Subgroups — C₄₁×#11347 #23787548

Order 10496.

666 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2296	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1742	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1740	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#339	C₂	1	2	yes	maximal
5	C₂×C₄₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
7	C₂×#117×C₄₁	C₂²	1	2	yes	—
8	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#389	—	2	2	no	—
12	C₄₁×#385	—	2	2	no	—
13	C₄×C₄⋊C₄×C₄₁	C₄	1	4	yes	complemented
14	C₄₁×#343	—	2	2	no	—
15	C₄₁×#342	—	2	2	no	—
16	C₄₁×#339	—	2	4	no	—
17	C₄₁×H(3,ℤ/4)	—	2	4	no	—
18	C₂³×C₄×C₄₁	—	2	1	no	—
19	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₄₁	—	2	2	no	—
21	C₂×M₁₆×C₄₁	—	2	2	no	—
22	C₂×M₁₆×C₄₁	—	2	2	no	—
23	C₂×M₁₆×C₄₁	—	2	2	no	—
24	C₂×M₁₆×C₄₁	—	2	2	no	—
25	C₂×M₁₆×C₄₁	Q₈	1	4	yes	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	#117×C₄₁	—	2	2	no	—
28	#117×C₄₁	—	2	2	no	—
29	#117×C₄₁	—	2	2	no	—
30	#117×C₄₁	—	2	2	no	—
31	#117×C₄₁	—	2	2	no	—
32	#117×C₄₁	—	2	2	no	—
33	#117×C₄₁	—	2	2	no	—
34	#117×C₄₁	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
42	C₂×C₄²×C₄₁	—	2	2	no	—
43	C₂×C₄²×C₄₁	—	2	2	no	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	—	2	1	no	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
49	#95×C₄₁	—	2	4	no	—
50	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	—	2	2	no	—
52	C₂²×C₄×C₄₁	—	2	2	no	—
53	C₂²×C₄×C₄₁	—	4	4	no	—
54	C₂²×C₄×C₄₁	—	4	2	no	—
55	C₂²×C₄×C₄₁	—	4	2	no	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₁	#41	1	2	yes	—
62	C₂²×C₄×C₄₁	#41	1	2	yes	—
63	C₂²×C₄×C₄₁	#41	1	2	yes	—
64	C₂²×C₄×C₄₁	#41	1	2	yes	—
65	M₁₆×C₄₁	—	2	4	no	—
66	M₁₆×C₄₁	—	2	4	no	—
67	M₁₆×C₄₁	—	4	2	no	—
68	M₁₆×C₄₁	—	4	2	no	—
69	C₂×C₈×C₄₁	—	2	4	no	—
70	C₂×C₈×C₄₁	—	4	2	no	—
71	C₂×C₈×C₄₁	—	4	2	no	—
72	C₄⋊C₄×C₄₁	—	2	4	no	—
73	C₄⋊C₄×C₄₁	—	2	4	no	—
74	C₄⋊C₄×C₄₁	—	4	8	no	—
75	C₄⋊C₄×C₄₁	—	4	1	no	—
76	C₄⋊C₄×C₄₁	—	4	2	no	—
77	C₄⋊C₄×C₄₁	—	4	1	no	—
78	C₂²⋊C₄×C₄₁	—	2	4	no	—
79	C₂²⋊C₄×C₄₁	—	2	4	no	—
80	C₂²⋊C₄×C₄₁	—	4	2	no	—
81	C₂²⋊C₄×C₄₁	—	4	2	no	—
82	C₄²×C₄₁	—	2	2	no	—
83	C₄²×C₄₁	—	2	2	no	—
84	C₄²×C₄₁	—	2	4	no	—
85	C₄²×C₄₁	—	2	4	no	—
86	C₄²×C₄₁	—	2	4	no	—
87	C₄²×C₄₁	—	4	1	no	—
88	C₄²×C₄₁	—	4	2	no	—
89	C₄²×C₄₁	—	4	1	no	—
90	C₂³×C₄₁	—	4	2	no	—
91	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₁	#121	1	2	yes	—
93	C₂³×C₄₁	#123	1	2	yes	—
94	C₂³×C₄₁	#124	1	1	yes	characteristic
95	C₂³×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₄₁	—	2	4	no	—
97	C₂×C₄×C₄₁	—	2	4	no	—
98	C₂×C₄×C₄₁	—	4	2	no	—
99	C₂×C₄×C₄₁	—	4	8	no	—
100	C₂×C₄×C₄₁	—	4	2	no	—
101	C₂×C₄×C₄₁	—	4	4	no	—
102	C₂×C₄×C₄₁	—	4	4	no	—
103	C₂×C₄×C₄₁	—	4	2	no	—
104	C₂×C₄×C₄₁	—	4	4	no	—
105	C₂×C₄×C₄₁	—	8	4	no	—
106	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₁	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₁	#123	1	2	yes	—
114	C₂×C₄×C₄₁	#123	1	2	yes	—
115	C₂×C₄×C₄₁	#123	1	2	yes	—
116	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
117	C₂×C₄×C₄₁	#125	1	2	yes	—
118	C₂×C₄×C₄₁	#125	1	2	yes	—
119	C₂×C₄×C₄₁	#126	1	2	yes	—
120	C₂×C₄×C₄₁	#126	1	2	yes	—
121	C₂×C₄×C₄₁	#127	1	1	yes	characteristic
122	C₈×C₄₁	—	4	4	no	—
123	#11347	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₁	—	2	2	no	—
125	C₂²×C₄₁	—	2	4	no	—
126	C₂²×C₄₁	—	4	2	no	—
127	C₂²×C₄₁	—	4	1	no	—
128	C₂²×C₄₁	—	4	1	no	—
129	C₂²×C₄₁	#381	1	1	yes	characteristic
130	C₂²×C₄₁	#390	1	1	yes	characteristic
