Subgroups — C₄₁×#9805 #23786006

Order 10496.

534 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#1965	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₄₁	C₄	1	4	yes	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#492	—	2	1	no	—
12	C₄₁×#492	—	2	1	no	—
13	C₄₁×#492	—	2	2	no	—
14	C₄₁×#431	C₂²	1	1	yes	characteristic
15	C₄₁×#423	C₂²	1	1	yes	characteristic
16	C₄₁×#423	C₂²	1	1	yes	characteristic
17	C₂×#100×C₄₁	—	2	2	no	—
18	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#343	—	2	2	no	—
21	C₄₁×#335	—	2	2	no	—
22	C₄₁×#333	—	2	2	no	—
23	C₄₁×#331	—	2	2	no	—
24	C₄₁×#322	—	2	1	no	—
25	C₄₁×#322	—	2	1	no	—
26	C₂²×Q₈×C₄₁	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₄₁	—	2	1	no	—
28	C₂²×D₄×C₄₁	—	2	1	no	—
29	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
30	C₂×SD₁₆×C₄₁	—	2	4	no	—
31	C₂×SD₁₆×C₄₁	—	2	4	no	—
32	C₂×SD₁₆×C₄₁	—	4	8	no	—
33	C₂×M₁₆×C₄₁	—	2	1	no	—
34	C₂×M₁₆×C₄₁	—	2	1	no	—
35	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
36	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
37	C₄⋊Q₈×C₄₁	—	2	1	no	—
38	C₄⋊Q₈×C₄₁	—	2	1	no	—
39	C₄⋊Q₈×C₄₁	—	2	1	no	—
40	C₄⋊Q₈×C₄₁	—	2	1	no	—
41	#127×C₄₁	—	2	1	no	—
42	#127×C₄₁	—	2	1	no	—
43	#127×C₄₁	—	2	1	no	—
44	#127×C₄₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
46	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₁	—	2	1	no	—
49	C₄⋊C₈×C₄₁	—	2	1	no	—
50	C₄⋊C₈×C₄₁	—	2	1	no	—
51	C₄⋊C₈×C₄₁	—	2	1	no	—
52	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
53	#100×C₄₁	—	4	4	no	—
54	C₂²⋊C₈×C₄₁	—	2	1	no	—
55	#97×C₄₁	—	2	1	no	—
56	C₄×C₈×C₄₁	—	2	1	no	—
57	C₄×C₈×C₄₁	—	2	1	no	—
58	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₄₁	D₄	1	2	yes	—
61	C₂⁴×C₄₁	—	2	2	no	—
62	C₂×Q₈×C₄₁	—	2	2	no	—
63	C₂×Q₈×C₄₁	—	2	2	no	—
64	C₂×Q₈×C₄₁	—	4	4	no	—
65	C₂×D₄×C₄₁	—	2	2	no	—
66	C₂×D₄×C₄₁	—	2	2	no	—
67	C₂×D₄×C₄₁	—	4	4	no	—
68	C₂×D₄×C₄₁	—	4	1	no	—
69	C₂×D₄×C₄₁	—	4	1	no	—
70	C₂×D₄×C₄₁	—	4	1	no	—
71	C₂×D₄×C₄₁	—	4	1	no	—
72	C₂×D₄×C₄₁	—	4	2	no	—
73	C₂×D₄×C₄₁	—	4	2	no	—
74	C₂×D₄×C₄₁	—	8	2	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
79	SD₁₆×C₄₁	—	4	8	no	—
80	SD₁₆×C₄₁	—	4	8	no	—
81	M₁₆×C₄₁	—	2	2	no	—
82	M₁₆×C₄₁	—	2	2	no	—
83	M₁₆×C₄₁	—	4	2	no	—
84	M₁₆×C₄₁	—	4	2	no	—
85	C₂×C₈×C₄₁	—	2	4	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	—	2	2	no	—
88	C₂×C₈×C₄₁	—	4	1	no	—
89	C₂×C₈×C₄₁	—	4	1	no	—
90	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
92	C₄⋊C₄×C₄₁	—	4	1	no	—
93	C₄⋊C₄×C₄₁	—	4	1	no	—
94	C₄⋊C₄×C₄₁	—	4	1	no	—
95	C₄⋊C₄×C₄₁	—	4	1	no	—
96	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₁	#41	1	1	yes	characteristic
100	C₂³×C₄₁	—	4	2	no	—
101	C₂³×C₄₁	—	4	4	no	—
102	C₂³×C₄₁	—	8	2	no	—
103	C₂³×C₄₁	—	8	2	no	—
104	C₂³×C₄₁	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₄₁	—	4	4	no	—
106	Q₈×C₄₁	—	8	2	no	—
107	D₄×C₄₁	—	4	4	no	—
108	D₄×C₄₁	—	8	2	no	—
109	D₄×C₄₁	—	8	2	no	—
110	D₄×C₄₁	—	8	2	no	—
111	D₄×C₄₁	—	8	2	no	—
112	D₄×C₄₁	—	8	1	no	—
113	D₄×C₄₁	—	8	1	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	8	2	no	—
118	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
123	C₂×C₄×C₄₁	#127	1	1	yes	characteristic
124	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₄₁	#137	1	2	yes	—
126	C₈×C₄₁	—	2	2	no	—
127	C₈×C₄₁	—	4	2	no	—
128	C₈×C₄₁	—	4	2	no	—
129	C₈×C₄₁	—	8	1	no	—
130	C₈×C₄₁	—	8	1	no	—
131	C₈×C₄₁	#100	1	4	yes	complemented
132	#9805	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₄₁	—	2	1	no	—
134	C₂²×C₄₁	—	2	1	no	—
135	C₂²×C₄₁	—	8	2	no	—
136	C₂²×C₄₁	—	8	2	no	—
137	C₂²×C₄₁	—	8	2	no	—
138	C₂²×C₄₁	—	16	4	no	—
139	C₂²×C₄₁	#390	1	1	yes	characteristic
140	C₂²×C₄₁	#440	1	2	yes	—
141	C₄×C₄₁	—	2	1	no	—
142	C₄×C₄₁	—	2	1	no	—
143	C₄×C₄₁	—	8	2	no	—
144	C₄×C₄₁	C₂×#100	1	2	yes	—
145	C₄×C₄₁	#468	1	2	yes	—
146	C₄×C₄₁	#471	1	2	yes	—
147	C₄×C₄₁	#496	1	1	yes	characteristic
