Subgroups — C₄₁×#3207 #19106987

Order 5248.

438 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#463	C₂	1	4	yes	maximal, complemented
8	C₄₁×#449	C₂	1	4	yes	maximal, complemented
9	C₄₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₄₁	—	2	1	no	—
11	C₂×#41×C₄₁	C₂²	1	2	yes	—
12	C₂×SD₁₆×C₄₁	C₂²	1	4	yes	—
13	C₂×D₈×C₄₁	C₂²	1	4	yes	—
14	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	#127×C₄₁	C₂²	1	1	yes	characteristic
17	#127×C₄₁	C₂²	1	1	yes	characteristic
18	#124×C₄₁	—	2	1	no	—
19	#124×C₄₁	C₂²	1	2	yes	—
20	#121×C₄₁	—	2	1	no	—
21	#121×C₄₁	—	2	1	no	—
22	#121×C₄₁	—	2	1	no	—
23	#121×C₄₁	C₂²	1	2	yes	—
24	C₄×D₄×C₄₁	—	2	1	no	—
25	C₄×D₄×C₄₁	—	2	1	no	—
26	C₄×D₄×C₄₁	—	2	1	no	—
27	C₄×D₄×C₄₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
30	#103×C₄₁	C₂²	1	4	yes	—
31	#102×C₄₁	C₂²	1	4	yes	—
32	C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
33	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
35	#41×C₄₁	—	4	4	no	—
36	#41×C₄₁	—	4	4	no	—
37	C₂×Q₈×C₄₁	—	2	1	no	—
38	C₂×Q₈×C₄₁	C₂³	1	2	yes	—
39	C₂×D₄×C₄₁	—	2	2	no	—
40	C₂×D₄×C₄₁	—	2	2	no	—
41	C₂×D₄×C₄₁	—	2	1	no	—
42	C₂×D₄×C₄₁	—	2	1	no	—
43	C₂×D₄×C₄₁	—	2	1	no	—
44	C₂×D₄×C₄₁	—	2	1	no	—
45	C₂×D₄×C₄₁	C₂³	1	2	yes	—
46	C₂×D₄×C₄₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₁	—	2	2	no	—
48	C₂²×C₄×C₄₁	—	2	2	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	2	yes	—
55	C₂×C₈×C₄₁	C₂³	1	2	yes	—
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	2	yes	—
60	C₂²⋊C₄×C₄₁	—	2	2	no	—
61	C₂²⋊C₄×C₄₁	—	2	2	no	—
62	C₂²⋊C₄×C₄₁	—	2	2	no	—
63	C₂²⋊C₄×C₄₁	—	2	2	no	—
64	C₄²×C₄₁	D₄	1	1	yes	characteristic
65	C₄²×C₄₁	D₄	1	1	yes	characteristic
66	C₄²×C₄₁	C₂³	1	1	yes	characteristic
67	C₄²×C₄₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄₁	—	2	2	no	—
69	C₂³×C₄₁	—	2	2	no	—
70	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₄₁	—	2	4	no	—
72	Q₈×C₄₁	—	4	2	no	—
73	D₄×C₄₁	—	2	4	no	—
74	D₄×C₄₁	—	2	4	no	—
75	D₄×C₄₁	—	4	4	no	—
76	D₄×C₄₁	—	4	4	no	—
77	D₄×C₄₁	—	4	2	no	—
78	D₄×C₄₁	—	4	2	no	—
79	D₄×C₄₁	—	4	2	no	—
80	D₄×C₄₁	—	4	2	no	—
81	C₂×C₄×C₄₁	—	2	2	no	—
82	C₂×C₄×C₄₁	—	2	2	no	—
83	C₂×C₄×C₄₁	—	2	2	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	2	no	—
86	C₂×C₄×C₄₁	—	4	4	no	—
87	C₂×C₄×C₄₁	—	4	4	no	—
88	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
94	C₈×C₄₁	—	2	2	no	—
95	C₈×C₄₁	—	2	2	no	—
96	C₂²×C₄₁	—	2	1	no	—
97	C₂²×C₄₁	—	2	1	no	—
98	C₂²×C₄₁	—	2	1	no	—
99	C₂²×C₄₁	—	4	2	no	—
100	C₂²×C₄₁	—	4	2	no	—
101	C₂²×C₄₁	—	4	2	no	—
102	C₂²×C₄₁	—	4	2	no	—
103	C₂²×C₄₁	—	4	2	no	—
104	C₂²×C₄₁	—	4	2	no	—
105	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₄₁	—	2	1	no	—
107	C₄×C₄₁	—	2	1	no	—
108	C₄×C₄₁	—	2	1	no	—
109	C₄×C₄₁	—	4	2	no	—
110	C₄×C₄₁	—	4	2	no	—
111	C₄×C₄₁	#135	1	2	yes	—
112	C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
113	C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
114	#3207	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₄₁	—	4	1	no	—
116	C₂×C₄₁	—	8	2	no	—
117	C₂×C₄₁	—	8	2	no	—
118	C₂×C₄₁	#535	1	1	yes	characteristic
119	C₂×C₄₁	C₂×#135	1	1	yes	characteristic
120	C₂×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	#532	C₂×C₄₁	1	1	yes	characteristic, complemented
