Subgroups — C₄₁×#33135 #23809336

Order 10496.

994 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2854	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1972	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1944	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1936	C₂	1	4	yes	maximal, complemented
7	C₄₁×#1893	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#123×C₄₁	—	2	2	no	—
13	C₂×#121×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#466	—	2	4	no	—
16	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#106×C₄₁	C₂²	1	2	yes	—
18	C₂×#102×C₄₁	C₂²	1	2	yes	—
19	C₂×#102×C₄₁	C₂²	1	4	yes	—
20	C₂×#97×C₄₁	—	2	4	no	—
21	C₂×#97×C₄₁	C₂²	1	2	yes	—
22	C₂×#97×C₄₁	C₂²	1	2	yes	—
23	C₄₁×#390	C₂²	1	4	yes	—
24	C₄₁×#389	C₂²	1	4	yes	—
25	C₄₁×#388	—	2	4	no	—
26	C₄₁×#386	—	2	2	no	—
27	C₄₁×#380	C₂²	1	2	yes	—
28	C₄₁×#340	C₂²	1	2	yes	—
29	C₄₁×#340	C₂²	1	4	yes	—
30	C₂²×D₄×C₄₁	—	2	2	no	—
31	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×D₈×C₄₁	—	2	2	no	—
36	C₂×D₈×C₄₁	—	2	4	no	—
37	C₂²×C₈×C₄₁	—	2	2	no	—
38	C₂²×C₈×C₄₁	—	2	4	no	—
39	C₂²×C₈×C₄₁	—	2	4	no	—
40	C₂²×C₈×C₄₁	D₄	1	4	yes	—
41	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
44	#123×C₄₁	—	4	8	no	—
45	#121×C₄₁	—	2	4	no	—
46	#121×C₄₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
54	C₂×C₄²×C₄₁	—	2	2	no	—
55	C₂×C₄²×C₄₁	—	2	2	no	—
56	C₂×C₄²×C₄₁	D₄	1	2	yes	—
57	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
58	#107×C₄₁	—	2	2	no	—
59	#106×C₄₁	—	2	4	no	—
60	#102×C₄₁	—	2	4	no	—
61	#102×C₄₁	—	2	8	no	—
62	#97×C₄₁	—	2	4	no	—
63	#97×C₄₁	—	2	4	no	—
64	#95×C₄₁	—	2	2	no	—
65	#95×C₄₁	—	2	4	no	—
66	C₂⁴×C₄₁	—	2	2	no	—
67	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
68	C₂×D₄×C₄₁	—	2	2	no	—
69	C₂×D₄×C₄₁	—	2	4	no	—
70	C₂×D₄×C₄₁	—	2	2	no	—
71	C₂×D₄×C₄₁	—	2	4	no	—
72	C₂×D₄×C₄₁	—	4	4	no	—
73	C₂×D₄×C₄₁	—	4	4	no	—
74	C₂×D₄×C₄₁	—	4	4	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	—	2	2	no	—
77	C₂²×C₄×C₄₁	—	2	2	no	—
78	C₂²×C₄×C₄₁	—	2	2	no	—
79	C₂²×C₄×C₄₁	—	2	4	no	—
80	C₂²×C₄×C₄₁	—	2	4	no	—
81	C₂²×C₄×C₄₁	—	2	2	no	—
82	C₂²×C₄×C₄₁	—	4	2	no	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₁	#41	1	2	yes	—
88	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
89	D₈×C₄₁	—	4	4	no	—
90	C₂×C₈×C₄₁	—	2	8	no	—
91	C₂×C₈×C₄₁	—	2	2	no	—
92	C₂×C₈×C₄₁	—	2	2	no	—
93	C₂×C₈×C₄₁	—	2	2	no	—
94	C₂×C₈×C₄₁	—	2	8	no	—
95	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₁	#41	1	4	yes	—
98	C₂×C₈×C₄₁	#41	1	4	yes	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₄⋊C₄×C₄₁	—	2	4	no	—
102	C₄⋊C₄×C₄₁	—	2	2	no	—
103	C₄⋊C₄×C₄₁	—	4	8	no	—
104	C₂²⋊C₄×C₄₁	—	4	8	no	—
105	C₂²⋊C₄×C₄₁	—	4	8	no	—
106	C₂²⋊C₄×C₄₁	—	4	4	no	—
107	C₂²⋊C₄×C₄₁	—	4	4	no	—
108	C₄²×C₄₁	—	2	4	no	—
109	C₄²×C₄₁	—	2	2	no	—
110	C₄²×C₄₁	—	2	2	no	—
111	C₂³×C₄₁	—	2	2	no	—
112	C₂³×C₄₁	—	2	2	no	—
113	C₂³×C₄₁	—	4	2	no	—
114	C₂³×C₄₁	—	4	4	no	—
115	C₂³×C₄₁	—	4	2	no	—
116	C₂³×C₄₁	—	4	2	no	—
117	C₂³×C₄₁	—	4	4	no	—
118	C₂³×C₄₁	#121	1	2	yes	—
119	C₂³×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
