Subgroups — C₂×C₄₁×#1394 #23788785

Order 10496.

818 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#101×C₄₁	C₂	1	2	yes	maximal
2	C₂×C₄₁×#391	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1394	C₂	1	4	yes	maximal, direct product factor
4	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₄₁	—	2	2	no	—
6	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₄₁	C₄	1	1	yes	characteristic
8	C₂×#101×C₄₁	—	2	2	no	—
9	C₂×#101×C₄₁	—	2	4	no	—
10	C₂×#101×C₄₁	C₄	1	8	yes	complemented
11	C₂×#101×C₄₁	C₂²	1	4	yes	—
12	C₄₁×#391	—	2	2	no	—
13	C₄₁×#391	C₄	1	2	yes	—
14	C₄₁×#391	C₂²	1	2	yes	—
15	C₂×#95×C₄₁	—	2	2	no	—
16	C₂²×Q₈×C₄₁	—	4	2	no	—
17	C₂²×Q₈×C₄₁	—	4	1	no	—
18	C₂²×Q₈×C₄₁	—	4	1	no	—
19	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
20	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
22	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
23	C₂²×Q₈×C₄₁	Q₈	1	2	yes	—
24	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	—	2	1	no	—
26	C₂³×C₄×C₄₁	—	2	1	no	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	D₄	1	2	yes	—
29	C₂×M₁₆×C₄₁	—	2	8	no	—
30	C₂×M₁₆×C₄₁	—	2	4	no	—
31	C₂×M₁₆×C₄₁	—	4	4	no	—
32	C₂×M₁₆×C₄₁	—	4	2	no	—
33	C₂²×C₈×C₄₁	—	4	2	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
36	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
37	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₁	—	2	1	no	—
39	C₂×C₄²×C₄₁	—	4	2	no	—
40	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
41	#101×C₄₁	—	2	8	no	—
42	#101×C₄₁	—	2	4	no	—
43	#101×C₄₁	C₂×C₄	1	8	yes	complemented
44	#95×C₄₁	—	2	4	no	—
45	#95×C₄₁	—	2	4	no	—
46	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Q₈×C₄₁	—	4	4	no	—
48	C₂×Q₈×C₄₁	—	4	4	no	—
49	C₂×Q₈×C₄₁	—	4	1	no	—
50	C₂×Q₈×C₄₁	—	4	1	no	—
51	C₂×Q₈×C₄₁	—	4	2	no	—
52	C₂×Q₈×C₄₁	—	4	2	no	—
53	C₂×Q₈×C₄₁	—	4	2	no	—
54	C₂×Q₈×C₄₁	—	4	2	no	—
55	C₂×Q₈×C₄₁	—	4	2	no	—
56	C₂×Q₈×C₄₁	—	4	2	no	—
57	C₂×Q₈×C₄₁	—	4	1	no	—
58	C₂×Q₈×C₄₁	—	4	1	no	—
59	C₂×Q₈×C₄₁	C₄²	1	2	yes	—
60	C₂×Q₈×C₄₁	C₂²⋊C₄	1	4	yes	—
61	C₂×Q₈×C₄₁	C₄⋊C₄	1	4	yes	—
62	C₂×Q₈×C₄₁	C₄⋊C₄	1	2	yes	—
63	C₂×Q₈×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	—	2	2	no	—
67	C₂²×C₄×C₄₁	—	2	1	no	—
68	C₂²×C₄×C₄₁	—	2	4	no	—
69	C₂²×C₄×C₄₁	—	2	2	no	—
70	C₂²×C₄×C₄₁	—	2	1	no	—
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	2	4	no	—
73	C₂²×C₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	4	2	no	—
76	C₂²×C₄×C₄₁	—	4	2	no	—
77	C₂²×C₄×C₄₁	—	4	4	no	—
78	C₂²×C₄×C₄₁	—	4	1	no	—
79	C₂²×C₄×C₄₁	—	4	1	no	—
80	C₂²×C₄×C₄₁	C₄²	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
84	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
85	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
86	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
88	M₁₆×C₄₁	—	2	8	no	—
89	M₁₆×C₄₁	—	4	8	no	—
90	M₁₆×C₄₁	—	4	4	no	—
91	C₂×C₈×C₄₁	—	4	8	no	—
92	C₂×C₈×C₄₁	—	4	4	no	—
93	C₄⋊C₄×C₄₁	—	4	2	no	—
94	C₄⋊C₄×C₄₁	—	4	2	no	—
95	C₄²×C₄₁	—	4	2	no	—
96	C₄²×C₄₁	—	4	2	no	—
97	C₂³×C₄₁	—	2	2	no	—
98	C₂³×C₄₁	—	2	2	no	—
