Subgroups — C₂×C₄₁×#2218 #23809802

Order 10496.

786 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	C₂²×D₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#466	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2218	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#121×C₄₁	—	2	2	no	—
10	C₂×D₁₆×C₄₁	—	2	4	no	—
11	C₂×D₁₆×C₄₁	C₂²	1	4	yes	complemented
12	C₂×M₃₂×C₄₁	—	2	4	no	—
13	C₂×M₃₂×C₄₁	C₂²	1	4	yes	complemented
14	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#466	—	2	4	no	—
17	C₄₁×#466	C₂²	1	8	yes	complemented
18	C₂×#107×C₄₁	C₂²	1	2	yes	—
19	C₂×#102×C₄₁	—	2	2	no	—
20	C₄₁×#367	C₂²	1	4	yes	complemented
21	C₄₁×#357	C₂²	1	8	yes	complemented
22	C₂²×D₄×C₄₁	—	2	2	no	—
23	C₂²×D₄×C₄₁	—	4	2	no	—
24	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
25	C₂×D₈×C₄₁	—	2	4	no	—
26	C₂×D₈×C₄₁	—	2	2	no	—
27	C₂×D₈×C₄₁	—	2	4	no	—
28	C₂×D₈×C₄₁	C₂³	1	4	yes	—
29	C₂²×C₈×C₄₁	—	2	2	no	—
30	C₂²×C₈×C₄₁	—	2	2	no	—
31	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₁	D₄	1	2	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
35	#121×C₄₁	—	2	8	no	—
36	#121×C₄₁	—	4	4	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₁	—	4	2	no	—
39	D₁₆×C₄₁	—	2	8	no	—
40	M₃₂×C₄₁	—	2	8	no	—
41	C₂×C₁₆×C₄₁	—	2	2	no	—
42	C₂×C₁₆×C₄₁	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
44	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
45	#107×C₄₁	—	2	2	no	—
46	#107×C₄₁	C₂³	1	4	yes	—
47	#102×C₄₁	—	2	8	no	—
48	C₂⁴×C₄₁	—	4	2	no	—
49	C₂⁴×C₄₁	D₈	1	1	yes	characteristic
50	C₂×D₄×C₄₁	—	2	4	no	—
51	C₂×D₄×C₄₁	—	4	8	no	—
52	C₂×D₄×C₄₁	—	4	4	no	—
53	C₂×D₄×C₄₁	—	4	2	no	—
54	C₂×D₄×C₄₁	—	4	4	no	—
55	C₂×D₄×C₄₁	—	8	8	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	2	2	no	—
58	C₂²×C₄×C₄₁	—	4	2	no	—
59	C₂²×C₄×C₄₁	D₈	1	2	yes	—
60	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
63	D₈×C₄₁	—	2	4	no	—
64	D₈×C₄₁	—	2	4	no	—
65	D₈×C₄₁	—	4	4	no	—
66	C₂×C₈×C₄₁	—	2	2	no	—
67	C₂×C₈×C₄₁	—	2	4	no	—
68	C₂×C₈×C₄₁	—	2	2	no	—
69	C₂×C₈×C₄₁	—	2	2	no	—
70	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₁	#41	1	2	yes	—
74	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₄₁	—	2	4	no	—
76	C₄⋊C₄×C₄₁	—	4	2	no	—
77	C₂²⋊C₄×C₄₁	—	4	8	no	—
78	C₁₆×C₄₁	—	2	2	no	—
79	C₁₆×C₄₁	C₂×D₄	1	4	yes	complemented
80	C₂³×C₄₁	—	2	2	no	—
81	C₂³×C₄₁	—	2	4	no	—
82	C₂³×C₄₁	—	4	4	no	—
83	C₂³×C₄₁	—	8	4	no	—
84	C₂³×C₄₁	—	8	4	no	—
85	C₂³×C₄₁	—	8	4	no	—
86	C₂³×C₄₁	C₂×D₈	1	2	yes	—
87	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
88	D₄×C₄₁	—	4	4	no	—
89	D₄×C₄₁	—	4	4	no	—
90	D₄×C₄₁	—	8	16	no	—
91	D₄×C₄₁	—	8	4	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	—	2	4	no	—
96	C₂×C₄×C₄₁	—	4	4	no	—
97	C₂×C₄×C₄₁	—	8	4	no	—
98	C₂×C₄×C₄₁	#121	1	2	yes	—
99	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₄₁	#135	1	2	yes	—
102	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
103	C₈×C₄₁	—	2	2	no	—
104	C₈×C₄₁	#121	1	2	yes	—
105	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
107	C₂×#2218	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₄₁	—	2	4	no	—
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	—	4	4	no	—
111	C₂²×C₄₁	—	8	4	no	—
112	C₂²×C₄₁	—	8	4	no	—
113	C₂²×C₄₁	—	8	4	no	—
114	C₂²×C₄₁	—	16	8	no	—
115	C₂²×C₄₁	#466	1	2	yes	—
116	C₂²×C₄₁	#509	1	4	yes	—
117	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₄₁	—	2	2	no	—
119	C₄×C₄₁	—	8	4	no	—
120	C₄×C₄₁	#466	1	2	yes	—
121	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
122	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
123	C₂²×D₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
124	C₂²×M₃₂	C₂×C₄₁	1	1	yes	characteristic, complemented
125	C₂×#466	C₂×C₄₁	1	2	yes	complemented
126	#2218	C₂×C₄₁	1	8	yes	direct product factor
127	C₂×#367	C₂×C₄₁	1	1	yes	characteristic, complemented
128	C₂×#357	C₂×C₄₁	1	2	yes	complemented
129	C₂×C₄₁	—	4	2	no	—
130	C₂×C₄₁	—	16	4	no	—
131	C₂×C₄₁	#2218	1	4	yes	direct product factor
132	C₂×C₄₁	C₂×#466	1	1	yes	characteristic
133	C₂×C₄₁	C₂×#509	1	2	yes	—
134	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
