Subgroups — C₄₁×#14343 #23790544

Order 10496.

1782 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#380	C₂	1	2	yes	maximal
3	C₄₁×#1445	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1440	C₂	1	4	yes	maximal
5	C₄₁×C₄⋊#95	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×#117×C₄₁	—	2	8	no	—
8	C₂×#117×C₄₁	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#380	—	2	8	no	—
15	C₂×#95×C₄₁	C₂²	1	4	yes	—
16	C₂×#95×C₄₁	C₂²	1	16	yes	—
17	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
18	C₂³×C₄×C₄₁	—	2	8	no	—
19	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₄₁	D₄	1	8	yes	—
21	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
22	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	#117×C₄₁	—	2	32	no	—
26	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
29	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
30	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
31	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
33	C₂×C₂²⋊C₄×C₄₁	—	4	8	no	—
34	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₄₁	—	2	8	no	—
36	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	—
37	#95×C₄₁	—	2	8	no	—
38	#95×C₄₁	—	4	16	no	—
39	C₂⁴×C₄₁	—	2	4	no	—
40	C₂⁴×C₄₁	—	2	4	no	—
41	C₂⁴×C₄₁	—	2	4	no	—
42	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
43	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
44	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
45	C₂²×C₄×C₄₁	—	2	4	no	—
46	C₂²×C₄×C₄₁	—	2	16	no	—
47	C₂²×C₄×C₄₁	—	2	32	no	—
48	C₂²×C₄×C₄₁	—	2	4	no	—
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	8	no	—
51	C₂²×C₄×C₄₁	—	2	8	no	—
52	C₂²×C₄×C₄₁	—	2	8	no	—
53	C₂²×C₄×C₄₁	—	2	16	no	—
54	C₂²×C₄×C₄₁	—	2	8	no	—
55	C₂²×C₄×C₄₁	—	2	16	no	—
56	C₂²×C₄×C₄₁	—	2	8	no	—
57	C₂²×C₄×C₄₁	C₂²⋊C₄	1	16	yes	—
58	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₁	#41	1	8	yes	—
64	C₄⋊C₄×C₄₁	—	2	16	no	—
65	C₄⋊C₄×C₄₁	—	4	16	no	—
66	C₂²⋊C₄×C₄₁	—	2	16	no	—
67	C₂²⋊C₄×C₄₁	—	4	16	no	—
68	C₄²×C₄₁	—	2	16	no	—
69	C₂³×C₄₁	—	2	4	no	—
70	C₂³×C₄₁	—	2	8	no	—
71	C₂³×C₄₁	—	2	8	no	—
72	C₂³×C₄₁	—	2	4	no	—
73	C₂³×C₄₁	—	2	4	no	—
74	C₂³×C₄₁	—	2	8	no	—
75	C₂³×C₄₁	—	4	4	no	—
76	C₂³×C₄₁	—	4	4	no	—
77	C₂³×C₄₁	—	4	8	no	—
78	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
79	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
80	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
81	C₂³×C₄₁	C₂×#41	1	4	yes	—
82	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
83	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
84	C₂×C₄×C₄₁	—	2	4	no	—
85	C₂×C₄×C₄₁	—	2	8	no	—
86	C₂×C₄×C₄₁	—	2	32	no	—
87	C₂×C₄×C₄₁	—	2	8	no	—
88	C₂×C₄×C₄₁	—	2	16	no	—
89	C₂×C₄×C₄₁	—	4	4	no	—
90	C₂×C₄×C₄₁	—	4	4	no	—
91	C₂×C₄×C₄₁	—	4	8	no	—
92	C₂×C₄×C₄₁	—	4	8	no	—
93	C₂×C₄×C₄₁	—	4	8	no	—
94	C₂×C₄×C₄₁	—	4	8	no	—
95	C₂×C₄×C₄₁	—	4	16	no	—
96	C₂×C₄×C₄₁	—	4	16	no	—
97	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₁	#117	1	8	yes	—
100	C₂×C₄×C₄₁	#123	1	16	yes	—
101	#14343	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₄₁	—	2	8	no	—
103	C₂²×C₄₁	—	2	4	no	—
104	C₂²×C₄₁	—	4	8	no	—
105	C₂²×C₄₁	—	4	8	no	—
106	C₂²×C₄₁	—	4	8	no	—
107	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×C₄₁	C₂×#117	1	2	yes	—
109	C₂²×C₄₁	#520	1	4	yes	—
110	C₂²×C₄₁	C₂×#123	1	4	yes	—
111	C₂²×C₄₁	#535	1	2	yes	—
112	C₂²×C₄₁	#536	1	2	yes	—
113	C₂²×C₄₁	#537	1	4	yes	—
114	C₂²×C₄₁	#540	1	4	yes	—
115	C₂²×C₄₁	#541	1	4	yes	—
116	C₂²×C₄₁	#542	1	8	yes	—
117	C₄×C₄₁	—	2	4	no	—
118	C₄×C₄₁	—	4	8	no	—
119	C₄×C₄₁	—	8	8	no	—
120	C₄×C₄₁	#381	1	8	yes	—
121	C₂²×#117	C₂×C₄₁	1	1	yes	characteristic
122	C₂×#380	C₂×C₄₁	1	2	yes	—
123	#1445	C₂×C₄₁	1	4	yes	complemented
124	#1440	C₂×C₄₁	1	4	yes	—
125	C₄⋊#95	C₂×C₄₁	1	4	yes	complemented
126	C₂×C₄₁	—	4	4	no	—
127	C₂×C₄₁	C₂×#381	1	1	yes	characteristic
128	C₂×C₄₁	#2311	1	2	yes	—
129	C₂×C₄₁	#2314	1	4	yes	—
130	C₂×C₄₁	#2624	1	4	yes	—
131	C₂×C₄₁	#2625	1	4	yes	—
132	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
133	C₂×#117	—	2	8	no	—
134	C₂×#117	C₄×C₄₁	1	8	yes	complemented
135	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
