Subgroups — C₄₁×#3232 #19107012

Order 5248.

334 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#499	C₂	1	2	yes	maximal, complemented
4	C₄₁×#488	C₂	1	2	yes	maximal, complemented
5	C₄₁×#484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#481	C₂	1	1	yes	maximal, characteristic
8	C₄₁×#480	C₂	1	1	yes	maximal, characteristic
9	C₄₁×#479	C₂	1	2	yes	maximal, complemented
10	C₄₁×#478	C₂	1	2	yes	maximal, complemented
11	C₄₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	#126×C₄₁	—	2	1	no	—
13	#125×C₄₁	—	2	1	no	—
14	#125×C₄₁	—	2	1	no	—
15	#125×C₄₁	C₂²	1	2	yes	—
16	#125×C₄₁	C₂²	1	2	yes	—
17	#125×C₄₁	C₂²	1	1	yes	characteristic
18	#124×C₄₁	C₂²	1	1	yes	characteristic
19	#123×C₄₁	—	2	1	no	—
20	C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	#121×C₄₁	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₁	—	2	1	no	—
24	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
25	#117×C₄₁	—	2	1	no	—
26	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
31	#107×C₄₁	C₂²	1	2	yes	—
32	#107×C₄₁	C₂²	1	2	yes	—
33	#106×C₄₁	C₂²	1	2	yes	—
34	#106×C₄₁	C₂²	1	2	yes	—
35	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
36	#103×C₄₁	C₂²	1	2	yes	—
37	#103×C₄₁	C₂²	1	2	yes	—
38	#102×C₄₁	C₂²	1	2	yes	—
39	#102×C₄₁	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₄₁	C₂²	1	2	yes	—
43	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₄₁	—	2	1	no	—
45	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₁	—	2	1	no	—
47	C₂²×C₄×C₄₁	—	2	1	no	—
48	C₂²×C₄×C₄₁	—	2	1	no	—
49	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₄₁	C₂³	1	2	yes	—
53	C₂×C₈×C₄₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₄₁	—	2	2	no	—
55	C₄⋊C₄×C₄₁	—	2	2	no	—
56	C₄⋊C₄×C₄₁	—	2	2	no	—
57	C₄⋊C₄×C₄₁	—	2	2	no	—
58	C₄⋊C₄×C₄₁	—	2	1	no	—
59	C₄⋊C₄×C₄₁	—	2	1	no	—
60	C₄⋊C₄×C₄₁	—	2	1	no	—
61	C₄⋊C₄×C₄₁	—	2	1	no	—
62	C₄⋊C₄×C₄₁	—	2	1	no	—
63	C₄⋊C₄×C₄₁	—	2	1	no	—
64	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
65	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₄₁	—	2	1	no	—
69	C₂²⋊C₄×C₄₁	—	2	1	no	—
70	C₂²⋊C₄×C₄₁	—	2	1	no	—
71	C₂²⋊C₄×C₄₁	—	2	1	no	—
72	C₂²⋊C₄×C₄₁	—	2	1	no	—
73	C₄²×C₄₁	—	2	1	no	—
74	C₄²×C₄₁	D₄	1	1	yes	characteristic
75	C₄²×C₄₁	D₄	1	1	yes	characteristic
76	C₄²×C₄₁	C₂³	1	1	yes	characteristic
77	C₄²×C₄₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄₁	—	2	1	no	—
79	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₄₁	—	2	2	no	—
81	D₄×C₄₁	—	2	2	no	—
82	D₄×C₄₁	—	4	2	no	—
83	C₂×C₄×C₄₁	—	2	2	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	—	4	1	no	—
91	C₂×C₄×C₄₁	—	4	2	no	—
92	C₂×C₄×C₄₁	—	4	2	no	—
93	C₂×C₄×C₄₁	—	4	2	no	—
94	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
100	C₈×C₄₁	—	2	2	no	—
101	C₈×C₄₁	—	2	2	no	—
102	C₂²×C₄₁	—	2	1	no	—
103	C₂²×C₄₁	—	2	1	no	—
104	C₂²×C₄₁	—	4	1	no	—
105	C₂²×C₄₁	—	4	1	no	—
106	C₂²×C₄₁	—	4	1	no	—
107	C₂²×C₄₁	#135	1	2	yes	—
108	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₄₁	—	2	1	no	—
110	C₄×C₄₁	—	2	1	no	—
111	C₄×C₄₁	—	2	1	no	—
112	C₄×C₄₁	—	2	1	no	—
113	C₄×C₄₁	—	2	1	no	—
114	C₄×C₄₁	—	4	1	no	—
115	C₄×C₄₁	—	4	2	no	—
116	C₄×C₄₁	—	4	1	no	—
117	C₄×C₄₁	—	4	1	no	—
118	C₄×C₄₁	—	4	2	no	—
119	C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
120	#3232	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₄₁	—	2	2	no	—
122	C₂×C₄₁	—	8	1	no	—
123	C₂×C₄₁	#536	1	1	yes	characteristic
124	C₂×C₄₁	C₂×#135	1	1	yes	characteristic
