Subgroups — C₄₁×#21690 #23797891

Order 10496.

1398 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×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#442	C₂	1	4	yes	maximal, complemented
3	C₄×#127×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1972	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1957	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#97×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₄₁	C₂²	1	4	yes	—
9	C₂×#127×C₄₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₄₁	—	2	4	no	—
11	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
12	C₄₁×#493	C₄	1	8	yes	complemented
13	C₄₁×#442	—	2	16	no	—
14	C₂×#106×C₄₁	C₂²	1	4	yes	—
15	C₂×#102×C₄₁	C₂²	1	8	yes	—
16	C₂×#97×C₄₁	C₂²	1	2	yes	—
17	C₂×#97×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#390	—	2	4	no	—
21	C₄₁×#390	C₂²	1	4	yes	—
22	C₄₁×#384	C₂²	1	2	yes	—
23	C₄³×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²×D₄×C₄₁	—	2	4	no	—
25	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₄₁	—	2	4	no	—
27	C₂×D₈×C₄₁	—	2	8	no	—
28	C₂×D₈×C₄₁	—	2	8	no	—
29	C₂×D₈×C₄₁	C₂×C₄	1	16	yes	complemented
30	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
32	#127×C₄₁	—	2	4	no	—
33	#127×C₄₁	C₂×C₄	1	8	yes	complemented
34	C₄×D₄×C₄₁	—	2	16	no	—
35	C₄×D₄×C₄₁	—	4	16	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
38	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
39	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
40	C₂×C₄²×C₄₁	D₄	1	4	yes	—
41	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
44	#106×C₄₁	—	2	8	no	—
45	#102×C₄₁	—	2	16	no	—
46	#97×C₄₁	—	2	4	no	—
47	#97×C₄₁	D₄	1	8	yes	complemented
48	C₄×C₈×C₄₁	—	2	2	no	—
49	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	complemented
50	C₂⁴×C₄₁	—	2	4	no	—
51	C₂×D₄×C₄₁	—	2	16	no	—
52	C₂×D₄×C₄₁	—	2	8	no	—
53	C₂×D₄×C₄₁	—	2	8	no	—
54	C₂×D₄×C₄₁	—	2	4	no	—
55	C₂×D₄×C₄₁	—	4	8	no	—
56	C₂×D₄×C₄₁	—	4	8	no	—
57	C₂×D₄×C₄₁	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	4	16	no	—
60	C₂²×C₄×C₄₁	—	4	8	no	—
61	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
65	D₈×C₄₁	—	2	32	no	—
66	C₂×C₈×C₄₁	—	2	2	no	—
67	C₂×C₈×C₄₁	—	2	2	no	—
68	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₄₁	#41	1	4	yes	—
72	C₄⋊C₄×C₄₁	—	2	8	no	—
73	C₄⋊C₄×C₄₁	—	4	8	no	—
74	C₂²⋊C₄×C₄₁	—	4	16	no	—
75	C₂²⋊C₄×C₄₁	—	4	8	no	—
76	C₄²×C₄₁	—	2	8	no	—
77	C₄²×C₄₁	—	2	2	no	—
78	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₁	C₂×D₄	1	4	yes	—
80	C₄²×C₄₁	#41	1	2	yes	—
81	C₂³×C₄₁	—	2	8	no	—
82	C₂³×C₄₁	—	4	4	no	—
83	C₂³×C₄₁	—	4	4	no	—
84	C₂³×C₄₁	—	4	8	no	—
85	C₂³×C₄₁	—	4	8	no	—
86	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
87	D₄×C₄₁	—	2	16	no	—
88	D₄×C₄₁	—	4	8	no	—
89	D₄×C₄₁	—	4	32	no	—
90	C₂×C₄×C₄₁	—	2	4	no	—
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	4	no	—
93	C₂×C₄×C₄₁	—	4	4	no	—
94	C₂×C₄×C₄₁	—	4	8	no	—
95	C₂×C₄×C₄₁	—	8	16	no	—
96	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₄₁	#127	1	4	yes	—
98	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	8	yes	—
99	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
102	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
103	C₈×C₄₁	—	2	4	no	—
104	C₈×C₄₁	C₄×D₄	1	8	yes	complemented
105	#21690	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₄₁	—	4	8	no	—
107	C₂²×C₄₁	—	4	16	no	—
108	C₂²×C₄₁	—	8	4	no	—
109	C₂²×C₄₁	—	8	4	no	—
110	C₂²×C₄₁	—	8	8	no	—
111	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
113	C₂²×C₄₁	#532	1	1	yes	characteristic
114	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
115	C₄×C₄₁	—	2	4	no	—
116	C₄×C₄₁	—	4	2	no	—
117	C₄×C₄₁	—	8	4	no	—
118	C₄×C₄₁	#442	1	8	yes	complemented
119	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
120	C₄×C₄₁	#532	1	2	yes	—
121	C₂×#493	C₂×C₄₁	1	1	yes	characteristic
122	C₂×#442	C₂×C₄₁	1	4	yes	complemented
123	C₄×#127	C₂×C₄₁	1	2	yes	complemented
124	#1972	C₂×C₄₁	1	4	yes	complemented
125	#1957	C₂×C₄₁	1	2	yes	complemented
126	#1844	C₂×C₄₁	1	1	yes	characteristic, complemented
127	C₄×#97	C₂×C₄₁	1	1	yes	characteristic, complemented
128	C₂×C₄₁	—	8	8	no	—
129	C₂×C₄₁	C₄×#127	1	1	yes	characteristic
130	C₂×C₄₁	C₂×#442	1	2	yes	—
131	C₂×C₄₁	#3151	1	2	yes	—
132	C₂×C₄₁	#3184	1	2	yes	—
133	C₂²×D₈	C₂²×C₄₁	1	4	yes	—
134	C₂×#127	C₂²×C₄₁	1	2	yes	—
135	C₂×C₄×D₄	—	2	4	no	—
136	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
