Subgroups — C₄₁×#10878 #23787079

Order 10496.

1062 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2296	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄³×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#339	C₂	1	2	yes	maximal
4	C₂×C₄₁×#339	C₂	1	2	yes	maximal
5	C₄₁×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
7	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₁	—	2	2	no	—
10	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#389	C₄	1	4	yes	complemented
13	C₄₁×#385	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
15	C₄³×C₄₁	—	2	4	no	—
16	C₄₁×#339	—	2	4	no	—
17	C₄₁×#339	C₄	1	8	yes	complemented
18	C₄₁×#336	—	2	2	no	—
19	C₂³×C₄×C₄₁	—	2	2	no	—
20	C₂³×C₄×C₄₁	D₄	1	2	yes	—
21	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₁	—	2	2	no	—
23	C₂×M₁₆×C₄₁	—	2	2	no	—
24	C₂×M₁₆×C₄₁	—	2	2	no	—
25	C₂×M₁₆×C₄₁	—	2	2	no	—
26	C₂×M₁₆×C₄₁	—	2	2	no	—
27	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₄₁	—	2	2	no	—
29	#117×C₄₁	—	2	2	no	—
30	#117×C₄₁	—	2	2	no	—
31	#117×C₄₁	—	2	2	no	—
32	#117×C₄₁	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
38	C₂×C₄²×C₄₁	—	2	16	no	—
39	C₂×C₄²×C₄₁	—	2	8	no	—
40	C₂×C₄²×C₄₁	—	2	2	no	—
41	C₂×C₄²×C₄₁	—	2	2	no	—
42	C₂×C₄²×C₄₁	—	2	1	no	—
43	C₂×C₄²×C₄₁	—	2	2	no	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	—	2	1	no	—
47	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₁	D₄	1	2	yes	—
50	C₂×C₄²×C₄₁	D₄	1	4	yes	—
51	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
52	#95×C₄₁	—	2	4	no	—
53	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	—	2	16	no	—
55	C₂²×C₄×C₄₁	—	2	4	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	2	2	no	—
60	C₂²×C₄×C₄₁	—	2	8	no	—
61	C₂²×C₄×C₄₁	—	2	2	no	—
62	C₂²×C₄×C₄₁	—	2	2	no	—
63	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₁	#41	1	2	yes	—
70	C₂²×C₄×C₄₁	#41	1	2	yes	—
71	C₂²×C₄×C₄₁	#41	1	2	yes	—
72	M₁₆×C₄₁	—	2	4	no	—
73	M₁₆×C₄₁	—	2	4	no	—
74	M₁₆×C₄₁	—	4	2	no	—
75	M₁₆×C₄₁	—	4	2	no	—
76	C₂×C₈×C₄₁	—	2	4	no	—
77	C₂×C₈×C₄₁	—	4	2	no	—
78	C₂×C₈×C₄₁	—	4	2	no	—
79	C₄⋊C₄×C₄₁	—	2	2	no	—
80	C₄⋊C₄×C₄₁	—	2	4	no	—
81	C₄⋊C₄×C₄₁	—	4	8	no	—
82	C₄⋊C₄×C₄₁	C₄²	1	4	yes	complemented
83	C₂²⋊C₄×C₄₁	—	2	4	no	—
84	C₂²⋊C₄×C₄₁	—	4	1	no	—
85	C₂²⋊C₄×C₄₁	—	4	1	no	—
86	C₄²×C₄₁	—	2	32	no	—
87	C₄²×C₄₁	—	2	4	no	—
88	C₄²×C₄₁	—	2	4	no	—
89	C₄²×C₄₁	—	2	4	no	—
90	C₄²×C₄₁	—	2	4	no	—
91	C₄²×C₄₁	—	2	4	no	—
92	C₄²×C₄₁	—	2	4	no	—
93	C₄²×C₄₁	—	4	2	no	—
94	C₄²×C₄₁	C₄⋊C₄	1	8	yes	complemented
95	C₂³×C₄₁	—	2	4	no	—
96	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₁	C₄×D₄	1	2	yes	—
98	C₂³×C₄₁	#120	1	1	yes	characteristic
99	C₂³×C₄₁	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄₁	#123	1	1	yes	characteristic
101	C₂×C₄×C₄₁	—	2	16	no	—
102	C₂×C₄×C₄₁	—	2	8	no	—
103	C₂×C₄×C₄₁	—	2	16	no	—
104	C₂×C₄×C₄₁	—	2	16	no	—
105	C₂×C₄×C₄₁	—	2	8	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	2	4	no	—
108	C₂×C₄×C₄₁	—	2	4	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	4	4	no	—
111	C₂×C₄×C₄₁	—	4	2	no	—
112	C₂×C₄×C₄₁	—	4	4	no	—
113	C₂×C₄×C₄₁	#104	1	8	yes	—
114	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₁	#117	1	2	yes	—
117	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₁	#121	1	2	yes	—
124	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₁	#123	1	2	yes	—
126	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₁	#125	1	2	yes	—
128	C₂×C₄×C₄₁	#126	1	2	yes	—
129	C₂×C₄×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
130	C₈×C₄₁	—	4	4	no	—
131	#10878	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₄₁	—	2	2	no	—
133	C₂²×C₄₁	—	2	2	no	—
134	C₂²×C₄₁	—	2	4	no	—
135	C₂²×C₄₁	—	2	4	no	—
136	C₂²×C₄₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₁	#385	1	2	yes	—
138	C₂²×C₄₁	#386	1	1	yes	characteristic
139	C₂²×C₄₁	#387	1	2	yes	—
140	C₂²×C₄₁	#391	1	1	yes	characteristic
141	C₂²×C₄₁	C₂×#104	1	4	yes	—
142	C₄×C₄₁	—	2	16	no	—
143	C₄×C₄₁	—	2	8	no	—
144	C₄×C₄₁	—	4	4	no	—
145	C₄×C₄₁	—	8	2	no	—