148	C₄×C₄₁	#497	1	1	yes	characteristic
149	C₂×#492	C₂×C₄₁	1	1	yes	characteristic
150	#1997	C₂×C₄₁	1	1	yes	characteristic, complemented
151	#1965	C₂×C₄₁	1	1	yes	characteristic
152	#1567	C₂×C₄₁	1	1	yes	characteristic, complemented
153	#1548	C₂×C₄₁	1	1	yes	characteristic, complemented
154	#1545	C₂×C₄₁	1	1	yes	characteristic
155	#1451	C₂×C₄₁	1	1	yes	characteristic, complemented
156	C₂×C₄₁	—	2	2	no	—
157	C₂×C₄₁	—	16	2	no	—
158	C₂×C₄₁	#1965	1	1	yes	characteristic
159	C₂×C₄₁	#1973	1	1	yes	characteristic
160	C₂×C₄₁	#1977	1	1	yes	characteristic
161	C₂²×SD₁₆	C₄×C₄₁	1	4	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
164	#492	—	2	1	no	—
165	#492	—	2	1	no	—
166	#492	—	2	2	no	—
167	#431	C₂²×C₄₁	1	1	yes	characteristic
168	#423	C₂²×C₄₁	1	1	yes	characteristic
169	#423	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×#100	—	2	2	no	—
171	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
173	#343	—	2	2	no	—
174	#335	—	2	2	no	—
175	#333	—	2	2	no	—
176	#331	—	2	2	no	—
177	#322	—	2	1	no	—
178	#322	—	2	1	no	—
179	C₄₁	#9805	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²×Q₈	C₂×C₄×C₄₁	1	2	yes	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
184	C₂×SD₁₆	—	2	4	no	—
185	C₂×SD₁₆	—	2	4	no	—
186	C₂×SD₁₆	—	4	8	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
190	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	#127	—	2	1	no	—
196	#127	—	2	1	no	—
197	#127	—	2	1	no	—
198	#127	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
207	#100	—	4	4	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	#97	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
213	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
214	C₄×C₈	D₄×C₄₁	1	2	yes	—
215	C₂⁴	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	8	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
233	SD₁₆	—	4	8	no	—
234	SD₁₆	—	4	8	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
245	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
253	C₄²	#41×C₄₁	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	8	2	no	—
257	C₂³	—	8	2	no	—
258	C₂³	C₄×D₄×C₄₁	1	1	yes	characteristic
259	Q₈	—	4	4	no	—
260	Q₈	—	8	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
274	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
275	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
276	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
277	C₂×C₄	#127×C₄₁	1	1	yes	characteristic
278	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
279	C₂×C₄	#137×C₄₁	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	8	1	no	—
284	C₈	—	8	1	no	—
285	C₈	#100×C₄₁	1	4	yes	complemented
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	16	4	no	—
292	C₂²	C₄₁×#390	1	1	yes	characteristic
293	C₂²	C₄₁×#440	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	2	no	—
297	C₄	C₂×#100×C₄₁	1	2	yes	—
298	C₄	C₄₁×#468	1	2	yes	—
299	C₄	C₄₁×#471	1	2	yes	—
300	C₄	C₄₁×#496	1	1	yes	characteristic
301	C₄	C₄₁×#497	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	2	no	—
304	C₂	C₄₁×#1965	1	1	yes	characteristic
305	C₂	C₄₁×#1973	1	1	yes	characteristic
306	C₂	C₄₁×#1977	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₂×#127×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
2. C₄₁×#1997	C₂	semidirect
4. C₄₁×#1567	C₂	semidirect
5. C₄₁×#1548	C₂	semidirect
7. C₄₁×#1451	C₂	semidirect
131. C₈×C₄₁	#100	semidirect
132. #9805	C₄₁	direct (N × H)
150. #1997	C₂×C₄₁	semidirect
152. #1567	C₂×C₄₁	semidirect
153. #1548	C₂×C₄₁	semidirect
155. #1451	C₂×C₄₁	semidirect
179. C₄₁	#9805	direct (N × H)
285. C₈	#100×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×#492	2
C₄₁×#1997	2
C₄₁×#1965	2
C₄₁×#1567	2
C₄₁×#1548	2
C₄₁×#1545	2
C₄₁×#1451	2
#9805	41

Subgroups — C41×#9805 #23786006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#9805 #23786006