Subgroups — C₄₁×#13753 #23789954

Order 10496.

2638 conjugacy classes of subgroups.

The subgroup lattice has 318 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×C₄×C₂²⋊C₄×C₄₁	C₂	1	2	yes	maximal
3	C₂×C₄₁×#376	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1434	C₂	1	4	yes	maximal, complemented
5	C₄₁×C₂²⋊#95	C₂	1	4	yes	maximal
6	C₄×#95×C₄₁	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
10	C₂²×C₄²×C₄₁	—	2	8	no	—
11	C₂²×C₄²×C₄₁	C₄	1	8	yes	complemented
12	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₁	C₂²	1	4	yes	—
15	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₄₁	C₄	1	16	yes	complemented
17	C₄₁×#376	—	2	8	no	—
18	C₂×#95×C₄₁	C₂²	1	8	yes	—
19	C₂×#95×C₄₁	C₂²	1	8	yes	—
20	C₂×#95×C₄₁	C₂²	1	4	yes	—
21	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	—	2	8	no	—
23	C₂³×C₄×C₄₁	—	2	8	no	—
24	C₂³×C₄×C₄₁	—	2	4	no	—
25	C₂³×C₄×C₄₁	—	2	8	no	—
26	C₂³×C₄×C₄₁	—	2	4	no	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
30	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
31	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
36	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
37	C₂×C₄²×C₄₁	—	2	8	no	—
38	C₂×C₄²×C₄₁	—	2	4	no	—
39	C₂×C₄²×C₄₁	—	2	8	no	—
40	C₂×C₄²×C₄₁	—	2	8	no	—
41	C₂×C₄²×C₄₁	—	2	32	no	—
42	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
43	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₁	C₂×C₄	1	16	yes	complemented
46	C₂×C₄²×C₄₁	D₄	1	8	yes	—
47	#95×C₄₁	—	2	16	no	—
48	#95×C₄₁	—	2	16	no	—
49	#95×C₄₁	C₂×C₄	1	16	yes	complemented
50	C₂⁴×C₄₁	—	2	2	no	—
51	C₂⁴×C₄₁	—	2	2	no	—
52	C₂⁴×C₄₁	—	2	4	no	—
53	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
55	C₂⁴×C₄₁	#41	1	8	yes	—
56	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	—	2	64	no	—
58	C₂²×C₄×C₄₁	—	2	16	no	—
59	C₂²×C₄×C₄₁	—	2	2	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	2	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	—	2	4	no	—
65	C₂²×C₄×C₄₁	—	2	16	no	—
66	C₂²×C₄×C₄₁	—	2	8	no	—
67	C₂²×C₄×C₄₁	—	2	16	no	—
68	C₂²×C₄×C₄₁	—	2	8	no	—
69	C₂²×C₄×C₄₁	—	2	32	no	—
70	C₂²×C₄×C₄₁	—	2	8	no	—
71	C₂²×C₄×C₄₁	—	2	16	no	—
72	C₂²×C₄×C₄₁	—	2	8	no	—
73	C₂²×C₄×C₄₁	—	2	16	no	—
74	C₂²×C₄×C₄₁	C₄²	1	16	yes	—
75	C₂²×C₄×C₄₁	C₂²×C₄	1	16	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	4	yes	—
79	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₄₁	#41	1	8	yes	—
85	C₂²×C₄×C₄₁	#41	1	4	yes	—
86	C₂²⋊C₄×C₄₁	—	2	32	no	—
87	C₄²×C₄₁	—	2	16	no	—
88	C₄²×C₄₁	—	2	8	no	—
89	C₄²×C₄₁	—	2	32	no	—
90	C₄²×C₄₁	C₂²⋊C₄	1	16	yes	complemented
91	C₂³×C₄₁	—	2	2	no	—
92	C₂³×C₄₁	—	2	2	no	—
93	C₂³×C₄₁	—	2	4	no	—
94	C₂³×C₄₁	—	2	8	no	—
95	C₂³×C₄₁	—	2	8	no	—
96	C₂³×C₄₁	—	2	16	no	—
97	C₂³×C₄₁	—	2	16	no	—
98	C₂³×C₄₁	C₂×C₄²	1	4	yes	—
99	C₂³×C₄₁	#117	1	16	yes	—
100	C₂³×C₄₁	#117	1	8	yes	—
101	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
102	C₂³×C₄₁	C₂³×C₄	1	2	yes	—
103	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
104	C₂³×C₄₁	C₂×#41	1	4	yes	—
105	C₂³×C₄₁	C₂×#41	1	2	yes	—
106	C₂³×C₄₁	C₂×#41	1	4	yes	—
107	C₂×C₄×C₄₁	—	2	64	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	2	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	8	no	—
115	C₂×C₄×C₄₁	—	2	16	no	—
116	C₂×C₄×C₄₁	—	2	16	no	—
117	C₂×C₄×C₄₁	—	2	32	no	—
118	C₂×C₄×C₄₁	—	2	16	no	—
119	C₂×C₄×C₄₁	—	2	16	no	—
120	C₂×C₄×C₄₁	—	2	16	no	—
121	C₂×C₄×C₄₁	—	4	16	no	—
122	C₂×C₄×C₄₁	—	4	16	no	—
123	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
125	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
