Subgroups — C₂×C₄₁×#1669 #23795301

Order 10496.

790 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#474	C₂	1	2	yes	maximal
3	C₂×C₈×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1669	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#332	C₂	1	2	yes	maximal, complemented
7	C₂²×Q₁₆×C₄₁	—	2	1	no	—
8	C₂²×Q₁₆×C₄₁	—	2	1	no	—
9	C₂³×C₈×C₄₁	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#494	—	2	2	no	—
13	C₄₁×#494	C₂²	1	4	yes	complemented
14	C₄₁×#474	—	2	4	no	—
15	C₄₁×#474	C₂²	1	8	yes	—
16	C₈×D₄×C₄₁	—	2	2	no	—
17	C₈×D₄×C₄₁	C₂²	1	4	yes	—
18	C₂×#107×C₄₁	—	2	2	no	—
19	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#102×C₄₁	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
23	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#335	C₂²	1	4	yes	complemented
25	C₄₁×#332	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₄₁	—	2	2	no	—
27	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	—	2	1	no	—
29	C₂×Q₁₆×C₄₁	—	2	4	no	—
30	C₂×Q₁₆×C₄₁	—	2	4	no	—
31	C₂×Q₁₆×C₄₁	—	4	4	no	—
32	C₂×Q₁₆×C₄₁	—	4	4	no	—
33	C₂²×C₈×C₄₁	—	2	1	no	—
34	C₂²×C₈×C₄₁	—	2	2	no	—
35	C₂²×C₈×C₄₁	—	2	4	no	—
36	C₂²×C₈×C₄₁	—	4	4	no	—
37	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
39	C₄⋊Q₈×C₄₁	—	2	2	no	—
40	C₄⋊Q₈×C₄₁	—	2	2	no	—
41	C₄⋊Q₈×C₄₁	—	2	2	no	—
42	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	—
43	C₄×D₄×C₄₁	—	2	1	no	—
44	C₄×D₄×C₄₁	—	2	1	no	—
45	C₄×D₄×C₄₁	D₄	1	2	yes	—
46	C₄×D₄×C₄₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	#107×C₄₁	—	2	8	no	—
53	C₄⋊C₈×C₄₁	—	2	1	no	—
54	C₄⋊C₈×C₄₁	—	2	2	no	—
55	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
56	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
57	#102×C₄₁	—	2	8	no	—
58	C₂²⋊C₈×C₄₁	—	2	4	no	—
59	C₄×C₈×C₄₁	D₄	1	2	yes	—
60	C₄×C₈×C₄₁	C₂³	1	2	yes	—
61	C₂⁴×C₄₁	—	2	1	no	—
62	C₂×Q₈×C₄₁	—	2	4	no	—
63	C₂×Q₈×C₄₁	—	4	2	no	—
64	C₂×Q₈×C₄₁	—	4	4	no	—
65	C₂×Q₈×C₄₁	—	4	4	no	—
66	C₂×D₄×C₄₁	—	2	1	no	—
67	C₂×D₄×C₄₁	—	2	2	no	—
68	C₂×D₄×C₄₁	Q₁₆	1	4	yes	complemented
69	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₁	—	2	1	no	—
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	4	2	no	—
74	C₂²×C₄×C₄₁	—	4	2	no	—
75	C₂²×C₄×C₄₁	—	4	2	no	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
79	Q₁₆×C₄₁	—	4	8	no	—
80	Q₁₆×C₄₁	—	4	8	no	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	—	2	4	no	—
84	C₂×C₈×C₄₁	—	4	2	no	—
85	C₂×C₈×C₄₁	—	4	4	no	—
86	C₂×C₈×C₄₁	—	4	8	no	—
87	C₂×C₈×C₄₁	D₈	1	4	yes	—
88	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₄₁	—	2	1	no	—
91	C₄⋊C₄×C₄₁	—	2	4	no	—
92	C₄⋊C₄×C₄₁	—	2	4	no	—
93	C₄⋊C₄×C₄₁	—	4	2	no	—
94	C₄⋊C₄×C₄₁	—	4	2	no	—
95	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
96	C₂²⋊C₄×C₄₁	—	2	2	no	—
97	C₂²⋊C₄×C₄₁	—	2	2	no	—
98	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₄₁	#41	1	1	yes	characteristic
101	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
102	C₂³×C₄₁	—	2	2	no	—
103	C₂³×C₄₁	—	2	4	no	—
104	C₂³×C₄₁	—	4	2	no	—
