Subgroups — C₄₁×#23628 #23799829

Order 10496.

1226 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#418	C₂	1	2	yes	maximal
2	C₂×C₄₁×#390	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2073	C₂	1	4	yes	maximal, complemented
5	C₄₁×#2071	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1736	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
10	C₂×#117×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₄₁	—	2	2	no	—
13	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#418	—	2	8	no	—
15	C₂×#102×C₄₁	C₂²	1	8	yes	—
16	C₄₁×#390	—	2	2	no	—
17	C₄₁×#390	C₂²	1	2	yes	—
18	C₄₁×#390	C₂²	1	2	yes	—
19	C₄₁×#384	C₂²	1	2	yes	complemented
20	C₄₁×#382	C₂²	1	2	yes	complemented
21	C₄₁×#380	—	2	2	no	—
22	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
23	C₄₁×#340	C₂²	1	8	yes	complemented
24	C₄₁×#336	C₂²	1	2	yes	complemented
25	C₄₁×#336	C₂²	1	2	yes	complemented
26	C₂⁵×C₄₁	—	2	1	no	—
27	C₂²×D₄×C₄₁	—	2	1	no	—
28	C₂²×D₄×C₄₁	—	2	4	no	—
29	C₂²×D₄×C₄₁	—	2	4	no	—
30	C₂²×D₄×C₄₁	—	2	2	no	—
31	C₂²×D₄×C₄₁	D₄	1	4	yes	—
32	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₁	—	2	1	no	—
34	C₂³×C₄×C₄₁	D₄	1	2	yes	—
35	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₁	—	2	4	no	—
37	C₂×M₁₆×C₄₁	—	2	4	no	—
38	C₂×M₁₆×C₄₁	—	2	4	no	—
39	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
40	#117×C₄₁	—	2	4	no	—
41	#117×C₄₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
51	C₂×C₂²⋊C₄×C₄₁	—	4	8	no	—
52	C₂×C₄²×C₄₁	—	2	4	no	—
53	C₂×C₄²×C₄₁	—	2	2	no	—
54	C₂×C₄²×C₄₁	D₄	1	2	yes	—
55	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
57	#102×C₄₁	—	2	16	no	—
58	#95×C₄₁	—	2	4	no	—
59	C₂⁴×C₄₁	—	2	4	no	—
60	C₂⁴×C₄₁	—	2	2	no	—
61	C₂⁴×C₄₁	—	4	4	no	—
62	C₂⁴×C₄₁	—	4	4	no	—
63	C₂⁴×C₄₁	—	4	2	no	—
64	C₂⁴×C₄₁	—	4	2	no	—
65	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
66	C₂×D₄×C₄₁	—	2	4	no	—
67	C₂×D₄×C₄₁	—	2	16	no	—
68	C₂×D₄×C₄₁	—	2	4	no	—
69	C₂×D₄×C₄₁	—	2	4	no	—
70	C₂×D₄×C₄₁	—	2	2	no	—
71	C₂×D₄×C₄₁	—	4	2	no	—
72	C₂×D₄×C₄₁	—	4	2	no	—
73	C₂×D₄×C₄₁	—	4	8	no	—
74	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₁	—	2	4	no	—
76	C₂²×C₄×C₄₁	—	2	4	no	—
77	C₂²×C₄×C₄₁	—	2	4	no	—
78	C₂²×C₄×C₄₁	—	2	2	no	—
79	C₂²×C₄×C₄₁	—	4	2	no	—
80	C₂²×C₄×C₄₁	—	4	4	no	—
81	C₂²×C₄×C₄₁	—	4	2	no	—
82	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₁	#41	1	2	yes	—
87	C₂²×C₄×C₄₁	#41	1	4	yes	—
88	C₂²×C₄×C₄₁	#41	1	4	yes	—
89	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₄₁	—	4	8	no	—
91	C₂×C₈×C₄₁	—	2	4	no	—
92	C₂×C₈×C₄₁	—	2	4	no	—
93	C₂×C₈×C₄₁	—	2	4	no	—
94	C₄⋊C₄×C₄₁	—	2	4	no	—
95	C₄⋊C₄×C₄₁	—	2	4	no	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	4	4	no	—
98	C₄⋊C₄×C₄₁	—	4	2	no	—
99	C₄⋊C₄×C₄₁	—	4	4	no	—
100	C₂²⋊C₄×C₄₁	—	4	8	no	—
101	C₂²⋊C₄×C₄₁	—	4	4	no	—
102	C₂²⋊C₄×C₄₁	—	4	8	no	—
103	C₄²×C₄₁	—	2	4	no	—
104	C₄²×C₄₁	—	2	2	no	—
105	C₄²×C₄₁	—	2	2	no	—
106	C₄²×C₄₁	—	4	4	no	—
107	C₂³×C₄₁	—	2	2	no	—
108	C₂³×C₄₁	—	2	2	no	—
109	C₂³×C₄₁	—	2	4	no	—
110	C₂³×C₄₁	—	4	16	no	—
111	C₂³×C₄₁	—	4	2	no	—
112	C₂³×C₄₁	—	4	2	no	—
113	C₂³×C₄₁	—	4	4	no	—
114	C₂³×C₄₁	—	4	4	no	—
115	C₂³×C₄₁	—	4	8	no	—
116	C₂³×C₄₁	—	8	8	no	—
117	C₂³×C₄₁	—	8	4	no	—
118	C₂³×C₄₁	—	8	4	no	—
119	C₂³×C₄₁	#121	1	2	yes	—
120	C₂³×C₄₁	#123	1	2	yes	—
121	C₂³×C₄₁	#124	1	2	yes	—
122	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
123	D₄×C₄₁	—	2	8	no	—
124	D₄×C₄₁	—	4	4	no	—
125	D₄×C₄₁	—	4	8	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	2	2	no	—
