Subgroups — C₄₁×#22316 #23798517

Order 10496.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2399	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1985	C₂	1	2	yes	maximal
3	C₄₁×#1984	C₂	1	2	yes	maximal
4	C₄₁×#1939	C₂	1	2	yes	maximal
5	C₄₁×#1917	C₂	1	2	yes	maximal
6	C₄₁×#1916	C₂	1	2	yes	maximal
7	C₄₁×#1832	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#1771	C₂	1	2	yes	maximal
9	C₄₁×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×#97×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#398	C₄	1	2	yes	—
20	C₄₁×#398	C₄	1	2	yes	—
21	C₄₁×#395	C₄	1	2	yes	—
22	C₄₁×#395	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₄₁×#376	C₂²	1	1	yes	characteristic
26	C₄₁×#336	C₂²	1	2	yes	—
27	C₄₁×#336	C₂²	1	2	yes	—
28	C₄₁×#336	C₂²	1	1	yes	characteristic
29	C₄₁×#336	C₂²	1	1	yes	characteristic
30	C₄₁×#336	C₂²	1	2	yes	—
31	C₄₁×#336	C₂²	1	2	yes	—
32	C₄₁×#336	C₂²	1	2	yes	—
33	C₄₁×#336	C₂²	1	2	yes	—
34	C₄₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
41	C₄×Q₈×C₄₁	—	4	2	no	—
42	C₄×Q₈×C₄₁	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₁	—	2	4	no	—
57	C₄⋊C₈×C₄₁	—	4	2	no	—
58	C₄⋊C₈×C₄₁	—	4	2	no	—
59	#97×C₄₁	—	4	2	no	—
60	#97×C₄₁	Q₈	1	4	yes	complemented
61	C₄×C₈×C₄₁	—	2	2	no	—
62	C₄×C₈×C₄₁	—	4	2	no	—
63	#95×C₄₁	C₂×C₄	1	2	yes	—
64	#95×C₄₁	C₂×C₄	1	2	yes	—
65	#95×C₄₁	C₂×C₄	1	2	yes	—
66	#95×C₄₁	C₂×C₄	1	2	yes	—
67	#95×C₄₁	C₂×C₄	1	2	yes	—
68	#95×C₄₁	C₂×C₄	1	2	yes	—
69	C₂×Q₈×C₄₁	—	2	4	no	—
70	C₂×Q₈×C₄₁	—	4	1	no	—
71	C₂×Q₈×C₄₁	—	4	1	no	—
72	C₂×Q₈×C₄₁	—	4	2	no	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	—	2	2	no	—
84	C₂×C₈×C₄₁	—	2	2	no	—
85	C₂×C₈×C₄₁	—	2	2	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	—	2	2	no	—
88	C₂×C₈×C₄₁	—	2	2	no	—
89	C₂×C₈×C₄₁	—	2	2	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	2	2	no	—
92	C₂×C₈×C₄₁	—	2	2	no	—
93	C₂×C₈×C₄₁	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₄₁	—	2	4	no	—
95	C₄⋊C₄×C₄₁	—	2	4	no	—
96	C₄⋊C₄×C₄₁	—	4	2	no	—
97	C₄⋊C₄×C₄₁	—	4	2	no	—
98	C₄⋊C₄×C₄₁	—	4	2	no	—
99	C₄⋊C₄×C₄₁	—	4	1	no	—
100	C₄⋊C₄×C₄₁	—	4	1	no	—
101	C₄⋊C₄×C₄₁	—	4	2	no	—
102	C₄²×C₄₁	—	2	2	no	—
103	C₄²×C₄₁	—	2	2	no	—
104	C₄²×C₄₁	—	2	2	no	—
105	C₄²×C₄₁	—	2	2	no	—
106	C₄²×C₄₁	—	2	2	no	—
107	C₄²×C₄₁	—	2	2	no	—
108	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
109	C₄²×C₄₁	#41	1	2	yes	—
110	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
111	Q₈×C₄₁	—	4	4	no	—
112	Q₈×C₄₁	—	4	4	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	2	no	—
117	C₂×C₄×C₄₁	—	2	1	no	—
118	C₂×C₄×C₄₁	—	2	1	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	2	2	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	2	2	no	—
128	C₂×C₄×C₄₁	—	2	2	no	—
129	C₂×C₄×C₄₁	—	2	2	no	—
130	C₂×C₄×C₄₁	—	2	2	no	—
131	C₂×C₄×C₄₁	—	2	2	no	—
132	C₂×C₄×C₄₁	—	2	2	no	—
133	C₂×C₄×C₄₁	C₄×Q₈	1	4	yes	—
134	C₂×C₄×C₄₁	#131	1	4	yes	—
135	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
139	C₈×C₄₁	—	2	4	no	—
140	C₈×C₄₁	—	4	2	no	—
141	C₈×C₄₁	—	4	2	no	—
142	C₈×C₄₁	—	4	2	no	—
143	#22316	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₁	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₄₁	#518	1	1	yes	characteristic
146	C₂²×C₄₁	#519	1	1	yes	characteristic
147	C₂²×C₄₁	C₂×#131	1	2	yes	—
148	C₂²×C₄₁	#568	1	2	yes	—
149	C₄×C₄₁	—	2	4	no	—
150	C₄×C₄₁	—	2	4	no	—
151	C₄×C₄₁	—	4	2	no	—
152	C₄×C₄₁	—	4	1	no	—
153	C₄×C₄₁	—	4	2	no	—
154	C₄×C₄₁	—	4	1	no	—
155	C₄×C₄₁	—	4	2	no	—
156	C₄×C₄₁	—	4	2	no	—
157	C₄×C₄₁	#542	1	2	yes	—
158	C₄×C₄₁	#544	1	2	yes	—
159	#2399	C₂×C₄₁	1	1	yes	characteristic
160	#1985	C₂×C₄₁	1	2	yes	—
