Subgroups — C₄₁×#7720 #23783921

Order 10496.

388 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1839	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1831	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1271	C₂	1	4	yes	maximal
5	C₂×#125×C₄₁	—	2	1	no	—
6	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
7	C₄₁×#389	—	2	1	no	—
8	C₄₁×#389	—	2	1	no	—
9	C₄₁×#389	C₄	1	2	yes	—
10	C₄₁×#389	C₂²	1	2	yes	—
11	C₄₁×#384	C₄	1	1	yes	characteristic
12	C₄₁×#384	C₄	1	1	yes	characteristic
13	C₄₁×#382	—	2	1	no	—
14	C₄³×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#336	—	2	1	no	—
16	C₄₁×#336	—	2	1	no	—
17	C₄₁×#336	—	2	1	no	—
18	C₄₁×#336	C₂²	1	2	yes	—
19	C₄₁×#336	C₂²	1	2	yes	—
20	C₂²×C₈×C₄₁	—	2	2	no	—
21	C₂²×C₈×C₄₁	—	2	2	no	—
22	#125×C₄₁	—	4	4	no	—
23	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
24	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
25	C₂×C₄⋊C₄×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	2	no	—
29	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₁	D₄	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	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₄₁	—	4	2	no	—
39	#95×C₄₁	—	2	2	no	—
40	#95×C₄₁	—	2	2	no	—
41	C₂²×C₄×C₄₁	—	2	2	no	—
42	C₂²×C₄×C₄₁	—	2	2	no	—
43	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₄₁	—	4	2	no	—
51	C₂×C₈×C₄₁	—	4	2	no	—
52	C₂×C₈×C₄₁	—	4	2	no	—
53	C₂×C₈×C₄₁	—	4	2	no	—
54	C₂×C₈×C₄₁	—	4	2	no	—
55	C₂×C₈×C₄₁	—	4	2	no	—
56	C₄⋊C₄×C₄₁	—	2	4	no	—
57	C₄⋊C₄×C₄₁	—	4	2	no	—
58	C₄⋊C₄×C₄₁	—	4	2	no	—
59	C₄⋊C₄×C₄₁	—	4	2	no	—
60	C₄⋊C₄×C₄₁	—	4	2	no	—
61	C₄⋊C₄×C₄₁	—	4	2	no	—
62	C₄⋊C₄×C₄₁	—	4	4	no	—
63	C₄⋊C₄×C₄₁	—	4	4	no	—
64	C₄²×C₄₁	—	2	2	no	—
65	C₄²×C₄₁	—	2	2	no	—
66	C₄²×C₄₁	—	2	2	no	—
67	C₄²×C₄₁	—	4	1	no	—
68	C₄²×C₄₁	—	4	1	no	—
69	C₄²×C₄₁	—	4	1	no	—
70	C₄²×C₄₁	SD₁₆	1	2	yes	—
71	C₄²×C₄₁	Q₁₆	1	2	yes	—
72	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₁	—	2	1	no	—
74	C₂×C₄×C₄₁	—	2	1	no	—
75	C₂×C₄×C₄₁	—	2	1	no	—
76	C₂×C₄×C₄₁	—	2	1	no	—
77	C₂×C₄×C₄₁	—	2	1	no	—
78	C₂×C₄×C₄₁	—	2	1	no	—
79	C₂×C₄×C₄₁	—	2	1	no	—
80	C₂×C₄×C₄₁	—	2	1	no	—
81	C₂×C₄×C₄₁	—	2	1	no	—
82	C₂×C₄×C₄₁	—	2	1	no	—
83	C₂×C₄×C₄₁	—	2	1	no	—
84	C₂×C₄×C₄₁	—	2	1	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	—	4	2	no	—
89	C₂×C₄×C₄₁	—	4	2	no	—
90	C₂×C₄×C₄₁	—	4	2	no	—
91	C₂×C₄×C₄₁	—	4	2	no	—
92	C₂×C₄×C₄₁	—	4	2	no	—
93	C₂×C₄×C₄₁	—	4	2	no	—
94	C₂×C₄×C₄₁	#101	1	2	yes	—
95	C₂×C₄×C₄₁	#103	1	2	yes	—
96	C₂×C₄×C₄₁	#103	1	2	yes	—
97	C₂×C₄×C₄₁	#131	1	2	yes	—
98	C₂×C₄×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
101	C₈×C₄₁	—	8	2	no	—
102	C₈×C₄₁	—	8	2	no	—
103	#7720	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	#408	1	1	yes	characteristic
105	C₂²×C₄₁	#410	1	1	yes	characteristic
106	C₂²×C₄₁	C₂×#101	1	1	yes	characteristic
107	C₂²×C₄₁	C₂×#103	1	1	yes	characteristic
108	C₂²×C₄₁	#418	1	1	yes	characteristic