136	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
137	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
138	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
139	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
140	#380	—	2	8	no	—
141	C₂×#95	C₂²×C₄₁	1	4	yes	—
142	C₂×#95	C₂²×C₄₁	1	16	yes	—
143	C₄₁	#14343	1	1	yes	characteristic, direct product factor, Sylow
144	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
145	C₂³×C₄	—	2	8	no	—
146	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
147	C₂³×C₄	D₄×C₄₁	1	8	yes	—
148	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
149	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
150	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
151	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
152	#117	—	2	32	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	16	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
158	C₂×C₂²⋊C₄	—	2	8	no	—
159	C₂×C₂²⋊C₄	—	2	8	no	—
160	C₂×C₂²⋊C₄	—	4	8	no	—
161	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
162	C₂×C₄²	—	2	8	no	—
163	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
164	#95	—	2	8	no	—
165	#95	—	4	16	no	—
166	C₂⁴	—	2	4	no	—
167	C₂⁴	—	2	4	no	—
168	C₂⁴	—	2	4	no	—
169	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
170	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
171	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	32	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	16	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	C₂²⋊C₄×C₄₁	1	16	yes	—
185	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
186	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
188	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
189	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
190	C₂²×C₄	#41×C₄₁	1	8	yes	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	4	16	no	—
193	C₂²⋊C₄	—	2	16	no	—
194	C₂²⋊C₄	—	4	16	no	—
195	C₄²	—	2	16	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	8	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	2	4	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	2	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	4	8	no	—
205	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
206	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
207	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
208	C₂³	C₂×#41×C₄₁	1	4	yes	—
209	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
210	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	32	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	16	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	8	no	—
222	C₂×C₄	—	4	16	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
226	C₂×C₄	#117×C₄₁	1	8	yes	—
227	C₂×C₄	#123×C₄₁	1	16	yes	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	8	no	—
230	C₂²	—	4	8	no	—
231	C₂²	—	4	8	no	—
232	C₂²	—	4	8	no	—
233	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
234	C₂²	C₂×#117×C₄₁	1	2	yes	—
235	C₂²	C₄₁×#520	1	4	yes	—
236	C₂²	C₂×#123×C₄₁	1	4	yes	—
237	C₂²	C₄₁×#535	1	2	yes	—
238	C₂²	C₄₁×#536	1	2	yes	—
239	C₂²	C₄₁×#537	1	4	yes	—
240	C₂²	C₄₁×#540	1	4	yes	—
241	C₂²	C₄₁×#541	1	4	yes	—
242	C₂²	C₄₁×#542	1	8	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	4	8	no	—
245	C₄	—	8	8	no	—
246	C₄	C₄₁×#381	1	8	yes	—
247	C₂	—	4	4	no	—
248	C₂	C₂×C₄₁×#381	1	1	yes	characteristic
249	C₂	C₄₁×#2311	1	2	yes	—
250	C₂	C₄₁×#2314	1	4	yes	—
251	C₂	C₄₁×#2624	1	4	yes	—
252	C₂	C₄₁×#2625	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
prime radical	C₂⁵×C₄₁	C₂³
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
3. C₄₁×#1445	C₂	semidirect
5. C₄₁×C₄⋊#95	C₂	semidirect
8. C₂×#117×C₄₁	C₄	semidirect
101. #14343	C₄₁	direct (N × H)
123. #1445	C₂×C₄₁	semidirect
125. C₄⋊#95	C₂×C₄₁	semidirect
134. C₂×#117	C₄×C₄₁	semidirect
143. C₄₁	#14343	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#117×C₄₁	2
C₂×C₄₁×#380	2
C₄₁×#1445	2
C₄₁×#1440	2
C₄₁×C₄⋊#95	2
#14343	41

Subgroups — C41×#14343 #23790544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14343 #23790544