Subgroups — C₂×C₄₁×#2185 #23809713

Order 10496.

1178 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×SD₃₂×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#466	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2185	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₄₁	—	2	1	no	—
8	C₂²×Q₁₆×C₄₁	—	2	8	no	—
9	C₂²×Q₁₆×C₄₁	—	2	1	no	—
10	C₂²×Q₁₆×C₄₁	—	2	2	no	—
11	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	complemented
13	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#121×C₄₁	—	2	1	no	—
16	C₂×SD₃₂×C₄₁	—	2	8	no	—
17	C₂×SD₃₂×C₄₁	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#466	—	2	2	no	—
20	C₄₁×#466	C₂²	1	4	yes	—
21	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#102×C₄₁	—	2	1	no	—
23	C₄₁×#358	C₂²	1	8	yes	complemented
24	C₄₁×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₄₁	—	2	8	no	—
26	C₂²×Q₈×C₄₁	—	2	2	no	—
27	C₂²×Q₈×C₄₁	—	2	4	no	—
28	C₂²×Q₈×C₄₁	—	4	4	no	—
29	C₂²×Q₈×C₄₁	—	4	2	no	—
30	C₂²×Q₈×C₄₁	—	4	1	no	—
31	C₂²×D₄×C₄₁	—	2	1	no	—
32	C₂²×D₄×C₄₁	—	4	1	no	—
33	C₂³×C₄×C₄₁	—	4	1	no	—
34	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₄₁	—	2	8	no	—
36	C₂×Q₁₆×C₄₁	—	2	16	no	—
37	C₂×Q₁₆×C₄₁	—	2	16	no	—
38	C₂×Q₁₆×C₄₁	—	2	2	no	—
39	C₂×Q₁₆×C₄₁	—	2	4	no	—
40	C₂×Q₁₆×C₄₁	—	2	4	no	—
41	C₂×Q₁₆×C₄₁	D₄	1	8	yes	complemented
42	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
43	C₂×D₈×C₄₁	—	2	4	no	—
44	C₂×D₈×C₄₁	—	2	1	no	—
45	C₂×D₈×C₄₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₄₁	—	2	4	no	—
47	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₁	D₄	1	2	yes	—
49	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
51	#121×C₄₁	—	2	4	no	—
52	#121×C₄₁	—	4	2	no	—
53	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₄₁	—	4	1	no	—
55	SD₃₂×C₄₁	—	2	16	no	—
56	C₂×C₁₆×C₄₁	—	2	4	no	—
57	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
58	#107×C₄₁	—	2	1	no	—
59	#107×C₄₁	C₂³	1	2	yes	—
60	#102×C₄₁	—	2	4	no	—
61	C₂⁴×C₄₁	—	4	1	no	—
62	C₂⁴×C₄₁	D₈	1	1	yes	characteristic
63	C₂×Q₈×C₄₁	—	2	16	no	—
64	C₂×Q₈×C₄₁	—	2	8	no	—
65	C₂×Q₈×C₄₁	—	2	4	no	—
66	C₂×Q₈×C₄₁	—	4	8	no	—
67	C₂×Q₈×C₄₁	—	4	8	no	—
68	C₂×Q₈×C₄₁	—	4	16	no	—
69	C₂×Q₈×C₄₁	—	4	4	no	—
70	C₂×Q₈×C₄₁	—	4	2	no	—
71	C₂×Q₈×C₄₁	—	4	4	no	—
72	C₂×D₄×C₄₁	—	2	2	no	—
73	C₂×D₄×C₄₁	—	4	4	no	—
74	C₂×D₄×C₄₁	—	4	2	no	—
75	C₂×D₄×C₄₁	—	4	1	no	—
76	C₂×D₄×C₄₁	—	4	2	no	—
77	C₂×D₄×C₄₁	—	8	4	no	—
78	C₂²×C₄×C₄₁	—	2	2	no	—
79	C₂²×C₄×C₄₁	—	2	2	no	—
80	C₂²×C₄×C₄₁	—	4	8	no	—
81	C₂²×C₄×C₄₁	—	4	4	no	—
82	C₂²×C₄×C₄₁	—	4	2	no	—
83	C₂²×C₄×C₄₁	—	4	1	no	—
84	C₂²×C₄×C₄₁	D₈	1	2	yes	—
85	C₂²×C₄×C₄₁	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₄₁	—	2	16	no	—
89	Q₁₆×C₄₁	—	2	8	no	—
90	Q₁₆×C₄₁	—	2	4	no	—
91	Q₁₆×C₄₁	C₂×D₄	1	8	yes	complemented
92	D₈×C₄₁	—	2	4	no	—
93	D₈×C₄₁	—	4	2	no	—
94	C₂×C₈×C₄₁	—	2	2	no	—
95	C₂×C₈×C₄₁	—	2	4	no	—
96	C₂×C₈×C₄₁	—	2	4	no	—
97	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₄⋊C₄×C₄₁	—	4	1	no	—
103	C₂²⋊C₄×C₄₁	—	4	4	no	—
104	C₁₆×C₄₁	—	2	4	no	—
105	C₂³×C₄₁	—	2	2	no	—
106	C₂³×C₄₁	—	2	2	no	—
107	C₂³×C₄₁	—	4	2	no	—
108	C₂³×C₄₁	—	8	2	no	—
109	C₂³×C₄₁	—	8	2	no	—
110	C₂³×C₄₁	—	8	2	no	—
111	C₂³×C₄₁	SD₃₂	1	4	yes	complemented
112	C₂³×C₄₁	C₂×D₈	1	2	yes	—
113	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₄₁	—	2	8	no	—
115	Q₈×C₄₁	—	4	16	no	—
116	Q₈×C₄₁	—	4	4	no	—
117	Q₈×C₄₁	—	4	8	no	—
118	D₄×C₄₁	—	4	2	no	—