128	C₂×C₄×C₄₁	—	2	2	no	—
129	C₂×C₄×C₄₁	—	2	2	no	—
130	C₂×C₄×C₄₁	—	2	2	no	—
131	C₂×C₄×C₄₁	—	4	8	no	—
132	C₂×C₄×C₄₁	—	4	4	no	—
133	C₂×C₄×C₄₁	—	4	2	no	—
134	C₂×C₄×C₄₁	—	4	4	no	—
135	C₂×C₄×C₄₁	—	4	4	no	—
136	C₂×C₄×C₄₁	—	4	4	no	—
137	C₂×C₄×C₄₁	—	8	4	no	—
138	C₂×C₄×C₄₁	#121	1	2	yes	—
139	C₂×C₄×C₄₁	#123	1	8	yes	—
140	C₂×C₄×C₄₁	#123	1	2	yes	—
141	C₂×C₄×C₄₁	#124	1	2	yes	—
142	C₂×C₄×C₄₁	#126	1	8	yes	—
143	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
144	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
146	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
147	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₄₁	—	4	4	no	—
149	#23628	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₄₁	—	2	4	no	—
151	C₂²×C₄₁	—	2	2	no	—
152	C₂²×C₄₁	—	2	2	no	—
153	C₂²×C₄₁	—	2	4	no	—
154	C₂²×C₄₁	—	4	4	no	—
155	C₂²×C₄₁	—	4	8	no	—
156	C₂²×C₄₁	—	8	16	no	—
157	C₂²×C₄₁	—	8	4	no	—
158	C₂²×C₄₁	—	8	2	no	—
159	C₂²×C₄₁	—	8	2	no	—
160	C₂²×C₄₁	#397	1	4	yes	—
161	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
162	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
163	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
164	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
165	C₂²×C₄₁	#575	1	2	yes	—
166	C₄×C₄₁	—	2	4	no	—
167	C₄×C₄₁	—	4	2	no	—
168	C₄×C₄₁	—	8	2	no	—
169	C₄×C₄₁	—	8	4	no	—
170	C₄×C₄₁	#397	1	4	yes	—
171	C₄×C₄₁	C₂×#123	1	2	yes	—
172	C₄×C₄₁	C₂×#126	1	2	yes	—
173	C₂×#418	C₂×C₄₁	1	2	yes	—
174	C₂×#390	C₂×C₄₁	1	1	yes	characteristic
175	#2285	C₂×C₄₁	1	1	yes	characteristic, complemented
176	#2073	C₂×C₄₁	1	4	yes	complemented
177	#2071	C₂×C₄₁	1	4	yes	complemented
178	#1736	C₂×C₄₁	1	2	yes	complemented
179	#1728	C₂×C₄₁	1	1	yes	characteristic, complemented
180	C₂×C₄₁	—	2	4	no	—
181	C₂×C₄₁	—	8	4	no	—
182	C₂×C₄₁	C₂×#397	1	1	yes	characteristic
183	C₂×C₄₁	#3036	1	2	yes	—
184	C₂×C₄₁	#3094	1	2	yes	—
185	C₂×C₄₁	#3138	1	1	yes	characteristic
186	C₂×C₄₁	#3140	1	1	yes	characteristic
187	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
188	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
189	C₂×#117	C₂²×C₄₁	1	2	yes	—
190	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	—	2	2	no	—
192	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
193	#418	—	2	8	no	—
194	C₂×#102	C₂²×C₄₁	1	8	yes	—
195	#390	—	2	2	no	—
196	#390	C₂²×C₄₁	1	2	yes	—
197	#390	C₂²×C₄₁	1	2	yes	—
198	#384	C₂²×C₄₁	1	2	yes	complemented
199	#382	C₂²×C₄₁	1	2	yes	complemented
200	#380	—	2	2	no	—
201	C₄×C₄⋊C₄	—	2	2	no	—
202	#340	C₂²×C₄₁	1	8	yes	complemented
203	#336	C₂²×C₄₁	1	2	yes	complemented
204	#336	C₂²×C₄₁	1	2	yes	complemented
205	C₄₁	#23628	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁵	—	2	1	no	—
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	4	no	—
209	C₂²×D₄	—	2	4	no	—
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	D₄×C₄₁	1	4	yes	—
212	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
213	C₂³×C₄	—	2	1	no	—
214	C₂³×C₄	D₄×C₄₁	1	2	yes	—
215	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
216	C₂×M₁₆	—	2	4	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	—	2	4	no	—
219	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
220	#117	—	2	4	no	—
221	#117	—	2	4	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₂²⋊C₄	—	4	8	no	—
232	C₂×C₄²	—	2	4	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	D₄×C₄₁	1	2	yes	—
235	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