105	C₂³×C₄₁	—	4	2	no	—
106	C₂³×C₄₁	#121	1	1	yes	characteristic
107	Q₈×C₄₁	—	4	4	no	—
108	Q₈×C₄₁	—	4	4	no	—
109	Q₈×C₄₁	—	8	4	no	—
110	D₄×C₄₁	—	2	2	no	—
111	D₄×C₄₁	—	2	4	no	—
112	D₄×C₄₁	C₂×Q₁₆	1	4	yes	complemented
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	4	2	no	—
116	C₂×C₄×C₄₁	—	4	2	no	—
117	C₂×C₄×C₄₁	—	4	2	no	—
118	C₂×C₄×C₄₁	—	4	4	no	—
119	C₂×C₄×C₄₁	—	4	4	no	—
120	C₂×C₄×C₄₁	—	8	4	no	—
121	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
126	C₂×C₄×C₄₁	#135	1	2	yes	—
127	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
131	C₈×C₄₁	—	2	2	no	—
132	C₈×C₄₁	—	4	2	no	—
133	C₈×C₄₁	—	4	4	no	—
134	C₈×C₄₁	C₂×D₈	1	4	yes	—
135	C₂×#1669	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₁	—	2	4	no	—
137	C₂²×C₄₁	—	4	8	no	—
138	C₂²×C₄₁	—	4	2	no	—
139	C₂²×C₄₁	—	4	2	no	—
140	C₂²×C₄₁	#459	1	2	yes	—
141	C₂²×C₄₁	#467	1	2	yes	—
142	C₂²×C₄₁	#473	1	2	yes	—
143	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
144	C₄×C₄₁	—	2	1	no	—
145	C₄×C₄₁	—	2	1	no	—
146	C₄×C₄₁	—	4	2	no	—
147	C₄×C₄₁	—	8	4	no	—
148	C₄×C₄₁	#459	1	2	yes	—
149	C₄×C₄₁	#467	1	2	yes	—
150	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
152	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
153	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×#494	C₂×C₄₁	1	1	yes	characteristic, complemented
155	C₂×#474	C₂×C₄₁	1	2	yes	—
156	C₂×C₈×D₄	C₂×C₄₁	1	1	yes	characteristic
157	#1669	C₂×C₄₁	1	8	yes	direct product factor
158	C₂×#335	C₂×C₄₁	1	1	yes	characteristic, complemented
159	C₂×#332	C₂×C₄₁	1	2	yes	complemented
160	C₂×C₄₁	—	4	4	no	—
161	C₂×C₄₁	#1669	1	4	yes	direct product factor
162	C₂×C₄₁	C₂×#459	1	1	yes	characteristic
163	C₂×C₄₁	C₂×#467	1	1	yes	characteristic
164	C₂×C₄₁	C₂×#473	1	1	yes	characteristic
165	C₂²×Q₁₆	—	2	1	no	—
166	C₂²×Q₁₆	—	2	1	no	—
167	C₂³×C₈	—	2	1	no	—
168	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
169	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
170	#494	—	2	2	no	—
171	#494	C₂²×C₄₁	1	4	yes	complemented
172	#474	—	2	4	no	—
173	#474	C₂²×C₄₁	1	8	yes	—
174	C₈×D₄	—	2	2	no	—
175	C₈×D₄	C₂²×C₄₁	1	4	yes	—
176	C₂×#107	—	2	2	no	—
177	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
179	C₂×#102	—	2	2	no	—
180	C₂×C₂²⋊C₈	—	2	1	no	—
181	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
182	#335	C₂²×C₄₁	1	4	yes	complemented
183	#332	C₂²×C₄₁	1	8	yes	complemented
184	C₄₁	C₂×#1669	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
187	C₂³×C₄	—	2	1	no	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	2	4	no	—
190	C₂×Q₁₆	—	4	4	no	—
191	C₂×Q₁₆	—	4	4	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	—	4	4	no	—
196	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	—
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	D₄×C₄₁	1	2	yes	—
205	C₄×D₄	C₂³×C₄₁	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
211	#107	—	2	8	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