119	D₄×C₄₁	—	4	2	no	—
120	D₄×C₄₁	—	8	8	no	—
121	D₄×C₄₁	—	8	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	2	4	no	—
126	C₂×C₄×C₄₁	—	4	16	no	—
127	C₂×C₄×C₄₁	—	4	4	no	—
128	C₂×C₄×C₄₁	—	4	8	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	8	2	no	—
131	C₂×C₄×C₄₁	#120	1	2	yes	—
132	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
136	C₈×C₄₁	—	2	2	no	—
137	C₈×C₄₁	#120	1	2	yes	—
138	C₈×C₄₁	C₂²×D₄	1	2	yes	—
139	C₂×#2185	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₁	—	2	2	no	—
141	C₂²×C₄₁	—	2	8	no	—
142	C₂²×C₄₁	—	2	2	no	—
143	C₂²×C₄₁	—	8	2	no	—
144	C₂²×C₄₁	—	8	2	no	—
145	C₂²×C₄₁	—	8	2	no	—
146	C₂²×C₄₁	—	16	4	no	—
147	C₂²×C₄₁	#447	1	2	yes	—
148	C₂²×C₄₁	C₂×SD₃₂	1	2	yes	—
149	C₂²×C₄₁	C₂×SD₃₂	1	4	yes	complemented
150	C₂²×C₄₁	#510	1	2	yes	—
151	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₄₁	—	2	2	no	—
153	C₄×C₄₁	—	4	8	no	—
154	C₄×C₄₁	—	8	2	no	—
155	C₄×C₄₁	#447	1	2	yes	—
156	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
157	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂³×Q₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
159	C₂²×SD₃₂	C₂×C₄₁	1	2	yes	complemented
160	C₂×#466	C₂×C₄₁	1	1	yes	characteristic
161	#2185	C₂×C₄₁	1	8	yes	direct product factor
162	C₂×#358	C₂×C₄₁	1	2	yes	complemented
163	C₂×C₂²⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
164	C₂×C₄₁	—	2	4	no	—
165	C₂×C₄₁	—	16	2	no	—
166	C₂×C₄₁	#2185	1	4	yes	direct product factor
167	C₂×C₄₁	C₂×#447	1	1	yes	characteristic
168	C₂×C₄₁	C₂²×SD₃₂	1	1	yes	characteristic
169	C₂×C₄₁	C₂×#510	1	1	yes	characteristic
170	C₂³×Q₈	—	2	1	no	—
171	C₂²×Q₁₆	—	2	8	no	—
172	C₂²×Q₁₆	—	2	1	no	—
173	C₂²×Q₁₆	—	2	2	no	—
174	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
175	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	complemented
176	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
177	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×#121	—	2	1	no	—
179	C₂×SD₃₂	—	2	8	no	—
180	C₂×SD₃₂	C₂²×C₄₁	1	8	yes	complemented
181	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
182	#466	—	2	2	no	—
183	#466	C₂²×C₄₁	1	4	yes	—
184	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
185	C₂×#102	—	2	1	no	—
186	#358	C₂²×C₄₁	1	8	yes	complemented
187	C₂²⋊C₁₆	C₂²×C₄₁	1	4	yes	complemented
188	C₄₁	C₂×#2185	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×Q₈	—	2	8	no	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	4	4	no	—
193	C₂²×Q₈	—	4	1	no	—
194	C₂²×Q₈	—	4	2	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	4	1	no	—
197	C₂³×C₄	—	4	1	no	—
198	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
199	C₂×Q₁₆	—	2	8	no	—
200	C₂×Q₁₆	—	2	16	no	—
201	C₂×Q₁₆	—	2	16	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	D₄×C₄₁	1	8	yes	complemented
206	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
207	C₂×D₈	—	2	4	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	C₂³×C₄₁	1	2	yes	—
210	C₂²×C₈	—	2	4	no	—
211	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
212	C₂²×C₈	D₄×C₄₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
214	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
215	#121	—	2	4	no	—
216	#121	—	4	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	4	1	no	—
219	SD₃₂	—	2	16	no	—
220	C₂×C₁₆	—	2	4	no	—
221	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