99	C₂³×C₄₁	#95	1	2	yes	—
100	C₂³×C₄₁	#99	1	1	yes	characteristic
101	C₂³×C₄₁	#99	1	2	yes	—
102	C₂³×C₄₁	#99	1	1	yes	characteristic
103	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₄₁	—	4	2	no	—
105	Q₈×C₄₁	—	4	4	no	—
106	Q₈×C₄₁	—	4	2	no	—
107	Q₈×C₄₁	—	4	4	no	—
108	Q₈×C₄₁	—	4	2	no	—
109	Q₈×C₄₁	—	4	1	no	—
110	Q₈×C₄₁	—	4	1	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	2	4	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	2	1	no	—
120	C₂×C₄×C₄₁	—	2	1	no	—
121	C₂×C₄×C₄₁	—	2	1	no	—
122	C₂×C₄×C₄₁	—	2	1	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	2	2	no	—
125	C₂×C₄×C₄₁	—	2	2	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	2	4	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	8	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	4	no	—
132	C₂×C₄×C₄₁	—	4	2	no	—
133	C₂×C₄×C₄₁	—	8	2	no	—
134	C₂×C₄×C₄₁	—	8	2	no	—
135	C₂×C₄×C₄₁	#95	1	2	yes	—
136	C₂×C₄×C₄₁	#95	1	4	yes	—
137	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₄₁	—	4	8	no	—
142	C₂×#1394	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₄₁	—	2	1	no	—
144	C₂²×C₄₁	—	2	1	no	—
145	C₂²×C₄₁	—	2	1	no	—
146	C₂²×C₄₁	—	2	1	no	—
147	C₂²×C₄₁	—	2	4	no	—
148	C₂²×C₄₁	—	2	4	no	—
149	C₂²×C₄₁	#342	1	2	yes	—
150	C₂²×C₄₁	#342	1	2	yes	—
151	C₂²×C₄₁	#355	1	2	yes	—
152	C₂²×C₄₁	#356	1	2	yes	—
153	C₂²×C₄₁	C₂×#95	1	1	yes	characteristic
154	C₂²×C₄₁	C₂×#99	1	1	yes	characteristic
155	C₂²×C₄₁	C₂×#99	1	1	yes	characteristic
156	C₄×C₄₁	—	2	2	no	—
157	C₄×C₄₁	—	2	4	no	—
158	C₄×C₄₁	—	2	2	no	—
159	C₄×C₄₁	—	2	1	no	—
160	C₄×C₄₁	—	2	1	no	—
161	C₄×C₄₁	—	2	2	no	—
162	C₄×C₄₁	—	4	4	no	—
163	C₄×C₄₁	—	8	2	no	—
164	C₂²×#101	C₂×C₄₁	1	2	yes	—
165	C₂×#391	C₂×C₄₁	1	1	yes	characteristic
166	#1394	C₂×C₄₁	1	4	yes	direct product factor
167	C₂×C₄₁	—	2	1	no	—
168	C₂×C₄₁	—	2	2	no	—
169	C₂×C₄₁	—	2	1	no	—
170	C₂×C₄₁	#1394	1	4	yes	direct product factor
171	C₂×C₄₁	C₂×#342	1	1	yes	characteristic
172	C₂×C₄₁	C₂×#355	1	1	yes	characteristic
173	C₂×C₄₁	C₂×#356	1	1	yes	characteristic
174	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
175	C₂²×M₁₆	—	2	2	no	—
176	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
177	C₂²×C₄²	C₄×C₄₁	1	1	yes	characteristic
178	C₂×#101	—	2	2	no	—
179	C₂×#101	—	2	4	no	—
180	C₂×#101	C₄×C₄₁	1	8	yes	complemented
181	C₂×#101	C₂²×C₄₁	1	4	yes	—
182	#391	—	2	2	no	—
183	#391	C₄×C₄₁	1	2	yes	—
184	#391	C₂²×C₄₁	1	2	yes	—
185	C₂×#95	—	2	2	no	—
186	C₄₁	C₂×#1394	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₈	—	4	2	no	—
188	C₂²×Q₈	—	4	1	no	—
189	C₂²×Q₈	—	4	1	no	—
190	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
191	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
192	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
193	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
194	C₂²×Q₈	Q₈×C₄₁	1	2	yes	—
195	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
196	C₂³×C₄	—	2	1	no	—
197	C₂³×C₄	—	2	1	no	—
198	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
199	C₂³×C₄	D₄×C₄₁	1	2	yes	—
200	C₂×M₁₆	—	2	8	no	—