215	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
216	#102	—	2	8	no	—
217	C₂²⋊C₈	—	2	4	no	—
218	C₄×C₈	D₄×C₄₁	1	2	yes	—
219	C₄×C₈	C₂³×C₄₁	1	2	yes	—
220	C₂⁴	—	2	1	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	Q₁₆×C₄₁	1	4	yes	complemented
228	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
237	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
238	Q₁₆	—	4	8	no	—
239	Q₁₆	—	4	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	4	8	no	—
246	C₂×C₈	D₈×C₄₁	1	4	yes	—
247	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
259	C₄²	#41×C₄₁	1	1	yes	characteristic
260	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	#121×C₄₁	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	8	4	no	—
269	D₄	—	2	2	no	—
270	D₄	—	2	4	no	—
271	D₄	C₂×Q₁₆×C₄₁	1	4	yes	complemented
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
281	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
282	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
284	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
285	C₂×C₄	#135×C₄₁	1	2	yes	—
286	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
287	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	4	no	—
293	C₈	C₂×D₈×C₄₁	1	4	yes	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	8	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₄₁×#459	1	2	yes	—
299	C₂²	C₄₁×#467	1	2	yes	—
300	C₂²	C₄₁×#473	1	2	yes	—
301	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	8	4	no	—
306	C₄	C₄₁×#459	1	2	yes	—
307	C₄	C₄₁×#467	1	2	yes	—
308	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
309	C₄	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
310	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
311	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
312	C₂	—	4	4	no	—
313	C₂	C₄₁×#1669	1	4	yes	direct product factor
314	C₂	C₂×C₄₁×#459	1	1	yes	characteristic
315	C₂	C₂×C₄₁×#467	1	1	yes	characteristic
316	C₂	C₂×C₄₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₂²×D₄×C₄₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#494	C₂	semidirect
4. C₄₁×#1669	C₂	direct (N × H)
5. C₂×C₄₁×#335	C₂	semidirect
6. C₂×C₄₁×#332	C₂	semidirect
13. C₄₁×#494	C₂²	semidirect
24. C₄₁×#335	C₂²	semidirect
25. C₄₁×#332	C₂²	semidirect
68. C₂×D₄×C₄₁	Q₁₆	semidirect
112. D₄×C₄₁	C₂×Q₁₆	semidirect
135. C₂×#1669	C₄₁	direct (N × H)
154. C₂×#494	C₂×C₄₁	semidirect
157. #1669	C₂×C₄₁	direct (N × H)
158. C₂×#335	C₂×C₄₁	semidirect
159. C₂×#332	C₂×C₄₁	semidirect
161. C₂×C₄₁	#1669	direct (N × H)
171. #494	C₂²×C₄₁	semidirect
182. #335	C₂²×C₄₁	semidirect
183. #332	C₂²×C₄₁	semidirect
184. C₄₁	C₂×#1669	direct (N × H)
227. C₂×D₄	Q₁₆×C₄₁	semidirect
271. D₄	C₂×Q₁₆×C₄₁	semidirect
313. C₂	C₄₁×#1669	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×#494	2
C₂×C₄₁×#474	2
C₂×C₈×D₄×C₄₁	2
C₄₁×#1669	2
C₂×C₄₁×#335	2
C₂×C₄₁×#332	2
C₂×#1669	41

Subgroups — C2×C41×#1669 #23795301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1669 #23795301