Subgroups — C₂²×C₄₁×#481 #23836705

Order 10496.

3006 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#481	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#106×C₄₁	C₂	1	2	yes	maximal, complemented
5	C₂²×#102×C₄₁	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₄₁	—	2	1	no	—
8	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₄₁	—	2	1	no	—
10	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
12	C₂×#121×C₄₁	—	2	6	no	—
13	C₂×#121×C₄₁	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₄₁	—	2	6	no	—
15	C₂²×C₄⋊C₄×C₄₁	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₁	C₂²	1	12	yes	complemented
19	C₂²×C₂²⋊C₄×C₄₁	—	2	1	no	—
20	C₄₁×#481	C₂²	1	64	yes	direct product factor
21	C₂×#106×C₄₁	C₂²	1	24	yes	complemented
22	C₂×#102×C₄₁	C₂²	1	24	yes	complemented
23	C₂×C₂²⋊C₈×C₄₁	C₂²	1	12	yes	complemented
24	C₂⁵×C₄₁	—	2	1	no	—
25	C₂⁵×C₄₁	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₄₁	—	2	12	no	—
27	C₂²×D₄×C₄₁	—	2	3	no	—
28	C₂²×D₄×C₄₁	—	2	8	no	—
29	C₂²×D₄×C₄₁	—	4	8	no	—
30	C₂²×D₄×C₄₁	C₂³	1	6	yes	—
31	C₂³×C₄×C₄₁	—	2	8	no	—
32	C₂³×C₄×C₄₁	—	2	12	no	—
33	C₂³×C₄×C₄₁	—	2	1	no	—
34	C₂³×C₄×C₄₁	—	2	2	no	—
35	C₂³×C₄×C₄₁	—	4	8	no	—
36	C₂³×C₄×C₄₁	D₄	1	6	yes	—
37	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₁	C₂³	1	6	yes	—
39	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₁	—	2	8	no	—
41	C₂²×C₈×C₄₁	C₂³	1	12	yes	—
42	#121×C₄₁	—	2	24	no	—
43	#121×C₄₁	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₄₁	—	2	6	no	—
45	C₂×C₄⋊C₄×C₄₁	—	2	24	no	—
46	C₂×C₄⋊C₄×C₄₁	—	2	12	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	3	no	—
48	C₂×C₄⋊C₄×C₄₁	C₂³	1	12	yes	—
49	C₂×C₄⋊C₄×C₄₁	C₂³	1	6	yes	—
50	C₂×C₄⋊C₄×C₄₁	C₂³	1	16	yes	complemented
51	C₂×C₂²⋊C₄×C₄₁	—	2	12	no	—
52	#106×C₄₁	C₂³	1	32	yes	complemented
53	#102×C₄₁	C₂³	1	32	yes	complemented
54	C₂²⋊C₈×C₄₁	C₂³	1	16	yes	complemented
55	C₂⁴×C₄₁	—	2	4	no	—
56	C₂⁴×C₄₁	—	2	4	no	—
57	C₂⁴×C₄₁	—	2	6	no	—
58	C₂⁴×C₄₁	—	4	4	no	—
59	C₂⁴×C₄₁	—	4	4	no	—
60	C₂⁴×C₄₁	—	4	4	no	—
61	C₂⁴×C₄₁	SD₁₆	1	8	yes	—
62	C₂⁴×C₄₁	C₂×D₄	1	6	yes	—
63	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
64	C₂×D₄×C₄₁	—	2	16	no	—
65	C₂×D₄×C₄₁	—	2	6	no	—
66	C₂×D₄×C₄₁	—	2	24	no	—
67	C₂×D₄×C₄₁	—	4	48	no	—
68	C₂×D₄×C₄₁	—	4	12	no	—
69	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
70	C₂²×C₄×C₄₁	—	2	4	no	—
71	C₂²×C₄×C₄₁	—	2	24	no	—
72	C₂²×C₄×C₄₁	—	2	24	no	—
73	C₂²×C₄×C₄₁	—	2	6	no	—
74	C₂²×C₄×C₄₁	—	2	12	no	—
75	C₂²×C₄×C₄₁	—	2	16	no	—
76	C₂²×C₄×C₄₁	—	4	48	no	—
77	C₂²×C₄×C₄₁	—	4	4	no	—
78	C₂²×C₄×C₄₁	—	4	8	no	—
79	C₂²×C₄×C₄₁	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	12	yes	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₄₁	#41	1	8	yes	—
