Subgroups — C₄₁×#1866 #19105646

Order 5248.

270 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#104×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#416	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₄₁	C₂²	1	1	yes	characteristic
9	#131×C₄₁	—	2	2	no	—
10	#131×C₄₁	—	2	2	no	—
11	C₂×M₁₆×C₄₁	—	2	1	no	—
12	C₂×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₁	—	2	1	no	—
15	C₂²×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	#117×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
19	#108×C₄₁	—	2	2	no	—
20	C₄⋊C₈×C₄₁	—	2	1	no	—
21	C₄⋊C₈×C₄₁	—	2	1	no	—
22	#104×C₄₁	—	2	2	no	—
23	#103×C₄₁	C₄	1	1	yes	characteristic, complemented
24	#103×C₄₁	C₄	1	1	yes	characteristic, complemented
25	#102×C₄₁	C₄	1	1	yes	characteristic, complemented
26	#102×C₄₁	C₄	1	1	yes	characteristic, complemented
27	#41×C₄₁	—	2	1	no	—
28	#41×C₄₁	—	2	1	no	—
29	#41×C₄₁	D₄	1	2	yes	—
30	#41×C₄₁	Q₈	1	2	yes	—
31	C₂×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
32	C₂×D₄×C₄₁	—	2	1	no	—
33	C₂×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₁	—	2	1	no	—
35	C₂²×C₄×C₄₁	—	2	1	no	—
36	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₄₁	—	2	4	no	—
38	M₁₆×C₄₁	—	2	1	no	—
39	M₁₆×C₄₁	—	2	1	no	—
40	M₁₆×C₄₁	—	2	1	no	—
41	M₁₆×C₄₁	D₄	1	2	yes	—
42	C₂×C₈×C₄₁	—	2	4	no	—
43	C₂×C₈×C₄₁	—	2	1	no	—
44	C₂×C₈×C₄₁	—	2	1	no	—
45	C₂×C₈×C₄₁	—	2	1	no	—
46	C₂×C₈×C₄₁	—	2	1	no	—
47	C₂×C₈×C₄₁	—	2	1	no	—
48	C₂×C₈×C₄₁	—	2	1	no	—
49	C₂×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₁	D₄	1	1	yes	characteristic
52	C₂×C₈×C₄₁	D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₄₁	—	2	1	no	—
56	C₄²×C₄₁	—	2	1	no	—
57	C₄²×C₄₁	—	2	1	no	—
58	C₄²×C₄₁	—	2	1	no	—
59	C₂³×C₄₁	—	2	1	no	—
60	C₂³×C₄₁	#41	1	1	yes	characteristic
61	Q₈×C₄₁	—	2	1	no	—
62	Q₈×C₄₁	C₄⋊C₄	1	2	yes	—
63	D₄×C₄₁	—	2	2	no	—
64	D₄×C₄₁	—	2	2	no	—
65	D₄×C₄₁	—	2	1	no	—
66	D₄×C₄₁	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₄₁	—	2	1	no	—
68	C₂×C₄×C₄₁	—	2	2	no	—
69	C₂×C₄×C₄₁	—	2	2	no	—
70	C₂×C₄×C₄₁	—	2	2	no	—
71	C₂×C₄×C₄₁	—	2	1	no	—
72	C₂×C₄×C₄₁	—	2	1	no	—
73	C₂×C₄×C₄₁	—	4	1	no	—
74	C₂×C₄×C₄₁	—	4	1	no	—
75	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₁	#41	1	1	yes	characteristic
81	C₈×C₄₁	—	2	1	no	—
82	C₈×C₄₁	—	2	1	no	—
83	C₈×C₄₁	—	2	1	no	—
84	C₈×C₄₁	—	2	2	no	—
85	C₈×C₄₁	—	2	1	no	—
86	C₈×C₄₁	—	4	1	no	—
87	C₂²×C₄₁	—	2	1	no	—
88	C₂²×C₄₁	—	2	1	no	—
89	C₂²×C₄₁	—	2	2	no	—
90	C₂²×C₄₁	—	4	1	no	—
91	C₂²×C₄₁	—	4	1	no	—
92	C₂²×C₄₁	C₄×D₄	1	1	yes	characteristic
93	C₂²×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂²×C₄₁	#123	1	1	yes	characteristic
95	C₄×C₄₁	—	2	2	no	—
96	C₄×C₄₁	—	4	2	no	—
97	C₄×C₄₁	—	4	1	no	—
98	C₄×C₄₁	—	4	1	no	—
99	C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
101	C₄×C₄₁	#120	1	1	yes	characteristic
102	C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
103	#1866	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₄₁	—	2	1	no	—
105	C₂×C₄₁	—	2	1	no	—
106	C₂×C₄₁	—	4	2	no	—
107	C₂×C₄₁	#385	1	1	yes	characteristic
108	C₂×C₄₁	#443	1	1	yes	characteristic
109	C₂×C₄₁	#444	1	1	yes	characteristic
110	C₂×#131	C₂×C₄₁	1	1	yes	characteristic
111	C₂×#108	C₂×C₄₁	1	1	yes	characteristic, complemented
112	#424	C₂×C₄₁	1	1	yes	characteristic, complemented
113	C₂×#104	C₂×C₄₁	1	1	yes	characteristic
114	#416	C₂×C₄₁	1	1	yes	characteristic
