Subgroups — C₄₁×#14685 #23790886

Order 10496.

2430 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#381	C₂	1	2	yes	maximal
3	C₂×C₄₁×#380	C₂	1	2	yes	maximal
4	C₄₁×C₄⋊#95	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1434	C₂	1	4	yes	maximal, complemented
6	C₄₁×C₂²⋊#95	C₂	1	2	yes	maximal
7	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
11	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₄²×C₄₁	—	2	8	no	—
13	C₂²×C₄²×C₄₁	C₄	1	8	yes	complemented
14	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#381	—	2	8	no	—
17	C₄₁×#380	—	2	8	no	—
18	C₂×#95×C₄₁	C₂²	1	4	yes	—
19	C₂×#95×C₄₁	C₂²	1	8	yes	—
20	C₂×#95×C₄₁	C₂²	1	8	yes	—
21	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	—	2	4	no	—
23	C₂³×C₄×C₄₁	—	2	4	no	—
24	C₂³×C₄×C₄₁	—	2	4	no	—
25	C₂³×C₄×C₄₁	—	2	8	no	—
26	C₂³×C₄×C₄₁	—	2	8	no	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₄₁	D₄	1	8	yes	—
31	C₂³×C₄×C₄₁	D₄	1	4	yes	—
32	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
36	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
40	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
41	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
42	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
43	C₂×C₄²×C₄₁	—	2	32	no	—
44	C₂×C₄²×C₄₁	—	2	8	no	—
45	C₂×C₄²×C₄₁	—	2	8	no	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
48	#95×C₄₁	—	2	8	no	—
49	#95×C₄₁	—	2	16	no	—
50	#95×C₄₁	—	2	16	no	—
51	C₂⁴×C₄₁	—	2	4	no	—
52	C₂⁴×C₄₁	—	2	4	no	—
53	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
54	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
55	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
56	C₂⁴×C₄₁	C₂×Q₈	1	2	yes	—
57	C₂⁴×C₄₁	#41	1	4	yes	—
58	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	—	2	32	no	—
60	C₂²×C₄×C₄₁	—	2	8	no	—
61	C₂²×C₄×C₄₁	—	2	8	no	—
62	C₂²×C₄×C₄₁	—	2	8	no	—
63	C₂²×C₄×C₄₁	—	2	16	no	—
64	C₂²×C₄×C₄₁	—	2	16	no	—
65	C₂²×C₄×C₄₁	—	2	4	no	—
66	C₂²×C₄×C₄₁	—	2	4	no	—
67	C₂²×C₄×C₄₁	—	2	4	no	—
68	C₂²×C₄×C₄₁	—	2	4	no	—
69	C₂²×C₄×C₄₁	—	2	4	no	—
70	C₂²×C₄×C₄₁	—	2	16	no	—
71	C₂²×C₄×C₄₁	—	2	16	no	—
72	C₂²×C₄×C₄₁	—	2	32	no	—
73	C₂²×C₄×C₄₁	—	2	8	no	—
74	C₂²×C₄×C₄₁	—	2	8	no	—
75	C₂²×C₄×C₄₁	—	2	16	no	—
76	C₂²×C₄×C₄₁	—	2	8	no	—
77	C₂²×C₄×C₄₁	—	2	16	no	—
78	C₂²×C₄×C₄₁	C₂²⋊C₄	1	16	yes	—
79	C₂²×C₄×C₄₁	C₄⋊C₄	1	16	yes	complemented
80	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₄₁	#41	1	8	yes	—
89	C₂²×C₄×C₄₁	#41	1	8	yes	—
90	C₄⋊C₄×C₄₁	—	4	16	no	—
91	C₂²⋊C₄×C₄₁	—	4	16	no	—
92	C₄²×C₄₁	—	2	16	no	—
93	C₄²×C₄₁	—	2	16	no	—
94	C₂³×C₄₁	—	2	4	no	—
95	C₂³×C₄₁	—	2	4	no	—
96	C₂³×C₄₁	—	2	8	no	—
97	C₂³×C₄₁	—	2	8	no	—
98	C₂³×C₄₁	—	2	16	no	—
99	C₂³×C₄₁	—	2	8	no	—
100	C₂³×C₄₁	—	2	8	no	—
101	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂³×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₄₁	#117	1	4	yes	—
104	C₂³×C₄₁	#121	1	8	yes	—
105	C₂³×C₄₁	C₂²⋊Q₈	1	8	yes	—
106	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
107	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
108	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
109	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂³×C₄₁	C₂×#41	1	2	yes	—
111	C₂³×C₄₁	C₂×#41	1	4	yes	—
112	C₂³×C₄₁	C₂×#41	1	4	yes	—
113	C₂×C₄×C₄₁	—	2	32	no	—
114	C₂×C₄×C₄₁	—	2	4	no	—
115	C₂×C₄×C₄₁	—	2	4	no	—
116	C₂×C₄×C₄₁	—	2	4	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	2	8	no	—
120	C₂×C₄×C₄₁	—	2	8	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	16	no	—
123	C₂×C₄×C₄₁	—	2	32	no	—
124	C₂×C₄×C₄₁	—	2	32	no	—
125	C₂×C₄×C₄₁	—	4	8	no	—
126	C₂×C₄×C₄₁	—	4	8	no	—
