Subgroups — C₂×C₄₁×#3259 #23837111

Order 10496.

2010 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#540	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3259	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄₁×#478	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₁×#469	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₁×#466	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#459	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
9	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#540	—	2	8	no	—
13	C₄₁×#540	C₂²	1	16	yes	complemented
14	C₂×#127×C₄₁	C₂²	1	2	yes	—
15	C₂×#125×C₄₁	C₂²	1	2	yes	—
16	C₂×#123×C₄₁	—	2	4	no	—
17	C₂×#121×C₄₁	—	2	4	no	—
18	C₂×#121×C₄₁	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₄₁	C₂²	1	4	yes	—
20	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#478	C₂²	1	32	yes	complemented
22	C₄₁×#469	C₂²	1	16	yes	complemented
23	C₄₁×#466	C₂²	1	16	yes	complemented
24	C₄₁×#459	C₂²	1	32	yes	complemented
25	C₂×#107×C₄₁	C₂²	1	2	yes	—
26	C₂×#106×C₄₁	C₂²	1	2	yes	—
27	C₄₁×#424	C₂²	1	8	yes	complemented
28	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
29	C₂×#102×C₄₁	C₂²	1	4	yes	—
30	C₂×#102×C₄₁	C₂²	1	4	yes	—
31	C₂²×D₄×C₄₁	—	2	4	no	—
32	C₂²×D₄×C₄₁	—	2	4	no	—
33	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₄₁	—	2	4	no	—
35	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	C₂×D₈×C₄₁	—	4	4	no	—
37	C₂×D₈×C₄₁	—	4	4	no	—
38	C₂×D₈×C₄₁	C₂³	1	8	yes	—
39	C₂×D₈×C₄₁	C₂³	1	8	yes	—
40	C₂×M₁₆×C₄₁	—	4	2	no	—
41	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
42	C₂²×C₈×C₄₁	—	4	2	no	—
43	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
46	#127×C₄₁	—	2	4	no	—
47	#127×C₄₁	C₂³	1	8	yes	—
48	#125×C₄₁	—	2	4	no	—
49	#125×C₄₁	C₂³	1	8	yes	—
50	#123×C₄₁	—	2	32	no	—
51	#121×C₄₁	—	2	8	no	—
52	#121×C₄₁	—	2	32	no	—
53	#121×C₄₁	C₂³	1	16	yes	—
54	C₄×D₄×C₄₁	—	2	8	no	—
55	C₄×D₄×C₄₁	C₂³	1	16	yes	—
56	#117×C₄₁	D₄	1	4	yes	—
57	#117×C₄₁	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
59	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
62	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
63	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
64	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
65	#107×C₄₁	C₂³	1	8	yes	—
66	#106×C₄₁	C₂³	1	8	yes	—
67	C₄⋊C₈×C₄₁	C₂³	1	16	yes	—
68	#102×C₄₁	C₂³	1	16	yes	—
69	#102×C₄₁	C₂³	1	16	yes	—
70	C₂⁴×C₄₁	—	2	4	no	—
71	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂×D₄×C₄₁	—	2	16	no	—
73	C₂×D₄×C₄₁	—	2	4	no	—
74	C₂×D₄×C₄₁	—	2	16	no	—
75	C₂×D₄×C₄₁	—	2	8	no	—
76	C₂×D₄×C₄₁	—	2	8	no	—
77	C₂×D₄×C₄₁	—	8	8	no	—
78	C₂×D₄×C₄₁	—	8	8	no	—
79	C₂×D₄×C₄₁	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₄₁	—	2	2	no	—
81	C₂²×C₄×C₄₁	—	2	2	no	—
82	C₂²×C₄×C₄₁	—	2	4	no	—
83	C₂²×C₄×C₄₁	—	2	16	no	—
84	C₂²×C₄×C₄₁	—	8	8	no	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
90	D₈×C₄₁	—	4	8	no	—
91	D₈×C₄₁	—	4	8	no	—
92	M₁₆×C₄₁	—	4	4	no	—
93	C₂×C₈×C₄₁	—	2	4	no	—
94	C₂×C₈×C₄₁	—	2	4	no	—
95	C₂×C₈×C₄₁	—	4	4	no	—
96	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
97	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
98	C₄⋊C₄×C₄₁	—	2	16	no	—
99	C₄⋊C₄×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
101	C₄⋊C₄×C₄₁	C₂⁴	1	8	yes	—
102	C₂²⋊C₄×C₄₁	—	2	16	no	—
103	C₂²⋊C₄×C₄₁	—	2	16	no	—
104	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
105	C₄²×C₄₁	C₂×D₄	1	4	yes	—
106	C₄²×C₄₁	C₂⁴	1	4	yes	—
107	C₂³×C₄₁	—	2	2	no	—
108	C₂³×C₄₁	—	2	4	no	—
109	C₂³×C₄₁	—	2	8	no	—
