Subgroups — C₄₁×#7980 #23784181

Order 10496.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2283	C₂	1	2	yes	maximal
3	C₄₁×#1289	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
5	C₄₁×#431	—	2	2	no	—
6	C₄₁×#431	—	2	2	no	—
7	C₂×C₄⋊C₈×C₄₁	C₄	1	2	yes	complemented
8	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₄₁	C₄	1	2	yes	complemented
10	C₄₁×#383	C₄	1	4	yes	complemented
11	C₄₁×#382	—	2	2	no	—
12	C₄₁×#381	—	2	2	no	—
13	C₄₁×#380	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
15	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
16	C₄₁×#376	C₄	1	4	yes	complemented
17	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
22	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
23	C₂×C₄²×C₄₁	—	2	4	no	—
24	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₁	D₄	1	2	yes	—
29	C₂×C₄²×C₄₁	D₄	1	2	yes	—
30	C₂×C₄²×C₄₁	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₄₁	Q₈	1	1	yes	characteristic
32	C₄⋊C₈×C₄₁	—	2	4	no	—
33	C₂²⋊C₈×C₄₁	—	2	4	no	—
34	C₄×C₈×C₄₁	—	2	2	no	—
35	C₄×C₈×C₄₁	—	2	2	no	—
36	#95×C₄₁	—	2	4	no	—
37	#95×C₄₁	—	2	4	no	—
38	#95×C₄₁	—	2	4	no	—
39	#95×C₄₁	C₂×C₄	1	8	yes	—
40	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₁	—	2	4	no	—
42	C₂²×C₄×C₄₁	—	2	8	no	—
43	C₂²×C₄×C₄₁	—	4	2	no	—
44	C₂²×C₄×C₄₁	—	4	2	no	—
45	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₄₁	—	2	4	no	—
58	C₂×C₈×C₄₁	—	4	4	no	—
59	C₄⋊C₄×C₄₁	—	4	8	no	—
60	C₂²⋊C₄×C₄₁	—	4	8	no	—
61	C₄²×C₄₁	—	4	8	no	—
62	C₄²×C₄₁	C₄²	1	1	yes	characteristic
63	C₄²×C₄₁	C₄²	1	2	yes	—
64	C₄²×C₄₁	C₄²	1	1	yes	characteristic
65	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
72	C₂³×C₄₁	—	4	1	no	—
73	C₂³×C₄₁	—	4	1	no	—
74	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
75	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₄₁	#135	1	4	yes	—
77	C₂×C₄×C₄₁	—	2	4	no	—
78	C₂×C₄×C₄₁	—	2	4	no	—
79	C₂×C₄×C₄₁	—	2	2	no	—
80	C₂×C₄×C₄₁	—	2	2	no	—
81	C₂×C₄×C₄₁	—	2	4	no	—
82	C₂×C₄×C₄₁	—	2	2	no	—
83	C₂×C₄×C₄₁	—	2	4	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	4	2	no	—
86	C₂×C₄×C₄₁	—	4	1	no	—
87	C₂×C₄×C₄₁	—	4	2	no	—
88	C₂×C₄×C₄₁	—	4	1	no	—
89	C₂×C₄×C₄₁	—	4	8	no	—
90	C₂×C₄×C₄₁	—	4	8	no	—
91	C₂×C₄×C₄₁	—	4	8	no	—
92	C₂×C₄×C₄₁	#95	1	1	yes	characteristic
93	C₂×C₄×C₄₁	#95	1	2	yes	—
94	C₂×C₄×C₄₁	#95	1	2	yes	—
95	C₂×C₄×C₄₁	#95	1	1	yes	characteristic
96	C₂×C₄×C₄₁	#95	1	1	yes	characteristic
97	C₂×C₄×C₄₁	#95	1	1	yes	characteristic
98	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₈×C₄₁	—	4	4	no	—
102	#7980	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₄₁	—	2	2	no	—
104	C₂²×C₄₁	—	2	1	no	—
105	C₂²×C₄₁	—	2	2	no	—
106	C₂²×C₄₁	—	2	1	no	—
107	C₂²×C₄₁	—	4	4	no	—
108	C₂²×C₄₁	C₂×#95	1	1	yes	characteristic
109	C₂²×C₄₁	#416	1	4	yes	—
110	C₂²×C₄₁	#427	1	2	yes	—
111	C₄×C₄₁	—	2	2	no	—
112	C₄×C₄₁	—	2	1	no	—
113	C₄×C₄₁	—	2	2	no	—
114	C₄×C₄₁	—	2	1	no	—
115	C₄×C₄₁	—	2	1	no	—
116	C₄×C₄₁	—	2	1	no	—
117	C₄×C₄₁	—	2	2	no	—
118	C₄×C₄₁	—	8	8	no	—
119	C₄×C₄₁	H(3,ℤ/4)	1	2	yes	—
120	C₄×C₄₁	#410	1	2	yes	—
121	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
122	#2283	C₂×C₄₁	1	2	yes	—
123	#1289	C₂×C₄₁	1	4	yes	—
124	C₂×C₄₁	—	2	4	no	—
125	C₂×C₄₁	—	2	2	no	—
126	C₂×C₄₁	C₂×H(3,ℤ/4)	1	1	yes	characteristic
127	C₂×C₄₁	#1735	1	1	yes	characteristic
128	C₂×C₄₁	#1740	1	1	yes	characteristic
129	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
130	#431	—	2	2	no	—
131	#431	—	2	2	no	—