83	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₄₁	C₂⁴	1	3	yes	—
85	C₂×C₈×C₄₁	—	2	24	no	—
86	C₂×C₈×C₄₁	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₄₁	—	2	12	no	—
88	C₄⋊C₄×C₄₁	—	2	16	no	—
89	C₄⋊C₄×C₄₁	—	2	6	no	—
90	C₄⋊C₄×C₄₁	C₂⁴	1	8	yes	—
91	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
92	C₂²⋊C₄×C₄₁	—	2	16	no	—
93	C₂³×C₄₁	—	2	12	no	—
94	C₂³×C₄₁	—	2	12	no	—
95	C₂³×C₄₁	—	2	32	no	—
96	C₂³×C₄₁	—	2	4	no	—
97	C₂³×C₄₁	—	4	12	no	—
98	C₂³×C₄₁	—	4	12	no	—
99	C₂³×C₄₁	—	4	12	no	—
100	C₂³×C₄₁	—	8	16	no	—
101	C₂³×C₄₁	#123	1	4	yes	—
102	C₂³×C₄₁	C₂×SD₁₆	1	24	yes	—
103	C₂³×C₄₁	C₂×SD₁₆	1	4	yes	—
104	C₂³×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
105	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
106	C₂³×C₄₁	C₂²×D₄	1	3	yes	—
107	D₄×C₄₁	—	2	8	no	—
108	D₄×C₄₁	—	4	32	no	—
109	D₄×C₄₁	—	4	12	no	—
110	C₂×C₄×C₄₁	—	2	12	no	—
111	C₂×C₄×C₄₁	—	2	24	no	—
112	C₂×C₄×C₄₁	—	2	8	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	8	no	—
115	C₂×C₄×C₄₁	—	4	32	no	—
116	C₂×C₄×C₄₁	—	4	12	no	—
117	C₂×C₄×C₄₁	—	4	24	no	—
118	C₂×C₄×C₄₁	#123	1	12	yes	—
119	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
120	C₂×C₄×C₄₁	C₂²×D₄	1	3	yes	—
121	C₂×C₄×C₄₁	C₂×#41	1	12	yes	—
122	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
123	C₈×C₄₁	—	2	8	no	—
124	C₂²×#481	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₄₁	—	2	48	no	—
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	2	4	no	—
128	C₂²×C₄₁	—	4	4	no	—
129	C₂²×C₄₁	—	4	4	no	—
130	C₂²×C₄₁	—	4	4	no	—
131	C₂²×C₄₁	—	8	24	no	—
132	C₂²×C₄₁	#481	1	16	yes	direct product factor
133	C₂²×C₄₁	C₂×#123	1	6	yes	—
134	C₂²×C₄₁	C₂²×SD₁₆	1	8	yes	—
135	C₂²×C₄₁	C₂²×SD₁₆	1	6	yes	—
136	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
137	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
138	C₄×C₄₁	—	2	4	no	—
139	C₄×C₄₁	—	4	4	no	—
140	C₄×C₄₁	—	4	8	no	—
141	C₄×C₄₁	C₂×#123	1	6	yes	—
142	C₄×C₄₁	C₂²×#41	1	2	yes	—
143	C₂²×#121	C₂×C₄₁	1	1	yes	characteristic
144	C₂³×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
145	C₂×#481	C₂×C₄₁	1	24	yes	direct product factor
146	C₂²×#106	C₂×C₄₁	1	2	yes	complemented
147	C₂²×#102	C₂×C₄₁	1	2	yes	complemented
148	C₂²×C₂²⋊C₈	C₂×C₄₁	1	1	yes	characteristic, complemented
149	C₂×C₄₁	—	2	8	no	—
150	C₂×C₄₁	—	8	4	no	—
151	C₂×C₄₁	C₂×#481	1	12	yes	direct product factor
152	C₂×C₄₁	C₂²×#123	1	1	yes	characteristic
153	C₂×C₄₁	C₂³×SD₁₆	1	1	yes	characteristic
154	C₂×C₄₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂³×D₄	—	2	1	no	—
156	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
157	C₂⁴×C₄	—	2	1	no	—
158	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
159	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