125	C₂×C₄₁	#575	1	1	yes	characteristic
126	#529	C₂×C₄₁	1	1	yes	characteristic
127	C₂×#125	C₂×C₄₁	1	1	yes	characteristic, complemented
128	#499	C₂×C₄₁	1	2	yes	complemented
129	#488	C₂×C₄₁	1	2	yes	complemented
130	#484	C₂×C₄₁	1	1	yes	characteristic, complemented
131	#483	C₂×C₄₁	1	1	yes	characteristic, complemented
132	#481	C₂×C₄₁	1	1	yes	characteristic
133	#480	C₂×C₄₁	1	1	yes	characteristic
134	#479	C₂×C₄₁	1	2	yes	complemented
135	#478	C₂×C₄₁	1	2	yes	complemented
136	#431	C₂×C₄₁	1	1	yes	characteristic, complemented
137	C₄₁	#3232	1	1	yes	characteristic, direct product factor, Sylow
138	#126	—	2	1	no	—
139	#125	—	2	1	no	—
140	#125	—	2	1	no	—
141	#125	C₂²×C₄₁	1	2	yes	—
142	#125	C₂²×C₄₁	1	2	yes	—
143	#125	C₂²×C₄₁	1	1	yes	characteristic
144	#124	C₂²×C₄₁	1	1	yes	characteristic
145	#123	—	2	1	no	—
146	C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
147	#121	C₂²×C₄₁	1	1	yes	characteristic
148	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
156	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
157	#107	C₂²×C₄₁	1	2	yes	—
158	#107	C₂²×C₄₁	1	2	yes	—
159	#106	C₂²×C₄₁	1	2	yes	—
160	#106	C₂²×C₄₁	1	2	yes	—
161	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
162	#103	C₂²×C₄₁	1	2	yes	—
163	#103	C₂²×C₄₁	1	2	yes	—
164	#102	C₂²×C₄₁	1	2	yes	—
165	#102	C₂²×C₄₁	1	2	yes	—
166	C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
167	C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
168	C₄×C₈	C₂²×C₄₁	1	2	yes	—
169	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
176	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₄₁	1	2	yes	—
179	C₂×C₈	C₂³×C₄₁	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
191	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
192	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
193	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
194	C₂²⋊C₄	—	2	1	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	D₄×C₄₁	1	1	yes	characteristic
201	C₄²	D₄×C₄₁	1	1	yes	characteristic
202	C₄²	C₂³×C₄₁	1	1	yes	characteristic
203	C₄²	C₂³×C₄₁	1	1	yes	characteristic
204	C₂³	—	2	1	no	—
205	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
225	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	1	no	—
233	C₂²	#135×C₄₁	1	2	yes	—
234	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
246	C₂	—	2	2	no	—
247	C₂	—	8	1	no	—
248	C₂	C₄₁×#536	1	1	yes	characteristic
249	C₂	C₂×#135×C₄₁	1	1	yes	characteristic
250	C₂	C₄₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
prime radical	#121×C₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#125×C₄₁	C₂	semidirect
3. C₄₁×#499	C₂	semidirect
4. C₄₁×#488	C₂	semidirect
5. C₄₁×#484	C₂	semidirect
6. C₄₁×#483	C₂	semidirect
9. C₄₁×#479	C₂	semidirect
10. C₄₁×#478	C₂	semidirect
11. C₄₁×#431	C₂	semidirect
120. #3232	C₄₁	direct (N × H)
127. C₂×#125	C₂×C₄₁	semidirect
128. #499	C₂×C₄₁	semidirect
129. #488	C₂×C₄₁	semidirect
130. #484	C₂×C₄₁	semidirect
131. #483	C₂×C₄₁	semidirect
134. #479	C₂×C₄₁	semidirect
135. #478	C₂×C₄₁	semidirect
136. #431	C₂×C₄₁	semidirect
137. C₄₁	#3232	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×#529	2
C₂×#125×C₄₁	2
C₄₁×#499	2
C₄₁×#488	2
C₄₁×#484	2
C₄₁×#483	2
C₄₁×#481	2
C₄₁×#480	2
C₄₁×#479	2
C₄₁×#478	2
C₄₁×#431	2
#3232	41

Subgroups — C41×#3232 #19107012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3232 #19107012