Subgroups — C₄₁×#3256 #19107036

Order 5248.

354 conjugacy classes of subgroups.

The subgroup lattice has 208 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	2	yes	maximal, complemented
2	C₄₁×#479	C₂	1	2	yes	maximal, complemented
3	C₄₁×#478	C₂	1	2	yes	maximal, complemented
4	C₄₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#469	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#468	C₂	1	2	yes	maximal, complemented
7	C₄₁×#464	C₂	1	2	yes	maximal, complemented
8	C₄₁×#463	C₂	1	2	yes	maximal, complemented
9	C₄₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₄₁	C₂²	1	2	yes	—
12	C₂×D₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
15	#126×C₄₁	—	2	2	no	—
16	#125×C₄₁	C₂²	1	2	yes	—
17	#124×C₄₁	C₂²	1	2	yes	—
18	#123×C₄₁	—	2	2	no	—
19	C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
20	#121×C₄₁	C₂²	1	2	yes	—
21	C₄×Q₈×C₄₁	C₂²	1	2	yes	—
22	C₄×D₄×C₄₁	—	2	2	no	—
23	C₄×D₄×C₄₁	C₂²	1	2	yes	—
24	#117×C₄₁	—	2	2	no	—
25	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
26	#107×C₄₁	C₂²	1	2	yes	—
27	#106×C₄₁	C₂²	1	1	yes	characteristic
28	#106×C₄₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
30	C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
31	#103×C₄₁	C₂²	1	2	yes	—
32	#103×C₄₁	C₂²	1	2	yes	—
33	#102×C₄₁	C₂²	1	2	yes	—
34	#102×C₄₁	C₂²	1	2	yes	—
35	C₂×Q₈×C₄₁	C₂³	1	2	yes	—
36	C₂×D₄×C₄₁	—	2	2	no	—
37	C₂×D₄×C₄₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₁	—	2	2	no	—
39	C₂²×C₄×C₄₁	D₄	1	2	yes	—
40	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
41	Q₁₆×C₄₁	—	4	1	no	—
42	SD₁₆×C₄₁	—	4	2	no	—
43	D₈×C₄₁	—	4	1	no	—
44	M₁₆×C₄₁	—	4	1	no	—
45	M₁₆×C₄₁	—	4	1	no	—
46	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₄₁	C₂³	1	2	yes	—
48	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₄₁	—	2	2	no	—
50	C₄⋊C₄×C₄₁	—	2	2	no	—
51	C₄⋊C₄×C₄₁	—	2	2	no	—
52	C₄⋊C₄×C₄₁	—	2	2	no	—
53	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₄₁	—	2	2	no	—
56	C₂²⋊C₄×C₄₁	—	2	2	no	—
57	C₂²⋊C₄×C₄₁	—	2	2	no	—
58	C₂²⋊C₄×C₄₁	—	2	2	no	—
59	C₂²⋊C₄×C₄₁	—	2	2	no	—
60	C₄²×C₄₁	—	2	2	no	—
61	C₄²×C₄₁	D₄	1	1	yes	characteristic
62	C₄²×C₄₁	D₄	1	1	yes	characteristic
63	C₄²×C₄₁	C₂³	1	2	yes	—
64	C₂³×C₄₁	—	2	2	no	—
65	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₄₁	—	2	2	no	—
67	Q₈×C₄₁	—	2	2	no	—
68	D₄×C₄₁	—	2	2	no	—
69	D₄×C₄₁	—	2	2	no	—
70	D₄×C₄₁	—	4	4	no	—
71	C₂×C₄×C₄₁	—	2	1	no	—
72	C₂×C₄×C₄₁	—	2	1	no	—
73	C₂×C₄×C₄₁	—	2	2	no	—
74	C₂×C₄×C₄₁	—	2	2	no	—
75	C₂×C₄×C₄₁	—	2	2	no	—
76	C₂×C₄×C₄₁	—	4	2	no	—
77	C₂×C₄×C₄₁	—	4	4	no	—
78	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
82	C₈×C₄₁	—	2	1	no	—
83	C₈×C₄₁	—	2	2	no	—
84	C₈×C₄₁	—	2	1	no	—
85	C₂²×C₄₁	—	2	1	no	—
86	C₂²×C₄₁	—	2	2	no	—
87	C₂²×C₄₁	—	4	2	no	—
88	C₂²×C₄₁	—	4	2	no	—
89	C₂²×C₄₁	—	4	2	no	—
90	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₄₁	—	2	1	no	—
92	C₄×C₄₁	—	2	2	no	—
93	C₄×C₄₁	—	2	2	no	—
94	C₄×C₄₁	—	4	2	no	—
95	C₄×C₄₁	—	4	2	no	—
96	C₄×C₄₁	—	4	2	no	—
97	C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
99	#3256	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₄₁	—	4	1	no	—
101	C₂×C₄₁	—	8	2	no	—
