Subgroups — C₂×C₄₁×#1566 #23793991

Order 10496.

854 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#423	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1566	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₁×#334	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₄₁	—	2	2	no	—
6	C₂³×C₈×C₄₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
8	C₄₁×#423	—	2	4	no	—
9	C₄₁×#423	C₂²	1	8	yes	—
10	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₁	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
17	C₄₁×#334	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₁	Q₈	1	1	yes	characteristic
21	C₂×M₁₆×C₄₁	—	2	8	no	—
22	C₂×M₁₆×C₄₁	—	4	8	no	—
23	C₂²×C₈×C₄₁	—	2	8	no	—
24	C₂²×C₈×C₄₁	—	2	4	no	—
25	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₁	D₄	1	2	yes	—
31	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
34	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₁	—	2	8	no	—
37	C₄⋊C₈×C₄₁	—	2	4	no	—
38	C₄⋊C₈×C₄₁	C₂³	1	8	yes	—
39	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₄₁	C₂³	1	4	yes	—
43	C₂⁴×C₄₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₁	—	2	4	no	—
45	C₂²×C₄×C₄₁	—	2	4	no	—
46	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
59	M₁₆×C₄₁	—	4	16	no	—
60	C₂×C₈×C₄₁	—	2	4	no	—
61	C₂×C₈×C₄₁	—	2	16	no	—
62	C₂×C₈×C₄₁	—	2	8	no	—
63	C₂×C₈×C₄₁	—	4	8	no	—
64	C₂×C₈×C₄₁	M₁₆	1	8	yes	complemented
65	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
67	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
72	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₁	C₂×D₄	1	2	yes	—
74	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
75	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	#106	1	4	yes	—
80	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₁	—	2	2	no	—
83	C₂×C₄×C₄₁	—	2	2	no	—
84	C₂×C₄×C₄₁	—	2	4	no	—
85	C₂×C₄×C₄₁	—	2	4	no	—
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	—	2	4	no	—
88	C₂×C₄×C₄₁	#106	1	8	yes	—
89	C₂×C₄×C₄₁	#106	1	4	yes	—
90	C₂×C₄×C₄₁	#108	1	4	yes	—
91	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₁	C₂×M₁₆	1	4	yes	—
98	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₄₁	C₂×SD₁₆	1	8	yes	—
100	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
103	C₈×C₄₁	—	2	4	no	—
104	C₈×C₄₁	—	4	8	no	—
105	C₈×C₄₁	C₂×M₁₆	1	8	yes	complemented
106	C₂×#1566	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₄₁	—	2	8	no	—
108	C₂²×C₄₁	—	2	2	no	—
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	#423	1	2	yes	—
111	C₂²×C₄₁	C₂×#106	1	4	yes	—
112	C₂²×C₄₁	C₂×#106	1	2	yes	—
113	C₂²×C₄₁	C₂×#108	1	2	yes	—
114	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₄₁	—	2	2	no	—
116	C₄×C₄₁	—	2	1	no	—
117	C₄×C₄₁	—	2	1	no	—
118	C₄×C₄₁	#423	1	2	yes	—
119	C₄×C₄₁	C₂×#106	1	4	yes	—
120	C₄×C₄₁	C₂×#106	1	2	yes	—
121	C₄×C₄₁	C₂×#108	1	4	yes	—
122	C₄×C₄₁	C₂²×M₁₆	1	2	yes	—
123	C₄×C₄₁	C₂²×SD₁₆	1	2	yes	—
124	C₂×#423	C₂×C₄₁	1	2	yes	—
125	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic
126	#1566	C₂×C₄₁	1	8	yes	direct product factor
127	C₂×#334	C₂×C₄₁	1	4	yes	complemented
128	C₂×C₄₁	—	2	4	no	—
129	C₂×C₄₁	#1566	1	4	yes	direct product factor
130	C₂×C₄₁	C₂×#423	1	1	yes	characteristic
131	C₂×C₄₁	C₂²×#106	1	1	yes	characteristic
132	C₂×C₄₁	C₂²×#108	1	1	yes	characteristic
133	C₂²×M₁₆	—	2	2	no	—
134	C₂³×C₈	C₄×C₄₁	1	2	yes	—
135	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
136	#423	—	2	4	no	—
137	#423	C₂²×C₄₁	1	8	yes	—
138	C₂×C₄⋊C₈	—	2	2	no	—
139	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
140	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
141	C₂×C₄×C₈	C₄×C₄₁	1	4	yes	—
142	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
145	#334	C₂²×C₄₁	1	16	yes	complemented
146	C₄₁	C₂×#1566	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
148	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
149	C₂³×C₄	Q₈×C₄₁	1	1	yes	characteristic
