Subgroups — C₄₁×#33230 #23809431

Order 10496.

1034 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#489	C₂	1	4	yes	maximal, complemented
2	C₄₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2858	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1959	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1957	C₂	1	2	yes	maximal
6	C₄₁×#1956	C₂	1	2	yes	maximal, complemented
7	C₄×#97×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₄₁	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
11	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#124×C₄₁	C₂²	1	2	yes	—
13	C₂×#124×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#489	—	2	16	no	—
15	C₂×#103×C₄₁	C₂²	1	8	yes	—
16	C₂×#102×C₄₁	C₂²	1	8	yes	—
17	C₂×#97×C₄₁	C₂²	1	2	yes	—
18	C₂×#97×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
20	C₄₁×#390	—	2	2	no	—
21	C₄₁×#390	—	2	2	no	—
22	C₄₁×#384	—	2	2	no	—
23	C₄₁×#384	C₂²	1	2	yes	—
24	C₄₁×#384	C₂²	1	2	yes	—
25	C₄³×C₄₁	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₄₁	—	2	2	no	—
27	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
28	C₂²×D₄×C₄₁	—	2	2	no	—
29	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₄₁	—	2	8	no	—
32	C₄⋊Q₈×C₄₁	—	2	4	no	—
33	C₄⋊Q₈×C₄₁	—	4	8	no	—
34	#127×C₄₁	—	2	4	no	—
35	#124×C₄₁	—	2	8	no	—
36	#124×C₄₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	D₄	1	2	yes	—
46	C₂×C₄²×C₄₁	D₄	1	2	yes	—
47	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
50	#103×C₄₁	—	2	16	no	—
51	#102×C₄₁	—	2	16	no	—
52	#97×C₄₁	—	2	4	no	—
53	#97×C₄₁	D₄	1	8	yes	complemented
54	C₄×C₈×C₄₁	—	2	4	no	—
55	C₂⁴×C₄₁	—	2	2	no	—
56	C₂×Q₈×C₄₁	—	2	4	no	—
57	C₂×Q₈×C₄₁	—	2	4	no	—
58	C₂×Q₈×C₄₁	—	2	4	no	—
59	C₂×Q₈×C₄₁	—	4	8	no	—
60	C₂×Q₈×C₄₁	—	4	4	no	—
61	C₂×D₄×C₄₁	—	2	4	no	—
62	C₂×D₄×C₄₁	—	2	4	no	—
63	C₂×D₄×C₄₁	—	2	4	no	—
64	C₂×D₄×C₄₁	—	4	4	no	—
65	C₂×D₄×C₄₁	—	4	4	no	—
66	C₂×D₄×C₄₁	—	4	4	no	—
67	C₂²×C₄×C₄₁	—	2	2	no	—
68	C₂²×C₄×C₄₁	—	2	4	no	—
69	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₄₁	—	2	4	no	—
75	C₂×C₈×C₄₁	—	2	4	no	—
76	C₂×C₈×C₄₁	C₂×D₄	1	8	yes	—
77	C₄⋊C₄×C₄₁	—	2	8	no	—
78	C₄⋊C₄×C₄₁	—	4	8	no	—
79	C₄⋊C₄×C₄₁	—	4	4	no	—
80	C₄⋊C₄×C₄₁	—	4	4	no	—
81	C₄⋊C₄×C₄₁	—	4	8	no	—
82	C₂²⋊C₄×C₄₁	—	4	8	no	—
83	C₂²⋊C₄×C₄₁	—	4	4	no	—
84	C₂²⋊C₄×C₄₁	—	4	4	no	—
85	C₄²×C₄₁	—	2	4	no	—
86	C₄²×C₄₁	—	2	4	no	—
87	C₄²×C₄₁	C₂×D₄	1	4	yes	—
88	C₄²×C₄₁	#41	1	4	yes	—
89	C₄²×C₄₁	#41	1	4	yes	—
90	C₂³×C₄₁	—	4	2	no	—
91	C₂³×C₄₁	—	4	2	no	—
92	C₂³×C₄₁	—	4	2	no	—
93	C₂³×C₄₁	—	4	4	no	—
94	C₂³×C₄₁	—	4	4	no	—
95	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₄₁	—	4	8	no	—
97	Q₈×C₄₁	—	8	8	no	—
98	D₄×C₄₁	—	4	8	no	—
99	D₄×C₄₁	—	8	8	no	—
100	C₂×C₄×C₄₁	—	2	2	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	4	2	no	—
105	C₂×C₄×C₄₁	—	4	2	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	—	4	4	no	—
108	C₂×C₄×C₄₁	—	4	8	no	—
109	C₂×C₄×C₄₁	#124	1	4	yes	—
110	C₂×C₄×C₄₁	#124	1	4	yes	—
111	C₂×C₄×C₄₁	#127	1	2	yes	—
112	C₂×C₄×C₄₁	#127	1	2	yes	—
113	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₄₁	#137	1	4	yes	—
115	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
118	C₈×C₄₁	—	2	8	no	—
119	#33230	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₄₁	—	8	2	no	—
121	C₂²×C₄₁	—	8	4	no	—
122	C₂²×C₄₁	—	8	4	no	—
123	C₂²×C₄₁	—	8	2	no	—
124	C₂²×C₄₁	—	8	2	no	—
125	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
126	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
127	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
128	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
129	C₂²×C₄₁	C₂×#137	1	2	yes	—
130	C₄×C₄₁	—	2	4	no	—
131	C₄×C₄₁	—	2	2	no	—
132	C₄×C₄₁	—	2	2	no	—
133	C₄×C₄₁	—	8	2	no	—
134	C₄×C₄₁	—	8	4	no	—
135	C₄×C₄₁	#489	1	8	yes	complemented
136	C₄×C₄₁	#532	1	4	yes	—
137	C₂×#489	C₂×C₄₁	1	4	yes	complemented
138	#2868	C₂×C₄₁	1	1	yes	characteristic, complemented
