Subgroups — C₂×C₄₁×#2861 #23812831

Order 10496.

2902 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2861	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄₁×#391	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₁×#390	C₂	1	2	yes	maximal
6	C₂×C₄₁×#388	C₂	1	8	yes	maximal, complemented
7	C₂×C₄³×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#125×C₄₁	—	2	8	no	—
11	C₂×#125×C₄₁	C₂²	1	8	yes	complemented
12	C₂×#124×C₄₁	—	2	8	no	—
13	C₂×#124×C₄₁	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
15	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
17	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
18	C₂²×C₄²×C₄₁	C₂²	1	4	yes	—
19	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#391	C₂²	1	16	yes	complemented
22	C₄₁×#390	C₂²	1	16	yes	—
23	C₄₁×#388	C₂²	1	64	yes	complemented
24	C₂×#95×C₄₁	C₂²	1	8	yes	—
25	C₄³×C₄₁	C₂²	1	8	yes	complemented
26	C₂⁵×C₄₁	C₂³	1	2	yes	—
27	C₂²×Q₈×C₄₁	—	2	4	no	—
28	C₂²×Q₈×C₄₁	—	2	8	no	—
29	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
30	C₂²×D₄×C₄₁	—	2	4	no	—
31	C₂²×D₄×C₄₁	—	2	8	no	—
32	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
35	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
36	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
37	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
38	#125×C₄₁	—	2	32	no	—
39	#124×C₄₁	—	2	32	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	16	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	16	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	32	no	—
46	C₂×C₂²⋊C₄×C₄₁	C₂³	1	16	yes	—
47	C₂×C₂²⋊C₄×C₄₁	C₂³	1	32	yes	—
48	C₂×C₄²×C₄₁	—	2	16	no	—
49	C₂×C₄²×C₄₁	—	2	8	no	—
50	C₂×C₄²×C₄₁	D₄	1	8	yes	—
51	C₂×C₄²×C₄₁	C₂³	1	16	yes	—
52	C₂×C₄²×C₄₁	C₂³	1	8	yes	—
53	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
54	#95×C₄₁	C₂³	1	32	yes	—
55	C₂⁴×C₄₁	—	2	4	no	—
56	C₂⁴×C₄₁	—	2	16	no	—
57	C₂⁴×C₄₁	—	2	8	no	—
58	C₂⁴×C₄₁	C₂⁴	1	4	yes	—
59	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₄₁	—	2	8	no	—
61	C₂×Q₈×C₄₁	—	2	16	no	—
62	C₂×Q₈×C₄₁	—	4	16	no	—
63	C₂×D₄×C₄₁	—	2	8	no	—
64	C₂×D₄×C₄₁	—	2	16	no	—
65	C₂×D₄×C₄₁	—	4	16	no	—
66	C₂²×C₄×C₄₁	—	2	8	no	—
67	C₂²×C₄×C₄₁	—	2	8	no	—
68	C₂²×C₄×C₄₁	—	2	8	no	—
69	C₂²×C₄×C₄₁	—	2	4	no	—
70	C₂²×C₄×C₄₁	—	2	8	no	—
71	C₂²×C₄×C₄₁	—	2	16	no	—
72	C₂²×C₄×C₄₁	—	2	8	no	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₁	#41	1	16	yes	—
76	C₂²×C₄×C₄₁	#41	1	8	yes	—
77	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
78	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
82	C₄⋊C₄×C₄₁	—	2	32	no	—
83	C₄⋊C₄×C₄₁	—	2	16	no	—
84	C₂²⋊C₄×C₄₁	—	2	32	no	—
85	C₂²⋊C₄×C₄₁	—	2	64	no	—
86	C₄²×C₄₁	—	2	32	no	—
87	C₄²×C₄₁	—	2	16	no	—
88	C₄²×C₄₁	C₂×D₄	1	16	yes	—
89	C₂³×C₄₁	—	2	4	no	—
90	C₂³×C₄₁	—	2	16	no	—
91	C₂³×C₄₁	—	2	8	no	—
92	C₂³×C₄₁	—	4	8	no	—
93	C₂³×C₄₁	—	4	32	no	—
94	C₂³×C₄₁	—	4	16	no	—
95	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
96	C₂³×C₄₁	C₂×#41	1	4	yes	—
97	C₂³×C₄₁	C₂×#41	1	8	yes	—
98	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
99	Q₈×C₄₁	—	4	16	no	—
100	D₄×C₄₁	—	4	16	no	—
101	C₂×C₄×C₄₁	—	2	32	no	—
102	C₂×C₄×C₄₁	—	2	16	no	—
103	C₂×C₄×C₄₁	—	2	8	no	—
104	C₂×C₄×C₄₁	—	2	8	no	—
105	C₂×C₄×C₄₁	—	2	8	no	—
