Subgroups — C₄₁×#2367 #19106147

Order 5248.

632 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#390	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#388	C₂	1	2	yes	maximal, complemented
7	C₄₁×#388	C₂	1	2	yes	maximal, complemented
8	C₄₁×#385	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₄₁	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₄₁	C₂²	1	2	yes	—
15	#124×C₄₁	C₄	1	8	yes	complemented
16	C₄×Q₈×C₄₁	—	2	4	no	—
17	C₄×D₄×C₄₁	—	2	4	no	—
18	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
30	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
33	#95×C₄₁	C₂²	1	1	yes	characteristic
34	#95×C₄₁	C₂²	1	2	yes	—
35	#95×C₄₁	C₂²	1	2	yes	—
36	#95×C₄₁	C₂²	1	1	yes	characteristic
37	#95×C₄₁	C₂²	1	2	yes	—
38	C₂⁴×C₄₁	C₂³	1	2	yes	—
39	C₂×Q₈×C₄₁	—	2	2	no	—
40	C₂×Q₈×C₄₁	—	2	2	no	—
41	C₂×Q₈×C₄₁	C₂×C₄	1	4	yes	complemented
42	C₂×D₄×C₄₁	—	2	2	no	—
43	C₂×D₄×C₄₁	—	2	2	no	—
44	C₂×D₄×C₄₁	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₄₁	—	2	4	no	—
46	C₂²×C₄×C₄₁	—	2	4	no	—
47	C₂²×C₄×C₄₁	—	4	2	no	—
48	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₄₁	—	2	4	no	—
59	C₄⋊C₄×C₄₁	—	2	2	no	—
60	C₄⋊C₄×C₄₁	—	2	2	no	—
61	C₄⋊C₄×C₄₁	—	2	2	no	—
62	C₄⋊C₄×C₄₁	D₄	1	4	yes	—
63	C₂²⋊C₄×C₄₁	—	2	4	no	—
64	C₂²⋊C₄×C₄₁	—	2	4	no	—
65	C₂²⋊C₄×C₄₁	—	4	2	no	—
66	C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
67	C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	complemented
68	C₄²×C₄₁	—	2	2	no	—
69	C₄²×C₄₁	—	2	4	no	—
70	C₄²×C₄₁	—	2	2	no	—
71	C₄²×C₄₁	—	4	2	no	—
72	C₄²×C₄₁	C₂×C₄	1	4	yes	complemented
73	C₂³×C₄₁	—	2	2	no	—
74	C₂³×C₄₁	—	2	2	no	—
75	C₂³×C₄₁	—	2	2	no	—
76	C₂³×C₄₁	—	2	2	no	—
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	C₂²×C₄	1	4	yes	—
80	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
81	Q₈×C₄₁	—	2	8	no	—
82	D₄×C₄₁	—	2	8	no	—
83	C₂×C₄×C₄₁	—	2	1	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	1	no	—
86	C₂×C₄×C₄₁	—	2	1	no	—
87	C₂×C₄×C₄₁	—	2	1	no	—
88	C₂×C₄×C₄₁	—	2	2	no	—
89	C₂×C₄×C₄₁	—	2	2	no	—
90	C₂×C₄×C₄₁	—	2	1	no	—
91	C₂×C₄×C₄₁	—	2	2	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	1	no	—
96	C₂×C₄×C₄₁	—	2	1	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	1	no	—
104	C₂×C₄×C₄₁	—	4	8	no	—
105	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₄₁	C₂×D₄	1	4	yes	—
110	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₄₁	#41	1	2	yes	—
112	C₂×C₄×C₄₁	#41	1	2	yes	—
113	C₂×C₄×C₄₁	#41	1	2	yes	—
114	C₂²×C₄₁	—	2	4	no	—
115	C₂²×C₄₁	—	2	4	no	—
116	C₂²×C₄₁	—	2	4	no	—
117	C₂²×C₄₁	—	4	2	no	—
118	C₂²×C₄₁	—	4	2	no	—
119	C₂²×C₄₁	—	4	2	no	—
120	C₂²×C₄₁	—	4	2	no	—
121	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
122	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
124	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
125	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
126	C₂²×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₄₁	—	2	4	no	—
129	C₄×C₄₁	—	2	2	no	—
130	C₄×C₄₁	—	2	2	no	—
131	C₄×C₄₁	—	2	2	no	—
132	C₄×C₄₁	—	2	4	no	—
133	C₄×C₄₁	—	2	2	no	—
134	C₄×C₄₁	—	4	1	no	—
135	C₄×C₄₁	—	4	1	no	—
136	C₄×C₄₁	—	4	2	no	—
137	C₄×C₄₁	C₄×D₄	1	4	yes	—
138	#2367	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₄₁	—	4	4	no	—
140	C₂×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₄₁	C₄×#41	1	1	yes	characteristic
142	C₂×C₄₁	#520	1	1	yes	characteristic
143	C₂×C₄₁	#546	1	1	yes	characteristic
144	C₂×C₄₁	#548	1	1	yes	characteristic
145	C₂×C₄₁	#556	1	1	yes	characteristic
146	C₂×C₄₁	#559	1	1	yes	characteristic
147	C₂×#124	C₂×C₄₁	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
149	C₂×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic
150	#391	C₂×C₄₁	1	1	yes	characteristic, complemented
151	#390	C₂×C₄₁	1	1	yes	characteristic
152	#388	C₂×C₄₁	1	2	yes	complemented
153	#388	C₂×C₄₁	1	2	yes	complemented
154	#385	C₂×C₄₁	1	2	yes	complemented
155	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
156	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
157	C₄×C₂²⋊C₄	C₂×C₄₁	1	2	yes	complemented