222	#107	—	2	1	no	—
223	#107	C₂³×C₄₁	1	2	yes	—
224	#102	—	2	4	no	—
225	C₂⁴	—	4	1	no	—
226	C₂⁴	D₈×C₄₁	1	1	yes	characteristic
227	C₂×Q₈	—	2	16	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	4	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×Q₈	—	4	16	no	—
233	C₂×Q₈	—	4	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	8	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	D₈×C₄₁	1	1	yes	characteristic
249	C₂²×C₄	D₈×C₄₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
252	Q₁₆	—	2	16	no	—
253	Q₁₆	—	2	8	no	—
254	Q₁₆	—	2	4	no	—
255	Q₁₆	C₂×D₄×C₄₁	1	8	yes	complemented
256	D₈	—	2	4	no	—
257	D₈	—	4	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
264	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₁₆	—	2	4	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	—	8	2	no	—
274	C₂³	—	8	2	no	—
275	C₂³	SD₃₂×C₄₁	1	4	yes	complemented
276	C₂³	C₂×D₈×C₄₁	1	2	yes	—
277	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	4	16	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	8	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	8	no	—
285	D₄	—	8	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	8	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	#120×C₄₁	1	2	yes	—
296	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
297	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
298	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
299	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
300	C₈	—	2	2	no	—
301	C₈	#120×C₄₁	1	2	yes	—
302	C₈	C₂²×D₄×C₄₁	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	16	4	no	—
310	C₂²	C₄₁×#447	1	2	yes	—
311	C₂²	C₂×SD₃₂×C₄₁	1	2	yes	—
312	C₂²	C₂×SD₃₂×C₄₁	1	4	yes	complemented
313	C₂²	C₄₁×#510	1	2	yes	—
314	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	C₄₁×#447	1	2	yes	—
319	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
320	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	—	16	2	no	—
323	C₂	C₄₁×#2185	1	4	yes	direct product factor
324	C₂	C₂×C₄₁×#447	1	1	yes	characteristic
325	C₂	C₂²×SD₃₂×C₄₁	1	1	yes	characteristic
326	C₂	C₂×C₄₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₄₁	C₂	semidirect
2. C₂²×SD₃₂×C₄₁	C₂	semidirect
4. C₄₁×#2185	C₂	direct (N × H)
5. C₂×C₄₁×#358	C₂	semidirect
6. C₂×C₄₁×C₂²⋊C₁₆	C₂	semidirect
12. C₂²×Q₁₆×C₄₁	C₂²	semidirect
17. C₂×SD₃₂×C₄₁	C₂²	semidirect
23. C₄₁×#358	C₂²	semidirect
24. C₄₁×C₂²⋊C₁₆	C₂²	semidirect
41. C₂×Q₁₆×C₄₁	D₄	semidirect
91. Q₁₆×C₄₁	C₂×D₄	semidirect
111. C₂³×C₄₁	SD₃₂	semidirect
139. C₂×#2185	C₄₁	direct (N × H)
149. C₂²×C₄₁	C₂×SD₃₂	semidirect
158. C₂³×Q₁₆	C₂×C₄₁	semidirect
159. C₂²×SD₃₂	C₂×C₄₁	semidirect
161. #2185	C₂×C₄₁	direct (N × H)
162. C₂×#358	C₂×C₄₁	semidirect
163. C₂×C₂²⋊C₁₆	C₂×C₄₁	semidirect
166. C₂×C₄₁	#2185	direct (N × H)
175. C₂²×Q₁₆	C₂²×C₄₁	semidirect
180. C₂×SD₃₂	C₂²×C₄₁	semidirect
186. #358	C₂²×C₄₁	semidirect
187. C₂²⋊C₁₆	C₂²×C₄₁	semidirect
188. C₄₁	C₂×#2185	direct (N × H)
205. C₂×Q₁₆	D₄×C₄₁	semidirect
255. Q₁₆	C₂×D₄×C₄₁	semidirect
275. C₂³	SD₃₂×C₄₁	semidirect
312. C₂²	C₂×SD₃₂×C₄₁	semidirect
323. C₂	C₄₁×#2185	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×Q₁₆×C₄₁	2
C₂²×SD₃₂×C₄₁	2
C₂×C₄₁×#466	2
C₄₁×#2185	2
C₂×C₄₁×#358	2
C₂×C₄₁×C₂²⋊C₁₆	2
C₂×#2185	41

Subgroups — C2×C41×#2185 #23809713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2185 #23809713