Subgroups — C₄₁×#33191 #23809392

Order 10496.

978 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#473	C₂	1	4	yes	maximal
3	C₄₁×#3039	C₂	1	2	yes	maximal
4	C₄×#131×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1979	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1966	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₄₁	C₂²	1	4	yes	—
9	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
10	C₂×#131×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
12	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#494	—	2	1	no	—
14	C₄₁×#494	—	2	1	no	—
15	C₄₁×#494	—	2	2	no	—
16	C₄₁×#473	—	2	16	no	—
17	C₄₁×#462	—	2	4	no	—
18	C₄₁×#461	—	2	4	no	—
19	C₂×#108×C₄₁	C₂²	1	4	yes	—
20	C₄₁×#423	C₂²	1	2	yes	—
21	C₄₁×#419	C₂²	1	4	yes	—
22	C₂×#101×C₄₁	C₂²	1	8	yes	—
23	C₄₁×#405	—	2	2	no	—
24	C₄₁×#405	C₂²	1	2	yes	—
25	C₄×M₁₆×C₄₁	—	2	1	no	—
26	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₁	—	2	1	no	—
28	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
30	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
31	#137×C₄₁	—	2	16	no	—
32	C₂×Q₁₆×C₄₁	—	2	4	no	—
33	C₂×Q₁₆×C₄₁	—	2	8	no	—
34	C₂×Q₁₆×C₄₁	—	2	4	no	—
35	C₂×Q₁₆×C₄₁	—	2	4	no	—
36	C₂×Q₁₆×C₄₁	—	2	8	no	—
37	C₂×SD₁₆×C₄₁	—	2	4	no	—
38	#131×C₄₁	—	2	4	no	—
39	#131×C₄₁	D₄	1	8	yes	—
40	C₂×M₁₆×C₄₁	—	2	2	no	—
41	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
42	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₄₁	—	2	2	no	—
44	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₄₁	—	2	4	no	—
46	C₄⋊Q₈×C₄₁	—	2	2	no	—
47	C₄⋊Q₈×C₄₁	—	2	2	no	—
48	C₄×Q₈×C₄₁	D₄	1	2	yes	—
49	C₄×D₄×C₄₁	—	2	2	no	—
50	C₄×D₄×C₄₁	D₄	1	2	yes	—
51	#117×C₄₁	—	2	1	no	—
52	#117×C₄₁	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
54	C₂×C₄²×C₄₁	—	2	1	no	—
55	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
56	#108×C₄₁	—	2	8	no	—
57	C₄⋊C₈×C₄₁	—	2	4	no	—
58	#103×C₄₁	—	2	8	no	—
59	#101×C₄₁	—	2	16	no	—
60	#97×C₄₁	—	2	2	no	—
61	#97×C₄₁	—	2	1	no	—
62	C₄×C₈×C₄₁	—	2	2	no	—
63	C₄×C₈×C₄₁	—	2	1	no	—
64	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₄₁	D₄	1	2	yes	—
67	#41×C₄₁	—	2	2	no	—
68	#41×C₄₁	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₄₁	—	2	8	no	—
70	C₂×Q₈×C₄₁	—	2	4	no	—
71	C₂×Q₈×C₄₁	—	2	4	no	—
72	C₂×Q₈×C₄₁	—	2	8	no	—
73	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₄₁	—	2	1	no	—
75	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	—	2	1	no	—
77	C₂²×C₄×C₄₁	—	2	2	no	—
78	C₂²×C₄×C₄₁	—	2	4	no	—
79	C₂²×C₄×C₄₁	#41	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₄₁	—	4	8	no	—
82	Q₁₆×C₄₁	—	4	8	no	—
83	Q₁₆×C₄₁	—	4	8	no	—
84	SD₁₆×C₄₁	—	4	8	no	—
85	M₁₆×C₄₁	—	2	2	no	—
86	M₁₆×C₄₁	—	2	8	no	—
87	M₁₆×C₄₁	—	2	8	no	—
88	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₄₁	—	2	2	no	—
90	C₂×C₈×C₄₁	—	2	8	no	—
91	C₂×C₈×C₄₁	—	2	4	no	—
92	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₄₁	#41	1	4	yes	—
96	C₄⋊C₄×C₄₁	—	2	2	no	—
97	C₄⋊C₄×C₄₁	—	4	4	no	—
98	C₄⋊C₄×C₄₁	—	4	4	no	—
99	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
100	C₂²⋊C₄×C₄₁	—	2	1	no	—
101	C₂²⋊C₄×C₄₁	—	2	2	no	—
102	C₄²×C₄₁	—	2	2	no	—
103	C₄²×C₄₁	—	2	1	no	—
104	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₁	C₂×D₄	1	2	yes	—
107	C₂³×C₄₁	—	2	1	no	—
108	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
109	Q₈×C₄₁	—	2	1	no	—
110	Q₈×C₄₁	—	4	8	no	—
111	Q₈×C₄₁	—	4	4	no	—
112	Q₈×C₄₁	—	4	4	no	—
113	Q₈×C₄₁	#127	1	2	yes	—
114	D₄×C₄₁	—	2	4	no	—
115	D₄×C₄₁	—	2	1	no	—
116	D₄×C₄₁	#127	1	2	yes	—
117	C₂×C₄×C₄₁	—	2	1	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	2	4	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	—	4	8	no	—
124	C₂×C₄×C₄₁	#121	1	2	yes	—
125	C₂×C₄×C₄₁	#121	1	2	yes	—
126	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
128	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
130	C₈×C₄₁	—	2	4	no	—
131	C₈×C₄₁	—	4	4	no	—
132	C₈×C₄₁	#121	1	4	yes	—
133	C₈×C₄₁	#121	1	4	yes	—
134	#33191	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₄₁	—	2	2	no	—
136	C₂²×C₄₁	—	2	2	no	—
137	C₂²×C₄₁	—	4	2	no	—
138	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
139	C₂²×C₄₁	#532	1	2	yes	—
140	C₄×C₄₁	—	2	2	no	—
141	C₄×C₄₁	—	2	2	no	—
142	C₄×C₄₁	—	4	2	no	—
143	C₄×C₄₁	—	8	4	no	—
144	C₄×C₄₁	#473	1	4	yes	complemented
