	8	yes	complemented
295	C₄	C₂×#121×C₄₁	1	2	yes	—
296	C₄	C₂×#124×C₄₁	1	1	yes	characteristic
297	C₄	C₂×#127×C₄₁	1	1	yes	characteristic
298	C₂	—	2	4	no	—
299	C₂	—	8	4	no	—
300	C₂	C₂×C₄₁×#390	1	1	yes	characteristic
301	C₂	C₂×C₄₁×#408	1	2	yes	—
302	C₂	C₄₁×#2931	1	4	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	C₂×C₄₁×#532	C₂
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
2. C₂×C₄₁×#423	C₂	semidirect
3. C₂×C₄₁×#408	C₂	semidirect
4. C₂²×C₄×C₈×C₄₁	C₂	semidirect
5. C₄₁×#1954	C₂	semidirect
38. C₂²×C₈×C₄₁	D₄	semidirect
130. #21187	C₄₁	direct (N × H)
143. C₄×C₄₁	#408	semidirect
148. C₂×#423	C₂×C₄₁	semidirect
149. C₂×#408	C₂×C₄₁	semidirect
150. C₂²×C₄×C₈	C₂×C₄₁	semidirect
151. #1954	C₂×C₄₁	semidirect
174. C₄₁	#21187	direct (N × H)
190. C₂²×C₈	D₄×C₄₁	semidirect
294. C₄	C₄₁×#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#532	2
C₂×C₄₁×#423	2
C₂×C₄₁×#408	2
C₂²×C₄×C₈×C₄₁	2
C₄₁×#1954	2
#21187	41

Subgroups — C41×#21187 #23797388

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21187 #23797388