Subgroups — C₄₁×#14681 #23790882

Order 10496.

6350 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	C₂⁵×C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#381	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₁×#379	C₂	1	2	yes	maximal
4	C₄₁×C₂²⋊#95	C₂	1	8	yes	maximal, complemented
5	C₂⁶×C₄₁	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₄₁	—	2	8	no	—
7	C₂⁴×C₄×C₄₁	—	2	16	no	—
8	C₂⁴×C₄×C₄₁	C₄	1	8	yes	complemented
9	C₂⁴×C₄×C₄₁	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
13	C₄₁×#381	—	2	16	no	—
14	C₄₁×#379	—	2	8	no	—
15	C₂×#95×C₄₁	C₂²	1	16	yes	complemented
16	C₂⁵×C₄₁	—	2	8	no	—
17	C₂⁵×C₄₁	—	2	16	no	—
18	C₂⁵×C₄₁	C₂×C₄	1	4	yes	—
19	C₂⁵×C₄₁	D₄	1	8	yes	—
20	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
21	C₂⁵×C₄₁	C₂³	1	2	yes	—
22	C₂³×C₄×C₄₁	—	2	128	no	—
23	C₂³×C₄×C₄₁	—	2	32	no	—
24	C₂³×C₄×C₄₁	—	2	16	no	—
25	C₂³×C₄×C₄₁	—	2	8	no	—
26	C₂³×C₄×C₄₁	—	2	8	no	—
27	C₂³×C₄×C₄₁	—	2	64	no	—
28	C₂³×C₄×C₄₁	—	2	32	no	—
29	C₂³×C₄×C₄₁	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₄₁	D₄	1	16	yes	—
32	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
34	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
35	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
36	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
37	C₂×C₂²⋊C₄×C₄₁	—	2	32	no	—
38	#95×C₄₁	—	2	32	no	—
39	C₂⁴×C₄₁	—	2	128	no	—
40	C₂⁴×C₄₁	—	2	16	no	—
41	C₂⁴×C₄₁	—	2	32	no	—
42	C₂⁴×C₄₁	—	2	32	no	—
43	C₂⁴×C₄₁	—	2	64	no	—
44	C₂⁴×C₄₁	—	2	8	no	—
45	C₂⁴×C₄₁	—	2	8	no	—
46	C₂⁴×C₄₁	—	2	8	no	—
47	C₂⁴×C₄₁	C₂²⋊C₄	1	16	yes	—
48	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
49	C₂⁴×C₄₁	C₂²×C₄	1	4	yes	—
50	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
51	C₂⁴×C₄₁	C₂×D₄	1	8	yes	—
52	C₂⁴×C₄₁	C₂×D₄	1	8	yes	—
53	C₂⁴×C₄₁	#41	1	16	yes	—
54	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	—	2	128	no	—
56	C₂²×C₄×C₄₁	—	2	256	no	—
57	C₂²×C₄×C₄₁	—	2	32	no	—
58	C₂²×C₄×C₄₁	—	2	16	no	—
59	C₂²×C₄×C₄₁	—	2	16	no	—
60	C₂²×C₄×C₄₁	—	2	64	no	—
61	C₂²×C₄×C₄₁	—	2	8	no	—
62	C₂²×C₄×C₄₁	—	2	8	no	—
63	C₂²×C₄×C₄₁	—	2	64	no	—
64	C₂²×C₄×C₄₁	—	2	16	no	—
65	C₂²×C₄×C₄₁	—	2	8	no	—
66	C₂²×C₄×C₄₁	—	2	32	no	—
67	C₂²×C₄×C₄₁	C₂²⋊C₄	1	32	yes	complemented
68	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₄₁	C₂×D₄	1	16	yes	—
70	C₂²⋊C₄×C₄₁	—	4	16	no	—
71	C₂²⋊C₄×C₄₁	—	4	32	no	—
72	C₂³×C₄₁	—	2	256	no	—
73	C₂³×C₄₁	—	2	128	no	—
74	C₂³×C₄₁	—	2	8	no	—
75	C₂³×C₄₁	—	2	8	no	—
76	C₂³×C₄₁	—	2	8	no	—
77	C₂³×C₄₁	—	2	16	no	—
78	C₂³×C₄₁	—	2	16	no	—
79	C₂³×C₄₁	—	2	16	no	—
80	C₂³×C₄₁	—	2	32	no	—
81	C₂³×C₄₁	—	2	64	no	—
82	C₂³×C₄₁	—	2	32	no	—
83	C₂³×C₄₁	—	2	64	no	—
84	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
85	C₂³×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
86	C₂³×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
87	C₂³×C₄₁	#117	1	16	yes	—
88	C₂³×C₄₁	#120	1	16	yes	—
89	C₂³×C₄₁	#123	1	32	yes	—
90	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
