Subgroups — C₄₁×#21826 #23798027

Order 10496.

1050 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2283	C₂	1	2	yes	maximal
3	C₄₁×#1996	C₂	1	8	yes	maximal
4	C₄₁×#1820	C₂	1	4	yes	maximal
5	C₄₁×#533	—	2	2	no	—
6	C₄₁×#533	—	2	2	no	—
7	C₄₁×#533	—	2	2	no	—
8	C₄₁×#533	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₄₁	C₄	1	2	yes	complemented
10	C₂×#125×C₄₁	C₄	1	2	yes	complemented
11	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	4	yes	—
12	C₂×C₄×Q₈×C₄₁	C₄	1	4	yes	complemented
13	C₂×#117×C₄₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#383	—	2	2	no	—
16	C₄₁×#382	—	2	2	no	—
17	C₄₁×#381	C₂²	1	4	yes	—
18	C₄₁×#380	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
20	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
21	C₄₁×#376	—	2	2	no	—
22	C₄₁×#342	C₂²	1	16	yes	—
23	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
26	C₄⋊Q₈×C₄₁	—	2	2	no	—
27	C₄⋊Q₈×C₄₁	—	2	2	no	—
28	C₄⋊Q₈×C₄₁	—	2	2	no	—
29	C₄⋊Q₈×C₄₁	—	2	2	no	—
30	#125×C₄₁	—	2	2	no	—
31	#125×C₄₁	—	2	4	no	—
32	#125×C₄₁	—	2	2	no	—
33	C₂²⋊Q₈×C₄₁	—	2	4	no	—
34	C₂²⋊Q₈×C₄₁	—	2	4	no	—
35	C₂²⋊Q₈×C₄₁	—	2	4	no	—
36	C₂²⋊Q₈×C₄₁	—	2	4	no	—
37	C₄×Q₈×C₄₁	—	2	4	no	—
38	C₄×Q₈×C₄₁	—	2	4	no	—
39	C₄×Q₈×C₄₁	—	2	4	no	—
40	C₄×Q₈×C₄₁	—	2	4	no	—
41	#117×C₄₁	—	2	4	no	—
42	#117×C₄₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
47	C₂×C₂²⋊C₄×C₄₁	C₂³	1	8	yes	—
48	C₂×C₂²⋊C₄×C₄₁	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₁	—	2	4	no	—
50	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₁	D₄	1	2	yes	—
55	C₂×C₄²×C₄₁	D₄	1	2	yes	—
56	C₂×C₄²×C₄₁	D₄	1	2	yes	—
57	C₂×C₄²×C₄₁	D₄	1	2	yes	—
58	#95×C₄₁	—	2	4	no	—
59	#95×C₄₁	—	2	4	no	—
60	#95×C₄₁	—	2	4	no	—
61	#95×C₄₁	—	2	4	no	—
62	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
63	C₂×Q₈×C₄₁	—	2	4	no	—
64	C₂×Q₈×C₄₁	—	2	4	no	—
65	C₂×Q₈×C₄₁	—	4	8	no	—
66	C₂²×C₄×C₄₁	—	2	4	no	—
67	C₂²×C₄×C₄₁	—	2	8	no	—
68	C₂²×C₄×C₄₁	—	4	4	no	—
69	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₄₁	—	2	4	no	—
81	C₄⋊C₄×C₄₁	—	2	4	no	—
82	C₄⋊C₄×C₄₁	—	2	8	no	—
83	C₄⋊C₄×C₄₁	—	2	8	no	—
84	C₄⋊C₄×C₄₁	—	2	4	no	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	4	8	no	—
87	C₂²⋊C₄×C₄₁	—	2	8	no	—
88	C₂²⋊C₄×C₄₁	—	2	16	no	—
89	C₄²×C₄₁	—	2	8	no	—
90	C₄²×C₄₁	—	4	8	no	—
91	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
95	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₁	—	4	2	no	—
102	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
103	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
104	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
105	Q₈×C₄₁	—	4	8	no	—
106	Q₈×C₄₁	—	4	8	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	8	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	8	no	—
119	C₂×C₄×C₄₁	—	4	8	no	—
120	C₂×C₄×C₄₁	—	4	8	no	—
121	C₂×C₄×C₄₁	—	4	8	no	—
122	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
129	#21826	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₄₁	—	2	2	no	—
131	C₂²×C₄₁	—	2	2	no	—
132	C₂²×C₄₁	—	2	2	no	—
133	C₂²×C₄₁	—	4	4	no	—
134	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₁	#519	1	2	yes	—
136	C₂²×C₄₁	#536	1	4	yes	—
137	C₄×C₄₁	—	2	2	no	—
138	C₄×C₄₁	—	2	1	no	—
139	C₄×C₄₁	—	2	1	no	—
140	C₄×C₄₁	—	2	2	no	—
141	C₄×C₄₁	—	2	2	no	—
142	C₄×C₄₁	—	2	2	no	—
143	C₄×C₄₁	—	4	8	no	—
144	C₄×C₄₁	—	8	8	no	—
145	C₄×C₄₁	#410	1	4	yes	—
146	C₂×#533	C₂×C₄₁	1	1	yes	characteristic
147	#2283	C₂×C₄₁	1	2	yes	—
148	#1996	C₂×C₄₁	1	8	yes	—
