Subgroups — C₂×C₄₁×#1820 #23796888

Order 10496.

1566 conjugacy classes of subgroups.

The subgroup lattice has 278 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×#381	C₂	1	2	yes	maximal
3	C₄₁×#1820	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₁×#342	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
6	C₂×#117×C₄₁	—	2	4	no	—
7	C₂×#117×C₄₁	—	2	4	no	—
8	C₂×#117×C₄₁	C₄	1	4	yes	complemented
9	C₂×#117×C₄₁	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₄	1	2	yes	complemented
11	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₁	C₄	1	2	yes	complemented
14	C₄₁×#381	—	2	4	no	—
15	C₄₁×#381	C₂²	1	8	yes	—
16	C₂×#95×C₄₁	—	2	4	no	—
17	C₄₁×#342	C₂²	1	16	yes	—
18	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₄₁	—	2	8	no	—
20	C₂³×C₄×C₄₁	—	2	4	no	—
21	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₄₁	D₄	1	4	yes	—
26	C₂³×C₄×C₄₁	D₄	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
28	#117×C₄₁	—	2	16	no	—
29	#117×C₄₁	—	2	16	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
33	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
36	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
39	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
40	C₂×C₄²×C₄₁	—	2	4	no	—
41	C₂×C₄²×C₄₁	—	2	4	no	—
42	C₂×C₄²×C₄₁	C₂×C₄	1	8	yes	complemented
43	#95×C₄₁	—	2	16	no	—
44	C₂⁴×C₄₁	—	4	4	no	—
45	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
47	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
48	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
49	C₂⁴×C₄₁	#41	1	8	yes	—
50	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	—	2	4	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	8	no	—
54	C₂²×C₄×C₄₁	—	2	16	no	—
55	C₂²×C₄×C₄₁	—	2	16	no	—
56	C₂²×C₄×C₄₁	—	2	8	no	—
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	2	4	no	—
59	C₂²×C₄×C₄₁	—	2	8	no	—
60	C₂²×C₄×C₄₁	—	4	8	no	—
61	C₂²×C₄×C₄₁	—	4	16	no	—
62	C₂²×C₄×C₄₁	—	4	8	no	—
63	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
66	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
72	C₄⋊C₄×C₄₁	—	2	16	no	—
73	C₂²⋊C₄×C₄₁	—	2	8	no	—
74	C₂²⋊C₄×C₄₁	—	2	8	no	—
75	C₂²⋊C₄×C₄₁	—	2	16	no	—
76	C₂²⋊C₄×C₄₁	—	4	8	no	—
77	C₄²×C₄₁	—	2	16	no	—
78	C₂³×C₄₁	—	2	4	no	—
79	C₂³×C₄₁	—	2	8	no	—
80	C₂³×C₄₁	—	4	2	no	—
81	C₂³×C₄₁	—	4	4	no	—
82	C₂³×C₄₁	—	4	16	no	—
83	C₂³×C₄₁	—	4	2	no	—
84	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₄₁	#117	1	4	yes	—
88	C₂³×C₄₁	#117	1	4	yes	—
89	C₂³×C₄₁	#123	1	8	yes	—
90	C₂³×C₄₁	#123	1	8	yes	—
91	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
92	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
93	C₂³×C₄₁	C₂×#41	1	4	yes	—
94	C₂×C₄×C₄₁	—	2	4	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	16	no	—
97	C₂×C₄×C₄₁	—	2	4	no	—
98	C₂×C₄×C₄₁	—	2	4	no	—
99	C₂×C₄×C₄₁	—	2	16	no	—
100	C₂×C₄×C₄₁	—	2	4	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	8	no	—
103	C₂×C₄×C₄₁	—	4	8	no	—
104	C₂×C₄×C₄₁	—	4	8	no	—
105	C₂×C₄×C₄₁	—	4	8	no	—
106	C₂×C₄×C₄₁	—	4	16	no	—
107	C₂×C₄×C₄₁	—	4	8	no	—
108	C₂×C₄×C₄₁	—	8	16	no	—
109	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×#1820	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₄₁	—	2	16	no	—
113	C₂²×C₄₁	—	2	8	no	—
114	C₂²×C₄₁	—	2	2	no	—
115	C₂²×C₄₁	—	2	2	no	—
116	C₂²×C₄₁	—	2	4	no	—
117	C₂²×C₄₁	—	2	4	no	—
118	C₂²×C₄₁	—	4	4	no	—
119	C₂²×C₄₁	—	4	8	no	—
120	C₂²×C₄₁	—	4	4	no	—
121	C₂²×C₄₁	#381	1	4	yes	—
122	C₂²×C₄₁	#381	1	2	yes	—
123	C₂²×C₄₁	#381	1	2	yes	—
124	C₂²×C₄₁	#410	1	4	yes	—
125	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₁	C₂×#117	1	2	yes	—
127	C₂²×C₄₁	C₂×#123	1	4	yes	—
128	C₄×C₄₁	—	2	4	no	—
129	C₄×C₄₁	—	2	4	no	—
130	C₄×C₄₁	—	4	8	no	—
131	C₄×C₄₁	—	8	8	no	—
132	C₂²×#117	C₂×C₄₁	1	1	yes	characteristic
133	C₂×#381	C₂×C₄₁	1	2	yes	—
134	#1820	C₂×C₄₁	1	8	yes	direct product factor
135	C₂×#342	C₂×C₄₁	1	4	yes	—
136	C₂×C₄₁	—	2	4	no	—
137	C₂×C₄₁	—	2	2	no	—
138	C₂×C₄₁	—	2	4	no	—
139	C₂×C₄₁	—	2	2	no	—
140	C₂×C₄₁	#1820	1	4	yes	direct product factor
141	C₂×C₄₁	C₂×#381	1	1	yes	characteristic
142	C₂×C₄₁	C₂×#410	1	2	yes	—
143	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
144	C₂×#117	—	2	4	no	—
145	C₂×#117	—	2	4	no	—
146	C₂×#117	C₄×C₄₁	1	4	yes	complemented