106	C₂×C₄×C₄₁	—	2	4	no	—
107	C₂×C₄×C₄₁	—	2	8	no	—
108	C₂×C₄×C₄₁	—	2	16	no	—
109	C₂×C₄×C₄₁	—	2	8	no	—
110	C₂×C₄×C₄₁	—	4	8	no	—
111	C₂×C₄×C₄₁	—	4	16	no	—
112	C₂×C₄×C₄₁	#124	1	16	yes	—
113	C₂×C₄×C₄₁	C₂²×D₄	1	8	yes	—
114	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
115	C₂×C₄×C₄₁	C₂×#41	1	16	yes	—
116	C₂×C₄×C₄₁	C₂×#41	1	8	yes	—
117	C₂×#2861	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	—	4	4	no	—
119	C₂²×C₄₁	—	4	16	no	—
120	C₂²×C₄₁	—	4	8	no	—
121	C₂²×C₄₁	—	8	16	no	—
122	C₂²×C₄₁	C₂×#124	1	4	yes	—
123	C₂²×C₄₁	#529	1	16	yes	—
124	C₂²×C₄₁	#532	1	8	yes	—
125	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
126	C₂²×C₄₁	C₂²×#41	1	2	yes	—
127	C₂²×C₄₁	C₂²×#41	1	4	yes	—
128	C₄×C₄₁	—	2	16	no	—
129	C₄×C₄₁	—	2	8	no	—
130	C₄×C₄₁	—	4	4	no	—
131	C₄×C₄₁	—	4	8	no	—
132	C₄×C₄₁	C₂×#124	1	8	yes	—
133	C₂²×#125	C₂×C₄₁	1	1	yes	characteristic, complemented
134	C₂²×#124	C₂×C₄₁	1	1	yes	characteristic
135	#2861	C₂×C₄₁	1	16	yes	direct product factor
136	C₂×#391	C₂×C₄₁	1	2	yes	complemented
137	C₂×#390	C₂×C₄₁	1	2	yes	—
138	C₂×#388	C₂×C₄₁	1	8	yes	complemented
139	C₂×C₄³	C₂×C₄₁	1	1	yes	characteristic, complemented
140	C₂×C₄₁	—	8	4	no	—
141	C₂×C₄₁	#2861	1	8	yes	direct product factor
142	C₂×C₄₁	C₂²×#124	1	1	yes	characteristic
143	C₂×C₄₁	C₂×#529	1	4	yes	—
144	C₂×C₄₁	C₂×#532	1	2	yes	—
145	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
146	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
147	C₂×#125	—	2	8	no	—
148	C₂×#125	C₂²×C₄₁	1	8	yes	complemented
149	C₂×#124	—	2	8	no	—
150	C₂×#124	C₂²×C₄₁	1	8	yes	—
151	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
152	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
153	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
154	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
155	C₂²×C₄²	C₂²×C₄₁	1	4	yes	—
156	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
157	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
158	#391	C₂²×C₄₁	1	16	yes	complemented
159	#390	C₂²×C₄₁	1	16	yes	—
160	#388	C₂²×C₄₁	1	64	yes	complemented
161	C₂×#95	C₂²×C₄₁	1	8	yes	—
162	C₄³	C₂²×C₄₁	1	8	yes	complemented
163	C₄₁	C₂×#2861	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁵	C₂³×C₄₁	1	2	yes	—
165	C₂²×Q₈	—	2	4	no	—
166	C₂²×Q₈	—	2	8	no	—
167	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
168	C₂²×D₄	—	2	4	no	—
169	C₂²×D₄	—	2	8	no	—
170	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
171	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
172	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
173	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
174	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
175	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
176	#125	—	2	32	no	—
177	#124	—	2	32	no	—
178	C₂×C₄⋊C₄	—	2	16	no	—
179	C₂×C₄⋊C₄	—	2	8	no	—
180	C₂×C₄⋊C₄	C₂³×C₄₁	1	16	yes	—
181	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
182	C₂×C₂²⋊C₄	—	2	16	no	—
183	C₂×C₂²⋊C₄	—	2	32	no	—
184	C₂×C₂²⋊C₄	C₂³×C₄₁	1	16	yes	—
185	C₂×C₂²⋊C₄	C₂³×C₄₁	1	32	yes	—
186	C₂×C₄²	—	2	16	no	—
187	C₂×C₄²	—	2	8	no	—
188	C₂×C₄²	D₄×C₄₁	1	8	yes	—
189	C₂×C₄²	C₂³×C₄₁	1	16	yes	—
190	C₂×C₄²	C₂³×C₄₁	1	8	yes	—
191	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
