Subgroups — C₄₁×#8233 #23784434

Order 10496.

516 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	C₄×C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#332	C₂	1	2	yes	maximal
5	C₄₁×#1296	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
8	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#97×C₄₁	—	2	1	no	—
11	C₂×#97×C₄₁	—	2	1	no	—
12	C₄₁×#391	—	2	1	no	—
13	C₄₁×#389	—	2	1	no	—
14	C₄₁×#389	C₄	1	2	yes	—
15	C₄₁×#384	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₄₁	C₄	1	2	yes	—
18	C₄³×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#336	—	2	1	no	—
20	C₄₁×#336	—	2	1	no	—
21	C₄₁×#332	C₄	1	8	yes	complemented
22	C₂²×Q₈×C₄₁	—	2	1	no	—
23	C₂²×C₈×C₄₁	—	2	2	no	—
24	C₂²×C₈×C₄₁	—	2	2	no	—
25	C₄⋊Q₈×C₄₁	—	2	2	no	—
26	C₄⋊Q₈×C₄₁	C₂×C₄	1	4	yes	—
27	C₄×Q₈×C₄₁	—	4	4	no	—
28	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₁	—	2	2	no	—
35	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₁	—	2	2	no	—
43	C₄⋊C₈×C₄₁	—	2	2	no	—
44	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
45	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
46	#97×C₄₁	—	4	2	no	—
47	#97×C₄₁	—	4	2	no	—
48	#95×C₄₁	—	2	2	no	—
49	#95×C₄₁	—	2	2	no	—
50	C₂×Q₈×C₄₁	—	2	4	no	—
51	C₂×Q₈×C₄₁	—	4	2	no	—
52	C₂×Q₈×C₄₁	—	4	1	no	—
53	C₂×Q₈×C₄₁	—	4	1	no	—
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₄₁	—	2	4	no	—
64	C₂×C₈×C₄₁	—	2	4	no	—
65	C₂×C₈×C₄₁	—	4	2	no	—
66	C₂×C₈×C₄₁	—	4	2	no	—
67	C₂×C₈×C₄₁	—	4	2	no	—
68	C₂×C₈×C₄₁	—	4	2	no	—
69	C₄⋊C₄×C₄₁	—	2	2	no	—
70	C₄⋊C₄×C₄₁	—	2	4	no	—
71	C₄⋊C₄×C₄₁	—	4	2	no	—
72	C₄⋊C₄×C₄₁	—	4	4	no	—
73	C₄⋊C₄×C₄₁	—	4	1	no	—
74	C₄⋊C₄×C₄₁	—	4	2	no	—
75	C₄⋊C₄×C₄₁	—	4	1	no	—
76	C₄⋊C₄×C₄₁	C₄²	1	4	yes	—
77	C₄²×C₄₁	—	2	2	no	—
78	C₄²×C₄₁	—	2	1	no	—
79	C₄²×C₄₁	—	2	1	no	—
80	C₄²×C₄₁	—	2	2	no	—
81	C₄²×C₄₁	—	2	2	no	—
82	C₄²×C₄₁	—	2	1	no	—
83	C₄²×C₄₁	—	2	1	no	—
84	C₄²×C₄₁	—	4	1	no	—
85	C₄²×C₄₁	—	4	4	no	—
86	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₁	#41	1	2	yes	—
88	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₄₁	—	4	4	no	—
90	Q₈×C₄₁	—	4	4	no	—
91	C₂×C₄×C₄₁	—	2	1	no	—
92	C₂×C₄×C₄₁	—	2	1	no	—
93	C₂×C₄×C₄₁	—	2	1	no	—
94	C₂×C₄×C₄₁	—	2	1	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	1	no	—
97	C₂×C₄×C₄₁	—	2	1	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	1	no	—
104	C₂×C₄×C₄₁	—	2	1	no	—
105	C₂×C₄×C₄₁	—	4	2	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	—	4	2	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	4	2	no	—
110	C₂×C₄×C₄₁	#101	1	2	yes	—
111	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
119	C₂×C₄×C₄₁	#137	1	2	yes	—
120	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
121	C₈×C₄₁	—	4	4	no	—
122	C₈×C₄₁	—	4	4	no	—
123	#8233	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₁	C₂×#101	1	1	yes	characteristic
126	C₂²×C₄₁	#413	1	1	yes	characteristic
127	C₂²×C₄₁	#417	1	1	yes	characteristic
128	C₂²×C₄₁	#417	1	1	yes	characteristic
129	C₂²×C₄₁	#418	1	1	yes	characteristic
130	C₂²×C₄₁	#419	1	1	yes	characteristic
131	C₄×C₄₁	—	2	2	no	—
132	C₄×C₄₁	—	2	1	no	—
133	C₄×C₄₁	—	2	1	no	—
134	C₄×C₄₁	—	4	1	no	—
135	C₄×C₄₁	—	4	1	no	—
136	C₄×C₄₁	—	4	1	no	—
137	C₄×C₄₁	—	4	4	no	—
138	C₄×C₄₁	—	8	2	no	—
139	C₄×C₄₁	#440	1	2	yes	—