161	#1984	C₂×C₄₁	1	2	yes	—
162	#1939	C₂×C₄₁	1	2	yes	—
163	#1917	C₂×C₄₁	1	2	yes	—
164	#1916	C₂×C₄₁	1	2	yes	—
165	#1832	C₂×C₄₁	1	1	yes	characteristic
166	#1771	C₂×C₄₁	1	2	yes	—
167	#1746	C₂×C₄₁	1	1	yes	characteristic
168	C₂×C₄₁	#2331	1	1	yes	characteristic
169	C₂×C₄₁	#2962	1	2	yes	—
170	C₂×C₄₁	#2977	1	1	yes	characteristic
171	C₂×C₄₁	#2978	1	1	yes	characteristic
172	C₂×C₄₁	#2985	1	2	yes	—
173	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
174	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
177	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
178	C₂×#97	C₂²×C₄₁	1	2	yes	—
179	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
182	#398	C₄×C₄₁	1	2	yes	—
183	#398	C₄×C₄₁	1	2	yes	—
184	#395	C₄×C₄₁	1	2	yes	—
185	#395	C₄×C₄₁	1	2	yes	—
186	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
188	#376	C₂²×C₄₁	1	1	yes	characteristic
189	#336	C₂²×C₄₁	1	1	yes	characteristic
190	#336	C₂²×C₄₁	1	1	yes	characteristic
191	#336	C₂²×C₄₁	1	2	yes	—
192	#336	C₂²×C₄₁	1	2	yes	—
193	#336	C₂²×C₄₁	1	2	yes	—
194	#336	C₂²×C₄₁	1	2	yes	—
195	#336	C₂²×C₄₁	1	2	yes	—
196	#336	C₂²×C₄₁	1	2	yes	—
197	#336	C₂²×C₄₁	1	2	yes	—
198	C₄₁	#22316	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
200	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
203	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
204	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
205	C₄×Q₈	—	4	2	no	—
206	C₄×Q₈	—	4	2	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
214	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
215	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
217	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
218	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
220	C₄⋊C₈	—	2	4	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	4	2	no	—
223	#97	—	4	2	no	—
224	#97	Q₈×C₄₁	1	4	yes	complemented
225	C₄×C₈	—	2	2	no	—
226	C₄×C₈	—	4	2	no	—
227	#95	C₂×C₄×C₄₁	1	2	yes	—
228	#95	C₂×C₄×C₄₁	1	2	yes	—
229	#95	C₂×C₄×C₄₁	1	2	yes	—
230	#95	C₂×C₄×C₄₁	1	2	yes	—
231	#95	C₂×C₄×C₄₁	1	2	yes	—
232	#95	C₂×C₄×C₄₁	1	2	yes	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	C₂×Q₈×C₄₁	1	4	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
273	C₄²	#41×C₄₁	1	2	yes	—
274	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
275	Q₈	—	4	4	no	—
276	Q₈	—	4	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	C₄×Q₈×C₄₁	1	4	yes	—
298	C₂×C₄	#131×C₄₁	1	4	yes	—
299	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
300	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
301	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
302	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	C₂×C₄×Q₈×C₄₁	1	1	yes	characteristic
308	C₂²	C₄₁×#518	1	1	yes	characteristic
309	C₂²	C₄₁×#519	1	1	yes	characteristic
310	C₂²	C₂×#131×C₄₁	1	2	yes	—
311	C₂²	C₄₁×#568	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	C₄₁×#542	1	2	yes	—
321	C₄	C₄₁×#544	1	2	yes	—
322	C₂	C₄₁×#2331	1	1	yes	characteristic
323	C₂	C₄₁×#2962	1	2	yes	—
324	C₂	C₄₁×#2977	1	1	yes	characteristic
325	C₂	C₄₁×#2978	1	1	yes	characteristic
326	C₂	C₄₁×#2985	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
60. #97×C₄₁	Q₈	semidirect
143. #22316	C₄₁	direct (N × H)
198. C₄₁	#22316	direct (N × H)
224. #97	Q₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#2399	2
C₄₁×#1985	2
C₄₁×#1984	2
C₄₁×#1939	2
C₄₁×#1917	2
C₄₁×#1916	2
C₄₁×#1832	2
C₄₁×#1771	2
C₄₁×#1746	2
#22316	41

Subgroups — C41×#22316 #23798517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22316 #23798517