115	#339	C₂×C₄₁	1	1	yes	characteristic, complemented
116	#336	C₂×C₄₁	1	1	yes	characteristic, complemented
117	C₄₁	#1866	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×#41	C₂²×C₄₁	1	1	yes	characteristic
119	#131	—	2	2	no	—
120	#131	—	2	2	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
123	C₂×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	C₂²×C₄₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×C₄₁	1	1	yes	characteristic
127	#117	C₂²×C₄₁	1	1	yes	characteristic
128	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
129	#108	—	2	2	no	—
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	#104	—	2	2	no	—
133	#103	C₄×C₄₁	1	1	yes	characteristic, complemented
134	#103	C₄×C₄₁	1	1	yes	characteristic, complemented
135	#102	C₄×C₄₁	1	1	yes	characteristic, complemented
136	#102	C₄×C₄₁	1	1	yes	characteristic, complemented
137	#41	—	2	1	no	—
138	#41	—	2	1	no	—
139	#41	D₄×C₄₁	1	2	yes	—
140	#41	Q₈×C₄₁	1	2	yes	—
141	C₂×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
147	M₁₆	—	2	4	no	—
148	M₁₆	—	2	1	no	—
149	M₁₆	—	2	1	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	D₄×C₄₁	1	2	yes	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
161	C₂×C₈	D₄×C₄₁	1	1	yes	characteristic
162	C₂×C₈	D₄×C₄₁	1	1	yes	characteristic
163	C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
164	C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
165	C₂²⋊C₄	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	#41×C₄₁	1	1	yes	characteristic
171	Q₈	—	2	1	no	—
172	Q₈	C₄⋊C₄×C₄₁	1	2	yes	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	1	no	—
176	D₄	C₄⋊C₄×C₄₁	1	2	yes	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
189	C₂×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
190	C₂×C₄	#41×C₄₁	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	—	4	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	1	no	—
202	C₂²	C₄×D₄×C₄₁	1	1	yes	characteristic
203	C₂²	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
204	C₂²	#123×C₄₁	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	1	no	—
209	C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
210	C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
211	C₄	#120×C₄₁	1	1	yes	characteristic
212	C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	4	2	no	—
216	C₂	C₄₁×#385	1	1	yes	characteristic
217	C₂	C₄₁×#443	1	1	yes	characteristic
218	C₂	C₄₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂³×C₄₁
socle	C₂²×C₄₁	C₄×D₄
prime radical	C₂×#41×C₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂²×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#108×C₄₁	C₂	semidirect
3. C₄₁×#424	C₂	semidirect
6. C₄₁×#339	C₂	semidirect
7. C₄₁×#336	C₂	semidirect
23. #103×C₄₁	C₄	semidirect
24. #103×C₄₁	C₄	semidirect
25. #102×C₄₁	C₄	semidirect
26. #102×C₄₁	C₄	semidirect
103. #1866	C₄₁	direct (N × H)
111. C₂×#108	C₂×C₄₁	semidirect
112. #424	C₂×C₄₁	semidirect
115. #339	C₂×C₄₁	semidirect
116. #336	C₂×C₄₁	semidirect
117. C₄₁	#1866	direct (N × H)
133. #103	C₄×C₄₁	semidirect
134. #103	C₄×C₄₁	semidirect
135. #102	C₄×C₄₁	semidirect
136. #102	C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₄₁	2
C₂×#108×C₄₁	2
C₄₁×#424	2
C₂×#104×C₄₁	2
C₄₁×#416	2
C₄₁×#339	2
C₄₁×#336	2
#1866	41

Subgroups — C41×#1866 #19105646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#1866 #19105646