109	C₂²×C₄₁	#421	1	1	yes	characteristic
110	C₂²×C₄₁	#421	1	1	yes	characteristic
111	C₄×C₄₁	—	2	2	no	—
112	C₄×C₄₁	—	2	2	no	—
113	C₄×C₄₁	—	2	2	no	—
114	C₄×C₄₁	—	2	2	no	—
115	C₄×C₄₁	—	2	1	no	—
116	C₄×C₄₁	—	2	1	no	—
117	C₄×C₄₁	—	2	2	no	—
118	C₄×C₄₁	—	4	1	no	—
119	C₄×C₄₁	—	8	2	no	—
120	C₄×C₄₁	—	8	2	no	—
121	#2859	C₂×C₄₁	1	1	yes	characteristic
122	#1839	C₂×C₄₁	1	1	yes	characteristic
123	#1831	C₂×C₄₁	1	1	yes	characteristic
124	#1271	C₂×C₄₁	1	4	yes	—
125	C₂×C₄₁	#1470	1	1	yes	characteristic
126	C₂×C₄₁	#1481	1	1	yes	characteristic
127	C₂×C₄₁	#1496	1	1	yes	characteristic
128	C₂×C₄₁	#1514	1	1	yes	characteristic
129	C₂×C₄₁	#1518	1	1	yes	characteristic
130	C₂×C₄₁	#1534	1	1	yes	characteristic
131	C₂×C₄₁	#1549	1	1	yes	characteristic
132	C₂×#125	—	2	1	no	—
133	C₂×C₄⋊C₈	—	2	1	no	—
134	#389	—	2	1	no	—
135	#389	—	2	1	no	—
136	#389	C₄×C₄₁	1	2	yes	—
137	#389	C₂²×C₄₁	1	2	yes	—
138	#384	C₄×C₄₁	1	1	yes	characteristic
139	#384	C₄×C₄₁	1	1	yes	characteristic
140	#382	—	2	1	no	—
141	C₄³	C₂²×C₄₁	1	1	yes	characteristic
142	#336	—	2	1	no	—
143	#336	—	2	1	no	—
144	#336	—	2	1	no	—
145	#336	C₂²×C₄₁	1	2	yes	—
146	#336	C₂²×C₄₁	1	2	yes	—
147	C₄₁	#7720	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	#125	—	4	4	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
159	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
161	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
166	C₄⋊C₈	—	4	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	SD₁₆×C₄₁	1	2	yes	—
199	C₄²	Q₁₆×C₄₁	1	2	yes	—
200	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	#101×C₄₁	1	2	yes	—
223	C₂×C₄	#103×C₄₁	1	2	yes	—
224	C₂×C₄	#103×C₄₁	1	2	yes	—
225	C₂×C₄	#131×C₄₁	1	2	yes	—
226	C₂×C₄	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
227	C₂×C₄	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
228	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
229	C₈	—	8	2	no	—
230	C₈	—	8	2	no	—
231	C₂²	C₄₁×#408	1	1	yes	characteristic
232	C₂²	C₄₁×#410	1	1	yes	characteristic
233	C₂²	C₂×#101×C₄₁	1	1	yes	characteristic
234	C₂²	C₂×#103×C₄₁	1	1	yes	characteristic
235	C₂²	C₄₁×#418	1	1	yes	characteristic
236	C₂²	C₄₁×#421	1	1	yes	characteristic
237	C₂²	C₄₁×#421	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	8	2	no	—
247	C₄	—	8	2	no	—
248	C₂	C₄₁×#1470	1	1	yes	characteristic
249	C₂	C₄₁×#1481	1	1	yes	characteristic
250	C₂	C₄₁×#1496	1	1	yes	characteristic
251	C₂	C₄₁×#1514	1	1	yes	characteristic
252	C₂	C₄₁×#1518	1	1	yes	characteristic
253	C₂	C₄₁×#1534	1	1	yes	characteristic
254	C₂	C₄₁×#1549	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
103. #7720	C₄₁	direct (N × H)
147. C₄₁	#7720	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×#2859	2
C₄₁×#1839	2
C₄₁×#1831	2
C₄₁×#1271	2
#7720	41

Subgroups — C41×#7720 #23783921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7720 #23783921