Subgroups — C₄₁×#7891 #23784092

Order 10496.

770 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₄₁	—	2	1	no	—
6	C₂×#117×C₄₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
8	C₄₁×#431	—	2	2	no	—
9	C₄₁×#431	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
11	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
12	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
13	C₂×C₄⋊C₈×C₄₁	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
21	C₄₁×#376	—	2	1	no	—
22	C₄³×C₄₁	—	2	1	no	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₁	—	2	4	no	—
27	C₂²×C₈×C₄₁	—	2	2	no	—
28	C₂²×C₈×C₄₁	—	2	2	no	—
29	C₂²×C₈×C₄₁	—	4	4	no	—
30	#117×C₄₁	—	2	4	no	—
31	#117×C₄₁	C₈	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
34	C₂×C₄²×C₄₁	—	2	2	no	—
35	C₂×C₄²×C₄₁	—	2	2	no	—
36	C₂×C₄²×C₄₁	—	2	2	no	—
37	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₁	D₄	1	2	yes	—
43	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
44	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₄₁	—	2	4	no	—
46	C₄⋊C₈×C₄₁	—	2	4	no	—
47	C₄⋊C₈×C₄₁	—	2	2	no	—
48	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₄₁	—	2	4	no	—
50	C₄×C₈×C₄₁	—	2	2	no	—
51	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
52	#95×C₄₁	—	2	2	no	—
53	#95×C₄₁	—	2	2	no	—
54	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	2	4	no	—
58	C₂²×C₄×C₄₁	—	4	2	no	—
59	C₂²×C₄×C₄₁	C₄²	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₁	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₄₁	M₁₆	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₁	—	2	4	no	—
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	—	4	4	no	—
76	C₂×C₈×C₄₁	—	4	8	no	—
77	C₂×C₈×C₄₁	—	4	4	no	—
78	C₄⋊C₄×C₄₁	—	2	4	no	—
79	C₄⋊C₄×C₄₁	C₂×C₈	1	8	yes	complemented
80	C₂²⋊C₄×C₄₁	—	2	8	no	—
81	C₄²×C₄₁	—	2	8	no	—
82	C₄²×C₄₁	—	4	4	no	—
83	C₄²×C₄₁	—	4	2	no	—
84	C₄²×C₄₁	—	4	2	no	—
85	C₄²×C₄₁	C₄²	1	2	yes	—
86	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
89	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
90	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
91	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
95	C₂³×C₄₁	—	2	2	no	—
96	C₂³×C₄₁	—	2	2	no	—
97	C₂³×C₄₁	#95	1	2	yes	—
98	C₂³×C₄₁	C₂²⋊C₈	1	4	yes	—
99	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	4	2	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	4	4	no	—
110	C₂×C₄×C₄₁	—	4	4	no	—
111	C₂×C₄×C₄₁	#95	1	2	yes	—
112	C₂×C₄×C₄₁	#95	1	2	yes	—
113	C₂×C₄×C₄₁	#95	1	2	yes	—
114	C₂×C₄×C₄₁	C₄×C₈	1	4	yes	—
115	C₂×C₄×C₄₁	#97	1	4	yes	—
116	C₂×C₄×C₄₁	C₂²⋊C₈	1	4	yes	—
117	C₂×C₄×C₄₁	C₄⋊C₈	1	8	yes	—
118	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₄₁	#137	1	2	yes	—
128	C₈×C₄₁	—	4	4	no	—
129	C₈×C₄₁	—	4	4	no	—
130	#7891	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₄₁	—	2	8	no	—
132	C₂²×C₄₁	—	2	2	no	—
133	C₂²×C₄₁	—	2	2	no	—
134	C₂²×C₄₁	#336	1	4	yes	—
135	C₂²×C₄₁	C₂×#95	1	1	yes	characteristic
136	C₂²×C₄₁	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₄₁	#417	1	2	yes	—
138	C₂²×C₄₁	#421	1	2	yes	—
139	C₄×C₄₁	—	2	1	no	—
140	C₄×C₄₁	—	2	2	no	—
141	C₄×C₄₁	—	2	1	no	—
142	C₄×C₄₁	—	2	2	no	—
143	C₄×C₄₁	—	4	8	no	—
144	C₄×C₄₁	#336	1	4	yes	—
145	C₄×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₄₁	C₂×#97	1	1	yes	characteristic
147	C₄×C₄₁	#418	1	1	yes	characteristic
148	C₄×C₄₁	#419	1	1	yes	characteristic
149	C₄×C₄₁	C₂×C₄⋊C₈	1	2	yes	—
150	C₄×C₄₁	#428	1	2	yes	—
151	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
152	C₂²×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
153	C₄×#117	C₂×C₄₁	1	1	yes	characteristic
154	#1279	C₂×C₄₁	1	4	yes	complemented
155	C₂×C₄₁	—	2	4	no	—
156	C₂×C₄₁	#1477	1	4	yes	—
157	C₂×C₄₁	C₂×#336	1	1	yes	characteristic
158	C₂×C₄₁	#1733	1	1	yes	characteristic
159	C₂×C₄₁	#1736	1	1	yes	characteristic
160	C₂³×C₈	—	2	1	no	—
161	C₂×#117	C₄×C₄₁	1	2	yes	—
162	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
163	#431	—	2	2	no	—