120	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
121	D₄×C₄₁	—	4	2	no	—
122	D₄×C₄₁	—	4	2	no	—
123	D₄×C₄₁	—	4	4	no	—
124	D₄×C₄₁	—	8	8	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	4	no	—
129	C₂×C₄×C₄₁	—	2	2	no	—
130	C₂×C₄×C₄₁	—	2	2	no	—
131	C₂×C₄×C₄₁	—	2	4	no	—
132	C₂×C₄×C₄₁	—	2	4	no	—
133	C₂×C₄×C₄₁	—	4	4	no	—
134	C₂×C₄×C₄₁	—	4	4	no	—
135	C₂×C₄×C₄₁	—	4	2	no	—
136	C₂×C₄×C₄₁	—	4	4	no	—
137	C₂×C₄×C₄₁	—	4	4	no	—
138	C₂×C₄×C₄₁	—	4	2	no	—
139	C₂×C₄×C₄₁	—	4	2	no	—
140	C₂×C₄×C₄₁	—	4	4	no	—
141	C₂×C₄×C₄₁	#121	1	2	yes	—
142	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
144	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
145	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
146	C₈×C₄₁	—	2	2	no	—
147	C₈×C₄₁	—	2	4	no	—
148	C₈×C₄₁	#123	1	4	yes	complemented
149	#33135	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₄₁	—	2	2	no	—
151	C₂²×C₄₁	—	2	8	no	—
152	C₂²×C₄₁	—	2	2	no	—
153	C₂²×C₄₁	—	8	2	no	—
154	C₂²×C₄₁	—	8	4	no	—
155	C₂²×C₄₁	—	8	4	no	—
156	C₂²×C₄₁	—	8	2	no	—
157	C₂²×C₄₁	—	8	2	no	—
158	C₂²×C₄₁	#469	1	4	yes	—
159	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
160	C₂²×C₄₁	#532	1	2	yes	—
161	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
162	C₂²×C₄₁	#575	1	2	yes	—
163	C₄×C₄₁	—	2	2	no	—
164	C₄×C₄₁	—	4	2	no	—
165	C₄×C₄₁	—	4	2	no	—
166	C₄×C₄₁	—	8	4	no	—
167	C₄×C₄₁	—	8	2	no	—
168	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
169	C₄×C₄₁	#525	1	1	yes	characteristic
170	C₄×C₄₁	#529	1	2	yes	—
171	C₂×#466	C₂×C₄₁	1	1	yes	characteristic
172	C₂²×#97	C₂×C₄₁	1	1	yes	characteristic, complemented
173	#2854	C₂×C₄₁	1	2	yes	complemented
174	#1972	C₂×C₄₁	1	2	yes	complemented
175	#1944	C₂×C₄₁	1	2	yes	complemented
176	#1936	C₂×C₄₁	1	4	yes	complemented
177	#1893	C₂×C₄₁	1	2	yes	complemented
178	#1807	C₂×C₄₁	1	1	yes	characteristic, complemented
179	C₂×C₄₁	—	2	4	no	—
180	C₂×C₄₁	—	16	2	no	—
181	C₂×C₄₁	#2854	1	1	yes	characteristic
182	C₂×C₄₁	C₂×#469	1	1	yes	characteristic
183	C₂×C₄₁	#3054	1	1	yes	characteristic
184	C₂×C₄₁	#3180	1	4	yes	—
185	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
186	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
187	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
188	C₂×#123	—	2	2	no	—
189	C₂×#121	C₂²×C₄₁	1	2	yes	—
190	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
191	#466	—	2	4	no	—
192	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
193	C₂×#106	C₂²×C₄₁	1	2	yes	—
194	C₂×#102	C₂²×C₄₁	1	4	yes	—
195	C₂×#102	C₂²×C₄₁	1	2	yes	—
196	C₂×#97	—	2	4	no	—
197	C₂×#97	C₂²×C₄₁	1	2	yes	—
198	C₂×#97	C₂²×C₄₁	1	2	yes	—
199	#390	C₂²×C₄₁	1	4	yes	—
200	#389	C₂²×C₄₁	1	4	yes	—
201	#388	—	2	4	no	—
202	#386	—	2	2	no	—
203	#380	C₂²×C₄₁	1	2	yes	—
204	#340	C₂²×C₄₁	1	2	yes	—
205	#340	C₂²×C₄₁	1	4	yes	—
206	C₄₁	#33135	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
209	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
210	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
211	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
212	C₂×D₈	—	2	2	no	—
