Subgroups — C₄₁×#2323 #19106103

Order 5248.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#389	C₂	1	2	yes	maximal, complemented
6	C₄₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#388	C₂	1	2	yes	maximal
8	C₄₁×#388	C₂	1	2	yes	maximal
9	C₄₁×#387	C₂	1	2	yes	maximal, complemented
10	C₄₁×#380	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₄₁	C₄	1	8	yes	complemented
15	C₄×Q₈×C₄₁	—	2	4	no	—
16	#117×C₄₁	—	2	4	no	—
17	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
31	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
33	#95×C₄₁	C₂²	1	2	yes	—
34	#95×C₄₁	C₂²	1	2	yes	—
35	#95×C₄₁	C₂²	1	2	yes	—
36	#95×C₄₁	C₂²	1	2	yes	—
37	#95×C₄₁	C₂²	1	2	yes	—
38	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₄₁	—	2	2	no	—
40	C₂×Q₈×C₄₁	—	2	2	no	—
41	C₂×Q₈×C₄₁	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₄₁	—	2	2	no	—
43	C₂²×C₄×C₄₁	—	2	2	no	—
44	C₂²×C₄×C₄₁	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₄₁	—	2	2	no	—
56	C₄⋊C₄×C₄₁	—	2	2	no	—
57	C₄⋊C₄×C₄₁	—	2	4	no	—
58	C₄⋊C₄×C₄₁	—	2	2	no	—
59	C₄⋊C₄×C₄₁	—	2	2	no	—
60	C₄⋊C₄×C₄₁	—	4	2	no	—
61	C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
62	C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
63	C₂²⋊C₄×C₄₁	—	2	2	no	—
64	C₂²⋊C₄×C₄₁	—	2	2	no	—
65	C₂²⋊C₄×C₄₁	—	4	2	no	—
66	C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
67	C₄²×C₄₁	—	2	2	no	—
68	C₄²×C₄₁	—	2	4	no	—
69	C₄²×C₄₁	—	4	2	no	—
70	C₄²×C₄₁	D₄	1	4	yes	—
71	C₂³×C₄₁	—	2	1	no	—
72	C₂³×C₄₁	—	2	1	no	—
73	C₂³×C₄₁	—	2	1	no	—
74	C₂³×C₄₁	—	2	1	no	—
75	C₂³×C₄₁	—	2	1	no	—
76	C₂³×C₄₁	—	2	1	no	—
77	C₂³×C₄₁	C₂²×C₄	1	2	yes	—
78	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
79	Q₈×C₄₁	—	2	8	no	—
80	C₂×C₄×C₄₁	—	2	2	no	—
81	C₂×C₄×C₄₁	—	2	1	no	—
82	C₂×C₄×C₄₁	—	2	1	no	—
83	C₂×C₄×C₄₁	—	2	2	no	—
84	C₂×C₄×C₄₁	—	2	1	no	—
85	C₂×C₄×C₄₁	—	2	2	no	—
86	C₂×C₄×C₄₁	—	2	2	no	—
87	C₂×C₄×C₄₁	—	2	1	no	—
88	C₂×C₄×C₄₁	—	2	1	no	—
89	C₂×C₄×C₄₁	—	2	1	no	—
90	C₂×C₄×C₄₁	—	2	8	no	—
91	C₂×C₄×C₄₁	—	2	1	no	—
92	C₂×C₄×C₄₁	—	2	1	no	—
93	C₂×C₄×C₄₁	—	2	2	no	—
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	2	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	2	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₄₁	#41	1	2	yes	—
111	C₂²×C₄₁	—	2	2	no	—
112	C₂²×C₄₁	—	2	2	no	—
113	C₂²×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	—	4	1	no	—
115	C₂²×C₄₁	—	4	1	no	—
116	C₂²×C₄₁	—	4	1	no	—
117	C₂²×C₄₁	—	4	1	no	—
118	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂²×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₄₁	—	2	2	no	—
126	C₄×C₄₁	—	2	4	no	—
127	C₄×C₄₁	—	2	4	no	—
128	C₄×C₄₁	—	2	2	no	—
129	C₄×C₄₁	—	2	2	no	—
130	C₄×C₄₁	—	4	1	no	—
131	C₄×C₄₁	—	4	1	no	—
132	C₄×C₄₁	—	4	2	no	—
133	C₄×C₄₁	—	4	2	no	—
134	C₄×C₄₁	C₄×D₄	1	4	yes	—
135	#2323	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₄₁	—	4	2	no	—
137	C₂×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₄₁	#519	1	1	yes	characteristic
139	C₂×C₄₁	#520	1	1	yes	characteristic
140	C₂×C₄₁	#536	1	1	yes	characteristic
141	C₂×C₄₁	#537	1	1	yes	characteristic
142	C₂×C₄₁	#541	1	1	yes	characteristic
143	C₂×C₄₁	#542	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
146	C₂×#117	C₂×C₄₁	1	1	yes	characteristic
147	#391	C₂×C₄₁	1	1	yes	characteristic, complemented
148	#389	C₂×C₄₁	1	2	yes	complemented
149	#389	C₂×C₄₁	1	1	yes	characteristic, complemented
150	#388	C₂×C₄₁	1	2	yes	—
151	#388	C₂×C₄₁	1	2	yes	—
152	#387	C₂×C₄₁	1	2	yes	complemented
153	#380	C₂×C₄₁	1	1	yes	characteristic