91	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
92	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
93	C₂³×C₄₁	C₂×#41	1	8	yes	—
94	C₂×C₄×C₄₁	—	2	128	no	—
95	C₂×C₄×C₄₁	—	2	8	no	—
96	C₂×C₄×C₄₁	—	2	8	no	—
97	C₂×C₄×C₄₁	—	2	32	no	—
98	C₂×C₄×C₄₁	—	2	64	no	—
99	C₂×C₄×C₄₁	—	4	8	no	—
100	C₂×C₄×C₄₁	—	4	16	no	—
101	C₂×C₄×C₄₁	—	4	32	no	—
102	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	16	yes	—
103	#14681	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	—	2	128	no	—
105	C₂²×C₄₁	—	2	32	no	—
106	C₂²×C₄₁	—	2	64	no	—
107	C₂²×C₄₁	—	2	8	no	—
108	C₂²×C₄₁	—	2	8	no	—
109	C₂²×C₄₁	—	2	8	no	—
110	C₂²×C₄₁	—	2	16	no	—
111	C₂²×C₄₁	—	2	32	no	—
112	C₂²×C₄₁	#379	1	8	yes	—
113	C₂²×C₄₁	#381	1	16	yes	complemented
114	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₄₁	C₂×#117	1	4	yes	—
117	C₂²×C₄₁	C₂×#120	1	4	yes	—
118	C₂²×C₄₁	C₂×#123	1	8	yes	—
119	C₂²×C₄₁	#525	1	16	yes	—
120	C₄×C₄₁	—	2	16	no	—
121	C₄×C₄₁	—	8	8	no	—
122	C₂⁵×C₄	C₂×C₄₁	1	1	yes	characteristic
123	C₂×#381	C₂×C₄₁	1	4	yes	complemented
124	C₂×#379	C₂×C₄₁	1	2	yes	—
125	C₂²⋊#95	C₂×C₄₁	1	8	yes	complemented
126	C₂×C₄₁	—	2	8	no	—
127	C₂×C₄₁	—	2	16	no	—
128	C₂×C₄₁	C₂×#379	1	1	yes	characteristic
129	C₂×C₄₁	C₂×#381	1	2	yes	—
130	C₂×C₄₁	#2280	1	4	yes	—
131	C₂×C₄₁	#2849	1	8	yes	—
132	C₂⁶	C₂²×C₄₁	1	1	yes	characteristic
133	C₂⁴×C₄	—	2	8	no	—
134	C₂⁴×C₄	—	2	16	no	—
135	C₂⁴×C₄	C₄×C₄₁	1	8	yes	complemented
136	C₂⁴×C₄	C₂²×C₄₁	1	4	yes	—
137	C₂⁴×C₄	C₂²×C₄₁	1	2	yes	—
138	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
139	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
140	#381	—	2	16	no	—
141	#379	—	2	8	no	—
142	C₂×#95	C₂²×C₄₁	1	16	yes	complemented
143	C₄₁	#14681	1	1	yes	characteristic, direct product factor, Sylow
144	C₂⁵	—	2	8	no	—
145	C₂⁵	—	2	16	no	—
146	C₂⁵	C₂×C₄×C₄₁	1	4	yes	—
147	C₂⁵	D₄×C₄₁	1	8	yes	—
148	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
149	C₂⁵	C₂³×C₄₁	1	2	yes	—
150	C₂³×C₄	—	2	128	no	—
151	C₂³×C₄	—	2	32	no	—
152	C₂³×C₄	—	2	16	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	64	no	—
156	C₂³×C₄	—	2	32	no	—
157	C₂³×C₄	C₂×C₄×C₄₁	1	16	yes	—
158	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
159	C₂³×C₄	D₄×C₄₁	1	16	yes	—
160	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
161	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
162	C₂×C₂²⋊C₄	—	2	8	no	—
163	C₂×C₂²⋊C₄	—	2	16	no	—
164	C₂×C₂²⋊C₄	—	2	16	no	—
165	C₂×C₂²⋊C₄	—	2	32	no	—
166	#95	—	2	32	no	—
167	C₂⁴	—	2	128	no	—
168	C₂⁴	—	2	16	no	—
169	C₂⁴	—	2	32	no	—
170	C₂⁴	—	2	32	no	—
171	C₂⁴	—	2	64	no	—
172	C₂⁴	—	2	8	no	—
173	C₂⁴	—	2	8	no	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	C₂²⋊C₄×C₄₁	1	16	yes	—
176	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
177	C₂⁴	C₂²×C₄×C₄₁	1	4	yes	—
178	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
179	C₂⁴	C₂×D₄×C₄₁	1	8	yes	—
180	C₂⁴	C₂×D₄×C₄₁	1	8	yes	—
181	C₂⁴	#41×C₄₁	1	16	yes	—