158	C₄₁	#2367	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²×Q₈	C₂²×C₄₁	1	1	yes	characteristic
160	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
161	C₂³×C₄	C₂²×C₄₁	1	2	yes	—
162	#124	C₄×C₄₁	1	8	yes	complemented
163	C₄×Q₈	—	2	4	no	—
164	C₄×D₄	—	2	4	no	—
165	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
167	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
175	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
177	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
179	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
180	#95	C₂²×C₄₁	1	1	yes	characteristic
181	#95	C₂²×C₄₁	1	2	yes	—
182	#95	C₂²×C₄₁	1	2	yes	—
183	#95	C₂²×C₄₁	1	1	yes	characteristic
184	#95	C₂²×C₄₁	1	2	yes	—
185	C₂⁴	C₂³×C₄₁	1	2	yes	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	C₂×C₄×C₄₁	1	4	yes	complemented
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	C₂×C₄×C₄₁	1	4	yes	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
197	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
201	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
202	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	D₄×C₄₁	1	4	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
214	C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	complemented
215	C₄²	—	2	2	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	C₂×C₄×C₄₁	1	4	yes	complemented
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂²×C₄×C₄₁	1	4	yes	—
227	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
228	Q₈	—	2	8	no	—
229	D₄	—	2	8	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
253	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
254	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
255	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
256	C₂×C₄	C₂×D₄×C₄₁	1	4	yes	—
257	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
258	C₂×C₄	#41×C₄₁	1	2	yes	—
259	C₂×C₄	#41×C₄₁	1	2	yes	—
260	C₂×C₄	#41×C₄₁	1	2	yes	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
269	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
270	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
271	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
272	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
273	C₂²	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
274	C₂²	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	1	no	—
282	C₄	—	4	1	no	—
283	C₄	—	4	2	no	—
284	C₄	C₄×D₄×C₄₁	1	4	yes	—
285	C₂	—	4	4	no	—
286	C₂	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
287	C₂	C₄×#41×C₄₁	1	1	yes	characteristic
288	C₂	C₄₁×#520	1	1	yes	characteristic
289	C₂	C₄₁×#546	1	1	yes	characteristic
290	C₂	C₄₁×#548	1	1	yes	characteristic
291	C₂	C₄₁×#556	1	1	yes	characteristic
292	C₂	C₄₁×#559	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₄₁	C₂	semidirect
4. C₄₁×#391	C₂	semidirect
6. C₄₁×#388	C₂	semidirect
7. C₄₁×#388	C₂	semidirect
8. C₄₁×#385	C₂	semidirect
9. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
10. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
11. C₄×C₂²⋊C₄×C₄₁	C₂	semidirect
15. #124×C₄₁	C₄	semidirect
41. C₂×Q₈×C₄₁	C₂×C₄	semidirect
66. C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
67. C₂²⋊C₄×C₄₁	C₂×C₄	semidirect
72. C₄²×C₄₁	C₂×C₄	semidirect
138. #2367	C₄₁	direct (N × H)
148. C₂×C₄×Q₈	C₂×C₄₁	semidirect
150. #391	C₂×C₄₁	semidirect
152. #388	C₂×C₄₁	semidirect
153. #388	C₂×C₄₁	semidirect
154. #385	C₂×C₄₁	semidirect
155. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
156. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
157. C₄×C₂²⋊C₄	C₂×C₄₁	semidirect
158. C₄₁	#2367	direct (N × H)
162. #124	C₄×C₄₁	semidirect
188. C₂×Q₈	C₂×C₄×C₄₁	semidirect
213. C₂²⋊C₄	C₂×C₄×C₄₁	semidirect
214. C₂²⋊C₄	C₂×C₄×C₄₁	semidirect
219. C₄²	C₂×C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#124×C₄₁	2
C₂×C₄×Q₈×C₄₁	2
C₂×C₄×D₄×C₄₁	2
C₄₁×#391	2
C₄₁×#390	2
C₄₁×#388	2
C₄₁×#385	2
C₄×C₄⋊C₄×C₄₁	2
C₄×C₂²⋊C₄×C₄₁	2
#2367	41

Subgroups — C41×#2367 #19106147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2367 #19106147