Subgroups — C₄₁×#33166 #23809367

Order 10496.

1238 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#463	C₂	1	3	yes	maximal, complemented
2	C₂×C₄₁×#459	C₂	1	3	yes	maximal, complemented
3	C₄₁×#2657	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2275	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1983	C₂	1	3	yes	maximal, complemented
6	C₄₁×#1982	C₂	1	3	yes	maximal, complemented
7	C₄₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₄₁	C₂²	1	3	yes	—
10	C₂×#127×C₄₁	C₂²	1	3	yes	—
11	C₂×#124×C₄₁	C₂²	1	3	yes	—
12	C₂×C₄×D₄×C₄₁	C₂²	1	3	yes	—
13	C₄₁×#463	—	2	12	no	—
14	C₄₁×#459	—	2	12	no	—
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	3	yes	—
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	3	yes	—
17	C₂×#103×C₄₁	C₂²	1	3	yes	—
18	C₂×#102×C₄₁	C₂²	1	3	yes	—
19	C₂×#102×C₄₁	C₂²	1	6	yes	—
20	C₄₁×#392	—	2	1	no	—
21	C₄₁×#392	—	2	3	no	—
22	C₄×C₄⋊C₄×C₄₁	C₂²	1	3	yes	—
23	C₄×C₂²⋊C₄×C₄₁	—	2	3	no	—
24	C₄₁×#376	C₂²	1	1	yes	characteristic
25	C₂×#95×C₄₁	—	2	1	no	—
26	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
27	C₂²×D₄×C₄₁	—	2	3	no	—
28	C₂²×D₄×C₄₁	—	2	3	no	—
29	C₂²×D₄×C₄₁	C₂³	1	3	yes	—
30	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₄₁	—	2	3	no	—
32	C₂×SD₁₆×C₄₁	—	2	6	no	—
33	C₂×D₈×C₄₁	—	2	12	no	—
34	C₂×D₈×C₄₁	—	2	6	no	—
35	C₂²×C₈×C₄₁	C₂³	1	3	yes	—
36	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
37	#127×C₄₁	—	2	6	no	—
38	#127×C₄₁	—	2	6	no	—
39	#124×C₄₁	—	2	6	no	—
40	#124×C₄₁	—	2	6	no	—
41	C₄×D₄×C₄₁	—	2	6	no	—
42	C₄×D₄×C₄₁	D₄	1	12	yes	complemented
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	3	yes	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	3	no	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	3	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	3	no	—
47	C₂×C₄²×C₄₁	—	2	3	no	—
48	C₂×C₄²×C₄₁	C₂³	1	3	yes	—
49	C₄⋊C₈×C₄₁	—	2	6	no	—
50	C₄⋊C₈×C₄₁	—	2	6	no	—
51	#103×C₄₁	—	2	6	no	—
52	#102×C₄₁	—	2	12	no	—
53	#102×C₄₁	—	2	6	no	—
54	#95×C₄₁	—	2	3	no	—
55	#95×C₄₁	D₄	1	1	yes	characteristic
56	#95×C₄₁	D₄	1	3	yes	—
57	C₂⁴×C₄₁	—	2	1	no	—
58	C₂⁴×C₄₁	—	2	3	no	—
59	C₂×Q₈×C₄₁	—	2	3	no	—
60	C₂×Q₈×C₄₁	—	2	3	no	—
61	C₂×D₄×C₄₁	—	2	3	no	—
62	C₂×D₄×C₄₁	—	2	6	no	—
63	C₂×D₄×C₄₁	—	2	3	no	—
64	C₂×D₄×C₄₁	—	2	3	no	—
65	C₂×D₄×C₄₁	—	2	6	no	—
