Subgroups — C₄₁×#23269 #23799470

Order 10496.

852 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1984	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1923	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1869	C₂	1	2	yes	maximal
5	C₈×C₂²⋊C₄×C₄₁	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1744	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
8	C₂×#124×C₄₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#391	C₄	1	1	yes	characteristic
17	C₄₁×#390	C₄	1	1	yes	characteristic
18	C₄₁×#388	C₄	1	4	yes	—
19	C₄₁×#382	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
21	C₄³×C₄₁	C₂²	1	1	yes	characteristic
22	C₄₁×#336	C₂²	1	4	yes	—
23	C₄₁×#336	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	4	yes	—
25	C₄₁×#336	C₂²	1	2	yes	—
26	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₄₁	—	2	8	no	—
30	C₂²×C₈×C₄₁	—	4	2	no	—
31	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
33	#124×C₄₁	C₈	1	8	yes	complemented
34	C₄×Q₈×C₄₁	—	4	1	no	—
35	C₄×Q₈×C₄₁	—	4	1	no	—
36	C₄×D₄×C₄₁	—	4	1	no	—
37	C₄×D₄×C₄₁	—	4	1	no	—
38	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
50	C₄⋊C₈×C₄₁	—	4	2	no	—
51	C₂²⋊C₈×C₄₁	—	2	8	no	—
52	C₂²⋊C₈×C₄₁	—	4	2	no	—
53	C₄×C₈×C₄₁	—	2	8	no	—
54	C₄×C₈×C₄₁	—	4	2	no	—
55	#95×C₄₁	C₂×C₄	1	4	yes	—
56	#95×C₄₁	C₂×C₄	1	4	yes	—
57	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
58	C₂×Q₈×C₄₁	—	4	1	no	—
59	C₂×Q₈×C₄₁	—	4	1	no	—
60	C₂×Q₈×C₄₁	C₂×C₈	1	4	yes	complemented
61	C₂×D₄×C₄₁	—	4	1	no	—
62	C₂×D₄×C₄₁	—	4	1	no	—
63	C₂×D₄×C₄₁	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₄₁	—	2	4	no	—
65	C₂²×C₄×C₄₁	—	2	4	no	—
66	C₂²×C₄×C₄₁	—	2	2	no	—
67	C₂²×C₄×C₄₁	—	2	2	no	—
68	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₄₁	—	2	4	no	—
77	C₂×C₈×C₄₁	—	2	4	no	—
78	C₂×C₈×C₄₁	—	2	4	no	—
79	C₂×C₈×C₄₁	—	2	4	no	—
80	C₂×C₈×C₄₁	—	2	4	no	—
81	C₂×C₈×C₄₁	—	4	8	no	—
82	C₂×C₈×C₄₁	#41	1	8	yes	—
83	C₄⋊C₄×C₄₁	—	4	1	no	—
84	C₄⋊C₄×C₄₁	—	4	1	no	—
85	C₄⋊C₄×C₄₁	—	4	2	no	—
86	C₂²⋊C₄×C₄₁	—	2	8	no	—
87	C₂²⋊C₄×C₄₁	—	4	2	no	—
88	C₂²⋊C₄×C₄₁	C₂×C₈	1	16	yes	complemented
89	C₄²×C₄₁	—	2	4	no	—
90	C₄²×C₄₁	—	2	4	no	—
91	C₄²×C₄₁	—	2	2	no	—
92	C₄²×C₄₁	—	2	2	no	—
93	C₄²×C₄₁	—	2	1	no	—
94	C₄²×C₄₁	—	2	1	no	—
95	C₄²×C₄₁	—	4	1	no	—
96	C₄²×C₄₁	C₂×C₈	1	4	yes	complemented
97	C₂³×C₄₁	—	2	4	no	—
98	C₂³×C₄₁	—	2	2	no	—
99	C₂³×C₄₁	—	2	4	no	—
100	C₂³×C₄₁	—	2	2	no	—
101	C₂³×C₄₁	C₂²×C₈	1	4	yes	—
102	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
103	Q₈×C₄₁	—	4	4	no	—
104	D₄×C₄₁	—	4	4	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	2	no	—
107	C₂×C₄×C₄₁	—	2	1	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	1	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	1	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	2	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	4	2	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	C₂²×C₈	1	4	yes	—
123	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₄₁	C₂²×C₈	1	4	yes	—
125	C₂×C₄×C₄₁	#131	1	4	yes	—
126	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
127	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
129	C₈×C₄₁	—	2	8	no	—
130	C₈×C₄₁	—	4	4	no	—
131	#23269	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₄₁	—	2	8	no	—
133	C₂²×C₄₁	—	2	4	no	—
134	C₂²×C₄₁	—	4	2	no	—
135	C₂²×C₄₁	—	4	4	no	—
136	C₂²×C₄₁	—	4	2	no	—
137	C₂²×C₄₁	C₄×#41	1	2	yes	—
138	C₂²×C₄₁	#518	1	1	yes	characteristic
139	C₂²×C₄₁	C₂³×C₈	1	1	yes	characteristic
140	C₂²×C₄₁	C₂×#131	1	2	yes	—
141	C₂²×C₄₁	#568	1	1	yes	characteristic
142	C₄×C₄₁	—	2	4	no	—
143	C₄×C₄₁	—	2	4	no	—
144	C₄×C₄₁	—	2	2	no	—
145	C₄×C₄₁	—	2	4	no	—
146	C₄×C₄₁	—	4	2	no	—
147	C₄×C₄₁	—	4	1	no	—
148	C₄×C₄₁	—	4	2	no	—
149	C₄×C₄₁	#551	1	2	yes	—
150	C₄×C₄₁	#555	1	2	yes	—
151	C₄×#124	C₂×C₄₁	1	1	yes	characteristic
152	#1984	C₂×C₄₁	1	2	yes	complemented
153	#1923	C₂×C₄₁	1	4	yes	complemented
154	#1869	C₂×C₄₁	1	2	yes	—
155	C₈×C₂²⋊C₄	C₂×C₄₁	1	4	yes	complemented
156	#1744	C₂×C₄₁	1	2	yes	complemented
157	C₂×C₄₁	—	4	4	no	—
158	C₂×C₄₁	#2372	1	1	yes	characteristic