154	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
155	C₄₁	#2323	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×Q₈	C₂²×C₄₁	1	1	yes	characteristic
157	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
158	C₂²⋊Q₈	C₄×C₄₁	1	8	yes	complemented
159	C₄×Q₈	—	2	4	no	—
160	#117	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
173	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
175	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
176	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
177	#95	C₂²×C₄₁	1	2	yes	—
178	#95	C₂²×C₄₁	1	2	yes	—
179	#95	C₂²×C₄₁	1	2	yes	—
180	#95	C₂²×C₄₁	1	2	yes	—
181	#95	C₂²×C₄₁	1	2	yes	—
182	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₂×C₄×C₄₁	1	4	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂×C₄×C₄₁	1	4	yes	—
189	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
191	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
193	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
194	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
195	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
206	C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	4	2	no	—
214	C₄²	D₄×C₄₁	1	4	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₂²×C₄×C₄₁	1	2	yes	—
222	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
223	Q₈	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	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	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
248	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
249	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
250	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
252	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
253	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
254	C₂×C₄	#41×C₄₁	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
263	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
264	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
265	C₂²	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
266	C₂²	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
267	C₂²	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
268	C₂²	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	2	no	—
278	C₄	C₄×D₄×C₄₁	1	4	yes	—
279	C₂	—	4	2	no	—
280	C₂	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
281	C₂	C₄₁×#519	1	1	yes	characteristic
282	C₂	C₄₁×#520	1	1	yes	characteristic
283	C₂	C₄₁×#536	1	1	yes	characteristic
284	C₂	C₄₁×#537	1	1	yes	characteristic
285	C₂	C₄₁×#541	1	1	yes	characteristic
286	C₂	C₄₁×#542	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₄₁
Frattini, cosocle	C₂³	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂³
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
5. C₄₁×#389	C₂	semidirect
6. C₄₁×#389	C₂	semidirect
9. C₄₁×#387	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
14. C₂²⋊Q₈×C₄₁	C₄	semidirect
135. #2323	C₄₁	direct (N × H)
145. C₂×C₄×Q₈	C₂×C₄₁	semidirect
147. #391	C₂×C₄₁	semidirect
148. #389	C₂×C₄₁	semidirect
149. #389	C₂×C₄₁	semidirect
152. #387	C₂×C₄₁	semidirect
154. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
155. C₄₁	#2323	direct (N × H)
158. C₂²⋊Q₈	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×C₂²⋊Q₈×C₄₁	2
C₂×C₄×Q₈×C₄₁	2
C₂×#117×C₄₁	2
C₄₁×#391	2
C₄₁×#389	2
C₄₁×#388	2
C₄₁×#387	2
C₄₁×#380	2
C₄×C₄⋊C₄×C₄₁	2
#2323	41

Subgroups — C41×#2323 #19106103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2323 #19106103