Subgroups — C₄₁×#20347 #23796548

Order 10496.

2270 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₄₁	C₂	1	2	yes	maximal
3	C₄₁×#1736	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1735	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₄₁	—	2	8	no	—
7	C₂²×M₁₆×C₄₁	C₄	1	8	yes	complemented
8	C₂²×M₁₆×C₄₁	C₂²	1	4	yes	—
9	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#117×C₄₁	—	2	8	no	—
11	C₂×#117×C₄₁	—	2	2	no	—
12	C₂×#117×C₄₁	—	2	2	no	—
13	C₂×#117×C₄₁	C₄	1	16	yes	complemented
14	C₂×#117×C₄₁	C₂²	1	4	yes	—
15	C₂×#117×C₄₁	C₂²	1	4	yes	—
16	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₄₁	—	2	2	no	—
18	C₂²×C₄²×C₄₁	—	2	2	no	—
19	C₄₁×#340	C₂²	1	16	yes	complemented
20	C₂⁵×C₄₁	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₄₁	—	2	4	no	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	D₄	1	2	yes	—
26	C₂³×C₄×C₄₁	D₄	1	8	yes	—
27	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₄₁	—	2	32	no	—
31	C₂×M₁₆×C₄₁	—	2	8	no	—
32	C₂×M₁₆×C₄₁	—	2	8	no	—
33	C₂×M₁₆×C₄₁	C₂×C₄	1	16	yes	complemented
34	C₂²×C₈×C₄₁	—	2	8	no	—
35	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	complemented
36	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
37	#117×C₄₁	—	2	16	no	—
38	#117×C₄₁	—	2	16	no	—
39	#117×C₄₁	—	2	8	no	—
40	#117×C₄₁	—	2	8	no	—
41	#117×C₄₁	C₂×C₄	1	16	yes	—
42	#117×C₄₁	C₂×C₄	1	16	yes	complemented
43	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
45	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	complemented
46	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
48	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
50	C₂×C₄²×C₄₁	—	2	16	no	—
51	C₂×C₄²×C₄₁	—	2	4	no	—
52	C₂×C₄²×C₄₁	—	2	4	no	—
53	C₂⁴×C₄₁	—	4	4	no	—
54	C₂⁴×C₄₁	C₂²⋊C₄	1	8	yes	—
55	C₂⁴×C₄₁	C₄⋊C₄	1	4	yes	—
56	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
57	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₁	—	2	16	no	—
59	C₂²×C₄×C₄₁	—	2	8	no	—
60	C₂²×C₄×C₄₁	—	4	16	no	—
61	C₂²×C₄×C₄₁	C₄²	1	16	yes	—
62	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
63	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	16	yes	—
66	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
68	C₂²×C₄×C₄₁	C₄⋊C₄	1	32	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₄₁	—	2	32	no	—
83	C₂×C₈×C₄₁	—	2	16	no	—
84	C₂×C₈×C₄₁	—	2	8	no	—
85	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₄₁	—	2	16	no	—
87	C₄⋊C₄×C₄₁	—	2	8	no	—
88	C₄⋊C₄×C₄₁	C₂²×C₄	1	16	yes	—
89	C₂²⋊C₄×C₄₁	—	2	16	no	—
90	C₂²⋊C₄×C₄₁	—	2	16	no	—
91	C₄²×C₄₁	—	2	16	no	—
92	C₄²×C₄₁	—	2	16	no	—
93	C₂³×C₄₁	—	2	8	no	—
94	C₂³×C₄₁	—	2	4	no	—
95	C₂³×C₄₁	—	4	16	no	—
96	C₂³×C₄₁	—	4	8	no	—
97	C₂³×C₄₁	#95	1	16	yes	—
98	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₁	—	2	16	no	—
105	C₂×C₄×C₄₁	—	4	32	no	—
106	C₂×C₄×C₄₁	—	4	16	no	—
107	C₂×C₄×C₄₁	#95	1	32	yes	—
108	C₂×C₄×C₄₁	#95	1	16	yes	—
109	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
110	C₂×C₄×C₄₁	C₂×C₄²	1	8	yes	—
111	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
113	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
118	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
119	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
124	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
125	C₈×C₄₁	—	2	8	no	—
126	#20347	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₁	—	2	8	no	—
128	C₂²×C₄₁	—	2	8	no	—
129	C₂²×C₄₁	—	2	8	no	—
130	C₂²×C₄₁	—	2	8	no	—
131	C₂²×C₄₁	—	2	4	no	—
132	C₂²×C₄₁	—	4	16	no	—
133	C₂²×C₄₁	C₂×#95	1	4	yes	—
134	C₂²×C₄₁	C₂×#95	1	8	yes	—
135	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₁	#575	1	4	yes	—
138	C₄×C₄₁	—	4	16	no	—
139	C₄×C₄₁	C₂×#95	1	4	yes	—
140	C₄×C₄₁	C₂×#95	1	8	yes	—
141	C₄×C₄₁	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₄₁	C₂²×C₄⋊C₄	1	2	yes	—
144	C₂³×M₁₆	C₂×C₄₁	1	1	yes	characteristic
145	C₂²×#117	C₂×C₄₁	1	2	yes	—
146	#1736	C₂×C₄₁	1	8	yes	complemented
147	#1735	C₂×C₄₁	1	4	yes	complemented
148	C₂×C₄₁	—	2	8	no	—
149	C₂×C₄₁	—	2	4	no	—
150	C₂×C₄₁	C₂²×#95	1	1	yes	characteristic
151	C₂×C₄₁	#2896	1	4	yes	—
152	C₂×C₄₁	#2911	1	2	yes	—
153	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
154	C₂²×M₁₆	—	2	8	no	—
155	C₂²×M₁₆	C₄×C₄₁	1	8	yes	complemented
156	C₂²×M₁₆	C₂²×C₄₁	1	4	yes	—
157	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
158	C₂×#117	—	2	8	no	—
159	C₂×#117	—	2	2	no	—
160	C₂×#117	—	2	2	no	—
161	C₂×#117	C₄×C₄₁	1	16	yes	complemented
162	C₂×#117	C₂²×C₄₁	1	4	yes	—
163	C₂×#117	C₂²×C₄₁	1	4	yes	—
164	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
165	C₂²×C₂²⋊C₄	—	2	2	no	—
166	C₂²×C₄²	—	2	2	no	—
167	#340	C₂²×C₄₁	1	16	yes	complemented
168	C₄₁	#20347	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁵	D₄×C₄₁	1	1	yes	characteristic
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
