Subgroups — C₄₁×#21967 #23798168

Order 10496.

754 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#424	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2314	C₂	1	2	yes	maximal
3	C₄₁×#1949	C₂	1	4	yes	maximal, complemented
4	C₄₁×#1948	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1809	C₂	1	2	yes	maximal
6	C₄₁×#1807	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#107×C₄₁	C₂²	1	2	yes	—
11	C₂×#106×C₄₁	C₂²	1	2	yes	—
12	C₂×#106×C₄₁	C₂²	1	2	yes	—
13	C₂×#106×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#424	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
16	C₄₁×#388	—	2	4	no	—
17	C₄₁×#387	—	2	4	no	—
18	C₄₁×#384	C₂²	1	2	yes	—
19	C₄₁×#382	C₂²	1	2	yes	—
20	C₄₁×#380	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
22	C₄₁×#340	C₂²	1	4	yes	—
23	C₄₁×#340	C₂²	1	4	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₄₁	—	4	1	no	—
26	C₂×M₁₆×C₄₁	—	4	1	no	—
27	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₄₁	—	4	1	no	—
29	C₂²×C₈×C₄₁	—	4	1	no	—
30	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
32	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
33	#117×C₄₁	C₂×C₄	1	4	yes	—
34	#117×C₄₁	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
41	C₂×C₄²×C₄₁	—	2	4	no	—
42	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
43	#107×C₄₁	—	4	2	no	—
44	#106×C₄₁	—	4	2	no	—
45	#106×C₄₁	—	4	2	no	—
46	#106×C₄₁	—	4	2	no	—
47	C₄⋊C₈×C₄₁	C₂×C₄	1	16	yes	—
48	#95×C₄₁	—	2	4	no	—
49	#95×C₄₁	—	2	4	no	—
50	#95×C₄₁	—	2	4	no	—
51	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	—	2	1	no	—
53	C₂²×C₄×C₄₁	—	2	1	no	—
54	C₂²×C₄×C₄₁	—	2	1	no	—
55	C₂²×C₄×C₄₁	—	2	1	no	—
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₄₁	—	4	4	no	—
64	C₂×C₈×C₄₁	—	2	2	no	—
65	C₂×C₈×C₄₁	—	2	2	no	—
66	C₂×C₈×C₄₁	—	2	2	no	—
67	C₂×C₈×C₄₁	—	2	2	no	—
68	C₂×C₈×C₄₁	—	4	4	no	—
69	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
71	C₄⋊C₄×C₄₁	—	2	4	no	—
72	C₄⋊C₄×C₄₁	—	2	4	no	—
73	C₄⋊C₄×C₄₁	—	2	4	no	—
74	C₄⋊C₄×C₄₁	—	2	4	no	—
75	C₄⋊C₄×C₄₁	—	8	4	no	—
76	C₄⋊C₄×C₄₁	C₄⋊C₄	1	8	yes	—
77	C₂²⋊C₄×C₄₁	—	8	4	no	—
78	C₂²⋊C₄×C₄₁	C₄⋊C₄	1	8	yes	—
79	C₄²×C₄₁	—	8	4	no	—
80	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
81	C₄²×C₄₁	C₂×Q₈	1	4	yes	—
82	C₂³×C₄₁	—	2	1	no	—
83	C₂³×C₄₁	—	2	1	no	—
84	C₂³×C₄₁	—	2	4	no	—
85	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₁	—	2	1	no	—
88	C₂×C₄×C₄₁	—	2	1	no	—
89	C₂×C₄×C₄₁	—	2	4	no	—
90	C₂×C₄×C₄₁	—	2	4	no	—
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	2	no	—
93	C₂×C₄×C₄₁	—	2	2	no	—
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	4	4	no	—
96	C₂×C₄×C₄₁	—	4	4	no	—
97	C₂×C₄×C₄₁	—	4	4	no	—
98	C₂×C₄×C₄₁	—	4	4	no	—
99	C₂×C₄×C₄₁	—	4	4	no	—
100	C₂×C₄×C₄₁	—	4	4	no	—
101	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
102	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
109	C₈×C₄₁	—	2	4	no	—
110	C₈×C₄₁	—	2	4	no	—
111	#21967	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₄₁	—	2	2	no	—
113	C₂²×C₄₁	—	2	4	no	—
114	C₂²×C₄₁	—	4	2	no	—
115	C₂²×C₄₁	—	4	2	no	—
116	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₁	#537	1	1	yes	characteristic
118	C₂²×C₄₁	#537	1	1	yes	characteristic
119	C₂²×C₄₁	#577	1	4	yes	—
120	C₄×C₄₁	—	2	2	no	—
121	C₄×C₄₁	—	2	8	no	—
122	C₄×C₄₁	—	8	4	no	—
123	C₄×C₄₁	—	8	4	no	—
124	C₄×C₄₁	#518	1	1	yes	characteristic
125	C₄×C₄₁	#519	1	1	yes	characteristic
126	C₄×C₄₁	#544	1	1	yes	characteristic
127	C₄×C₄₁	#544	1	1	yes	characteristic
128	C₂×#424	C₂×C₄₁	1	1	yes	characteristic
129	#2314	C₂×C₄₁	1	2	yes	—
130	#1949	C₂×C₄₁	1	4	yes	complemented
131	#1948	C₂×C₄₁	1	4	yes	complemented
132	#1809	C₂×C₄₁	1	2	yes	—
133	#1807	C₂×C₄₁	1	2	yes	—
134	C₂×C₄₁	—	4	2	no	—