192	#95	C₂³×C₄₁	1	32	yes	—
193	C₂⁴	—	2	4	no	—
194	C₂⁴	—	2	16	no	—
195	C₂⁴	—	2	8	no	—
196	C₂⁴	C₂⁴×C₄₁	1	4	yes	—
197	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
198	C₂×Q₈	—	2	8	no	—
199	C₂×Q₈	—	2	16	no	—
200	C₂×Q₈	—	4	16	no	—
201	C₂×D₄	—	2	8	no	—
202	C₂×D₄	—	2	16	no	—
203	C₂×D₄	—	4	16	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
212	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
213	C₂²×C₄	#41×C₄₁	1	16	yes	—
214	C₂²×C₄	#41×C₄₁	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
216	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
217	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
219	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
220	C₄⋊C₄	—	2	32	no	—
221	C₄⋊C₄	—	2	16	no	—
222	C₂²⋊C₄	—	2	32	no	—
223	C₂²⋊C₄	—	2	64	no	—
224	C₄²	—	2	32	no	—
225	C₄²	—	2	16	no	—
226	C₄²	C₂×D₄×C₄₁	1	16	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	16	no	—
229	C₂³	—	2	8	no	—
230	C₂³	—	4	8	no	—
231	C₂³	—	4	32	no	—
232	C₂³	—	4	16	no	—
233	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
234	C₂³	C₂×#41×C₄₁	1	4	yes	—
235	C₂³	C₂×#41×C₄₁	1	8	yes	—
236	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
237	Q₈	—	4	16	no	—
238	D₄	—	4	16	no	—
239	C₂×C₄	—	2	32	no	—
240	C₂×C₄	—	2	16	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	16	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	#124×C₄₁	1	16	yes	—
251	C₂×C₄	C₂²×D₄×C₄₁	1	8	yes	—
252	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
253	C₂×C₄	C₂×#41×C₄₁	1	16	yes	—
254	C₂×C₄	C₂×#41×C₄₁	1	8	yes	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	8	16	no	—
259	C₂²	C₂×#124×C₄₁	1	4	yes	—
260	C₂²	C₄₁×#529	1	16	yes	—
261	C₂²	C₄₁×#532	1	8	yes	—
262	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
263	C₂²	C₂²×#41×C₄₁	1	2	yes	—
264	C₂²	C₂²×#41×C₄₁	1	4	yes	—
265	C₄	—	2	16	no	—
266	C₄	—	2	8	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	8	no	—
269	C₄	C₂×#124×C₄₁	1	8	yes	—
270	C₂	—	8	4	no	—
271	C₂	C₄₁×#2861	1	8	yes	direct product factor
272	C₂	C₂²×#124×C₄₁	1	1	yes	characteristic
273	C₂	C₂×C₄₁×#529	1	4	yes	—
274	C₂	C₂×C₄₁×#532	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₁
prime radical	C₂³×D₄×C₄₁	C₂²
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
1. C₂²×#125×C₄₁	C₂	semidirect
3. C₄₁×#2861	C₂	direct (N × H)
4. C₂×C₄₁×#391	C₂	semidirect
6. C₂×C₄₁×#388	C₂	semidirect
7. C₂×C₄³×C₄₁	C₂	semidirect
11. C₂×#125×C₄₁	C₂²	semidirect
21. C₄₁×#391	C₂²	semidirect
23. C₄₁×#388	C₂²	semidirect
25. C₄³×C₄₁	C₂²	semidirect
117. C₂×#2861	C₄₁	direct (N × H)
133. C₂²×#125	C₂×C₄₁	semidirect
135. #2861	C₂×C₄₁	direct (N × H)
136. C₂×#391	C₂×C₄₁	semidirect
138. C₂×#388	C₂×C₄₁	semidirect
139. C₂×C₄³	C₂×C₄₁	semidirect
141. C₂×C₄₁	#2861	direct (N × H)
148. C₂×#125	C₂²×C₄₁	semidirect
158. #391	C₂²×C₄₁	semidirect
160. #388	C₂²×C₄₁	semidirect
162. C₄³	C₂²×C₄₁	semidirect
163. C₄₁	C₂×#2861	direct (N × H)
271. C₂	C₄₁×#2861	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×#125×C₄₁	2
C₂²×#124×C₄₁	2
C₄₁×#2861	2
C₂×C₄₁×#391	2
C₂×C₄₁×#390	2
C₂×C₄₁×#388	2
C₂×C₄³×C₄₁	2
C₂×#2861	41

Subgroups — C2×C41×#2861 #23812831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2861 #23812831