66	C₂×D₄×C₄₁	—	4	6	no	—
67	C₂×D₄×C₄₁	—	4	6	no	—
68	C₂×D₄×C₄₁	—	8	3	no	—
69	C₂×D₄×C₄₁	—	8	6	no	—
70	C₂×D₄×C₄₁	—	8	3	no	—
71	C₂×D₄×C₄₁	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₄₁	—	2	3	no	—
73	C₂²×C₄×C₄₁	—	2	12	no	—
74	C₂²×C₄×C₄₁	—	2	1	no	—
75	C₂²×C₄×C₄₁	—	8	6	no	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
79	SD₁₆×C₄₁	—	4	4	no	—
80	D₈×C₄₁	—	4	12	no	—
81	C₂×C₈×C₄₁	—	2	6	no	—
82	C₂×C₈×C₄₁	—	2	3	no	—
83	C₂×C₈×C₄₁	—	2	3	no	—
84	C₄⋊C₄×C₄₁	—	2	3	no	—
85	C₄⋊C₄×C₄₁	C₂×D₄	1	6	yes	—
86	C₂²⋊C₄×C₄₁	—	2	12	no	—
87	C₂²⋊C₄×C₄₁	—	4	6	no	—
88	C₂²⋊C₄×C₄₁	—	4	6	no	—
89	C₄²×C₄₁	—	8	3	no	—
90	C₄²×C₄₁	C₂×D₄	1	6	yes	—
91	C₄²×C₄₁	#41	1	6	yes	—
92	C₂³×C₄₁	—	2	6	no	—
93	C₂³×C₄₁	—	4	3	no	—
94	C₂³×C₄₁	—	4	1	no	—
95	C₂³×C₄₁	—	4	3	no	—
96	C₂³×C₄₁	—	4	6	no	—
97	C₂³×C₄₁	—	4	3	no	—
98	C₂³×C₄₁	—	4	6	no	—
99	C₂³×C₄₁	—	4	3	no	—
100	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₄₁	—	4	1	no	—
102	Q₈×C₄₁	—	4	3	no	—
103	D₄×C₄₁	—	2	6	no	—
104	D₄×C₄₁	—	4	3	no	—
105	D₄×C₄₁	—	4	3	no	—
106	D₄×C₄₁	—	4	6	no	—
107	D₄×C₄₁	—	8	12	no	—
108	D₄×C₄₁	—	8	12	no	—
109	D₄×C₄₁	#127	1	4	yes	—
110	C₂×C₄×C₄₁	—	2	3	no	—
111	C₂×C₄×C₄₁	—	2	6	no	—
112	C₂×C₄×C₄₁	—	2	3	no	—
113	C₂×C₄×C₄₁	—	2	6	no	—
114	C₂×C₄×C₄₁	—	4	6	no	—
115	C₂×C₄×C₄₁	—	4	3	no	—
116	C₂×C₄×C₄₁	—	8	12	no	—
117	C₂×C₄×C₄₁	#121	1	6	yes	—
118	C₂×C₄×C₄₁	#121	1	6	yes	—
119	C₂×C₄×C₄₁	C₂²×D₄	1	3	yes	—
120	C₂×C₄×C₄₁	C₂×#41	1	3	yes	—
121	C₈×C₄₁	—	4	3	no	—
122	C₈×C₄₁	—	4	1	no	—
123	#33166	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	—	2	4	no	—
125	C₂²×C₄₁	—	4	6	no	—
126	C₂²×C₄₁	—	4	6	no	—
127	C₂²×C₄₁	—	8	3	no	—
128	C₂²×C₄₁	—	8	6	no	—
129	C₂²×C₄₁	—	8	3	no	—
130	C₂²×C₄₁	—	8	6	no	—
131	C₂²×C₄₁	—	8	3	no	—
132	C₂²×C₄₁	C₂×#121	1	3	yes	—
133	C₂²×C₄₁	#576	1	3	yes	—
134	C₂²×C₄₁	#577	1	1	yes	characteristic
135	C₄×C₄₁	—	2	6	no	—
136	C₄×C₄₁	—	2	6	no	—
137	C₄×C₄₁	—	4	6	no	—
138	C₄×C₄₁	—	8	1	no	—
139	C₄×C₄₁	C₂×#127	1	1	yes	characteristic
140	C₄×C₄₁	#532	1	3	yes	—
141	C₂×#463	C₂×C₄₁	1	3	yes	complemented
142	C₂×#459	C₂×C₄₁	1	3	yes	complemented
143	#2657	C₂×C₄₁	1	1	yes	characteristic, complemented
144	#2275	C₂×C₄₁	1	1	yes	characteristic, complemented
145	#1983	C₂×C₄₁	1	3	yes	complemented
146	#1982	C₂×C₄₁	1	3	yes	complemented
147	#1843	C₂×C₄₁	1	1	yes	characteristic, complemented
