Subgroups — C₄₁×#33190 #23809391

Order 10496.

1082 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, complemented
2	C₂×C₄₁×#472	C₂	1	4	yes	maximal, complemented
3	C₄₁×#3038	C₂	1	2	yes	maximal, complemented
4	C₄×#131×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1978	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1965	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	—
9	C₂²×D₈×C₄₁	C₂²	1	4	yes	—
10	C₂×#131×C₄₁	C₂²	1	2	yes	—
11	C₂×#127×C₄₁	C₂²	1	2	yes	—
12	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#493	—	2	1	no	—
14	C₄₁×#493	—	2	1	no	—
15	C₄₁×#493	—	2	2	no	—
16	C₄₁×#472	—	2	16	no	—
17	C₄₁×#460	—	2	4	no	—
18	C₄₁×#459	—	2	4	no	—
19	C₂×#108×C₄₁	C₂²	1	4	yes	—
20	C₄₁×#423	C₂²	1	2	yes	—
21	C₄₁×#418	C₂²	1	4	yes	—
22	C₂×#100×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×D₄×C₄₁	—	2	4	no	—
31	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
32	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
33	C₂×SD₁₆×C₄₁	—	2	4	no	—
34	C₂×D₈×C₄₁	—	2	4	no	—
35	C₂×D₈×C₄₁	—	2	8	no	—
36	C₂×D₈×C₄₁	—	2	4	no	—
37	C₂×D₈×C₄₁	—	2	4	no	—
38	C₂×D₈×C₄₁	—	2	8	no	—
39	#131×C₄₁	—	2	4	no	—
40	#131×C₄₁	D₄	1	8	yes	complemented
41	C₂×M₁₆×C₄₁	—	2	2	no	—
42	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
43	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₁	—	2	2	no	—
45	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
46	#127×C₄₁	—	2	4	no	—
47	#127×C₄₁	—	2	2	no	—
48	#127×C₄₁	—	2	2	no	—
49	C₄×Q₈×C₄₁	D₄	1	2	yes	—
50	C₄×D₄×C₄₁	—	2	2	no	—
51	C₄×D₄×C₄₁	D₄	1	2	yes	—
52	#117×C₄₁	—	2	1	no	—
53	#117×C₄₁	C₂³	1	1	yes	characteristic
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	#102×C₄₁	—	2	8	no	—
59	#100×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	C₂⁴×C₄₁	—	2	4	no	—
68	#41×C₄₁	—	2	2	no	—
69	#41×C₄₁	C₂×D₄	1	4	yes	—
70	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂×D₄×C₄₁	—	2	1	no	—
72	C₂×D₄×C₄₁	—	2	8	no	—
73	C₂×D₄×C₄₁	—	2	4	no	—
74	C₂×D₄×C₄₁	—	2	4	no	—
75	C₂×D₄×C₄₁	—	2	8	no	—
76	C₂×D₄×C₄₁	—	4	4	no	—
77	C₂×D₄×C₄₁	—	4	4	no	—
78	C₂×D₄×C₄₁	—	8	8	no	—
79	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₁	—	2	1	no	—
81	C₂²×C₄×C₄₁	—	2	2	no	—
82	C₂²×C₄×C₄₁	#41	1	2	yes	—
83	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₄₁	—	4	8	no	—
85	D₈×C₄₁	—	4	8	no	—
86	D₈×C₄₁	—	4	8	no	—
87	D₈×C₄₁	—	4	8	no	—
88	M₁₆×C₄₁	—	2	2	no	—
89	M₁₆×C₄₁	—	2	8	no	—
90	M₁₆×C₄₁	—	2	8	no	—
91	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₄₁	—	2	2	no	—
93	C₂×C₈×C₄₁	—	2	8	no	—
94	C₂×C₈×C₄₁	—	2	4	no	—
95	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₄₁	#41	1	4	yes	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
101	C₂²⋊C₄×C₄₁	—	2	1	no	—
102	C₂²⋊C₄×C₄₁	—	2	2	no	—
103	C₄²×C₄₁	—	2	2	no	—
104	C₄²×C₄₁	—	2	1	no	—
105	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₁	C₂×D₄	1	2	yes	—
108	C₂³×C₄₁	—	2	1	no	—
109	C₂³×C₄₁	—	4	4	no	—
110	C₂³×C₄₁	—	4	8	no	—
111	C₂³×C₄₁	—	8	8	no	—
112	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
