Subgroups — C₄₁×#7693 #23783894

Order 10496.

390 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2863	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1745	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1272	C₂	1	2	yes	maximal
5	C₄₁×#1271	C₂	1	2	yes	maximal
6	C₂×#97×C₄₁	—	2	1	no	—
7	C₂×C₄×C₈×C₄₁	—	2	1	no	—
8	C₄₁×#391	C₄	1	1	yes	characteristic
9	C₄₁×#391	C₄	1	1	yes	characteristic
10	C₄₁×#391	C₂²	1	1	yes	characteristic
11	C₄₁×#389	C₄	1	1	yes	characteristic
12	C₄₁×#389	C₄	1	1	yes	characteristic
13	C₄₁×#389	C₂²	1	1	yes	characteristic
14	C₄₁×#387	—	2	1	no	—
15	C₄₁×#387	—	2	1	no	—
16	C₄₁×#387	—	2	1	no	—
17	C₄₁×#383	—	2	1	no	—
18	C₄³×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#336	—	2	1	no	—
20	C₄₁×#336	—	2	1	no	—
21	C₄₁×#336	C₂²	1	2	yes	—
22	C₄₁×#336	C₂²	1	2	yes	—
23	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₁	—	2	2	no	—
25	C₂²×C₈×C₄₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
39	#97×C₄₁	—	2	4	no	—
40	C₄×C₈×C₄₁	—	2	4	no	—
41	#95×C₄₁	—	2	1	no	—
42	#95×C₄₁	—	2	1	no	—
43	#95×C₄₁	—	2	1	no	—
44	#95×C₄₁	—	2	1	no	—
45	#95×C₄₁	—	2	1	no	—
46	#95×C₄₁	—	2	1	no	—
47	#95×C₄₁	—	2	1	no	—
48	#95×C₄₁	—	2	1	no	—
49	#95×C₄₁	—	2	1	no	—
50	C₂×Q₈×C₄₁	—	2	2	no	—
51	C₂×Q₈×C₄₁	—	4	1	no	—
52	C₂×Q₈×C₄₁	—	4	1	no	—
53	C₂²×C₄×C₄₁	—	2	1	no	—
54	C₂²×C₄×C₄₁	—	2	1	no	—
55	C₂²×C₄×C₄₁	—	2	1	no	—
56	C₂²×C₄×C₄₁	—	2	1	no	—
57	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₄₁	—	2	4	no	—
65	C₂×C₈×C₄₁	—	2	4	no	—
66	C₂×C₈×C₄₁	—	4	2	no	—
67	C₂×C₈×C₄₁	—	4	2	no	—
68	C₂×C₈×C₄₁	—	4	2	no	—
69	C₂×C₈×C₄₁	—	4	2	no	—
70	C₄⋊C₄×C₄₁	—	2	2	no	—
71	C₄⋊C₄×C₄₁	—	4	1	no	—
72	C₄⋊C₄×C₄₁	—	4	1	no	—
73	C₄⋊C₄×C₄₁	—	4	2	no	—
74	C₄⋊C₄×C₄₁	—	4	2	no	—
75	C₄⋊C₄×C₄₁	—	4	2	no	—
76	C₄²×C₄₁	—	2	2	no	—
77	C₄²×C₄₁	—	2	2	no	—
78	C₄²×C₄₁	—	2	2	no	—
79	C₄²×C₄₁	—	2	2	no	—
80	C₄²×C₄₁	—	2	2	no	—
81	C₄²×C₄₁	—	4	1	no	—
82	C₄²×C₄₁	—	4	1	no	—
83	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₄₁	—	8	2	no	—
85	C₂×C₄×C₄₁	—	2	1	no	—
86	C₂×C₄×C₄₁	—	2	1	no	—
87	C₂×C₄×C₄₁	—	2	1	no	—
88	C₂×C₄×C₄₁	—	2	1	no	—
89	C₂×C₄×C₄₁	—	2	1	no	—
90	C₂×C₄×C₄₁	—	2	1	no	—
91	C₂×C₄×C₄₁	—	2	1	no	—
92	C₂×C₄×C₄₁	—	2	1	no	—
93	C₂×C₄×C₄₁	—	2	1	no	—
94	C₂×C₄×C₄₁	—	2	1	no	—
95	C₂×C₄×C₄₁	—	2	1	no	—
96	C₂×C₄×C₄₁	—	2	1	no	—
97	C₂×C₄×C₄₁	—	2	1	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	4	1	no	—
102	C₂×C₄×C₄₁	—	4	1	no	—
103	C₂×C₄×C₄₁	—	4	1	no	—
104	C₂×C₄×C₄₁	—	4	1	no	—
105	C₂×C₄×C₄₁	—	4	1	no	—
106	C₂×C₄×C₄₁	—	4	1	no	—
107	C₂×C₄×C₄₁	—	4	1	no	—
108	C₂×C₄×C₄₁	—	4	1	no	—
109	C₂×C₄×C₄₁	—	4	1	no	—
110	C₂×C₄×C₄₁	—	4	1	no	—
111	C₂×C₄×C₄₁	—	4	1	no	—
112	C₂×C₄×C₄₁	—	4	1	no	—
113	C₂×C₄×C₄₁	#104	1	2	yes	—
114	C₂×C₄×C₄₁	#104	1	2	yes	—
115	C₂×C₄×C₄₁	#131	1	2	yes	—
116	C₂×C₄×C₄₁	#135	1	2	yes	—
117	C₂×C₄×C₄₁	#137	1	2	yes	—
118	C₈×C₄₁	—	4	4	no	—
119	C₈×C₄₁	—	4	4	no	—
120	#7693	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₄₁	#408	1	1	yes	characteristic
122	C₂²×C₄₁	#410	1	1	yes	characteristic