110	C₂³×C₄₁	—	4	8	no	—
111	C₂³×C₄₁	—	4	8	no	—
112	C₂³×C₄₁	—	4	8	no	—
113	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
114	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₄₁	—	2	8	no	—
116	D₄×C₄₁	—	2	8	no	—
117	D₄×C₄₁	—	4	8	no	—
118	D₄×C₄₁	—	8	16	no	—
119	D₄×C₄₁	—	8	16	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	2	8	no	—
126	C₂×C₄×C₄₁	—	4	8	no	—
127	C₂×C₄×C₄₁	—	8	16	no	—
128	C₂×C₄×C₄₁	C₂²×D₄	1	8	yes	—
129	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
132	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
133	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
134	C₈×C₄₁	—	2	4	no	—
135	C₈×C₄₁	—	2	4	no	—
136	C₂×#3259	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₁	—	2	2	no	—
138	C₂²×C₄₁	—	2	4	no	—
139	C₂²×C₄₁	—	4	4	no	—
140	C₂²×C₄₁	—	4	8	no	—
141	C₂²×C₄₁	—	4	8	no	—
142	C₂²×C₄₁	—	4	8	no	—
143	C₂²×C₄₁	—	8	16	no	—
144	C₂²×C₄₁	#537	1	2	yes	—
145	C₂²×C₄₁	#576	1	4	yes	—
146	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
147	C₄×C₄₁	—	2	2	no	—
148	C₄×C₄₁	—	2	8	no	—
149	C₄×C₄₁	—	4	8	no	—
150	C₄×C₄₁	#537	1	2	yes	—
151	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
152	C₄×C₄₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×#540	C₂×C₄₁	1	2	yes	complemented
154	#3259	C₂×C₄₁	1	16	yes	direct product factor
155	C₂×#478	C₂×C₄₁	1	4	yes	complemented
156	C₂×#469	C₂×C₄₁	1	2	yes	complemented
157	C₂×#466	C₂×C₄₁	1	2	yes	complemented
158	C₂×#459	C₂×C₄₁	1	4	yes	complemented
159	C₂×#424	C₂×C₄₁	1	1	yes	characteristic, complemented
160	C₂×C₄₁	—	4	2	no	—
161	C₂×C₄₁	—	8	8	no	—
162	C₂×C₄₁	#3259	1	4	yes	direct product factor
163	C₂×C₄₁	C₂×#537	1	1	yes	characteristic
164	C₂×C₄₁	C₂×#576	1	2	yes	—
165	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
166	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
167	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
168	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
169	#540	—	2	8	no	—
170	#540	C₂²×C₄₁	1	16	yes	complemented
171	C₂×#127	C₂²×C₄₁	1	2	yes	—
172	C₂×#125	C₂²×C₄₁	1	2	yes	—
173	C₂×#123	—	2	4	no	—
174	C₂×#121	—	2	4	no	—
175	C₂×#121	C₂²×C₄₁	1	4	yes	—
176	C₂×C₄×D₄	C₂²×C₄₁	1	4	yes	—
177	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
178	#478	C₂²×C₄₁	1	32	yes	complemented
179	#469	C₂²×C₄₁	1	16	yes	complemented
180	#466	C₂²×C₄₁	1	16	yes	complemented
181	#459	C₂²×C₄₁	1	32	yes	complemented
182	C₂×#107	C₂²×C₄₁	1	2	yes	—
183	C₂×#106	C₂²×C₄₁	1	2	yes	—
184	#424	C₂²×C₄₁	1	8	yes	complemented
185	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
186	C₂×#102	C₂²×C₄₁	1	4	yes	—
187	C₂×#102	C₂²×C₄₁	1	4	yes	—
188	C₄₁	C₂×#3259	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×D₄	—	2	4	no	—
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
192	C₂³×C₄	—	2	4	no	—
193	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
194	C₂×D₈	—	4	4	no	—
195	C₂×D₈	—	4	4	no	—
196	C₂×D₈	C₂³×C₄₁	1	8	yes	—
197	C₂×D₈	C₂³×C₄₁	1	8	yes	—
198	C₂×M₁₆	—	4	2	no	—
199	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
200	C₂²×C₈	—	4	2	no	—
201	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
203	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
204	#127	—	2	4	no	—
205	#127	C₂³×C₄₁	1	8	yes	—
206	#125	—	2	4	no	—
207	#125	C₂³×C₄₁	1	8	yes	—
208	#123	—	2	32	no	—
209	#121	—	2	8	no	—
210	#121	—	2	32	no	—
211	#121	C₂³×C₄₁	1	16	yes	—
212	C₄×D₄	—	2	8	no	—
213	C₄×D₄	C₂³×C₄₁	1	16	yes	—
214	#117	D₄×C₄₁	1	4	yes	—
215	#117	C₂³×C₄₁	1	4	yes	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₂²⋊C₄	—	2	4	no	—