113	Q₈×C₄₁	—	2	1	no	—
114	Q₈×C₄₁	#127	1	2	yes	—
115	D₄×C₄₁	—	2	4	no	—
116	D₄×C₄₁	—	2	1	no	—
117	D₄×C₄₁	—	4	8	no	—
118	D₄×C₄₁	—	4	4	no	—
119	D₄×C₄₁	—	4	4	no	—
120	D₄×C₄₁	—	8	8	no	—
121	D₄×C₄₁	—	8	8	no	—
122	D₄×C₄₁	#127	1	2	yes	—
123	C₂×C₄×C₄₁	—	2	1	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	4	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	4	4	no	—
128	C₂×C₄×C₄₁	#121	1	2	yes	—
129	C₂×C₄×C₄₁	#121	1	2	yes	—
130	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
134	C₈×C₄₁	—	2	4	no	—
135	C₈×C₄₁	—	4	4	no	—
136	C₈×C₄₁	#121	1	4	yes	—
137	C₈×C₄₁	#121	1	4	yes	—
138	#33190	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₁	—	2	2	no	—
140	C₂²×C₄₁	—	2	2	no	—
141	C₂²×C₄₁	—	4	2	no	—
142	C₂²×C₄₁	—	8	4	no	—
143	C₂²×C₄₁	—	8	8	no	—
144	C₂²×C₄₁	—	16	8	no	—
145	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
146	C₂²×C₄₁	#532	1	2	yes	—
147	C₄×C₄₁	—	2	2	no	—
148	C₄×C₄₁	—	2	2	no	—
149	C₄×C₄₁	—	4	2	no	—
150	C₄×C₄₁	#472	1	4	yes	complemented
151	C₄×C₄₁	C₂×#121	1	2	yes	—
152	C₄×C₄₁	C₂×#127	1	1	yes	characteristic
153	C₄×C₄₁	#532	1	1	yes	characteristic
154	C₂×#493	C₂×C₄₁	1	1	yes	characteristic, complemented
155	C₂×#472	C₂×C₄₁	1	4	yes	complemented
156	#3038	C₂×C₄₁	1	2	yes	complemented
157	C₄×#131	C₂×C₄₁	1	1	yes	characteristic, complemented
158	#1978	C₂×C₄₁	1	4	yes	complemented
159	#1965	C₂×C₄₁	1	2	yes	complemented
160	#1848	C₂×C₄₁	1	1	yes	characteristic, complemented
161	C₂×C₄₁	—	2	2	no	—
162	C₂×C₄₁	—	4	2	no	—
163	C₂×C₄₁	—	16	4	no	—
164	C₂×C₄₁	#2856	1	1	yes	characteristic
165	C₂×C₄₁	C₂×#472	1	2	yes	—
166	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	—
167	C₂²×D₈	C₂²×C₄₁	1	4	yes	—
168	C₂×#131	C₂²×C₄₁	1	2	yes	—
169	C₂×#127	C₂²×C₄₁	1	2	yes	—
170	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
171	#493	—	2	1	no	—
172	#493	—	2	1	no	—
173	#493	—	2	2	no	—
174	#472	—	2	16	no	—
175	#460	—	2	4	no	—
176	#459	—	2	4	no	—
177	C₂×#108	C₂²×C₄₁	1	4	yes	—
178	#423	C₂²×C₄₁	1	2	yes	—
179	#418	C₂²×C₄₁	1	4	yes	—
180	C₂×#100	C₂²×C₄₁	1	8	yes	—
181	#405	—	2	2	no	—
182	#405	C₂²×C₄₁	1	2	yes	—
183	C₄×M₁₆	—	2	1	no	—
184	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
185	C₂×C₄×C₈	—	2	1	no	—
186	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
187	C₄₁	#33190	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
189	C₂²×D₄	—	2	4	no	—
190	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
191	AGL(1,ℤ/8)	—	2	16	no	—
192	C₂×SD₁₆	—	2	4	no	—
193	C₂×D₈	—	2	4	no	—
194	C₂×D₈	—	2	8	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	2	4	no	—
197	C₂×D₈	—	2	8	no	—
198	#131	—	2	4	no	—
199	#131	D₄×C₄₁	1	8	yes	complemented
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
202	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
205	#127	—	2	4	no	—
206	#127	—	2	2	no	—
207	#127	—	2	2	no	—
208	C₄×Q₈	D₄×C₄₁	1	2	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	D₄×C₄₁	1	2	yes	—
211	#117	—	2	1	no	—
212	#117	C₂³×C₄₁	1	1	yes	characteristic