149	#1820	C₂×C₄₁	1	4	yes	—
150	C₂×C₄₁	—	2	4	no	—
151	C₂×C₄₁	—	2	2	no	—
152	C₂×C₄₁	#2310	1	1	yes	characteristic
153	C₂×C₄₁	C₂×#410	1	2	yes	—
154	#533	—	2	2	no	—
155	#533	—	2	2	no	—
156	#533	—	2	2	no	—
157	#533	—	2	2	no	—
158	C₂×C₄⋊Q₈	C₄×C₄₁	1	2	yes	complemented
159	C₂×#125	C₄×C₄₁	1	2	yes	complemented
160	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	4	yes	—
161	C₂×C₄×Q₈	C₄×C₄₁	1	4	yes	complemented
162	C₂×#117	C₂²×C₄₁	1	2	yes	—
163	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
164	#383	—	2	2	no	—
165	#382	—	2	2	no	—
166	#381	C₂²×C₄₁	1	4	yes	—
167	#380	C₂²×C₄₁	1	4	yes	—
168	C₄×C₄⋊C₄	—	2	2	no	—
169	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
170	#376	—	2	2	no	—
171	#342	C₂²×C₄₁	1	16	yes	—
172	C₄₁	#21826	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
174	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	#125	—	2	2	no	—
181	#125	—	2	4	no	—
182	#125	—	2	2	no	—
183	C₂²⋊Q₈	—	2	4	no	—
184	C₂²⋊Q₈	—	2	4	no	—
185	C₂²⋊Q₈	—	2	4	no	—
186	C₂²⋊Q₈	—	2	4	no	—
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	2	4	no	—
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	4	no	—
191	#117	—	2	4	no	—
192	#117	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
196	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
197	C₂×C₂²⋊C₄	C₂³×C₄₁	1	8	yes	—
198	C₂×C₂²⋊C₄	C₂³×C₄₁	1	4	yes	—
199	C₂×C₄²	—	2	4	no	—
200	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
204	C₂×C₄²	D₄×C₄₁	1	2	yes	—
205	C₂×C₄²	D₄×C₄₁	1	2	yes	—
206	C₂×C₄²	D₄×C₄₁	1	2	yes	—
207	C₂×C₄²	D₄×C₄₁	1	2	yes	—
208	#95	—	2	4	no	—
209	#95	—	2	4	no	—
210	#95	—	2	4	no	—
211	#95	—	2	4	no	—
212	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	8	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
223	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	2	16	no	—
239	C₄²	—	2	8	no	—
240	C₄²	—	4	8	no	—
241	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
242	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
243	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
244	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
245	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
246	C₄²	C₂²⋊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	C₂²⋊C₄×C₄₁	1	2	yes	—
250	C₄²	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
251	C₂³	—	4	2	no	—
252	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
253	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
254	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
255	Q₈	—	4	8	no	—
256	Q₈	—	4	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
273	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
284	C₂²	C₄₁×#519	1	2	yes	—
285	C₂²	C₄₁×#536	1	4	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	8	no	—
293	C₄	—	8	8	no	—
294	C₄	C₄₁×#410	1	4	yes	—
295	C₂	—	2	4	no	—
296	C₂	—	2	2	no	—
297	C₂	C₄₁×#2310	1	1	yes	characteristic
298	C₂	C₂×C₄₁×#410	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Q₈×C₄₁	C₄	semidirect
10. C₂×#125×C₄₁	C₄	semidirect
12. C₂×C₄×Q₈×C₄₁	C₄	semidirect
129. #21826	C₄₁	direct (N × H)
158. C₂×C₄⋊Q₈	C₄×C₄₁	semidirect
159. C₂×#125	C₄×C₄₁	semidirect
161. C₂×C₄×Q₈	C₄×C₄₁	semidirect
172. C₄₁	#21826	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×#533	2
C₄₁×#2283	2
C₄₁×#1996	2
C₄₁×#1820	2
#21826	41

Subgroups — C41×#21826 #23798027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21826 #23798027