221	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
222	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
223	#107	C₂³×C₄₁	1	8	yes	—
224	#106	C₂³×C₄₁	1	8	yes	—
225	C₄⋊C₈	C₂³×C₄₁	1	16	yes	—
226	#102	C₂³×C₄₁	1	16	yes	—
227	#102	C₂³×C₄₁	1	16	yes	—
228	C₂⁴	—	2	4	no	—
229	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
230	C₂×D₄	—	2	16	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	8	8	no	—
236	C₂×D₄	—	8	8	no	—
237	C₂×D₄	C₂⁴×C₄₁	1	8	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	16	no	—
242	C₂²×C₄	—	8	8	no	—
243	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
248	D₈	—	4	8	no	—
249	D₈	—	4	8	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
255	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
256	C₄⋊C₄	—	2	16	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
259	C₄⋊C₄	C₂⁴×C₄₁	1	8	yes	—
260	C₂²⋊C₄	—	2	16	no	—
261	C₂²⋊C₄	—	2	16	no	—
262	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
263	C₄²	C₂×D₄×C₄₁	1	4	yes	—
264	C₄²	C₂⁴×C₄₁	1	4	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	4	8	no	—
269	C₂³	—	4	8	no	—
270	C₂³	—	4	8	no	—
271	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
272	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
273	D₄	—	2	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	4	8	no	—
276	D₄	—	8	16	no	—
277	D₄	—	8	16	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	8	16	no	—
286	C₂×C₄	C₂²×D₄×C₄₁	1	8	yes	—
287	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
288	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
289	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
290	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
291	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	—	8	16	no	—
301	C₂²	C₄₁×#537	1	2	yes	—
302	C₂²	C₄₁×#576	1	4	yes	—
303	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
304	C₄	—	2	8	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	8	no	—
307	C₄	C₄₁×#537	1	2	yes	—
308	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
309	C₄	C₂×2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	8	8	no	—
312	C₂	C₄₁×#3259	1	4	yes	direct product factor
313	C₂	C₂×C₄₁×#537	1	1	yes	characteristic
314	C₂	C₂×C₄₁×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#540	C₂	semidirect
2. C₄₁×#3259	C₂	direct (N × H)
3. C₂×C₄₁×#478	C₂	semidirect
4. C₂×C₄₁×#469	C₂	semidirect
5. C₂×C₄₁×#466	C₂	semidirect
6. C₂×C₄₁×#459	C₂	semidirect
7. C₂×C₄₁×#424	C₂	semidirect
13. C₄₁×#540	C₂²	semidirect
21. C₄₁×#478	C₂²	semidirect
22. C₄₁×#469	C₂²	semidirect
23. C₄₁×#466	C₂²	semidirect
24. C₄₁×#459	C₂²	semidirect
27. C₄₁×#424	C₂²	semidirect
136. C₂×#3259	C₄₁	direct (N × H)
153. C₂×#540	C₂×C₄₁	semidirect
154. #3259	C₂×C₄₁	direct (N × H)
155. C₂×#478	C₂×C₄₁	semidirect
156. C₂×#469	C₂×C₄₁	semidirect
157. C₂×#466	C₂×C₄₁	semidirect
158. C₂×#459	C₂×C₄₁	semidirect
159. C₂×#424	C₂×C₄₁	semidirect
162. C₂×C₄₁	#3259	direct (N × H)
170. #540	C₂²×C₄₁	semidirect
178. #478	C₂²×C₄₁	semidirect
179. #469	C₂²×C₄₁	semidirect
180. #466	C₂²×C₄₁	semidirect
181. #459	C₂²×C₄₁	semidirect
184. #424	C₂²×C₄₁	semidirect
188. C₄₁	C₂×#3259	direct (N × H)
312. C₂	C₄₁×#3259	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×#540	2
C₄₁×#3259	2
C₂×C₄₁×#478	2
C₂×C₄₁×#469	2
C₂×C₄₁×#466	2
C₂×C₄₁×#459	2
C₂×C₄₁×#424	2
C₂×#3259	41

Subgroups — C2×C41×#3259 #23837111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3259 #23837111