122	#532	C₂×C₄₁	1	1	yes	characteristic, complemented
123	#493	C₂×C₄₁	1	1	yes	characteristic, complemented
124	#492	C₂×C₄₁	1	1	yes	characteristic, complemented
125	#487	C₂×C₄₁	1	1	yes	characteristic, complemented
126	#486	C₂×C₄₁	1	1	yes	characteristic, complemented
127	#463	C₂×C₄₁	1	4	yes	complemented
128	#449	C₂×C₄₁	1	4	yes	complemented
129	#431	C₂×C₄₁	1	1	yes	characteristic, complemented
130	C₄₁	#3207	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#41	—	2	1	no	—
132	C₂×#41	C₂²×C₄₁	1	2	yes	—
133	C₂×SD₁₆	C₂²×C₄₁	1	4	yes	—
134	C₂×D₈	C₂²×C₄₁	1	4	yes	—
135	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
137	#127	C₂²×C₄₁	1	1	yes	characteristic
138	#127	C₂²×C₄₁	1	1	yes	characteristic
139	#124	—	2	1	no	—
140	#124	C₂²×C₄₁	1	2	yes	—
141	#121	—	2	1	no	—
142	#121	—	2	1	no	—
143	#121	—	2	1	no	—
144	#121	C₂²×C₄₁	1	2	yes	—
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	—	2	1	no	—
148	C₄×D₄	C₂²×C₄₁	1	2	yes	—
149	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
151	#103	C₂²×C₄₁	1	4	yes	—
152	#102	C₂²×C₄₁	1	4	yes	—
153	C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
154	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
156	#41	—	4	4	no	—
157	#41	—	4	4	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	C₂³×C₄₁	1	2	yes	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	C₂³×C₄₁	1	2	yes	—
167	C₂×D₄	C₂³×C₄₁	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
171	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
173	SD₁₆	—	4	4	no	—
174	D₈	—	4	4	no	—
175	C₂×C₈	C₂³×C₄₁	1	2	yes	—
176	C₂×C₈	C₂³×C₄₁	1	2	yes	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₄²	D₄×C₄₁	1	1	yes	characteristic
186	C₄²	D₄×C₄₁	1	1	yes	characteristic
187	C₄²	C₂³×C₄₁	1	1	yes	characteristic
188	C₄²	C₂³×C₄₁	1	1	yes	characteristic
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
192	Q₈	—	2	4	no	—
193	Q₈	—	4	2	no	—
194	D₄	—	2	4	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	#135×C₄₁	1	2	yes	—
233	C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
234	C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
235	C₂	—	4	1	no	—
236	C₂	—	8	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₄₁×#535	1	1	yes	characteristic
239	C₂	C₂×#135×C₄₁	1	1	yes	characteristic
240	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. C₄₁×#532	C₂	semidirect
2. C₄₁×#532	C₂	semidirect
3. C₄₁×#493	C₂	semidirect
4. C₄₁×#492	C₂	semidirect
5. C₄₁×#487	C₂	semidirect
6. C₄₁×#486	C₂	semidirect
7. C₄₁×#463	C₂	semidirect
8. C₄₁×#449	C₂	semidirect
9. C₄₁×#431	C₂	semidirect
114. #3207	C₄₁	direct (N × H)
121. #532	C₂×C₄₁	semidirect
122. #532	C₂×C₄₁	semidirect
123. #493	C₂×C₄₁	semidirect
124. #492	C₂×C₄₁	semidirect
125. #487	C₂×C₄₁	semidirect
126. #486	C₂×C₄₁	semidirect
127. #463	C₂×C₄₁	semidirect
128. #449	C₂×C₄₁	semidirect
129. #431	C₂×C₄₁	semidirect
130. C₄₁	#3207	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×#532	2
C₄₁×#493	2
C₄₁×#492	2
C₄₁×#487	2
C₄₁×#486	2
C₄₁×#463	2
C₄₁×#449	2
C₄₁×#431	2
#3207	41

Subgroups — C41×#3207 #19106987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3207 #19106987