126	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₄₁	#117	1	8	yes	—
128	C₂×C₄×C₄₁	#117	1	4	yes	—
129	C₂×C₄×C₄₁	C₄×D₄	1	16	yes	—
130	#13753	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₄₁	—	2	4	no	—
132	C₂²×C₄₁	—	2	4	no	—
133	C₂²×C₄₁	—	2	8	no	—
134	C₂²×C₄₁	—	2	8	no	—
135	C₂²×C₄₁	—	2	32	no	—
136	C₂²×C₄₁	#376	1	8	yes	complemented
137	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₁	C₂×#117	1	2	yes	—
140	C₂²×C₄₁	C₂×#117	1	4	yes	—
141	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
142	C₂²×C₄₁	C₂×#117	1	2	yes	—
143	C₂²×C₄₁	C₂×C₄×D₄	1	4	yes	—
144	C₂²×C₄₁	C₄×#41	1	4	yes	—
145	C₂²×C₄₁	#525	1	8	yes	—
146	C₂²×C₄₁	#529	1	8	yes	—
147	C₄×C₄₁	—	2	4	no	—
148	C₄×C₄₁	—	2	8	no	—
149	C₄×C₄₁	—	2	8	no	—
150	C₄×C₄₁	—	4	16	no	—
151	C₄×C₄₁	C₄×C₂²⋊C₄	1	8	yes	—
152	C₂³×C₄²	C₂×C₄₁	1	1	yes	characteristic
153	C₂×C₄×C₂²⋊C₄	C₂×C₄₁	1	2	yes	—
154	C₂×#376	C₂×C₄₁	1	2	yes	complemented
155	#1434	C₂×C₄₁	1	4	yes	complemented
156	C₂²⋊#95	C₂×C₄₁	1	4	yes	—
157	C₄×#95	C₂×C₄₁	1	2	yes	complemented
158	C₂×C₄₁	—	2	8	no	—
159	C₂×C₄₁	C₂×#376	1	1	yes	characteristic
160	C₂×C₄₁	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄₁	C₄×#117	1	1	yes	characteristic
162	C₂×C₄₁	#2280	1	2	yes	—
163	C₂×C₄₁	#2283	1	2	yes	—
164	C₂×C₄₁	#2290	1	8	yes	—
165	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
166	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
167	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
168	C₂²×C₄²	—	2	8	no	—
169	C₂²×C₄²	C₄×C₄₁	1	8	yes	complemented
170	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
171	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
172	C₂²×C₄²	C₂²×C₄₁	1	4	yes	—
173	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
174	C₄×C₂²⋊C₄	C₄×C₄₁	1	16	yes	complemented
175	#376	—	2	8	no	—
176	C₂×#95	C₂²×C₄₁	1	8	yes	—
177	C₂×#95	C₂²×C₄₁	1	8	yes	—
178	C₂×#95	C₂²×C₄₁	1	4	yes	—
179	C₄₁	#13753	1	1	yes	characteristic, direct product factor, Sylow
180	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
181	C₂³×C₄	—	2	8	no	—
182	C₂³×C₄	—	2	8	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	2	4	no	—
186	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
187	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
188	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
189	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
190	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
191	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
192	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
193	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	—	2	16	no	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
196	C₂×C₄²	—	2	8	no	—
197	C₂×C₄²	—	2	8	no	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	—	2	8	no	—
200	C₂×C₄²	—	2	32	no	—
201	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
202	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
203	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
204	C₂×C₄²	C₂×C₄×C₄₁	1	16	yes	complemented
205	C₂×C₄²	D₄×C₄₁	1	8	yes	—
206	#95	—	2	16	no	—
207	#95	—	2	16	no	—
208	#95	C₂×C₄×C₄₁	1	16	yes	complemented
209	C₂⁴	—	2	2	no	—
210	C₂⁴	—	2	2	no	—
211	C₂⁴	—	2	4	no	—
212	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
213	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
214	C₂⁴	#41×C₄₁	1	8	yes	—
215	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
216	C₂²×C₄	—	2	64	no	—