146	C₄×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₄₁	#387	1	2	yes	—
148	C₄×C₄₁	#388	1	2	yes	—
149	C₄×C₄₁	#389	1	2	yes	—
150	#2296	C₂×C₄₁	1	1	yes	characteristic
151	C₂×C₄³	C₂×C₄₁	1	1	yes	characteristic
152	C₂×#339	C₂×C₄₁	1	2	yes	—
153	C₂×#339	C₂×C₄₁	1	2	yes	—
154	#1728	C₂×C₄₁	1	1	yes	characteristic
155	C₂×C₄₁	—	2	2	no	—
156	C₂×C₄₁	—	2	2	no	—
157	C₂×C₄₁	#1436	1	1	yes	characteristic
158	C₂×C₄₁	C₄×#104	1	2	yes	—
159	C₂×C₄₁	#1896	1	2	yes	—
160	C₂×C₄₁	#1994	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
162	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
164	C₂²×C₄²	—	2	2	no	—
165	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
166	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
167	#389	C₄×C₄₁	1	4	yes	complemented
168	#385	—	2	2	no	—
169	C₄×C₄⋊C₄	—	2	2	no	—
170	C₄³	—	2	4	no	—
171	#339	—	2	4	no	—
172	#339	C₄×C₄₁	1	8	yes	complemented
173	#336	—	2	2	no	—
174	C₄₁	#10878	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	D₄×C₄₁	1	2	yes	—
177	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	C₂×C₄×C₄₁	1	4	yes	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₂²⋊C₄	—	2	1	no	—
194	C₂×C₄²	—	2	8	no	—
195	C₂×C₄²	—	2	16	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
204	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
205	C₂×C₄²	D₄×C₄₁	1	2	yes	—
206	C₂×C₄²	D₄×C₄₁	1	4	yes	—
207	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
208	#95	—	2	4	no	—
209	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
225	C₂²×C₄	#41×C₄₁	1	2	yes	—
226	C₂²×C₄	#41×C₄₁	1	2	yes	—
227	C₂²×C₄	#41×C₄₁	1	2	yes	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	C₄²×C₄₁	1	4	yes	complemented
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	1	no	—
242	C₄²	—	2	32	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₄⋊C₄×C₄₁	1	8	yes	complemented
251	C₂³	—	2	4	no	—
252	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
253	C₂³	C₄×D₄×C₄₁	1	2	yes	—
254	C₂³	#120×C₄₁	1	1	yes	characteristic
255	C₂³	C₂²⋊Q₈×C₄₁	1	2	yes	—
256	C₂³	#123×C₄₁	1	1	yes	characteristic
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	16	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	#104×C₄₁	1	8	yes	—
270	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
272	C₂×C₄	#117×C₄₁	1	2	yes	—
273	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
275	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
277	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
278	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
279	C₂×C₄	#121×C₄₁	1	2	yes	—
280	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
281	C₂×C₄	#123×C₄₁	1	2	yes	—
282	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
283	C₂×C₄	#125×C₄₁	1	2	yes	—
284	C₂×C₄	#126×C₄₁	1	2	yes	—
285	C₂×C₄	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
292	C₂²	C₄₁×#385	1	2	yes	—
293	C₂²	C₄₁×#386	1	1	yes	characteristic
294	C₂²	C₄₁×#387	1	2	yes	—
295	C₂²	C₄₁×#391	1	1	yes	characteristic
296	C₂²	C₂×#104×C₄₁	1	4	yes	—
297	C₄	—	2	16	no	—
298	C₄	—	2	8	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	2	no	—
301	C₄	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
302	C₄	C₄₁×#387	1	2	yes	—
303	C₄	C₄₁×#388	1	2	yes	—
304	C₄	C₄₁×#389	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	C₄₁×#1436	1	1	yes	characteristic
308	C₂	C₄×#104×C₄₁	1	2	yes	—
309	C₂	C₄₁×#1896	1	2	yes	—
310	C₂	C₄₁×#1994	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
12. C₄₁×#389	C₄	semidirect
17. C₄₁×#339	C₄	semidirect
82. C₄⋊C₄×C₄₁	C₄²	semidirect
94. C₄²×C₄₁	C₄⋊C₄	semidirect
131. #10878	C₄₁	direct (N × H)
167. #389	C₄×C₄₁	semidirect
172. #339	C₄×C₄₁	semidirect
174. C₄₁	#10878	direct (N × H)
238. C₄⋊C₄	C₄²×C₄₁	semidirect
250. C₄²	C₄⋊C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#2296	2
C₂×C₄³×C₄₁	2
C₂×C₄₁×#339	2
C₄₁×#1728	2
#10878	41

Subgroups — C41×#10878 #23787079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#10878 #23787079