201	C₂×M₁₆	—	2	4	no	—
202	C₂×M₁₆	—	4	4	no	—
203	C₂×M₁₆	—	4	2	no	—
204	C₂²×C₈	—	4	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	4	1	no	—
207	C₂×C₄⋊C₄	—	4	1	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	4	2	no	—
211	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
212	#101	—	2	8	no	—
213	#101	—	2	4	no	—
214	#101	C₂×C₄×C₄₁	1	8	yes	complemented
215	#95	—	2	4	no	—
216	#95	—	2	4	no	—
217	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	C₄²×C₄₁	1	2	yes	—
231	C₂×Q₈	C₂²⋊C₄×C₄₁	1	4	yes	—
232	C₂×Q₈	C₄⋊C₄×C₄₁	1	4	yes	—
233	C₂×Q₈	C₄⋊C₄×C₄₁	1	2	yes	—
234	C₂×Q₈	C₂²×C₄×C₄₁	1	2	yes	—
235	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂×Q₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	C₄²×C₄₁	1	2	yes	—
252	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
253	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
255	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
256	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
257	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
259	M₁₆	—	2	8	no	—
260	M₁₆	—	4	8	no	—
261	M₁₆	—	4	4	no	—
262	C₂×C₈	—	4	8	no	—
263	C₂×C₈	—	4	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	#95×C₄₁	1	2	yes	—
271	C₂³	#99×C₄₁	1	1	yes	characteristic
272	C₂³	#99×C₄₁	1	2	yes	—
273	C₂³	#99×C₄₁	1	1	yes	characteristic
274	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	4	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	4	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	#95×C₄₁	1	2	yes	—
307	C₂×C₄	#95×C₄₁	1	4	yes	—
308	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
310	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
312	C₈	—	4	8	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	C₄₁×#342	1	2	yes	—
320	C₂²	C₄₁×#342	1	2	yes	—
321	C₂²	C₄₁×#355	1	2	yes	—
322	C₂²	C₄₁×#356	1	2	yes	—
323	C₂²	C₂×#95×C₄₁	1	1	yes	characteristic
324	C₂²	C₂×#99×C₄₁	1	1	yes	characteristic
325	C₂²	C₂×#99×C₄₁	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	2	no	—
334	C₂	—	2	1	no	—
335	C₂	—	2	2	no	—
336	C₂	—	2	1	no	—
337	C₂	C₄₁×#1394	1	4	yes	direct product factor
338	C₂	C₂×C₄₁×#342	1	1	yes	characteristic
339	C₂	C₂×C₄₁×#355	1	1	yes	characteristic
340	C₂	C₂×C₄₁×#356	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#99
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×Q₈	C₂³×C₄₁
prime radical	C₂⁴×C₄₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂⁴×C₄₁ ◁ C₂³×Q₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1394	C₂	direct (N × H)
10. C₂×#101×C₄₁	C₄	semidirect
43. #101×C₄₁	C₂×C₄	semidirect
142. C₂×#1394	C₄₁	direct (N × H)
166. #1394	C₂×C₄₁	direct (N × H)
170. C₂×C₄₁	#1394	direct (N × H)
180. C₂×#101	C₄×C₄₁	semidirect
186. C₄₁	C₂×#1394	direct (N × H)
214. #101	C₂×C₄×C₄₁	semidirect
337. C₂	C₄₁×#1394	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×#101×C₄₁	2
C₂×C₄₁×#391	2
C₄₁×#1394	2
C₂×#1394	41

Subgroups — C2×C41×#1394 #23788785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1394 #23788785