217	C₂²×C₄	—	2	16	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	16	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	16	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	32	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	2	16	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	16	no	—
233	C₂²×C₄	C₄²×C₄₁	1	16	yes	—
234	C₂²×C₄	C₂²×C₄×C₄₁	1	16	yes	—
235	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
236	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
239	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
241	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
242	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
243	C₂²×C₄	#41×C₄₁	1	8	yes	—
244	C₂²×C₄	#41×C₄₁	1	4	yes	—
245	C₂²⋊C₄	—	2	32	no	—
246	C₄²	—	2	16	no	—
247	C₄²	—	2	8	no	—
248	C₄²	—	2	32	no	—
249	C₄²	C₂²⋊C₄×C₄₁	1	16	yes	complemented
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	8	no	—
254	C₂³	—	2	8	no	—
255	C₂³	—	2	16	no	—
256	C₂³	—	2	16	no	—
257	C₂³	C₂×C₄²×C₄₁	1	4	yes	—
258	C₂³	#117×C₄₁	1	16	yes	—
259	C₂³	#117×C₄₁	1	8	yes	—
260	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
261	C₂³	C₂³×C₄×C₄₁	1	2	yes	—
262	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
263	C₂³	C₂×#41×C₄₁	1	4	yes	—
264	C₂³	C₂×#41×C₄₁	1	2	yes	—
265	C₂³	C₂×#41×C₄₁	1	4	yes	—
266	C₂×C₄	—	2	64	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	32	no	—
277	C₂×C₄	—	2	16	no	—
278	C₂×C₄	—	2	16	no	—
279	C₂×C₄	—	2	16	no	—
280	C₂×C₄	—	4	16	no	—
281	C₂×C₄	—	4	16	no	—
282	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
283	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
285	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
286	C₂×C₄	#117×C₄₁	1	8	yes	—
287	C₂×C₄	#117×C₄₁	1	4	yes	—
288	C₂×C₄	C₄×D₄×C₄₁	1	16	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	32	no	—
294	C₂²	C₄₁×#376	1	8	yes	complemented
295	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
296	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
297	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
298	C₂²	C₂×#117×C₄₁	1	2	yes	—
299	C₂²	C₂×#117×C₄₁	1	2	yes	—
300	C₂²	C₂×#117×C₄₁	1	4	yes	—
301	C₂²	C₂×C₄×D₄×C₄₁	1	4	yes	—
302	C₂²	C₄×#41×C₄₁	1	4	yes	—
303	C₂²	C₄₁×#525	1	8	yes	—
304	C₂²	C₄₁×#529	1	8	yes	—
305	C₄	—	2	4	no	—
306	C₄	—	2	8	no	—
307	C₄	—	2	8	no	—
308	C₄	—	4	16	no	—
309	C₄	C₄×C₂²⋊C₄×C₄₁	1	8	yes	—
310	C₂	—	2	8	no	—
311	C₂	C₂×C₄₁×#376	1	1	yes	characteristic
312	C₂	C₂×C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
313	C₂	C₄×#117×C₄₁	1	1	yes	characteristic
314	C₂	C₄₁×#2280	1	2	yes	—
315	C₂	C₄₁×#2283	1	2	yes	—
316	C₂	C₄₁×#2290	1	8	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₁	C₂³
commutator	C₂²	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
socle	C₂⁴×C₄₁	C₂⁴
prime radical	C₂⁵×C₄₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄₁×#376	C₂	semidirect
4. C₄₁×#1434	C₂	semidirect
6. C₄×#95×C₄₁	C₂	semidirect
11. C₂²×C₄²×C₄₁	C₄	semidirect
16. C₄×C₂²⋊C₄×C₄₁	C₄	semidirect
45. C₂×C₄²×C₄₁	C₂×C₄	semidirect
49. #95×C₄₁	C₂×C₄	semidirect
90. C₄²×C₄₁	C₂²⋊C₄	semidirect
130. #13753	C₄₁	direct (N × H)
136. C₂²×C₄₁	#376	semidirect
154. C₂×#376	C₂×C₄₁	semidirect
155. #1434	C₂×C₄₁	semidirect
157. C₄×#95	C₂×C₄₁	semidirect
169. C₂²×C₄²	C₄×C₄₁	semidirect
174. C₄×C₂²⋊C₄	C₄×C₄₁	semidirect
179. C₄₁	#13753	direct (N × H)
204. C₂×C₄²	C₂×C₄×C₄₁	semidirect
208. #95	C₂×C₄×C₄₁	semidirect
249. C₄²	C₂²⋊C₄×C₄₁	semidirect
294. C₂²	C₄₁×#376	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×C₄×C₂²⋊C₄×C₄₁	2
C₂×C₄₁×#376	2
C₄₁×#1434	2
C₄₁×C₂²⋊#95	2
C₄×#95×C₄₁	2
#13753	41

Subgroups — C41×#13753 #23789954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13753 #23789954