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
215	#108	—	2	8	no	—
216	C₄⋊C₈	—	2	4	no	—
217	#102	—	2	8	no	—
218	#100	—	2	16	no	—
219	#97	—	2	2	no	—
220	#97	—	2	1	no	—
221	C₄×C₈	—	2	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
224	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
225	C₄×C₈	D₄×C₄₁	1	2	yes	—
226	C₂⁴	—	2	4	no	—
227	#41	—	2	2	no	—
228	#41	C₂×D₄×C₄₁	1	4	yes	—
229	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	8	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	8	8	no	—
238	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	#41×C₄₁	1	2	yes	—
242	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
243	SD₁₆	—	4	8	no	—
244	D₈	—	4	8	no	—
245	D₈	—	4	8	no	—
246	D₈	—	4	8	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	8	no	—
249	M₁₆	—	2	8	no	—
250	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
256	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
257	C₂×C₈	#41×C₄₁	1	4	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
266	C₄²	C₂×D₄×C₄₁	1	2	yes	—
267	C₂³	—	2	1	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	8	no	—
270	C₂³	—	8	8	no	—
271	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	#127×C₄₁	1	2	yes	—
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	—	4	8	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	8	8	no	—
280	D₄	—	8	8	no	—
281	D₄	#127×C₄₁	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	#121×C₄₁	1	2	yes	—
288	C₂×C₄	#121×C₄₁	1	2	yes	—
289	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
292	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	4	4	no	—
295	C₈	#121×C₄₁	1	4	yes	—
296	C₈	#121×C₄₁	1	4	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
304	C₂²	C₄₁×#532	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	C₄₁×#472	1	4	yes	complemented
309	C₄	C₂×#121×C₄₁	1	2	yes	—
310	C₄	C₂×#127×C₄₁	1	1	yes	characteristic
311	C₄	C₄₁×#532	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	—	16	4	no	—
315	C₂	C₄₁×#2856	1	1	yes	characteristic
316	C₂	C₂×C₄₁×#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
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×#493	C₂	semidirect
2. C₂×C₄₁×#472	C₂	semidirect
3. C₄₁×#3038	C₂	semidirect
4. C₄×#131×C₄₁	C₂	semidirect
5. C₄₁×#1978	C₂	semidirect
6. C₄₁×#1965	C₂	semidirect
7. C₄₁×#1848	C₂	semidirect
40. #131×C₄₁	D₄	semidirect
138. #33190	C₄₁	direct (N × H)
150. C₄×C₄₁	#472	semidirect
154. C₂×#493	C₂×C₄₁	semidirect
155. C₂×#472	C₂×C₄₁	semidirect
156. #3038	C₂×C₄₁	semidirect
157. C₄×#131	C₂×C₄₁	semidirect
158. #1978	C₂×C₄₁	semidirect
159. #1965	C₂×C₄₁	semidirect
160. #1848	C₂×C₄₁	semidirect
187. C₄₁	#33190	direct (N × H)
199. #131	D₄×C₄₁	semidirect
308. C₄	C₄₁×#472	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×#472	2
C₄₁×#3038	2
C₄×#131×C₄₁	2
C₄₁×#1978	2
C₄₁×#1965	2
C₄₁×#1848	2
#33190	41

Subgroups — C41×#33190 #23809391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33190 #23809391