135	C₂×C₄₁	#2316	1	1	yes	characteristic
136	C₂×C₄₁	#2912	1	2	yes	—
137	C₂×C₄₁	#3260	1	2	yes	—
138	C₂×C₄₁	#3263	1	2	yes	—
139	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
140	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
141	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
142	C₂×#107	C₂²×C₄₁	1	2	yes	—
143	C₂×#106	C₂²×C₄₁	1	2	yes	—
144	C₂×#106	C₂²×C₄₁	1	2	yes	—
145	C₂×#106	C₂²×C₄₁	1	2	yes	—
146	#424	C₄×C₄₁	1	8	yes	complemented
147	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
148	#388	—	2	4	no	—
149	#387	—	2	4	no	—
150	#384	C₂²×C₄₁	1	2	yes	—
151	#382	C₂²×C₄₁	1	2	yes	—
152	#380	C₂²×C₄₁	1	4	yes	—
153	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
154	#340	C₂²×C₄₁	1	4	yes	—
155	#340	C₂²×C₄₁	1	4	yes	—
156	C₄₁	#21967	1	1	yes	characteristic, direct product factor, Sylow
157	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
158	C₂×M₁₆	—	4	1	no	—
159	C₂×M₁₆	—	4	1	no	—
160	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
161	C₂²×C₈	—	4	1	no	—
162	C₂²×C₈	—	4	1	no	—
163	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
164	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
165	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
166	#117	C₂×C₄×C₄₁	1	4	yes	—
167	#117	Q₈×C₄₁	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
170	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
171	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
176	#107	—	4	2	no	—
177	#106	—	4	2	no	—
178	#106	—	4	2	no	—
179	#106	—	4	2	no	—
180	C₄⋊C₈	C₂×C₄×C₄₁	1	16	yes	—
181	#95	—	2	4	no	—
182	#95	—	2	4	no	—
183	#95	—	2	4	no	—
184	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
194	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
196	M₁₆	—	4	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
203	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	8	4	no	—
209	C₄⋊C₄	C₄⋊C₄×C₄₁	1	8	yes	—
210	C₂²⋊C₄	—	8	4	no	—
211	C₂²⋊C₄	C₄⋊C₄×C₄₁	1	8	yes	—
212	C₄²	—	8	4	no	—
213	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
214	C₄²	C₂×Q₈×C₄₁	1	4	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	4	no	—
218	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
219	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
236	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
237	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
238	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
239	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
240	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
241	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
249	C₂²	C₄₁×#537	1	1	yes	characteristic
250	C₂²	C₄₁×#537	1	1	yes	characteristic
251	C₂²	C₄₁×#577	1	4	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	8	no	—
254	C₄	—	8	4	no	—
255	C₄	—	8	4	no	—
256	C₄	C₄₁×#518	1	1	yes	characteristic
257	C₄	C₄₁×#519	1	1	yes	characteristic
258	C₄	C₄₁×#544	1	1	yes	characteristic
259	C₄	C₄₁×#544	1	1	yes	characteristic
260	C₂	—	4	2	no	—
261	C₂	C₄₁×#2316	1	1	yes	characteristic
262	C₂	C₄₁×#2912	1	2	yes	—
263	C₂	C₄₁×#3260	1	2	yes	—
264	C₂	C₄₁×#3263	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	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1949	C₂	semidirect
4. C₄₁×#1948	C₂	semidirect
14. C₄₁×#424	C₄	semidirect
111. #21967	C₄₁	direct (N × H)
130. #1949	C₂×C₄₁	semidirect
131. #1948	C₂×C₄₁	semidirect
146. #424	C₄×C₄₁	semidirect
156. C₄₁	#21967	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×#424	2
C₄₁×#2314	2
C₄₁×#1949	2
C₄₁×#1948	2
C₄₁×#1809	2
C₄₁×#1807	2
#21967	41

Subgroups — C41×#21967 #23798168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21967 #23798168