182	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
183	C₂²×C₄	—	2	128	no	—
184	C₂²×C₄	—	2	256	no	—
185	C₂²×C₄	—	2	32	no	—
186	C₂²×C₄	—	2	16	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	2	64	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	64	no	—
192	C₂²×C₄	—	2	16	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	32	no	—
195	C₂²×C₄	C₂²⋊C₄×C₄₁	1	32	yes	complemented
196	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
197	C₂²×C₄	C₂×D₄×C₄₁	1	16	yes	—
198	C₂²⋊C₄	—	4	16	no	—
199	C₂²⋊C₄	—	4	32	no	—
200	C₂³	—	2	128	no	—
201	C₂³	—	2	256	no	—
202	C₂³	—	2	8	no	—
203	C₂³	—	2	8	no	—
204	C₂³	—	2	8	no	—
205	C₂³	—	2	16	no	—
206	C₂³	—	2	16	no	—
207	C₂³	—	2	16	no	—
208	C₂³	—	2	32	no	—
209	C₂³	—	2	64	no	—
210	C₂³	—	2	32	no	—
211	C₂³	—	2	64	no	—
212	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
213	C₂³	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
214	C₂³	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
215	C₂³	#117×C₄₁	1	16	yes	—
216	C₂³	#120×C₄₁	1	16	yes	—
217	C₂³	#123×C₄₁	1	32	yes	—
218	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
219	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
220	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
221	C₂³	C₂×#41×C₄₁	1	8	yes	—
222	C₂×C₄	—	2	128	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	32	no	—
226	C₂×C₄	—	2	64	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	16	no	—
229	C₂×C₄	—	4	32	no	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	16	yes	—
231	C₂²	—	2	128	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	16	no	—
236	C₂²	—	2	32	no	—
237	C₂²	—	2	32	no	—
238	C₂²	—	2	64	no	—
239	C₂²	C₄₁×#379	1	8	yes	—
240	C₂²	C₄₁×#381	1	16	yes	complemented
241	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
242	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	2	yes	—
243	C₂²	C₂×#117×C₄₁	1	4	yes	—
244	C₂²	C₂×#120×C₄₁	1	4	yes	—
245	C₂²	C₂×#123×C₄₁	1	8	yes	—
246	C₂²	C₄₁×#525	1	16	yes	—
247	C₄	—	2	16	no	—
248	C₄	—	8	8	no	—
249	C₂	—	2	8	no	—
250	C₂	—	2	16	no	—
251	C₂	C₂×C₄₁×#379	1	1	yes	characteristic
252	C₂	C₂×C₄₁×#381	1	2	yes	—
253	C₂	C₄₁×#2280	1	4	yes	—
254	C₂	C₄₁×#2849	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
prime radical	C₂⁶×C₄₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#381	C₂	semidirect
4. C₄₁×C₂²⋊#95	C₂	semidirect
8. C₂⁴×C₄×C₄₁	C₄	semidirect
15. C₂×#95×C₄₁	C₂²	semidirect
67. C₂²×C₄×C₄₁	C₂²⋊C₄	semidirect
103. #14681	C₄₁	direct (N × H)
113. C₂²×C₄₁	#381	semidirect
123. C₂×#381	C₂×C₄₁	semidirect
125. C₂²⋊#95	C₂×C₄₁	semidirect
135. C₂⁴×C₄	C₄×C₄₁	semidirect
142. C₂×#95	C₂²×C₄₁	semidirect
143. C₄₁	#14681	direct (N × H)
195. C₂²×C₄	C₂²⋊C₄×C₄₁	semidirect
240. C₂²	C₄₁×#381	semidirect

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₄₁	2
C₂×C₄₁×#381	2
C₂×C₄₁×#379	2
C₄₁×C₂²⋊#95	2
#14681	41

Subgroups — C41×#14681 #23790882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14681 #23790882