160	C₂×#121	—	2	6	no	—
161	C₂×#121	C₂²×C₄₁	1	12	yes	—
162	C₂²×C₄⋊C₄	—	2	6	no	—
163	C₂²×C₄⋊C₄	—	2	1	no	—
164	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
165	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
166	C₂²×C₄⋊C₄	C₂²×C₄₁	1	12	yes	complemented
167	C₂²×C₂²⋊C₄	—	2	1	no	—
168	#481	C₂²×C₄₁	1	64	yes	direct product factor
169	C₂×#106	C₂²×C₄₁	1	24	yes	complemented
170	C₂×#102	C₂²×C₄₁	1	24	yes	complemented
171	C₂×C₂²⋊C₈	C₂²×C₄₁	1	12	yes	complemented
172	C₄₁	C₂²×#481	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁵	—	2	1	no	—
174	C₂⁵	D₄×C₄₁	1	1	yes	characteristic
175	C₂²×D₄	—	2	12	no	—
176	C₂²×D₄	—	2	3	no	—
177	C₂²×D₄	—	2	8	no	—
178	C₂²×D₄	—	4	8	no	—
179	C₂²×D₄	C₂³×C₄₁	1	6	yes	—
180	C₂³×C₄	—	2	8	no	—
181	C₂³×C₄	—	2	1	no	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	2	12	no	—
184	C₂³×C₄	—	4	8	no	—
185	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₄₁	1	6	yes	—
187	C₂³×C₄	C₂³×C₄₁	1	6	yes	—
188	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
189	C₂²×C₈	—	2	8	no	—
190	C₂²×C₈	C₂³×C₄₁	1	12	yes	—
191	#121	—	2	24	no	—
192	#121	C₂³×C₄₁	1	16	yes	—
193	C₂×C₄⋊C₄	—	2	6	no	—
194	C₂×C₄⋊C₄	—	2	24	no	—
195	C₂×C₄⋊C₄	—	2	12	no	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	C₂³×C₄₁	1	12	yes	—
198	C₂×C₄⋊C₄	C₂³×C₄₁	1	6	yes	—
199	C₂×C₄⋊C₄	C₂³×C₄₁	1	16	yes	complemented
200	C₂×C₂²⋊C₄	—	2	12	no	—
201	#106	C₂³×C₄₁	1	32	yes	complemented
202	#102	C₂³×C₄₁	1	32	yes	complemented
203	C₂²⋊C₈	C₂³×C₄₁	1	16	yes	complemented
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	—	4	4	no	—
209	C₂⁴	—	4	4	no	—
210	C₂⁴	SD₁₆×C₄₁	1	8	yes	—
211	C₂⁴	C₂×D₄×C₄₁	1	6	yes	—
212	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
213	C₂×D₄	—	2	16	no	—
214	C₂×D₄	—	2	6	no	—
215	C₂×D₄	—	2	24	no	—
216	C₂×D₄	—	4	48	no	—
217	C₂×D₄	—	4	12	no	—
218	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	24	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	16	no	—
225	C₂²×C₄	—	4	48	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	8	no	—
228	C₂²×C₄	C₂×D₄×C₄₁	1	6	yes	—
229	C₂²×C₄	C₂×D₄×C₄₁	1	12	yes	—
230	C₂²×C₄	C₂×D₄×C₄₁	1	3	yes	—
231	C₂²×C₄	#41×C₄₁	1	8	yes	—
232	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
233	C₂²×C₄	C₂⁴×C₄₁	1	3	yes	—
234	C₂×C₈	—	2	24	no	—
235	C₂×C₈	C₂⁴×C₄₁	1	8	yes	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	C₂⁴×C₄₁	1	8	yes	—
240	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
241	C₂²⋊C₄	—	2	16	no	—
242	C₂³	—	2	12	no	—
243	C₂³	—	2	12	no	—
244	C₂³	—	2	32	no	—
245	C₂³	—	2	4	no	—
246	C₂³	—	4	12	no	—
247	C₂³	—	4	12	no	—
248	C₂³	—	4	12	no	—
249	C₂³	—	8	16	no	—
250	C₂³	#123×C₄₁	1	4	yes	—
251	C₂³	C₂×SD₁₆×C₄₁	1	24	yes	—
252	C₂³	C₂×SD₁₆×C₄₁	1	4	yes	—