172	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
173	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₄₁	1	2	yes	—
175	C₂³×C₄	D₄×C₄₁	1	8	yes	—
176	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
177	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
178	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
179	C₂×M₁₆	—	2	32	no	—
180	C₂×M₁₆	—	2	8	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂×M₁₆	C₂×C₄×C₄₁	1	16	yes	complemented
183	C₂²×C₈	—	2	8	no	—
184	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	complemented
185	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
186	#117	—	2	16	no	—
187	#117	—	2	16	no	—
188	#117	—	2	8	no	—
189	#117	—	2	8	no	—
190	#117	C₂×C₄×C₄₁	1	16	yes	—
191	#117	C₂×C₄×C₄₁	1	16	yes	complemented
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	complemented
195	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
196	C₂×C₂²⋊C₄	—	2	16	no	—
197	C₂×C₂²⋊C₄	—	2	4	no	—
198	C₂×C₂²⋊C₄	—	2	4	no	—
199	C₂×C₄²	—	2	16	no	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂⁴	—	4	4	no	—
203	C₂⁴	C₂²⋊C₄×C₄₁	1	8	yes	—
204	C₂⁴	C₄⋊C₄×C₄₁	1	4	yes	—
205	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
206	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
207	C₂²×C₄	—	2	16	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	4	16	no	—
210	C₂²×C₄	C₄²×C₄₁	1	16	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
212	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₄₁	1	16	yes	—
215	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
216	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
217	C₂²×C₄	C₄⋊C₄×C₄₁	1	32	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
227	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
228	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
229	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
231	M₁₆	—	2	32	no	—
232	C₂×C₈	—	2	16	no	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
235	C₄⋊C₄	—	2	16	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	C₂²×C₄×C₄₁	1	16	yes	—
238	C₂²⋊C₄	—	2	16	no	—
239	C₂²⋊C₄	—	2	16	no	—
240	C₄²	—	2	16	no	—
241	C₄²	—	2	16	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	8	no	—
246	C₂³	#95×C₄₁	1	16	yes	—
247	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
249	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
250	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
251	C₂³	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
252	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
253	C₂×C₄	—	2	16	no	—
254	C₂×C₄	—	4	32	no	—
255	C₂×C₄	—	4	16	no	—
256	C₂×C₄	#95×C₄₁	1	32	yes	—
257	C₂×C₄	#95×C₄₁	1	16	yes	—
258	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
259	C₂×C₄	C₂×C₄²×C₄₁	1	8	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
266	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
267	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
268	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
271	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
273	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	16	no	—
281	C₂²	C₂×#95×C₄₁	1	4	yes	—
282	C₂²	C₂×#95×C₄₁	1	8	yes	—
283	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	2	yes	—
284	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
285	C₂²	C₄₁×#575	1	4	yes	—
286	C₄	—	4	16	no	—
287	C₄	C₂×#95×C₄₁	1	4	yes	—
288	C₄	C₂×#95×C₄₁	1	8	yes	—
289	C₄	C₂²×C₄²×C₄₁	1	1	yes	characteristic
290	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
291	C₄	C₂²×C₄⋊C₄×C₄₁	1	2	yes	—
292	C₂	—	2	8	no	—
293	C₂	—	2	4	no	—
294	C₂	C₂²×#95×C₄₁	1	1	yes	characteristic
295	C₂	C₄₁×#2896	1	4	yes	—
296	C₂	C₄₁×#2911	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₄	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂²×D₄
prime radical	C₂⁵×C₄₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂⁴×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1736	C₂	semidirect
4. C₄₁×#1735	C₂	semidirect
7. C₂²×M₁₆×C₄₁	C₄	semidirect
13. C₂×#117×C₄₁	C₄	semidirect
19. C₄₁×#340	C₂²	semidirect
33. C₂×M₁₆×C₄₁	C₂×C₄	semidirect
35. C₂²×C₈×C₄₁	C₂×C₄	semidirect
42. #117×C₄₁	C₂×C₄	semidirect
45. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
126. #20347	C₄₁	direct (N × H)
146. #1736	C₂×C₄₁	semidirect
147. #1735	C₂×C₄₁	semidirect
155. C₂²×M₁₆	C₄×C₄₁	semidirect
161. C₂×#117	C₄×C₄₁	semidirect
167. #340	C₂²×C₄₁	semidirect
168. C₄₁	#20347	direct (N × H)
182. C₂×M₁₆	C₂×C₄×C₄₁	semidirect
184. C₂²×C₈	C₂×C₄×C₄₁	semidirect
191. #117	C₂×C₄×C₄₁	semidirect
194. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₄₁	2
C₂²×#117×C₄₁	2
C₄₁×#1736	2
C₄₁×#1735	2
#20347	41

Subgroups — C41×#20347 #23796548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20347 #23796548