131	C₂²×C₄₁	#391	1	1	yes	characteristic
132	C₂²×C₄₁	#397	1	2	yes	—
133	C₂²×C₄₁	#398	1	2	yes	—
134	C₄×C₄₁	—	4	4	no	—
135	C₄×C₄₁	—	4	4	no	—
136	C₄×C₄₁	—	8	2	no	—
137	C₄×C₄₁	—	8	4	no	—
138	C₄×C₄₁	#387	1	2	yes	—
139	C₄×C₄₁	#387	1	2	yes	—
140	C₄×C₄₁	#399	1	2	yes	—
141	C₄×C₄₁	#400	1	2	yes	—
142	#2296	C₂×C₄₁	1	1	yes	characteristic
143	#1742	C₂×C₄₁	1	1	yes	characteristic
144	#1740	C₂×C₄₁	1	1	yes	characteristic
145	C₂×#339	C₂×C₄₁	1	2	yes	—
146	C₂×H(3,ℤ/4)	C₂×C₄₁	1	2	yes	—
147	C₂×C₄₁	—	2	1	no	—
148	C₂×C₄₁	—	2	2	no	—
149	C₂×C₄₁	—	2	1	no	—
150	C₂×C₄₁	—	2	2	no	—
151	C₂×C₄₁	#1445	1	1	yes	characteristic
152	C₂×C₄₁	#1827	1	2	yes	—
153	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
154	C₂×#117	C₂²×C₄₁	1	2	yes	—
155	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
157	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
158	#389	—	2	2	no	—
159	#385	—	2	2	no	—
160	C₄×C₄⋊C₄	C₄×C₄₁	1	4	yes	complemented
161	#343	—	2	2	no	—
162	#342	—	2	2	no	—
163	#339	—	2	4	no	—
164	H(3,ℤ/4)	—	2	4	no	—
165	C₄₁	#11347	1	1	yes	characteristic, direct product factor, Sylow
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	Q₈×C₄₁	1	4	yes	—
174	C₂²×C₈	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	D₄×C₄₁	1	4	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	1	no	—
194	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
197	#95	—	2	4	no	—
198	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
209	C₂²×C₄	#41×C₄₁	1	2	yes	—
210	C₂²×C₄	#41×C₄₁	1	2	yes	—
211	C₂²×C₄	#41×C₄₁	1	2	yes	—
212	C₂²×C₄	#41×C₄₁	1	2	yes	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	1	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
240	C₂³	#121×C₄₁	1	2	yes	—
241	C₂³	#123×C₄₁	1	2	yes	—
242	C₂³	#124×C₄₁	1	1	yes	characteristic
243	C₂³	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
255	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
257	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
258	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
259	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
260	C₂×C₄	C₂²⋊Q₈×C₄₁	1	2	yes	—
261	C₂×C₄	#123×C₄₁	1	2	yes	—
262	C₂×C₄	#123×C₄₁	1	2	yes	—
263	C₂×C₄	#123×C₄₁	1	2	yes	—
264	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
265	C₂×C₄	#125×C₄₁	1	2	yes	—
266	C₂×C₄	#125×C₄₁	1	2	yes	—
267	C₂×C₄	#126×C₄₁	1	2	yes	—
268	C₂×C₄	#126×C₄₁	1	2	yes	—
269	C₂×C₄	#127×C₄₁	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	C₄₁×#381	1	1	yes	characteristic
277	C₂²	C₄₁×#390	1	1	yes	characteristic
278	C₂²	C₄₁×#391	1	1	yes	characteristic
279	C₂²	C₄₁×#397	1	2	yes	—
280	C₂²	C₄₁×#398	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₄₁×#387	1	2	yes	—
286	C₄	C₄₁×#387	1	2	yes	—
287	C₄	C₄₁×#399	1	2	yes	—
288	C₄	C₄₁×#400	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	C₄₁×#1445	1	1	yes	characteristic
294	C₂	C₄₁×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	#117
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
socle	C₂²×C₄₁	#381
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×#117×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₄⋊C₄×C₄₁	C₄	semidirect
123. #11347	C₄₁	direct (N × H)
160. C₄×C₄⋊C₄	C₄×C₄₁	semidirect
165. C₄₁	#11347	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2296	2
C₄₁×#1742	2
C₄₁×#1740	2
C₂×C₄₁×#339	2
C₂×C₄₁×H(3,ℤ/4)	2
#11347	41

Subgroups — C41×#11347 #23787548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#11347 #23787548