127	C₂×C₄×C₄₁	—	4	8	no	—
128	C₂×C₄×C₄₁	—	4	16	no	—
129	C₂×C₄×C₄₁	—	4	16	no	—
130	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
131	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
132	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
133	C₂×C₄×C₄₁	#117	1	8	yes	—
134	C₂×C₄×C₄₁	#117	1	8	yes	—
135	C₂×C₄×C₄₁	#123	1	16	yes	—
136	#14685	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₁	—	2	16	no	—
138	C₂²×C₄₁	—	2	16	no	—
139	C₂²×C₄₁	—	2	4	no	—
140	C₂²×C₄₁	—	2	4	no	—
141	C₂²×C₄₁	—	2	8	no	—
142	C₂²×C₄₁	—	2	8	no	—
143	C₂²×C₄₁	#380	1	8	yes	—
144	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
147	C₂²×C₄₁	C₂×#117	1	2	yes	—
148	C₂²×C₄₁	C₂×#117	1	2	yes	—
149	C₂²×C₄₁	C₂×#121	1	2	yes	—
150	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
151	C₂²×C₄₁	C₂×#123	1	4	yes	—
152	C₂²×C₄₁	#525	1	4	yes	—
153	C₂²×C₄₁	#529	1	8	yes	—
154	C₂²×C₄₁	#532	1	4	yes	—
155	C₂²×C₄₁	#533	1	4	yes	—
156	C₄×C₄₁	—	2	8	no	—
157	C₄×C₄₁	—	2	4	no	—
158	C₄×C₄₁	—	2	8	no	—
159	C₄×C₄₁	—	8	8	no	—
160	C₄×C₄₁	#381	1	8	yes	complemented
161	C₂³×C₄²	C₂×C₄₁	1	1	yes	characteristic
162	C₂×#381	C₂×C₄₁	1	2	yes	—
163	C₂×#380	C₂×C₄₁	1	2	yes	—
164	C₄⋊#95	C₂×C₄₁	1	4	yes	complemented
165	#1434	C₂×C₄₁	1	4	yes	complemented
166	C₂²⋊#95	C₂×C₄₁	1	2	yes	—
167	C₂×C₄₁	—	2	8	no	—
168	C₂×C₄₁	C₂×#380	1	1	yes	characteristic
169	C₂×C₄₁	C₂×#381	1	1	yes	characteristic
170	C₂×C₄₁	#2280	1	1	yes	characteristic
171	C₂×C₄₁	#2283	1	2	yes	—
172	C₂×C₄₁	#2285	1	2	yes	—
173	C₂×C₄₁	#2854	1	4	yes	—
174	C₂×C₄₁	#2855	1	4	yes	—
175	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
176	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
177	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
179	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
180	C₂²×C₄²	—	2	8	no	—
181	C₂²×C₄²	C₄×C₄₁	1	8	yes	complemented
182	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
183	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
184	#381	—	2	8	no	—
185	#380	—	2	8	no	—
186	C₂×#95	C₂²×C₄₁	1	8	yes	—
187	C₂×#95	C₂²×C₄₁	1	4	yes	—
188	C₂×#95	C₂²×C₄₁	1	8	yes	—
189	C₄₁	#14685	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
191	C₂³×C₄	—	2	4	no	—
192	C₂³×C₄	—	2	4	no	—
193	C₂³×C₄	—	2	4	no	—
194	C₂³×C₄	—	2	8	no	—
195	C₂³×C₄	—	2	8	no	—
196	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
197	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
198	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
199	C₂³×C₄	D₄×C₄₁	1	8	yes	—
200	C₂³×C₄	D₄×C₄₁	1	4	yes	—
201	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
202	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
203	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
205	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
206	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	2	8	no	—
208	C₂×C₄⋊C₄	—	2	16	no	—
209	C₂×C₂²⋊C₄	—	2	8	no	—
210	C₂×C₂²⋊C₄	—	2	8	no	—
211	C₂×C₂²⋊C₄	—	2	8	no	—
212	C₂×C₄²	—	2	32	no	—
213	C₂×C₄²	—	2	8	no	—
214	C₂×C₄²	—	2	8	no	—
215	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
216	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
217	#95	—	2	8	no	—
218	#95	—	2	16	no	—
219	#95	—	2	16	no	—
220	C₂⁴	—	2	4	no	—
221	C₂⁴	—	2	4	no	—
222	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
223	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
224	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
225	C₂⁴	C₂×Q₈×C₄₁	1	2	yes	—
226	C₂⁴	#41×C₄₁	1	4	yes	—
227	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	—	2	32	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	16	no	—
