Subgroups — C₄₁×#21697 #23797898

Order 10496.

1238 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	C₂×C₄₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#442	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2328	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1972	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1970	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1961	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
11	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
12	C₂×#127×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
14	C₄₁×#493	C₄	1	8	yes	complemented
15	C₄₁×#442	—	2	8	no	—
16	C₄×D₈×C₄₁	—	2	8	no	—
17	C₂×#107×C₄₁	C₂²	1	2	yes	—
18	C₂×#106×C₄₁	C₂²	1	2	yes	—
19	C₂×#102×C₄₁	C₂²	1	4	yes	—
20	C₂×#102×C₄₁	C₂²	1	4	yes	—
21	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#390	C₂²	1	4	yes	—
28	C₄₁×#386	—	2	8	no	—
29	C₄₁×#382	C₂²	1	2	yes	—
30	C₄₁×#376	C₂²	1	1	yes	characteristic
31	C₂²×D₄×C₄₁	—	2	4	no	—
32	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₄₁	—	2	4	no	—
34	C₂×D₈×C₄₁	—	2	4	no	—
35	C₂×D₈×C₄₁	—	2	4	no	—
36	C₂×D₈×C₄₁	—	2	4	no	—
37	C₂×D₈×C₄₁	—	2	4	no	—
38	C₂×D₈×C₄₁	C₂×C₄	1	8	yes	complemented
39	C₂×D₈×C₄₁	C₂×C₄	1	8	yes	complemented
40	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
44	#127×C₄₁	—	2	4	no	—
45	#127×C₄₁	C₂×C₄	1	8	yes	complemented
46	C₄×D₄×C₄₁	—	2	8	no	—
47	C₄×D₄×C₄₁	—	2	8	no	—
48	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	#107×C₄₁	—	2	4	no	—
55	#106×C₄₁	—	2	4	no	—
56	#102×C₄₁	—	2	8	no	—
57	#102×C₄₁	—	2	8	no	—
58	#97×C₄₁	—	2	2	no	—
59	#97×C₄₁	—	2	2	no	—
60	#97×C₄₁	D₄	1	4	yes	complemented
61	C₄×C₈×C₄₁	—	2	2	no	—
62	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	complemented
63	C₄×C₈×C₄₁	D₄	1	4	yes	complemented
64	#95×C₄₁	—	2	4	no	—
65	#95×C₄₁	D₄	1	4	yes	—
66	C₂⁴×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	4	no	—
71	C₂×D₄×C₄₁	—	2	4	no	—
72	C₂×D₄×C₄₁	—	2	4	no	—
73	C₂×D₄×C₄₁	—	8	8	no	—
74	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₁	—	2	4	no	—
76	C₂²×C₄×C₄₁	—	4	8	no	—
77	C₂²×C₄×C₄₁	—	8	8	no	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
82	D₈×C₄₁	—	2	16	no	—
83	D₈×C₄₁	—	2	16	no	—
84	C₂×C₈×C₄₁	—	2	1	no	—
85	C₂×C₈×C₄₁	—	2	1	no	—
86	C₂×C₈×C₄₁	—	2	1	no	—
87	C₂×C₈×C₄₁	—	2	1	no	—
88	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₁	#41	1	2	yes	—
95	C₂×C₈×C₄₁	#41	1	2	yes	—
96	C₄⋊C₄×C₄₁	—	2	4	no	—
97	C₄⋊C₄×C₄₁	—	2	4	no	—
98	C₂²⋊C₄×C₄₁	—	4	8	no	—
99	C₂²⋊C₄×C₄₁	—	8	8	no	—
100	C₄²×C₄₁	—	2	1	no	—
101	C₄²×C₄₁	—	2	1	no	—
102	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₁	C₂×D₄	1	2	yes	—
104	C₄²×C₄₁	C₂×D₄	1	2	yes	—
105	C₄²×C₄₁	#41	1	2	yes	—
106	C₂³×C₄₁	—	2	8	no	—
107	C₂³×C₄₁	—	4	4	no	—
108	C₂³×C₄₁	—	4	4	no	—
109	C₂³×C₄₁	—	4	8	no	—
110	C₂³×C₄₁	—	4	8	no	—
111	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₄₁	—	2	8	no	—
113	D₄×C₄₁	—	2	8	no	—
114	D₄×C₄₁	—	4	4	no	—
115	D₄×C₄₁	—	4	4	no	—
116	D₄×C₄₁	—	8	16	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	4	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	—	4	8	no	—