159	C₂×C₄₁	C₈×#41	1	2	yes	—
160	C₂×C₄₁	#2968	1	2	yes	—
161	C₂×C₄₁	#2972	1	1	yes	characteristic
162	C₂×C₄₁	#2979	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
164	C₂×#124	C₄×C₄₁	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
166	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
171	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
172	#391	C₄×C₄₁	1	1	yes	characteristic
173	#390	C₄×C₄₁	1	1	yes	characteristic
174	#388	C₄×C₄₁	1	4	yes	—
175	#382	C₄×C₄₁	1	1	yes	characteristic
176	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
177	C₄³	C₂²×C₄₁	1	1	yes	characteristic
178	#336	C₂²×C₄₁	1	4	yes	—
179	#336	C₂²×C₄₁	1	2	yes	—
180	#336	C₂²×C₄₁	1	4	yes	—
181	#336	C₂²×C₄₁	1	2	yes	—
182	C₄₁	#23269	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
184	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
186	C₂²×C₈	—	2	8	no	—
187	C₂²×C₈	—	4	2	no	—
188	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
189	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
190	#124	C₈×C₄₁	1	8	yes	complemented
191	C₄×Q₈	—	4	1	no	—
192	C₄×Q₈	—	4	1	no	—
193	C₄×D₄	—	4	1	no	—
194	C₄×D₄	—	4	1	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
198	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
206	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
207	C₄⋊C₈	—	4	2	no	—
208	C₂²⋊C₈	—	2	8	no	—
209	C₂²⋊C₈	—	4	2	no	—
210	C₄×C₈	—	2	8	no	—
211	C₄×C₈	—	4	2	no	—
212	#95	C₂×C₄×C₄₁	1	4	yes	—
213	#95	C₂×C₄×C₄₁	1	4	yes	—
214	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	C₂×C₈×C₄₁	1	4	yes	complemented
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	C₂×C₈×C₄₁	1	4	yes	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₄₁	1	2	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	C₂⁴×C₄₁	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	8	no	—
239	C₂×C₈	#41×C₄₁	1	8	yes	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	C₂×C₈×C₄₁	1	16	yes	complemented
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	1	no	—
253	C₄²	C₂×C₈×C₄₁	1	4	yes	complemented
254	C₂³	—	2	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂²×C₈×C₄₁	1	4	yes	—
259	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	4	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	C₂²×C₈×C₄₁	1	4	yes	—
280	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
281	C₂×C₄	C₂²×C₈×C₄₁	1	4	yes	—
282	C₂×C₄	#131×C₄₁	1	4	yes	—
283	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
284	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
285	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
286	C₈	—	2	8	no	—
287	C₈	—	4	4	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₄×#41×C₄₁	1	2	yes	—
294	C₂²	C₄₁×#518	1	1	yes	characteristic
295	C₂²	C₂³×C₈×C₄₁	1	1	yes	characteristic
296	C₂²	C₂×#131×C₄₁	1	2	yes	—
297	C₂²	C₄₁×#568	1	1	yes	characteristic
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	C₄₁×#551	1	2	yes	—
306	C₄	C₄₁×#555	1	2	yes	—
307	C₂	—	4	4	no	—
308	C₂	C₄₁×#2372	1	1	yes	characteristic
309	C₂	C₈×#41×C₄₁	1	2	yes	—
310	C₂	C₄₁×#2968	1	2	yes	—
311	C₂	C₄₁×#2972	1	1	yes	characteristic
312	C₂	C₄₁×#2979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂²×D₄×C₄₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1984	C₂	semidirect
3. C₄₁×#1923	C₂	semidirect
5. C₈×C₂²⋊C₄×C₄₁	C₂	semidirect
6. C₄₁×#1744	C₂	semidirect
33. #124×C₄₁	C₈	semidirect
60. C₂×Q₈×C₄₁	C₂×C₈	semidirect
88. C₂²⋊C₄×C₄₁	C₂×C₈	semidirect
96. C₄²×C₄₁	C₂×C₈	semidirect
131. #23269	C₄₁	direct (N × H)
152. #1984	C₂×C₄₁	semidirect
153. #1923	C₂×C₄₁	semidirect
155. C₈×C₂²⋊C₄	C₂×C₄₁	semidirect
156. #1744	C₂×C₄₁	semidirect
182. C₄₁	#23269	direct (N × H)
190. #124	C₈×C₄₁	semidirect
217. C₂×Q₈	C₂×C₈×C₄₁	semidirect
245. C₂²⋊C₄	C₂×C₈×C₄₁	semidirect
253. C₄²	C₂×C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#124×C₄₁	2
C₄₁×#1984	2
C₄₁×#1923	2
C₄₁×#1869	2
C₈×C₂²⋊C₄×C₄₁	2
C₄₁×#1744	2
#23269	41

Subgroups — C41×#23269 #23799470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23269 #23799470