164	#431	C₄×C₄₁	1	4	yes	complemented
165	C₂×C₄⋊C₈	—	2	1	no	—
166	C₂×C₄⋊C₈	—	2	2	no	—
167	C₂×C₄⋊C₈	—	2	1	no	—
168	C₂×C₄⋊C₈	C₄×C₄₁	1	4	yes	complemented
169	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
172	C₄×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
173	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
174	C₄×C₂²⋊C₄	—	2	1	no	—
175	C₄×C₂²⋊C₄	—	2	1	no	—
176	#376	—	2	1	no	—
177	C₄³	—	2	1	no	—
178	C₄₁	#7891	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
180	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
181	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	4	4	no	—
186	#117	—	2	4	no	—
187	#117	C₈×C₄₁	1	8	yes	complemented
188	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
196	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₄₁	1	2	yes	—
199	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
200	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	4	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
205	C₂²⋊C₈	—	2	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₄²×C₄₁	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
221	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
222	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂×C₈×C₄₁	1	4	yes	—
225	C₂²×C₄	M₁₆×C₄₁	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	8	no	—
232	C₂×C₈	—	4	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	C₂×C₈×C₄₁	1	8	yes	complemented
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	8	no	—
238	C₄²	—	4	4	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₄²×C₄₁	1	2	yes	—
242	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
243	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
244	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
245	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
246	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
247	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
248	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
250	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	#95×C₄₁	1	2	yes	—
254	C₂³	C₂²⋊C₈×C₄₁	1	4	yes	—
255	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	#95×C₄₁	1	2	yes	—
268	C₂×C₄	#95×C₄₁	1	2	yes	—
269	C₂×C₄	#95×C₄₁	1	2	yes	—
270	C₂×C₄	C₄×C₈×C₄₁	1	4	yes	—
271	C₂×C₄	#97×C₄₁	1	4	yes	—
272	C₂×C₄	C₂²⋊C₈×C₄₁	1	4	yes	—
273	C₂×C₄	C₄⋊C₈×C₄₁	1	8	yes	—
274	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
282	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
283	C₂×C₄	#137×C₄₁	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	4	4	no	—
286	C₂²	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	C₄₁×#336	1	4	yes	—
290	C₂²	C₂×#95×C₄₁	1	1	yes	characteristic
291	C₂²	C₂×C₂²⋊C₈×C₄₁	1	2	yes	—
292	C₂²	C₄₁×#417	1	2	yes	—
293	C₂²	C₄₁×#421	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	8	no	—
299	C₄	C₄₁×#336	1	4	yes	—
300	C₄	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
301	C₄	C₂×#97×C₄₁	1	1	yes	characteristic
302	C₄	C₄₁×#418	1	1	yes	characteristic
303	C₄	C₄₁×#419	1	1	yes	characteristic
304	C₄	C₂×C₄⋊C₈×C₄₁	1	2	yes	—
305	C₄	C₄₁×#428	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	C₄₁×#1477	1	4	yes	—
308	C₂	C₂×C₄₁×#336	1	1	yes	characteristic
309	C₂	C₄₁×#1733	1	1	yes	characteristic
310	C₂	C₄₁×#1736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂³×C₄₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1279	C₂	semidirect
9. C₄₁×#431	C₄	semidirect
13. C₂×C₄⋊C₈×C₄₁	C₄	semidirect
31. #117×C₄₁	C₈	semidirect
79. C₄⋊C₄×C₄₁	C₂×C₈	semidirect
130. #7891	C₄₁	direct (N × H)
154. #1279	C₂×C₄₁	semidirect
164. #431	C₄×C₄₁	semidirect
168. C₂×C₄⋊C₈	C₄×C₄₁	semidirect
178. C₄₁	#7891	direct (N × H)
187. #117	C₈×C₄₁	semidirect
235. C₄⋊C₄	C₂×C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#431	2
C₂²×C₄⋊C₈×C₄₁	2
C₄×#117×C₄₁	2
C₄₁×#1279	2
#7891	41

Subgroups — C41×#7891 #23784092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7891 #23784092