139	#2858	C₂×C₄₁	1	1	yes	characteristic
140	#1959	C₂×C₄₁	1	4	yes	complemented
141	#1957	C₂×C₄₁	1	2	yes	—
142	#1956	C₂×C₄₁	1	2	yes	complemented
143	C₄×#97	C₂×C₄₁	1	1	yes	characteristic, complemented
144	C₂×C₄₁	—	16	2	no	—
145	C₂×C₄₁	#2858	1	1	yes	characteristic
146	C₂×C₄₁	C₂×#489	1	2	yes	—
147	C₂×C₄₁	#3141	1	2	yes	—
148	C₂×C₄₁	#3151	1	1	yes	characteristic
149	C₂×C₄₁	#3152	1	1	yes	characteristic
150	C₂×C₄⋊Q₈	—	2	2	no	—
151	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
153	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
154	C₂×#124	C₂²×C₄₁	1	2	yes	—
155	C₂×#124	C₂²×C₄₁	1	2	yes	—
156	#489	—	2	16	no	—
157	C₂×#103	C₂²×C₄₁	1	8	yes	—
158	C₂×#102	C₂²×C₄₁	1	8	yes	—
159	C₂×#97	C₂²×C₄₁	1	2	yes	—
160	C₂×#97	C₂²×C₄₁	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
162	#390	—	2	2	no	—
163	#390	—	2	2	no	—
164	#384	—	2	2	no	—
165	#384	C₂²×C₄₁	1	2	yes	—
166	#384	C₂²×C₄₁	1	2	yes	—
167	C₄³	C₂²×C₄₁	1	1	yes	characteristic
168	C₄₁	#33230	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
171	C₂²×D₄	—	2	2	no	—
172	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
173	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
174	C₄⋊Q₈	—	2	8	no	—
175	C₄⋊Q₈	—	2	4	no	—
176	C₄⋊Q₈	—	4	8	no	—
177	#127	—	2	4	no	—
178	#124	—	2	8	no	—
179	#124	—	2	8	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	4	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	D₄×C₄₁	1	2	yes	—
189	C₂×C₄²	D₄×C₄₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
193	#103	—	2	16	no	—
194	#102	—	2	16	no	—
195	#97	—	2	4	no	—
196	#97	D₄×C₄₁	1	8	yes	complemented
197	C₄×C₈	—	2	4	no	—
198	C₂⁴	—	2	2	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	8	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂×D₄×C₄₁	1	8	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	4	no	—
230	C₄²	C₂×D₄×C₄₁	1	4	yes	—
231	C₄²	#41×C₄₁	1	4	yes	—
232	C₄²	#41×C₄₁	1	4	yes	—
233	C₂³	—	4	2	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
239	Q₈	—	4	8	no	—
240	Q₈	—	8	8	no	—
241	D₄	—	4	8	no	—
242	D₄	—	8	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	#124×C₄₁	1	4	yes	—
253	C₂×C₄	#124×C₄₁	1	4	yes	—
254	C₂×C₄	#127×C₄₁	1	2	yes	—
255	C₂×C₄	#127×C₄₁	1	2	yes	—
256	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
257	C₂×C₄	#137×C₄₁	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
259	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
260	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
261	C₈	—	2	8	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
268	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
269	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
270	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
271	C₂²	C₂×#137×C₄₁	1	2	yes	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₄₁×#489	1	8	yes	complemented
278	C₄	C₄₁×#532	1	4	yes	—
279	C₂	—	16	2	no	—
280	C₂	C₄₁×#2858	1	1	yes	characteristic
281	C₂	C₂×C₄₁×#489	1	2	yes	—
282	C₂	C₄₁×#3141	1	2	yes	—
283	C₂	C₄₁×#3151	1	1	yes	characteristic
284	C₂	C₄₁×#3152	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×#127×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#489	C₂	semidirect
2. C₄₁×#2868	C₂	semidirect
4. C₄₁×#1959	C₂	semidirect
6. C₄₁×#1956	C₂	semidirect
7. C₄×#97×C₄₁	C₂	semidirect
53. #97×C₄₁	D₄	semidirect
119. #33230	C₄₁	direct (N × H)
135. C₄×C₄₁	#489	semidirect
137. C₂×#489	C₂×C₄₁	semidirect
138. #2868	C₂×C₄₁	semidirect
140. #1959	C₂×C₄₁	semidirect
142. #1956	C₂×C₄₁	semidirect
143. C₄×#97	C₂×C₄₁	semidirect
168. C₄₁	#33230	direct (N × H)
196. #97	D₄×C₄₁	semidirect
277. C₄	C₄₁×#489	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#489	2
C₄₁×#2868	2
C₄₁×#2858	2
C₄₁×#1959	2
C₄₁×#1957	2
C₄₁×#1956	2
C₄×#97×C₄₁	2
#33230	41

Subgroups — C41×#33230 #23809431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33230 #23809431