237	#102	—	2	16	no	—
238	#95	—	2	4	no	—
239	C₂⁴	—	2	4	no	—
240	C₂⁴	—	2	2	no	—
241	C₂⁴	—	4	4	no	—
242	C₂⁴	—	4	4	no	—
243	C₂⁴	—	4	2	no	—
244	C₂⁴	—	4	2	no	—
245	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	16	no	—
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	8	no	—
254	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
264	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
266	C₂²×C₄	#41×C₄₁	1	2	yes	—
267	C₂²×C₄	#41×C₄₁	1	4	yes	—
268	C₂²×C₄	#41×C₄₁	1	4	yes	—
269	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
270	M₁₆	—	4	8	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	8	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	8	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	4	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	4	no	—
290	C₂³	—	4	16	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	4	8	no	—
296	C₂³	—	8	8	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	8	4	no	—
299	C₂³	#121×C₄₁	1	2	yes	—
300	C₂³	#123×C₄₁	1	2	yes	—
301	C₂³	#124×C₄₁	1	2	yes	—
302	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
303	D₄	—	2	8	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	8	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	#121×C₄₁	1	2	yes	—
319	C₂×C₄	#123×C₄₁	1	8	yes	—
320	C₂×C₄	#123×C₄₁	1	2	yes	—
321	C₂×C₄	#124×C₄₁	1	2	yes	—
322	C₂×C₄	#126×C₄₁	1	8	yes	—
323	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
324	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
325	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
326	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
327	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
328	C₈	—	4	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	16	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₄₁×#397	1	4	yes	—
340	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
341	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
342	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
343	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
344	C₂²	C₄₁×#575	1	2	yes	—
345	C₄	—	2	4	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	2	no	—
349	C₄	C₄₁×#397	1	4	yes	—
350	C₄	C₂×#123×C₄₁	1	2	yes	—
351	C₄	C₂×#126×C₄₁	1	2	yes	—
352	C₂	—	2	4	no	—
353	C₂	—	8	4	no	—
354	C₂	C₂×C₄₁×#397	1	1	yes	characteristic
355	C₂	C₄₁×#3036	1	2	yes	—
356	C₂	C₄₁×#3094	1	2	yes	—
357	C₂	C₄₁×#3138	1	1	yes	characteristic
358	C₂	C₄₁×#3140	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂³×D₄×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#2285	C₂	semidirect
4. C₄₁×#2073	C₂	semidirect
5. C₄₁×#2071	C₂	semidirect
6. C₄₁×#1736	C₂	semidirect
7. C₄₁×#1728	C₂	semidirect
19. C₄₁×#384	C₂²	semidirect
20. C₄₁×#382	C₂²	semidirect
23. C₄₁×#340	C₂²	semidirect
24. C₄₁×#336	C₂²	semidirect
25. C₄₁×#336	C₂²	semidirect
149. #23628	C₄₁	direct (N × H)
175. #2285	C₂×C₄₁	semidirect
176. #2073	C₂×C₄₁	semidirect
177. #2071	C₂×C₄₁	semidirect
178. #1736	C₂×C₄₁	semidirect
179. #1728	C₂×C₄₁	semidirect
198. #384	C₂²×C₄₁	semidirect
199. #382	C₂²×C₄₁	semidirect
202. #340	C₂²×C₄₁	semidirect
203. #336	C₂²×C₄₁	semidirect
204. #336	C₂²×C₄₁	semidirect
205. C₄₁	#23628	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×C₄₁×#418	2
C₂×C₄₁×#390	2
C₄₁×#2285	2
C₄₁×#2073	2
C₄₁×#2071	2
C₄₁×#1736	2
C₄₁×#1728	2
#23628	41

Subgroups — C41×#23628 #23799829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23628 #23799829