148	C₂×C₄₁	—	4	4	no	—
149	C₂×C₄₁	—	16	3	no	—
150	C₂×C₄₁	#2856	1	1	yes	characteristic
151	C₂×C₄₁	#3041	1	3	yes	—
152	C₂×C₄₁	#3045	1	3	yes	—
153	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
154	C₂²×D₈	C₂²×C₄₁	1	3	yes	—
155	C₂×#127	C₂²×C₄₁	1	3	yes	—
156	C₂×#124	C₂²×C₄₁	1	3	yes	—
157	C₂×C₄×D₄	C₂²×C₄₁	1	3	yes	—
158	#463	—	2	12	no	—
159	#459	—	2	12	no	—
160	C₂×C₄⋊C₈	C₂²×C₄₁	1	3	yes	—
161	C₂×C₄⋊C₈	C₂²×C₄₁	1	3	yes	—
162	C₂×#103	C₂²×C₄₁	1	3	yes	—
163	C₂×#102	C₂²×C₄₁	1	3	yes	—
164	C₂×#102	C₂²×C₄₁	1	6	yes	—
165	#392	—	2	1	no	—
166	#392	—	2	3	no	—
167	C₄×C₄⋊C₄	C₂²×C₄₁	1	3	yes	—
168	C₄×C₂²⋊C₄	—	2	3	no	—
169	#376	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×#95	—	2	1	no	—
171	C₄₁	#33166	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
173	C₂²×D₄	—	2	3	no	—
174	C₂²×D₄	—	2	3	no	—
175	C₂²×D₄	C₂³×C₄₁	1	3	yes	—
176	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
177	C₂³×C₄	—	2	3	no	—
178	C₂×SD₁₆	—	2	6	no	—
179	C₂×D₈	—	2	12	no	—
180	C₂×D₈	—	2	6	no	—
181	C₂²×C₈	C₂³×C₄₁	1	3	yes	—
182	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
183	#127	—	2	6	no	—
184	#127	—	2	6	no	—
185	#124	—	2	6	no	—
186	#124	—	2	6	no	—
187	C₄×D₄	—	2	6	no	—
188	C₄×D₄	D₄×C₄₁	1	12	yes	complemented
189	C₂×C₄⋊C₄	C₂³×C₄₁	1	3	yes	—
190	C₂×C₂²⋊C₄	—	2	3	no	—
191	C₂×C₂²⋊C₄	—	2	3	no	—
192	C₂×C₂²⋊C₄	—	2	3	no	—
193	C₂×C₄²	—	2	3	no	—
194	C₂×C₄²	C₂³×C₄₁	1	3	yes	—
195	C₄⋊C₈	—	2	6	no	—
196	C₄⋊C₈	—	2	6	no	—
197	#103	—	2	6	no	—
198	#102	—	2	12	no	—
199	#102	—	2	6	no	—
200	#95	—	2	3	no	—
201	#95	D₄×C₄₁	1	1	yes	characteristic
202	#95	D₄×C₄₁	1	3	yes	—
203	C₂⁴	—	2	1	no	—
204	C₂⁴	—	2	3	no	—
205	C₂×Q₈	—	2	3	no	—
206	C₂×Q₈	—	2	3	no	—
207	C₂×D₄	—	2	3	no	—
208	C₂×D₄	—	2	6	no	—
209	C₂×D₄	—	2	3	no	—
210	C₂×D₄	—	2	3	no	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	4	6	no	—
213	C₂×D₄	—	4	6	no	—
214	C₂×D₄	—	8	3	no	—
215	C₂×D₄	—	8	6	no	—
216	C₂×D₄	—	8	3	no	—
217	C₂×D₄	C₂×D₄×C₄₁	1	6	yes	—
218	C₂²×C₄	—	2	3	no	—
219	C₂²×C₄	—	2	12	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	8	6	no	—
222	C₂²×C₄	C₂×D₄×C₄₁	1	3	yes	—
223	C₂²×C₄	C₂×D₄×C₄₁	1	3	yes	—
224	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
225	SD₁₆	—	4	4	no	—
226	D₈	—	4	12	no	—
227	C₂×C₈	—	2	6	no	—
228	C₂×C₈	—	2	3	no	—
229	C₂×C₈	—	2	3	no	—
230	C₄⋊C₄	—	2	3	no	—
