Subgroups — C₄₁×#3217 #19106997

Order 5248.

490 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#496	C₂	1	2	yes	maximal, complemented
4	C₄₁×#490	C₂	1	2	yes	maximal, complemented
5	C₄₁×#460	C₂	1	2	yes	maximal, complemented
6	C₄₁×#459	C₂	1	2	yes	maximal, complemented
7	C₄₁×#450	C₂	1	2	yes	maximal, complemented
8	C₄₁×#447	C₂	1	2	yes	maximal, complemented
9	C₄₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₄₁	—	2	1	no	—
11	C₂²×D₄×C₄₁	—	2	1	no	—
12	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₄₁	C₂²	1	4	yes	complemented
14	C₂×D₈×C₄₁	C₂²	1	4	yes	complemented
15	#127×C₄₁	—	2	1	no	—
16	#127×C₄₁	—	2	1	no	—
17	#127×C₄₁	C₂²	1	2	yes	—
18	#127×C₄₁	C₂²	1	2	yes	—
19	#126×C₄₁	—	2	1	no	—
20	#125×C₄₁	C₂²	1	1	yes	characteristic, complemented
21	#124×C₄₁	C₂²	1	1	yes	characteristic, complemented
22	#123×C₄₁	—	2	1	no	—
23	C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	#121×C₄₁	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
26	C₄×D₄×C₄₁	—	2	1	no	—
27	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
28	#117×C₄₁	—	2	1	no	—
29	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₄₁	C₂²	1	2	yes	complemented
31	#102×C₄₁	C₂²	1	4	yes	complemented
32	#102×C₄₁	C₂²	1	4	yes	complemented
33	C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
34	#97×C₄₁	C₂²	1	2	yes	complemented
35	C₂⁴×C₄₁	—	2	2	no	—
36	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
37	C₂×D₄×C₄₁	—	2	1	no	—
38	C₂×D₄×C₄₁	—	2	2	no	—
39	C₂×D₄×C₄₁	—	2	2	no	—
40	C₂×D₄×C₄₁	—	2	2	no	—
41	C₂×D₄×C₄₁	—	2	2	no	—
42	C₂×D₄×C₄₁	—	2	2	no	—
43	C₂×D₄×C₄₁	—	4	4	no	—
44	C₂×D₄×C₄₁	—	4	4	no	—
45	C₂×D₄×C₄₁	—	4	4	no	—
46	C₂×D₄×C₄₁	C₂³	1	4	yes	—
47	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
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	SD₁₆×C₄₁	—	4	4	no	—
53	D₈×C₄₁	—	4	4	no	—
54	C₂×C₈×C₄₁	C₂³	1	4	yes	—
55	C₄⋊C₄×C₄₁	—	2	1	no	—
56	C₄⋊C₄×C₄₁	—	2	1	no	—
57	C₄⋊C₄×C₄₁	—	2	1	no	—
58	C₄⋊C₄×C₄₁	—	2	1	no	—
59	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²⋊C₄×C₄₁	—	2	1	no	—
62	C₂²⋊C₄×C₄₁	—	2	1	no	—
63	C₂²⋊C₄×C₄₁	—	2	1	no	—
64	C₂²⋊C₄×C₄₁	—	2	1	no	—
65	C₂²⋊C₄×C₄₁	—	2	1	no	—
66	C₄²×C₄₁	—	2	1	no	—
67	C₄²×C₄₁	D₄	1	1	yes	characteristic
68	C₄²×C₄₁	D₄	1	1	yes	characteristic
69	C₄²×C₄₁	C₂³	1	1	yes	characteristic
70	C₄²×C₄₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄₁	—	2	1	no	—
72	C₂³×C₄₁	—	2	4	no	—
73	C₂³×C₄₁	—	4	4	no	—
74	C₂³×C₄₁	—	4	4	no	—
75	C₂³×C₄₁	—	4	4	no	—
76	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₄₁	—	2	2	no	—
78	D₄×C₄₁	—	2	2	no	—
79	D₄×C₄₁	—	2	4	no	—
80	D₄×C₄₁	—	2	4	no	—
81	D₄×C₄₁	—	4	4	no	—
82	D₄×C₄₁	—	4	4	no	—
83	D₄×C₄₁	—	4	4	no	—
84	D₄×C₄₁	—	4	4	no	—
85	D₄×C₄₁	—	4	2	no	—
86	D₄×C₄₁	—	4	4	no	—
87	C₂×C₄×C₄₁	—	2	2	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	1	no	—
92	C₂×C₄×C₄₁	—	2	1	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	4	no	—
101	C₂²×C₄₁	—	2	1	no	—
102	C₂²×C₄₁	—	2	1	no	—
103	C₂²×C₄₁	—	4	4	no	—
104	C₂²×C₄₁	—	4	1	no	—
105	C₂²×C₄₁	—	4	1	no	—
106	C₂²×C₄₁	—	4	4	no	—
107	C₂²×C₄₁	—	4	4	no	—
108	C₂²×C₄₁	—	4	1	no	—
109	C₂²×C₄₁	—	8	8	no	—
110	C₂²×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₄₁	—	2	1	no	—
113	C₄×C₄₁	—	2	1	no	—
114	C₄×C₄₁	—	2	1	no	—
115	C₄×C₄₁	—	2	1	no	—
116	C₄×C₄₁	—	4	1	no	—
117	C₄×C₄₁	—	4	1	no	—
118	C₄×C₄₁	—	4	1	no	—
119	C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
120	C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
121	#3217	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₄₁	—	2	2	no	—
123	C₂×C₄₁	—	8	1	no	—
124	C₂×C₄₁	—	8	4	no	—
125	C₂×C₄₁	#535	1	1	yes	characteristic
126	C₂×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×#127	C₂×C₄₁	1	1	yes	characteristic, complemented
129	#529	C₂×C₄₁	1	1	yes	characteristic, complemented
130	#496	C₂×C₄₁	1	2	yes	complemented
131	#490	C₂×C₄₁	1	2	yes	complemented
132	#460	C₂×C₄₁	1	2	yes	complemented
133	#459	C₂×C₄₁	1	2	yes	complemented
134	#450	C₂×C₄₁	1	2	yes	complemented