135	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
136	C₂×#121	—	2	2	no	—
137	C₂×D₁₆	—	2	4	no	—
138	C₂×D₁₆	C₂²×C₄₁	1	4	yes	complemented
139	C₂×M₃₂	—	2	4	no	—
140	C₂×M₃₂	C₂²×C₄₁	1	4	yes	complemented
141	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
143	#466	—	2	4	no	—
144	#466	C₂²×C₄₁	1	8	yes	complemented
145	C₂×#107	C₂²×C₄₁	1	2	yes	—
146	C₂×#102	—	2	2	no	—
147	#367	C₂²×C₄₁	1	4	yes	complemented
148	#357	C₂²×C₄₁	1	8	yes	complemented
149	C₄₁	C₂×#2218	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×D₄	—	2	2	no	—
151	C₂²×D₄	—	4	2	no	—
152	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
153	C₂×D₈	—	2	4	no	—
154	C₂×D₈	—	2	2	no	—
155	C₂×D₈	—	2	4	no	—
156	C₂×D₈	C₂³×C₄₁	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
160	C₂²×C₈	D₄×C₄₁	1	2	yes	—
161	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
162	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
163	#121	—	2	8	no	—
164	#121	—	4	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₂²⋊C₄	—	4	2	no	—
167	D₁₆	—	2	8	no	—
168	M₃₂	—	2	8	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	D₄×C₄₁	1	4	yes	complemented
171	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
172	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
173	#107	—	2	2	no	—
174	#107	C₂³×C₄₁	1	4	yes	—
175	#102	—	2	8	no	—
176	C₂⁴	—	4	2	no	—
177	C₂⁴	D₈×C₄₁	1	1	yes	characteristic
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	4	8	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	8	8	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	D₈×C₄₁	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
191	D₈	—	2	4	no	—
192	D₈	—	2	4	no	—
193	D₈	—	4	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
201	C₂×C₈	#41×C₄₁	1	2	yes	—
202	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₁₆	—	2	2	no	—
207	C₁₆	C₂×D₄×C₄₁	1	4	yes	complemented
208	C₂³	—	2	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	—	8	4	no	—
214	C₂³	C₂×D₈×C₄₁	1	2	yes	—
215	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	8	16	no	—
219	D₄	—	8	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	#121×C₄₁	1	2	yes	—
227	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
229	C₂×C₄	#135×C₄₁	1	2	yes	—
230	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	#121×C₄₁	1	2	yes	—
233	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
234	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	4	no	—
241	C₂²	—	16	8	no	—
242	C₂²	C₄₁×#466	1	2	yes	—
243	C₂²	C₄₁×#509	1	4	yes	—
244	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	4	no	—
247	C₄	C₄₁×#466	1	2	yes	—
248	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
249	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	16	4	no	—
252	C₂	C₄₁×#2218	1	4	yes	direct product factor
253	C₂	C₂×C₄₁×#466	1	1	yes	characteristic
254	C₂	C₂×C₄₁×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆×C₄₁	C₂	semidirect
2. C₂²×M₃₂×C₄₁	C₂	semidirect
3. C₂×C₄₁×#466	C₂	semidirect
4. C₄₁×#2218	C₂	direct (N × H)
5. C₂×C₄₁×#367	C₂	semidirect
6. C₂×C₄₁×#357	C₂	semidirect
11. C₂×D₁₆×C₄₁	C₂²	semidirect
13. C₂×M₃₂×C₄₁	C₂²	semidirect
17. C₄₁×#466	C₂²	semidirect
20. C₄₁×#367	C₂²	semidirect
21. C₄₁×#357	C₂²	semidirect
42. C₂×C₁₆×C₄₁	D₄	semidirect
79. C₁₆×C₄₁	C₂×D₄	semidirect
107. C₂×#2218	C₄₁	direct (N × H)
123. C₂²×D₁₆	C₂×C₄₁	semidirect
124. C₂²×M₃₂	C₂×C₄₁	semidirect
125. C₂×#466	C₂×C₄₁	semidirect
126. #2218	C₂×C₄₁	direct (N × H)
127. C₂×#367	C₂×C₄₁	semidirect
128. C₂×#357	C₂×C₄₁	semidirect
131. C₂×C₄₁	#2218	direct (N × H)
138. C₂×D₁₆	C₂²×C₄₁	semidirect
140. C₂×M₃₂	C₂²×C₄₁	semidirect
144. #466	C₂²×C₄₁	semidirect
147. #367	C₂²×C₄₁	semidirect
148. #357	C₂²×C₄₁	semidirect
149. C₄₁	C₂×#2218	direct (N × H)
170. C₂×C₁₆	D₄×C₄₁	semidirect
207. C₁₆	C₂×D₄×C₄₁	semidirect
252. C₂	C₄₁×#2218	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×D₁₆×C₄₁	2
C₂²×M₃₂×C₄₁	2
C₂×C₄₁×#466	2
C₄₁×#2218	2
C₂×C₄₁×#367	2
C₂×C₄₁×#357	2
C₂×#2218	41

Subgroups — C2×C41×#2218 #23809802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2218 #23809802