137	#493	C₄×C₄₁	1	8	yes	complemented
138	#442	—	2	16	no	—
139	C₂×#106	C₂²×C₄₁	1	4	yes	—
140	C₂×#102	C₂²×C₄₁	1	8	yes	—
141	C₂×#97	C₂²×C₄₁	1	2	yes	—
142	C₂×#97	C₂²×C₄₁	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
145	#390	—	2	4	no	—
146	#390	C₂²×C₄₁	1	4	yes	—
147	#384	C₂²×C₄₁	1	2	yes	—
148	C₄³	C₂²×C₄₁	1	1	yes	characteristic
149	C₄₁	#21690	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×D₄	—	2	4	no	—
151	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
152	C₂³×C₄	—	2	4	no	—
153	C₂×D₈	—	2	8	no	—
154	C₂×D₈	—	2	8	no	—
155	C₂×D₈	C₂×C₄×C₄₁	1	16	yes	complemented
156	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
157	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
158	#127	—	2	4	no	—
159	#127	C₂×C₄×C₄₁	1	8	yes	complemented
160	C₄×D₄	—	2	16	no	—
161	C₄×D₄	—	4	16	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
164	C₂×C₂²⋊C₄	—	2	8	no	—
165	C₂×C₂²⋊C₄	—	2	4	no	—
166	C₂×C₄²	D₄×C₄₁	1	4	yes	—
167	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
170	#106	—	2	8	no	—
171	#102	—	2	16	no	—
172	#97	—	2	4	no	—
173	#97	D₄×C₄₁	1	8	yes	complemented
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	complemented
176	C₂⁴	—	2	4	no	—
177	C₂×D₄	—	2	16	no	—
178	C₂×D₄	—	2	8	no	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	4	8	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂×D₄	C₂²×C₄×C₄₁	1	8	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	4	16	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
190	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
191	D₈	—	2	32	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
195	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
196	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
197	C₂×C₈	#41×C₄₁	1	4	yes	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₂²⋊C₄	—	4	16	no	—
201	C₂²⋊C₄	—	4	8	no	—
202	C₄²	—	2	8	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
205	C₄²	C₂×D₄×C₄₁	1	4	yes	—
206	C₄²	#41×C₄₁	1	2	yes	—
207	C₂³	—	2	8	no	—
208	C₂³	—	4	4	no	—
209	C₂³	—	4	4	no	—
210	C₂³	—	4	8	no	—
211	C₂³	—	4	8	no	—
212	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
213	D₄	—	2	16	no	—
214	D₄	—	4	8	no	—
215	D₄	—	4	32	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	8	16	no	—
222	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
223	C₂×C₄	#127×C₄₁	1	4	yes	—
224	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	8	yes	—
225	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
227	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
228	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	C₄×D₄×C₄₁	1	8	yes	complemented
231	C₂²	—	4	8	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	8	no	—
236	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
237	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
238	C₂²	C₄₁×#532	1	1	yes	characteristic
239	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	8	4	no	—
243	C₄	C₄₁×#442	1	8	yes	complemented
244	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
245	C₄	C₄₁×#532	1	2	yes	—
246	C₂	—	8	8	no	—
247	C₂	C₄×#127×C₄₁	1	1	yes	characteristic
248	C₂	C₂×C₄₁×#442	1	2	yes	—
249	C₂	C₄₁×#3151	1	2	yes	—
250	C₂	C₄₁×#3184	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#442	C₂	semidirect
3. C₄×#127×C₄₁	C₂	semidirect
4. C₄₁×#1972	C₂	semidirect
5. C₄₁×#1957	C₂	semidirect
6. C₄₁×#1844	C₂	semidirect
7. C₄×#97×C₄₁	C₂	semidirect
12. C₄₁×#493	C₄	semidirect
29. C₂×D₈×C₄₁	C₂×C₄	semidirect
33. #127×C₄₁	C₂×C₄	semidirect
47. #97×C₄₁	D₄	semidirect
49. C₄×C₈×C₄₁	C₂×C₄	semidirect
104. C₈×C₄₁	C₄×D₄	semidirect
105. #21690	C₄₁	direct (N × H)
118. C₄×C₄₁	#442	semidirect
122. C₂×#442	C₂×C₄₁	semidirect
123. C₄×#127	C₂×C₄₁	semidirect
124. #1972	C₂×C₄₁	semidirect
125. #1957	C₂×C₄₁	semidirect
126. #1844	C₂×C₄₁	semidirect
127. C₄×#97	C₂×C₄₁	semidirect
137. #493	C₄×C₄₁	semidirect
149. C₄₁	#21690	direct (N × H)
155. C₂×D₈	C₂×C₄×C₄₁	semidirect
159. #127	C₂×C₄×C₄₁	semidirect
173. #97	D₄×C₄₁	semidirect
175. C₄×C₈	C₂×C₄×C₄₁	semidirect
230. C₈	C₄×D₄×C₄₁	semidirect
243. C₄	C₄₁×#442	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×#442	2
C₄×#127×C₄₁	2
C₄₁×#1972	2
C₄₁×#1957	2
C₄₁×#1844	2
C₄×#97×C₄₁	2
#21690	41

Subgroups — C41×#21690 #23797891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21690 #23797891