231	C₄⋊C₄	C₂×D₄×C₄₁	1	6	yes	—
232	C₂²⋊C₄	—	2	12	no	—
233	C₂²⋊C₄	—	4	6	no	—
234	C₂²⋊C₄	—	4	6	no	—
235	C₄²	—	8	3	no	—
236	C₄²	C₂×D₄×C₄₁	1	6	yes	—
237	C₄²	#41×C₄₁	1	6	yes	—
238	C₂³	—	2	6	no	—
239	C₂³	—	4	3	no	—
240	C₂³	—	4	1	no	—
241	C₂³	—	4	3	no	—
242	C₂³	—	4	6	no	—
243	C₂³	—	4	3	no	—
244	C₂³	—	4	6	no	—
245	C₂³	—	4	3	no	—
246	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
247	Q₈	—	4	1	no	—
248	Q₈	—	4	3	no	—
249	D₄	—	2	6	no	—
250	D₄	—	4	3	no	—
251	D₄	—	4	3	no	—
252	D₄	—	4	6	no	—
253	D₄	—	8	12	no	—
254	D₄	—	8	12	no	—
255	D₄	#127×C₄₁	1	4	yes	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	6	no	—
258	C₂×C₄	—	2	3	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	4	6	no	—
261	C₂×C₄	—	4	3	no	—
262	C₂×C₄	—	8	12	no	—
263	C₂×C₄	#121×C₄₁	1	6	yes	—
264	C₂×C₄	#121×C₄₁	1	6	yes	—
265	C₂×C₄	C₂²×D₄×C₄₁	1	3	yes	—
266	C₂×C₄	C₂×#41×C₄₁	1	3	yes	—
267	C₈	—	4	3	no	—
268	C₈	—	4	1	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	6	no	—
271	C₂²	—	4	6	no	—
272	C₂²	—	8	6	no	—
273	C₂²	—	8	3	no	—
274	C₂²	—	8	6	no	—
275	C₂²	—	8	3	no	—
276	C₂²	—	8	3	no	—
277	C₂²	C₂×#121×C₄₁	1	3	yes	—
278	C₂²	C₄₁×#576	1	3	yes	—
279	C₂²	C₄₁×#577	1	1	yes	characteristic
280	C₄	—	2	6	no	—
281	C₄	—	2	6	no	—
282	C₄	—	4	6	no	—
283	C₄	—	8	1	no	—
284	C₄	C₂×#127×C₄₁	1	1	yes	characteristic
285	C₄	C₄₁×#532	1	3	yes	—
286	C₂	—	4	4	no	—
287	C₂	—	16	3	no	—
288	C₂	C₄₁×#2856	1	1	yes	characteristic
289	C₂	C₄₁×#3041	1	3	yes	—
290	C₂	C₄₁×#3045	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#463	C₂	semidirect
2. C₂×C₄₁×#459	C₂	semidirect
3. C₄₁×#2657	C₂	semidirect
4. C₄₁×#2275	C₂	semidirect
5. C₄₁×#1983	C₂	semidirect
6. C₄₁×#1982	C₂	semidirect
7. C₄₁×#1843	C₂	semidirect
42. C₄×D₄×C₄₁	D₄	semidirect
123. #33166	C₄₁	direct (N × H)
141. C₂×#463	C₂×C₄₁	semidirect
142. C₂×#459	C₂×C₄₁	semidirect
143. #2657	C₂×C₄₁	semidirect
144. #2275	C₂×C₄₁	semidirect
145. #1983	C₂×C₄₁	semidirect
146. #1982	C₂×C₄₁	semidirect
147. #1843	C₂×C₄₁	semidirect
171. C₄₁	#33166	direct (N × H)
188. C₄×D₄	D₄×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×#463	2
C₂×C₄₁×#459	2
C₄₁×#2657	2
C₄₁×#2275	2
C₄₁×#1983	2
C₄₁×#1982	2
C₄₁×#1843	2
#33166	41

Subgroups — C41×#33166 #23809367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33166 #23809367