123	C₂×C₄×C₄₁	—	8	16	no	—
124	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
125	C₂×C₄×C₄₁	#127	1	4	yes	—
126	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
132	C₈×C₄₁	—	2	2	no	—
133	C₈×C₄₁	—	2	2	no	—
134	C₈×C₄₁	C₄×D₄	1	4	yes	—
135	C₈×C₄₁	C₄×D₄	1	4	yes	—
136	#21697	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₁	—	4	8	no	—
138	C₂²×C₄₁	—	4	16	no	—
139	C₂²×C₄₁	—	8	4	no	—
140	C₂²×C₄₁	—	8	4	no	—
141	C₂²×C₄₁	—	8	8	no	—
142	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
144	C₂²×C₄₁	#532	1	1	yes	characteristic
145	C₂²×C₄₁	#576	1	4	yes	—
146	C₄×C₄₁	—	2	4	no	—
147	C₄×C₄₁	—	2	4	no	—
148	C₄×C₄₁	—	4	2	no	—
149	C₄×C₄₁	—	8	4	no	—
150	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₄₁	#532	1	1	yes	characteristic
153	C₄×C₄₁	#532	1	1	yes	characteristic
154	C₂×#493	C₂×C₄₁	1	1	yes	characteristic
155	C₂×#442	C₂×C₄₁	1	2	yes	complemented
156	C₂×C₄×D₈	C₂×C₄₁	1	2	yes	complemented
157	#2328	C₂×C₄₁	1	2	yes	complemented
158	#1972	C₂×C₄₁	1	2	yes	complemented
159	#1970	C₂×C₄₁	1	2	yes	complemented
160	#1961	C₂×C₄₁	1	2	yes	complemented
161	#1846	C₂×C₄₁	1	1	yes	characteristic, complemented
162	#1726	C₂×C₄₁	1	1	yes	characteristic, complemented
163	C₂×C₄₁	—	8	8	no	—
164	C₂×C₄₁	C₄×#127	1	1	yes	characteristic
165	C₂×C₄₁	#2948	1	2	yes	—
166	C₂×C₄₁	#3155	1	2	yes	—
167	C₂×C₄₁	#3176	1	2	yes	—
168	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
169	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
170	C₂×#127	C₂²×C₄₁	1	2	yes	—
171	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
172	#493	C₄×C₄₁	1	8	yes	complemented
173	#442	—	2	8	no	—
174	C₄×D₈	—	2	8	no	—
175	C₂×#107	C₂²×C₄₁	1	2	yes	—
176	C₂×#106	C₂²×C₄₁	1	2	yes	—
177	C₂×#102	C₂²×C₄₁	1	4	yes	—
178	C₂×#102	C₂²×C₄₁	1	4	yes	—
179	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
180	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
181	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
185	#390	C₂²×C₄₁	1	4	yes	—
186	#386	—	2	8	no	—
187	#382	C₂²×C₄₁	1	2	yes	—
188	#376	C₂²×C₄₁	1	1	yes	characteristic
189	C₄₁	#21697	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
192	C₂³×C₄	—	2	4	no	—
193	C₂×D₈	—	2	4	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	2	4	no	—
197	C₂×D₈	C₂×C₄×C₄₁	1	8	yes	complemented
198	C₂×D₈	C₂×C₄×C₄₁	1	8	yes	complemented
199	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
203	#127	—	2	4	no	—
204	#127	C₂×C₄×C₄₁	1	8	yes	complemented
205	C₄×D₄	—	2	8	no	—
206	C₄×D₄	—	2	8	no	—
207	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₂²⋊C₄	—	2	8	no	—
210	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
213	#107	—	2	4	no	—
214	#106	—	2	4	no	—
215	#102	—	2	8	no	—
216	#102	—	2	8	no	—
217	#97	—	2	2	no	—
218	#97	—	2	2	no	—
219	#97	D₄×C₄₁	1	4	yes	complemented
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	complemented
222	C₄×C₈	D₄×C₄₁	1	4	yes	complemented
223	#95	—	2	4	no	—
224	#95	D₄×C₄₁	1	4	yes	—
225	C₂⁴	—	2	4	no	—
226	C₂×D₄	—	2	16	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	8	8	no	—