135	#447	C₂×C₄₁	1	2	yes	complemented
136	#432	C₂×C₄₁	1	1	yes	characteristic, complemented
137	C₄₁	#3217	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×D₄	—	2	1	no	—
139	C₂²×D₄	—	2	1	no	—
140	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
141	C₂×SD₁₆	C₂²×C₄₁	1	4	yes	complemented
142	C₂×D₈	C₂²×C₄₁	1	4	yes	complemented
143	#127	—	2	1	no	—
144	#127	—	2	1	no	—
145	#127	C₂²×C₄₁	1	2	yes	—
146	#127	C₂²×C₄₁	1	2	yes	—
147	#126	—	2	1	no	—
148	#125	C₂²×C₄₁	1	1	yes	characteristic, complemented
149	#124	C₂²×C₄₁	1	1	yes	characteristic, complemented
150	#123	—	2	1	no	—
151	C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
152	#121	C₂²×C₄₁	1	1	yes	characteristic
153	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
154	C₄×D₄	—	2	1	no	—
155	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
156	#117	—	2	1	no	—
157	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
158	C₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
159	#102	C₂²×C₄₁	1	4	yes	complemented
160	#102	C₂²×C₄₁	1	4	yes	complemented
161	C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
162	#97	C₂²×C₄₁	1	2	yes	complemented
163	C₂⁴	—	2	2	no	—
164	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	C₂³×C₄₁	1	4	yes	—
175	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
178	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
180	SD₁₆	—	4	4	no	—
181	D₈	—	4	4	no	—
182	C₂×C₈	C₂³×C₄₁	1	4	yes	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	D₄×C₄₁	1	1	yes	characteristic
196	C₄²	D₄×C₄₁	1	1	yes	characteristic
197	C₄²	C₂³×C₄₁	1	1	yes	characteristic
198	C₄²	C₂³×C₄₁	1	1	yes	characteristic
199	C₂³	—	2	1	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	4	no	—
208	D₄	—	2	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
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	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
227	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	4	4	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	8	8	no	—
238	C₂²	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
239	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	1	no	—
247	C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
248	C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
249	C₂	—	2	2	no	—
250	C₂	—	8	4	no	—
251	C₂	—	8	1	no	—
252	C₂	C₄₁×#535	1	1	yes	characteristic
253	C₂	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
254	C₂	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
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
1. C₂×#127×C₄₁	C₂	semidirect
2. C₄₁×#529	C₂	semidirect
3. C₄₁×#496	C₂	semidirect
4. C₄₁×#490	C₂	semidirect
5. C₄₁×#460	C₂	semidirect
6. C₄₁×#459	C₂	semidirect
7. C₄₁×#450	C₂	semidirect
8. C₄₁×#447	C₂	semidirect
9. C₄₁×#432	C₂	semidirect
13. C₂×SD₁₆×C₄₁	C₂²	semidirect
14. C₂×D₈×C₄₁	C₂²	semidirect
20. #125×C₄₁	C₂²	semidirect
21. #124×C₄₁	C₂²	semidirect
25. C₄×Q₈×C₄₁	C₂²	semidirect
27. C₄×D₄×C₄₁	C₂²	semidirect
30. C₄⋊C₈×C₄₁	C₂²	semidirect
31. #102×C₄₁	C₂²	semidirect
32. #102×C₄₁	C₂²	semidirect
34. #97×C₄₁	C₂²	semidirect
121. #3217	C₄₁	direct (N × H)
128. C₂×#127	C₂×C₄₁	semidirect
129. #529	C₂×C₄₁	semidirect
130. #496	C₂×C₄₁	semidirect
131. #490	C₂×C₄₁	semidirect
132. #460	C₂×C₄₁	semidirect
133. #459	C₂×C₄₁	semidirect
134. #450	C₂×C₄₁	semidirect
135. #447	C₂×C₄₁	semidirect
136. #432	C₂×C₄₁	semidirect
137. C₄₁	#3217	direct (N × H)
141. C₂×SD₁₆	C₂²×C₄₁	semidirect
142. C₂×D₈	C₂²×C₄₁	semidirect
148. #125	C₂²×C₄₁	semidirect
149. #124	C₂²×C₄₁	semidirect
153. C₄×Q₈	C₂²×C₄₁	semidirect
155. C₄×D₄	C₂²×C₄₁	semidirect
158. C₄⋊C₈	C₂²×C₄₁	semidirect
159. #102	C₂²×C₄₁	semidirect
160. #102	C₂²×C₄₁	semidirect
162. #97	C₂²×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₄₁	2
C₄₁×#529	2
C₄₁×#496	2
C₄₁×#490	2
C₄₁×#460	2
C₄₁×#459	2
C₄₁×#450	2
C₄₁×#447	2
C₄₁×#432	2
#3217	41

Subgroups — C41×#3217 #19106997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3217 #19106997