145	C₄×C₄₁	C₂×#121	1	2	yes	—
146	C₄×C₄₁	C₂×#127	1	1	yes	characteristic
147	C₄×C₄₁	#532	1	1	yes	characteristic
148	C₂×#494	C₂×C₄₁	1	1	yes	characteristic, complemented
149	C₂×#473	C₂×C₄₁	1	4	yes	—
150	#3039	C₂×C₄₁	1	2	yes	—
151	C₄×#131	C₂×C₄₁	1	1	yes	characteristic
152	#1979	C₂×C₄₁	1	4	yes	complemented
153	#1966	C₂×C₄₁	1	2	yes	complemented
154	#1848	C₂×C₄₁	1	1	yes	characteristic, complemented
155	C₂×C₄₁	—	2	2	no	—
156	C₂×C₄₁	—	4	2	no	—
157	C₂×C₄₁	#2856	1	1	yes	characteristic
158	C₂×C₄₁	C₂×#473	1	2	yes	—
159	C₂×#137	C₂²×C₄₁	1	4	yes	—
160	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
161	C₂×#131	C₂²×C₄₁	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
163	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
164	#494	—	2	1	no	—
165	#494	—	2	1	no	—
166	#494	—	2	2	no	—
167	#473	—	2	16	no	—
168	#462	—	2	4	no	—
169	#461	—	2	4	no	—
170	C₂×#108	C₂²×C₄₁	1	4	yes	—
171	#423	C₂²×C₄₁	1	2	yes	—
172	#419	C₂²×C₄₁	1	4	yes	—
173	C₂×#101	C₂²×C₄₁	1	8	yes	—
174	#405	—	2	2	no	—
175	#405	C₂²×C₄₁	1	2	yes	—
176	C₄×M₁₆	—	2	1	no	—
177	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×C₄×C₈	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
180	C₄₁	#33191	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
182	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
183	#137	—	2	16	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×Q₁₆	—	2	8	no	—
186	C₂×Q₁₆	—	2	4	no	—
187	C₂×Q₁₆	—	2	4	no	—
188	C₂×Q₁₆	—	2	8	no	—
189	C₂×SD₁₆	—	2	4	no	—
190	#131	—	2	4	no	—
191	#131	D₄×C₄₁	1	8	yes	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
194	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄×Q₈	D₄×C₄₁	1	2	yes	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	D₄×C₄₁	1	2	yes	—
203	#117	—	2	1	no	—
204	#117	C₂³×C₄₁	1	1	yes	characteristic
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
208	#108	—	2	8	no	—
209	C₄⋊C₈	—	2	4	no	—
210	#103	—	2	8	no	—
211	#101	—	2	16	no	—
212	#97	—	2	2	no	—
213	#97	—	2	1	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
217	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
218	C₄×C₈	D₄×C₄₁	1	2	yes	—
219	#41	—	2	2	no	—
220	#41	C₂×D₄×C₄₁	1	4	yes	—
221	C₂×Q₈	—	2	8	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	#41×C₄₁	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
233	Q₁₆	—	4	8	no	—
234	Q₁₆	—	4	8	no	—
235	Q₁₆	—	4	8	no	—
236	SD₁₆	—	4	8	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	8	no	—
239	M₁₆	—	2	8	no	—
240	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
245	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
247	C₂×C₈	#41×C₄₁	1	4	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₄₁	1	2	yes	—
259	C₂³	—	2	1	no	—
260	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	—	4	8	no	—
263	Q₈	—	4	4	no	—
264	Q₈	—	4	4	no	—
265	Q₈	#127×C₄₁	1	2	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	#127×C₄₁	1	2	yes	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	#121×C₄₁	1	2	yes	—
277	C₂×C₄	#121×C₄₁	1	2	yes	—
278	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
280	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₈	#121×C₄₁	1	4	yes	—
285	C₈	#121×C₄₁	1	4	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
290	C₂²	C₄₁×#532	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₄₁×#473	1	4	yes	complemented
296	C₄	C₂×#121×C₄₁	1	2	yes	—
297	C₄	C₂×#127×C₄₁	1	1	yes	characteristic
298	C₄	C₄₁×#532	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	2	no	—
301	C₂	C₄₁×#2856	1	1	yes	characteristic
302	C₂	C₂×C₄₁×#473	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×#41×C₄₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#494	C₂	semidirect
5. C₄₁×#1979	C₂	semidirect
6. C₄₁×#1966	C₂	semidirect
7. C₄₁×#1848	C₂	semidirect
134. #33191	C₄₁	direct (N × H)
144. C₄×C₄₁	#473	semidirect
148. C₂×#494	C₂×C₄₁	semidirect
152. #1979	C₂×C₄₁	semidirect
153. #1966	C₂×C₄₁	semidirect
154. #1848	C₂×C₄₁	semidirect
180. C₄₁	#33191	direct (N × H)
295. C₄	C₄₁×#473	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#494	2
C₂×C₄₁×#473	2
C₄₁×#3039	2
C₄×#131×C₄₁	2
C₄₁×#1979	2
C₄₁×#1966	2
C₄₁×#1848	2
#33191	41

Subgroups — C41×#33191 #23809392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33191 #23809392