213	C₂×D₈	—	2	4	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	2	4	no	—
217	C₂²×C₈	D₄×C₄₁	1	4	yes	—
218	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
220	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
221	#123	—	4	8	no	—
222	#121	—	2	4	no	—
223	#121	—	2	4	no	—
224	C₂×C₄⋊C₄	—	2	4	no	—
225	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	D₄×C₄₁	1	2	yes	—
234	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
235	#107	—	2	2	no	—
236	#106	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	8	no	—
239	#97	—	2	4	no	—
240	#97	—	2	4	no	—
241	#95	—	2	2	no	—
242	#95	—	2	4	no	—
243	C₂⁴	—	2	2	no	—
244	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
264	C₂²×C₄	#41×C₄₁	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
266	D₈	—	4	4	no	—
267	C₂×C₈	—	2	8	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	8	no	—
272	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
274	C₂×C₈	#41×C₄₁	1	4	yes	—
275	C₂×C₈	#41×C₄₁	1	4	yes	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	8	no	—
281	C₂²⋊C₄	—	4	8	no	—
282	C₂²⋊C₄	—	4	8	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	4	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	#121×C₄₁	1	2	yes	—
296	C₂³	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
297	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	#121×C₄₁	1	2	yes	—
319	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
321	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
322	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	4	no	—
325	C₈	#123×C₄₁	1	4	yes	complemented
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	4	no	—
333	C₂²	—	8	4	no	—
334	C₂²	C₄₁×#469	1	4	yes	—
335	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
336	C₂²	C₄₁×#532	1	2	yes	—
337	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
338	C₂²	C₄₁×#575	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	4	no	—
344	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
345	C₄	C₄₁×#525	1	1	yes	characteristic
346	C₄	C₄₁×#529	1	2	yes	—
347	C₂	—	2	4	no	—
348	C₂	—	16	2	no	—
349	C₂	C₄₁×#2854	1	1	yes	characteristic
350	C₂	C₂×C₄₁×#469	1	1	yes	characteristic
351	C₂	C₄₁×#3054	1	1	yes	characteristic
352	C₂	C₄₁×#3180	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#97×C₄₁	C₂	semidirect
3. C₄₁×#2854	C₂	semidirect
4. C₄₁×#1972	C₂	semidirect
5. C₄₁×#1944	C₂	semidirect
6. C₄₁×#1936	C₂	semidirect
7. C₄₁×#1893	C₂	semidirect
8. C₄₁×#1807	C₂	semidirect
148. C₈×C₄₁	#123	semidirect
149. #33135	C₄₁	direct (N × H)
172. C₂²×#97	C₂×C₄₁	semidirect
173. #2854	C₂×C₄₁	semidirect
174. #1972	C₂×C₄₁	semidirect
175. #1944	C₂×C₄₁	semidirect
176. #1936	C₂×C₄₁	semidirect
177. #1893	C₂×C₄₁	semidirect
178. #1807	C₂×C₄₁	semidirect
206. C₄₁	#33135	direct (N × H)
325. C₈	#123×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×#466	2
C₂²×#97×C₄₁	2
C₄₁×#2854	2
C₄₁×#1972	2
C₄₁×#1944	2
C₄₁×#1936	2
C₄₁×#1893	2
C₄₁×#1807	2
#33135	41

Subgroups — C41×#33135 #23809336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33135 #23809336