140	C₄×C₄₁	#440	1	2	yes	—
141	C₄×C₄₁	#441	1	2	yes	—
142	C₄×C₄₁	#442	1	2	yes	—
143	C₄×C₄⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
144	#1767	C₂×C₄₁	1	1	yes	characteristic
145	#1767	C₂×C₄₁	1	1	yes	characteristic
146	C₂×#332	C₂×C₄₁	1	2	yes	—
147	#1296	C₂×C₄₁	1	2	yes	—
148	C₂×C₄₁	#1556	1	1	yes	characteristic
149	C₂×C₄₁	#1557	1	1	yes	characteristic
150	C₂×C₄₁	#1559	1	1	yes	characteristic
151	C₂×C₄₁	#1561	1	1	yes	characteristic
152	C₂×C₄₁	C₄×#101	1	1	yes	characteristic
153	C₂×C₄₁	#1762	1	1	yes	characteristic
154	C₂×C₄₁	#1763	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×C₄×Q₈	—	2	1	no	—
157	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
159	C₂×#97	—	2	1	no	—
160	C₂×#97	—	2	1	no	—
161	#391	—	2	1	no	—
162	#389	—	2	1	no	—
163	#389	C₄×C₄₁	1	2	yes	—
164	#384	C₂²×C₄₁	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	2	1	no	—
166	C₄×C₄⋊C₄	C₄×C₄₁	1	2	yes	—
167	C₄³	C₂²×C₄₁	1	1	yes	characteristic
168	#336	—	2	1	no	—
169	#336	—	2	1	no	—
170	#332	C₄×C₄₁	1	8	yes	complemented
171	C₄₁	#8233	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	C₂×C₄×C₄₁	1	4	yes	—
177	C₄×Q₈	—	4	4	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
195	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
196	#97	—	4	2	no	—
197	#97	—	4	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	C₄²×C₄₁	1	4	yes	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	4	no	—
236	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
237	C₄²	#41×C₄₁	1	2	yes	—
238	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	#101×C₄₁	1	2	yes	—
261	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
262	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
265	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
266	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
267	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
268	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
269	C₂×C₄	#137×C₄₁	1	2	yes	—
270	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
271	C₈	—	4	4	no	—
272	C₈	—	4	4	no	—
273	C₂²	C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
274	C₂²	C₂×#101×C₄₁	1	1	yes	characteristic
275	C₂²	C₄₁×#413	1	1	yes	characteristic
276	C₂²	C₄₁×#417	1	1	yes	characteristic
277	C₂²	C₄₁×#417	1	1	yes	characteristic
278	C₂²	C₄₁×#418	1	1	yes	characteristic
279	C₂²	C₄₁×#419	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	4	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	8	2	no	—
288	C₄	C₄₁×#440	1	2	yes	—
289	C₄	C₄₁×#440	1	2	yes	—
290	C₄	C₄₁×#441	1	2	yes	—
291	C₄	C₄₁×#442	1	2	yes	—
292	C₂	C₄₁×#1556	1	1	yes	characteristic
293	C₂	C₄₁×#1557	1	1	yes	characteristic
294	C₂	C₄₁×#1559	1	1	yes	characteristic
295	C₂	C₄₁×#1561	1	1	yes	characteristic
296	C₂	C₄×#101×C₄₁	1	1	yes	characteristic
297	C₂	C₄₁×#1762	1	1	yes	characteristic
298	C₂	C₄₁×#1763	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₄₁×#332	C₄	semidirect
123. #8233	C₄₁	direct (N × H)
170. #332	C₄×C₄₁	semidirect
171. C₄₁	#8233	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₄₁	2
C₄₁×#1767	2
C₂×C₄₁×#332	2
C₄₁×#1296	2
#8233	41

Subgroups — C41×#8233 #23784434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8233 #23784434