253	C₂³	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
254	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
255	C₂³	C₂²×D₄×C₄₁	1	3	yes	—
256	D₄	—	2	8	no	—
257	D₄	—	4	32	no	—
258	D₄	—	4	12	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	32	no	—
265	C₂×C₄	—	4	12	no	—
266	C₂×C₄	—	4	24	no	—
267	C₂×C₄	#123×C₄₁	1	12	yes	—
268	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
269	C₂×C₄	C₂²×D₄×C₄₁	1	3	yes	—
270	C₂×C₄	C₂×#41×C₄₁	1	12	yes	—
271	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	48	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	8	24	no	—
280	C₂²	C₄₁×#481	1	16	yes	direct product factor
281	C₂²	C₂×#123×C₄₁	1	6	yes	—
282	C₂²	C₂²×SD₁₆×C₄₁	1	8	yes	—
283	C₂²	C₂²×SD₁₆×C₄₁	1	6	yes	—
284	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	6	yes	—
285	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	8	no	—
289	C₄	C₂×#123×C₄₁	1	6	yes	—
290	C₄	C₂²×#41×C₄₁	1	2	yes	—
291	C₂	—	2	8	no	—
292	C₂	—	8	4	no	—
293	C₂	C₂×C₄₁×#481	1	12	yes	direct product factor
294	C₂	C₂²×#123×C₄₁	1	1	yes	characteristic
295	C₂	C₂³×SD₁₆×C₄₁	1	1	yes	characteristic
296	C₂	C₂²×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
prime radical	C₂²×#121×C₄₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂⁴×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊C₄×C₄₁	C₂	semidirect
3. C₂×C₄₁×#481	C₂	direct (N × H)
4. C₂²×#106×C₄₁	C₂	semidirect
5. C₂²×#102×C₄₁	C₂	semidirect
6. C₂²×C₂²⋊C₈×C₄₁	C₂	semidirect
18. C₂²×C₄⋊C₄×C₄₁	C₂²	semidirect
20. C₄₁×#481	C₂²	direct (N × H)
21. C₂×#106×C₄₁	C₂²	semidirect
22. C₂×#102×C₄₁	C₂²	semidirect
23. C₂×C₂²⋊C₈×C₄₁	C₂²	semidirect
50. C₂×C₄⋊C₄×C₄₁	C₂³	semidirect
52. #106×C₄₁	C₂³	semidirect
53. #102×C₄₁	C₂³	semidirect
54. C₂²⋊C₈×C₄₁	C₂³	semidirect
124. C₂²×#481	C₄₁	direct (N × H)
132. C₂²×C₄₁	#481	direct (N × H)
144. C₂³×C₄⋊C₄	C₂×C₄₁	semidirect
145. C₂×#481	C₂×C₄₁	direct (N × H)
146. C₂²×#106	C₂×C₄₁	semidirect
147. C₂²×#102	C₂×C₄₁	semidirect
148. C₂²×C₂²⋊C₈	C₂×C₄₁	semidirect
151. C₂×C₄₁	C₂×#481	direct (N × H)
166. C₂²×C₄⋊C₄	C₂²×C₄₁	semidirect
168. #481	C₂²×C₄₁	direct (N × H)
169. C₂×#106	C₂²×C₄₁	semidirect
170. C₂×#102	C₂²×C₄₁	semidirect
171. C₂×C₂²⋊C₈	C₂²×C₄₁	semidirect
172. C₄₁	C₂²×#481	direct (N × H)
199. C₂×C₄⋊C₄	C₂³×C₄₁	semidirect
201. #106	C₂³×C₄₁	semidirect
202. #102	C₂³×C₄₁	semidirect
203. C₂²⋊C₈	C₂³×C₄₁	semidirect
280. C₂²	C₄₁×#481	direct (N × H)
293. C₂	C₂×C₄₁×#481	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×#121×C₄₁	2
C₂³×C₄⋊C₄×C₄₁	2
C₂×C₄₁×#481	2
C₂²×#106×C₄₁	2
C₂²×#102×C₄₁	2
C₂²×C₂²⋊C₈×C₄₁	2
C₂²×#481	41

Subgroups — C22×C41×#481 #23836705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#481 #23836705