123	C₂²×C₄₁	#413	1	1	yes	characteristic
124	C₂²×C₄₁	#416	1	1	yes	characteristic
125	C₂²×C₄₁	#419	1	1	yes	characteristic
126	C₂²×C₄₁	C₂×#104	1	1	yes	characteristic
127	C₂²×C₄₁	C₂×#104	1	1	yes	characteristic
128	C₄×C₄₁	—	2	2	no	—
129	C₄×C₄₁	—	2	2	no	—
130	C₄×C₄₁	—	2	2	no	—
131	C₄×C₄₁	—	2	2	no	—
132	C₄×C₄₁	—	2	2	no	—
133	C₄×C₄₁	—	4	1	no	—
134	C₄×C₄₁	—	4	1	no	—
135	C₄×C₄₁	—	8	1	no	—
136	C₄×C₄₁	—	8	1	no	—
137	C₄×C₄₁	—	8	1	no	—
138	C₄×C₄₁	—	8	1	no	—
139	#2863	C₂×C₄₁	1	1	yes	characteristic
140	#1745	C₂×C₄₁	1	1	yes	characteristic
141	#1744	C₂×C₄₁	1	1	yes	characteristic
142	#1272	C₂×C₄₁	1	2	yes	—
143	#1271	C₂×C₄₁	1	2	yes	—
144	C₂×C₄₁	#1473	1	1	yes	characteristic
145	C₂×C₄₁	#1485	1	1	yes	characteristic
146	C₂×C₄₁	#1494	1	1	yes	characteristic
147	C₂×C₄₁	#1500	1	1	yes	characteristic
148	C₂×C₄₁	#1506	1	1	yes	characteristic
149	C₂×C₄₁	#1524	1	1	yes	characteristic
150	C₂×C₄₁	#1554	1	1	yes	characteristic
151	C₂×#97	—	2	1	no	—
152	C₂×C₄×C₈	—	2	1	no	—
153	#391	C₄×C₄₁	1	1	yes	characteristic
154	#391	C₄×C₄₁	1	1	yes	characteristic
155	#391	C₂²×C₄₁	1	1	yes	characteristic
156	#389	C₄×C₄₁	1	1	yes	characteristic
157	#389	C₄×C₄₁	1	1	yes	characteristic
158	#389	C₂²×C₄₁	1	1	yes	characteristic
159	#387	—	2	1	no	—
160	#387	—	2	1	no	—
161	#387	—	2	1	no	—
162	#383	—	2	1	no	—
163	C₄³	C₂²×C₄₁	1	1	yes	characteristic
164	#336	—	2	1	no	—
165	#336	—	2	1	no	—
166	#336	C₂²×C₄₁	1	2	yes	—
167	#336	C₂²×C₄₁	1	2	yes	—
168	C₄₁	#7693	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
185	#97	—	2	4	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	—	2	1	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	4	1	no	—
228	C₄²	—	4	1	no	—
229	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
230	Q₈	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	#104×C₄₁	1	2	yes	—
260	C₂×C₄	#104×C₄₁	1	2	yes	—
261	C₂×C₄	#131×C₄₁	1	2	yes	—
262	C₂×C₄	#135×C₄₁	1	2	yes	—
263	C₂×C₄	#137×C₄₁	1	2	yes	—
264	C₈	—	4	4	no	—
265	C₈	—	4	4	no	—
266	C₂²	C₄₁×#408	1	1	yes	characteristic
267	C₂²	C₄₁×#410	1	1	yes	characteristic
268	C₂²	C₄₁×#413	1	1	yes	characteristic
269	C₂²	C₄₁×#416	1	1	yes	characteristic
270	C₂²	C₄₁×#419	1	1	yes	characteristic
271	C₂²	C₂×#104×C₄₁	1	1	yes	characteristic
272	C₂²	C₂×#104×C₄₁	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	8	1	no	—
281	C₄	—	8	1	no	—
282	C₄	—	8	1	no	—
283	C₄	—	8	1	no	—
284	C₂	C₄₁×#1473	1	1	yes	characteristic
285	C₂	C₄₁×#1485	1	1	yes	characteristic
286	C₂	C₄₁×#1494	1	1	yes	characteristic
287	C₂	C₄₁×#1500	1	1	yes	characteristic
288	C₂	C₄₁×#1506	1	1	yes	characteristic
289	C₂	C₄₁×#1524	1	1	yes	characteristic
290	C₂	C₄₁×#1554	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
120. #7693	C₄₁	direct (N × H)
168. C₄₁	#7693	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2863	2
C₄₁×#1745	2
C₄₁×#1744	2
C₄₁×#1272	2
C₄₁×#1271	2
#7693	41

Subgroups — C41×#7693 #23783894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7693 #23783894