132	C₂×C₄⋊C₈	C₄×C₄₁	1	2	yes	complemented
133	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
134	C₂×C₄×C₈	C₄×C₄₁	1	2	yes	complemented
135	#383	C₄×C₄₁	1	4	yes	complemented
136	#382	—	2	2	no	—
137	#381	—	2	2	no	—
138	#380	C₂²×C₄₁	1	4	yes	—
139	C₄×C₄⋊C₄	—	2	2	no	—
140	C₄×C₂²⋊C₄	—	2	2	no	—
141	#376	C₄×C₄₁	1	4	yes	complemented
142	C₄₁	#7980	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
144	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
145	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₂²⋊C₄	—	2	4	no	—
149	C₂×C₄²	—	2	4	no	—
150	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
153	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
154	C₂×C₄²	D₄×C₄₁	1	2	yes	—
155	C₂×C₄²	D₄×C₄₁	1	2	yes	—
156	C₂×C₄²	Q₈×C₄₁	1	1	yes	characteristic
157	C₂×C₄²	Q₈×C₄₁	1	1	yes	characteristic
158	C₄⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	2	4	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	#95	—	2	4	no	—
163	#95	—	2	4	no	—
164	#95	—	2	4	no	—
165	#95	C₂×C₄×C₄₁	1	8	yes	—
166	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
172	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
173	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
175	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
182	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	4	no	—
185	C₄⋊C₄	—	4	8	no	—
186	C₂²⋊C₄	—	4	8	no	—
187	C₄²	—	4	8	no	—
188	C₄²	C₄²×C₄₁	1	1	yes	characteristic
189	C₄²	C₄²×C₄₁	1	2	yes	—
190	C₄²	C₄²×C₄₁	1	1	yes	characteristic
191	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
192	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
193	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
194	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
195	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
196	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
197	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
201	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
202	C₂³	#135×C₄₁	1	4	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	#95×C₄₁	1	1	yes	characteristic
219	C₂×C₄	#95×C₄₁	1	2	yes	—
220	C₂×C₄	#95×C₄₁	1	1	yes	characteristic
221	C₂×C₄	#95×C₄₁	1	1	yes	characteristic
222	C₂×C₄	#95×C₄₁	1	2	yes	—
223	C₂×C₄	#95×C₄₁	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
225	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×#95×C₄₁	1	1	yes	characteristic
234	C₂²	C₄₁×#416	1	4	yes	—
235	C₂²	C₄₁×#427	1	2	yes	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	—	8	8	no	—
244	C₄	C₄₁×H(3,ℤ/4)	1	2	yes	—
245	C₄	C₄₁×#410	1	2	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	2	no	—
248	C₂	C₂×C₄₁×H(3,ℤ/4)	1	1	yes	characteristic
249	C₂	C₄₁×#1735	1	1	yes	characteristic
250	C₂	C₄₁×#1740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	#95
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×#95
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊C₈×C₄₁	C₄	semidirect
9. C₂×C₄×C₈×C₄₁	C₄	semidirect
10. C₄₁×#383	C₄	semidirect
16. C₄₁×#376	C₄	semidirect
102. #7980	C₄₁	direct (N × H)
132. C₂×C₄⋊C₈	C₄×C₄₁	semidirect
134. C₂×C₄×C₈	C₄×C₄₁	semidirect
135. #383	C₄×C₄₁	semidirect
141. #376	C₄×C₄₁	semidirect
142. C₄₁	#7980	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#431	2
C₄₁×#2283	2
C₄₁×#1289	2
#7980	41

Subgroups — C41×#7980 #23784181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7980 #23784181