233	C₂²×C₄	—	2	16	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	16	no	—
240	C₂²×C₄	—	2	16	no	—
241	C₂²×C₄	—	2	32	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	16	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	16	no	—
247	C₂²×C₄	C₂²⋊C₄×C₄₁	1	16	yes	—
248	C₂²×C₄	C₄⋊C₄×C₄₁	1	16	yes	complemented
249	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
251	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
252	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
254	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
255	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
256	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
257	C₂²×C₄	#41×C₄₁	1	8	yes	—
258	C₂²×C₄	#41×C₄₁	1	8	yes	—
259	C₄⋊C₄	—	4	16	no	—
260	C₂²⋊C₄	—	4	16	no	—
261	C₄²	—	2	16	no	—
262	C₄²	—	2	16	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	8	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	16	no	—
268	C₂³	—	2	8	no	—
269	C₂³	—	2	8	no	—
270	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
271	C₂³	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
272	C₂³	#117×C₄₁	1	4	yes	—
273	C₂³	#121×C₄₁	1	8	yes	—
274	C₂³	C₂²⋊Q₈×C₄₁	1	8	yes	—
275	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
276	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
277	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
278	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
279	C₂³	C₂×#41×C₄₁	1	2	yes	—
280	C₂³	C₂×#41×C₄₁	1	4	yes	—
281	C₂³	C₂×#41×C₄₁	1	4	yes	—
282	C₂×C₄	—	2	32	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	16	no	—
292	C₂×C₄	—	2	32	no	—
293	C₂×C₄	—	2	32	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	16	no	—
298	C₂×C₄	—	4	16	no	—
299	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
301	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
302	C₂×C₄	#117×C₄₁	1	8	yes	—
303	C₂×C₄	#117×C₄₁	1	8	yes	—
304	C₂×C₄	#123×C₄₁	1	16	yes	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	8	no	—
308	C₂²	—	2	8	no	—
309	C₂²	—	2	16	no	—
310	C₂²	—	2	16	no	—
311	C₂²	C₄₁×#380	1	8	yes	—
312	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
313	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
314	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
315	C₂²	C₂×#117×C₄₁	1	2	yes	—
316	C₂²	C₂×#117×C₄₁	1	2	yes	—
317	C₂²	C₂×#121×C₄₁	1	2	yes	—
318	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
319	C₂²	C₂×#123×C₄₁	1	4	yes	—
320	C₂²	C₄₁×#525	1	4	yes	—
321	C₂²	C₄₁×#529	1	8	yes	—
322	C₂²	C₄₁×#532	1	4	yes	—
323	C₂²	C₄₁×#533	1	4	yes	—
324	C₄	—	2	8	no	—
325	C₄	—	2	4	no	—
326	C₄	—	2	8	no	—
327	C₄	—	8	8	no	—
328	C₄	C₄₁×#381	1	8	yes	complemented
329	C₂	—	2	8	no	—
330	C₂	C₂×C₄₁×#380	1	1	yes	characteristic
331	C₂	C₂×C₄₁×#381	1	1	yes	characteristic
332	C₂	C₄₁×#2280	1	1	yes	characteristic
333	C₂	C₄₁×#2283	1	2	yes	—
334	C₂	C₄₁×#2285	1	2	yes	—
335	C₂	C₄₁×#2854	1	4	yes	—
336	C₂	C₄₁×#2855	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₁×C₄⋊#95	C₂	semidirect
5. C₄₁×#1434	C₂	semidirect
13. C₂²×C₄²×C₄₁	C₄	semidirect
79. C₂²×C₄×C₄₁	C₄⋊C₄	semidirect
136. #14685	C₄₁	direct (N × H)
160. C₄×C₄₁	#381	semidirect
164. C₄⋊#95	C₂×C₄₁	semidirect
165. #1434	C₂×C₄₁	semidirect
181. C₂²×C₄²	C₄×C₄₁	semidirect
189. C₄₁	#14685	direct (N × H)
248. C₂²×C₄	C₄⋊C₄×C₄₁	semidirect
328. C₄	C₄₁×#381	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₄₁	2
C₂×C₄₁×#381	2
C₂×C₄₁×#380	2
C₄₁×C₄⋊#95	2
C₄₁×#1434	2
C₄₁×C₂²⋊#95	2
#14685	41

Subgroups — C41×#14685 #23790886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14685 #23790886