233	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	8	no	—
236	C₂²×C₄	—	8	8	no	—
237	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
240	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
241	D₈	—	2	16	no	—
242	D₈	—	2	16	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
253	C₂×C₈	#41×C₄₁	1	2	yes	—
254	C₂×C₈	#41×C₄₁	1	2	yes	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	8	no	—
258	C₂²⋊C₄	—	8	8	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
262	C₄²	C₂×D₄×C₄₁	1	2	yes	—
263	C₄²	C₂×D₄×C₄₁	1	2	yes	—
264	C₄²	#41×C₄₁	1	2	yes	—
265	C₂³	—	2	8	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	8	no	—
269	C₂³	—	4	8	no	—
270	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
271	D₄	—	2	8	no	—
272	D₄	—	2	8	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	8	16	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	8	16	no	—
283	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
284	C₂×C₄	#127×C₄₁	1	4	yes	—
285	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	C₄×D₄×C₄₁	1	4	yes	—
294	C₈	C₄×D₄×C₄₁	1	4	yes	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	16	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	8	8	no	—
300	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
301	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
302	C₂²	C₄₁×#532	1	1	yes	characteristic
303	C₂²	C₄₁×#576	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	4	no	—
308	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
309	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
310	C₄	C₄₁×#532	1	1	yes	characteristic
311	C₄	C₄₁×#532	1	1	yes	characteristic
312	C₂	—	8	8	no	—
313	C₂	C₄×#127×C₄₁	1	1	yes	characteristic
314	C₂	C₄₁×#2948	1	2	yes	—
315	C₂	C₄₁×#3155	1	2	yes	—
316	C₂	C₄₁×#3176	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#442	C₂	semidirect
3. C₂×C₄×D₈×C₄₁	C₂	semidirect
4. C₄₁×#2328	C₂	semidirect
5. C₄₁×#1972	C₂	semidirect
6. C₄₁×#1970	C₂	semidirect
7. C₄₁×#1961	C₂	semidirect
8. C₄₁×#1846	C₂	semidirect
9. C₄₁×#1726	C₂	semidirect
14. C₄₁×#493	C₄	semidirect
38. C₂×D₈×C₄₁	C₂×C₄	semidirect
39. C₂×D₈×C₄₁	C₂×C₄	semidirect
45. #127×C₄₁	C₂×C₄	semidirect
60. #97×C₄₁	D₄	semidirect
62. C₄×C₈×C₄₁	C₂×C₄	semidirect
63. C₄×C₈×C₄₁	D₄	semidirect
136. #21697	C₄₁	direct (N × H)
155. C₂×#442	C₂×C₄₁	semidirect
156. C₂×C₄×D₈	C₂×C₄₁	semidirect
157. #2328	C₂×C₄₁	semidirect
158. #1972	C₂×C₄₁	semidirect
159. #1970	C₂×C₄₁	semidirect
160. #1961	C₂×C₄₁	semidirect
161. #1846	C₂×C₄₁	semidirect
162. #1726	C₂×C₄₁	semidirect
172. #493	C₄×C₄₁	semidirect
189. C₄₁	#21697	direct (N × H)
197. C₂×D₈	C₂×C₄×C₄₁	semidirect
198. C₂×D₈	C₂×C₄×C₄₁	semidirect
204. #127	C₂×C₄×C₄₁	semidirect
219. #97	D₄×C₄₁	semidirect
221. C₄×C₈	C₂×C₄×C₄₁	semidirect
222. C₄×C₈	D₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#493	2
C₂×C₄₁×#442	2
C₂×C₄×D₈×C₄₁	2
C₄₁×#2328	2
C₄₁×#1972	2
C₄₁×#1970	2
C₄₁×#1961	2
C₄₁×#1846	2
C₄₁×#1726	2
#21697	41

Subgroups — C41×#21697 #23797898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21697 #23797898