150	C₂×M₁₆	—	2	8	no	—
151	C₂×M₁₆	—	4	8	no	—
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
155	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	—
156	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
158	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
159	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
160	C₂×C₄²	D₄×C₄₁	1	2	yes	—
161	C₂×C₄²	Q₈×C₄₁	1	1	yes	characteristic
162	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
163	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
165	C₄⋊C₈	—	2	8	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	C₂³×C₄₁	1	8	yes	—
168	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
169	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
170	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
171	C₄×C₈	C₂³×C₄₁	1	4	yes	—
172	C₂⁴	C₄⋊C₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
178	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
181	C₂²×C₄	SD₁₆×C₄₁	1	8	yes	—
182	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
186	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
188	M₁₆	—	4	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	16	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	M₁₆×C₄₁	1	8	yes	complemented
194	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
195	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
196	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
197	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
198	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
199	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
200	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
201	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₄₁	1	2	yes	—
203	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
204	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
205	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	#106×C₄₁	1	4	yes	—
209	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
210	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	#106×C₄₁	1	8	yes	—
218	C₂×C₄	#106×C₄₁	1	4	yes	—
219	C₂×C₄	#108×C₄₁	1	4	yes	—
220	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
225	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
226	C₂×C₄	C₂×M₁₆×C₄₁	1	4	yes	—
227	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
228	C₂×C₄	C₂×SD₁₆×C₄₁	1	8	yes	—
229	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
231	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	4	8	no	—
234	C₈	C₂×M₁₆×C₄₁	1	8	yes	complemented
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₄₁×#423	1	2	yes	—
239	C₂²	C₂×#106×C₄₁	1	4	yes	—
240	C₂²	C₂×#106×C₄₁	1	2	yes	—
241	C₂²	C₂×#108×C₄₁	1	2	yes	—
242	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	C₄₁×#423	1	2	yes	—
247	C₄	C₂×#106×C₄₁	1	2	yes	—
248	C₄	C₂×#106×C₄₁	1	4	yes	—
249	C₄	C₂×#108×C₄₁	1	4	yes	—
250	C₄	C₂²×M₁₆×C₄₁	1	2	yes	—
251	C₄	C₂²×SD₁₆×C₄₁	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₄₁×#1566	1	4	yes	direct product factor
254	C₂	C₂×C₄₁×#423	1	1	yes	characteristic
255	C₂	C₂²×#106×C₄₁	1	1	yes	characteristic
256	C₂	C₂²×#108×C₄₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1566	C₂	direct (N × H)
4. C₂×C₄₁×#334	C₂	semidirect
17. C₄₁×#334	C₂²	semidirect
64. C₂×C₈×C₄₁	M₁₆	semidirect
105. C₈×C₄₁	C₂×M₁₆	semidirect
106. C₂×#1566	C₄₁	direct (N × H)
126. #1566	C₂×C₄₁	direct (N × H)
127. C₂×#334	C₂×C₄₁	semidirect
129. C₂×C₄₁	#1566	direct (N × H)
145. #334	C₂²×C₄₁	semidirect
146. C₄₁	C₂×#1566	direct (N × H)
193. C₂×C₈	M₁₆×C₄₁	semidirect
234. C₈	C₂×M₁₆×C₄₁	semidirect
253. C₂	C₄₁×#1566	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×#423	2
C₂²×C₄×C₈×C₄₁	2
C₄₁×#1566	2
C₂×C₄₁×#334	2
C₂×#1566	41

Subgroups — C2×C41×#1566 #23793991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1566 #23793991