102	C₂×C₄₁	#537	1	1	yes	characteristic
103	C₂×C₄₁	#575	1	2	yes	—
104	#529	C₂×C₄₁	1	2	yes	complemented
105	#479	C₂×C₄₁	1	2	yes	complemented
106	#478	C₂×C₄₁	1	2	yes	complemented
107	#470	C₂×C₄₁	1	1	yes	characteristic, complemented
108	#469	C₂×C₄₁	1	1	yes	characteristic
109	#468	C₂×C₄₁	1	2	yes	complemented
110	#464	C₂×C₄₁	1	2	yes	complemented
111	#463	C₂×C₄₁	1	2	yes	complemented
112	#423	C₂×C₄₁	1	1	yes	characteristic, complemented
113	C₄₁	#3256	1	1	yes	characteristic, direct product factor, Sylow
114	C₂×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
115	C₂×SD₁₆	C₂²×C₄₁	1	2	yes	—
116	C₂×D₈	C₂²×C₄₁	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
118	C₂×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
119	#126	—	2	2	no	—
120	#125	C₂²×C₄₁	1	2	yes	—
121	#124	C₂²×C₄₁	1	2	yes	—
122	#123	—	2	2	no	—
123	C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
124	#121	C₂²×C₄₁	1	2	yes	—
125	C₄×Q₈	C₂²×C₄₁	1	2	yes	—
126	C₄×D₄	—	2	2	no	—
127	C₄×D₄	C₂²×C₄₁	1	2	yes	—
128	#117	—	2	2	no	—
129	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
130	#107	C₂²×C₄₁	1	2	yes	—
131	#106	C₂²×C₄₁	1	1	yes	characteristic
132	#106	C₂²×C₄₁	1	1	yes	characteristic
133	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
134	C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
135	#103	C₂²×C₄₁	1	2	yes	—
136	#103	C₂²×C₄₁	1	2	yes	—
137	#102	C₂²×C₄₁	1	2	yes	—
138	#102	C₂²×C₄₁	1	2	yes	—
139	C₂×Q₈	C₂³×C₄₁	1	2	yes	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₂³×C₄₁	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	D₄×C₄₁	1	2	yes	—
144	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	4	2	no	—
147	D₈	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	4	1	no	—
150	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
151	C₂×C₈	C₂³×C₄₁	1	2	yes	—
152	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
158	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	D₄×C₄₁	1	1	yes	characteristic
166	C₄²	D₄×C₄₁	1	1	yes	characteristic
167	C₄²	C₂³×C₄₁	1	2	yes	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
185	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
202	C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	C₄₁×#537	1	1	yes	characteristic
206	C₂	C₄₁×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#469	C₂
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×#529	C₂	semidirect
2. C₄₁×#479	C₂	semidirect
3. C₄₁×#478	C₂	semidirect
4. C₄₁×#470	C₂	semidirect
6. C₄₁×#468	C₂	semidirect
7. C₄₁×#464	C₂	semidirect
8. C₄₁×#463	C₂	semidirect
9. C₄₁×#423	C₂	semidirect
99. #3256	C₄₁	direct (N × H)
104. #529	C₂×C₄₁	semidirect
105. #479	C₂×C₄₁	semidirect
106. #478	C₂×C₄₁	semidirect
107. #470	C₂×C₄₁	semidirect
109. #468	C₂×C₄₁	semidirect
110. #464	C₂×C₄₁	semidirect
111. #463	C₂×C₄₁	semidirect
112. #423	C₂×C₄₁	semidirect
113. C₄₁	#3256	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
C₄₁×#479	2
C₄₁×#478	2
C₄₁×#470	2
C₄₁×#469	2
C₄₁×#468	2
C₄₁×#464	2
C₄₁×#463	2
C₄₁×#423	2
#3256	41

Subgroups — C41×#3256 #19107036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3256 #19107036