147	C₂×#117	C₂²×C₄₁	1	4	yes	—
148	C₂²×C₄⋊C₄	C₄×C₄₁	1	2	yes	complemented
149	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
150	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
151	C₂²×C₄²	C₄×C₄₁	1	2	yes	complemented
152	#381	—	2	4	no	—
153	#381	C₂²×C₄₁	1	8	yes	—
154	C₂×#95	—	2	4	no	—
155	#342	C₂²×C₄₁	1	16	yes	—
156	C₄₁	C₂×#1820	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
162	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
164	C₂³×C₄	D₄×C₄₁	1	4	yes	—
165	C₂³×C₄	D₄×C₄₁	1	4	yes	—
166	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
167	#117	—	2	16	no	—
168	#117	—	2	16	no	—
169	C₂×C₄⋊C₄	—	2	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
172	C₂×C₂²⋊C₄	—	2	4	no	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	—	2	16	no	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
178	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	complemented
182	#95	—	2	16	no	—
183	C₂⁴	—	4	4	no	—
184	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
185	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
186	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
187	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
188	C₂⁴	#41×C₄₁	1	8	yes	—
189	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	16	no	—
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	4	8	no	—
200	C₂²×C₄	—	4	16	no	—
201	C₂²×C₄	—	4	8	no	—
202	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
203	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
204	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
205	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
207	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
210	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
211	C₄⋊C₄	—	2	16	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	2	8	no	—
214	C₂²⋊C₄	—	2	16	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₄²	—	2	16	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	8	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	16	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
224	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
225	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
226	C₂³	#117×C₄₁	1	4	yes	—
227	C₂³	#117×C₄₁	1	4	yes	—
228	C₂³	#123×C₄₁	1	8	yes	—
229	C₂³	#123×C₄₁	1	8	yes	—
230	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
231	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
232	C₂³	C₂×#41×C₄₁	1	4	yes	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	16	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	8	16	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
250	C₂²	—	2	16	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	4	no	—
259	C₂²	C₄₁×#381	1	4	yes	—
260	C₂²	C₄₁×#381	1	2	yes	—
261	C₂²	C₄₁×#381	1	2	yes	—
262	C₂²	C₄₁×#410	1	4	yes	—
263	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
264	C₂²	C₂×#117×C₄₁	1	2	yes	—
265	C₂²	C₂×#123×C₄₁	1	4	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	2	4	no	—
268	C₄	—	4	8	no	—
269	C₄	—	8	8	no	—
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	—	2	4	no	—
273	C₂	—	2	2	no	—
274	C₂	C₄₁×#1820	1	4	yes	direct product factor
275	C₂	C₂×C₄₁×#381	1	1	yes	characteristic
276	C₂	C₂×C₄₁×#410	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
prime radical	C₂⁵×C₄₁	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂⁴×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1820	C₂	direct (N × H)
8. C₂×#117×C₄₁	C₄	semidirect
10. C₂²×C₄⋊C₄×C₄₁	C₄	semidirect
13. C₂²×C₄²×C₄₁	C₄	semidirect
32. C₂×C₄⋊C₄×C₄₁	C₂×C₄	semidirect
42. C₂×C₄²×C₄₁	C₂×C₄	semidirect
111. C₂×#1820	C₄₁	direct (N × H)
134. #1820	C₂×C₄₁	direct (N × H)
140. C₂×C₄₁	#1820	direct (N × H)
146. C₂×#117	C₄×C₄₁	semidirect
148. C₂²×C₄⋊C₄	C₄×C₄₁	semidirect
151. C₂²×C₄²	C₄×C₄₁	semidirect
156. C₄₁	C₂×#1820	direct (N × H)
171. C₂×C₄⋊C₄	C₂×C₄×C₄₁	semidirect
181. C₂×C₄²	C₂×C₄×C₄₁	semidirect
274. C₂	C₄₁×#1820	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×#381	2
C₄₁×#1820	2
C₂×C₄₁×#342	2
C₂×#1820	41

Subgroups — C2×C41×#1820 #23796888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1820 #23796888