Subgroups — C₄×C₄₁×#504 #23809526

Order 10496.

834 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#504	C₂	1	2	yes	maximal
2	C₄×#131×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2108	C₂	1	6	yes	maximal, complemented
4	C₄×M₃₂×C₄₁	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₄₁	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₄₁	C₄	1	1	yes	characteristic
7	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
8	C₄×#41×C₄₁	C₄	1	1	yes	characteristic
9	C₄₁×#504	C₄	1	16	yes	direct product factor
10	C₂×M₃₂×C₄₁	C₂²	1	6	yes	—
11	C₂²×C₁₆×C₄₁	C₂²	1	6	yes	—
12	C₄₁×#405	C₄	1	3	yes	—
13	C₄₁×#405	C₂²	1	3	yes	—
14	C₄×M₁₆×C₄₁	C₄	1	3	yes	—
15	C₂×C₄×C₈×C₄₁	C₂²	1	3	yes	—
16	C₄₁×#346	C₂²	1	6	yes	—
17	C₄×C₁₆×C₄₁	C₂²	1	3	yes	—
18	C₄×C₁₆×C₄₁	C₂²	1	6	yes	—
19	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
21	#131×C₄₁	C₈	1	8	yes	—
22	#131×C₄₁	C₂×C₄	1	8	yes	—
23	C₂×M₁₆×C₄₁	C₂×C₄	1	3	yes	—
24	C₂×M₁₆×C₄₁	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₄₁	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₄₁	C₂³	1	3	yes	—
27	C₄×Q₈×C₄₁	C₈	1	2	yes	—
28	C₄×D₄×C₄₁	C₈	1	6	yes	—
29	#117×C₄₁	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₄₁	C₂×C₄	1	3	yes	—
31	M₃₂×C₄₁	C₂×C₄	1	24	yes	complemented
32	C₂×C₁₆×C₄₁	C₂×C₄	1	24	yes	complemented
33	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
34	C₂×C₁₆×C₄₁	C₂³	1	6	yes	—
35	#97×C₄₁	C₂×C₄	1	6	yes	—
36	C₄×C₈×C₄₁	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₄₁	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₄₁	C₂³	1	3	yes	—
40	#41×C₄₁	C₄²	1	4	yes	—
41	#41×C₄₁	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₄₁	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₄₁	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₄₁	C₂×C₈	1	6	yes	—
45	C₂²×C₄×C₄₁	C₂²×C₄	1	3	yes	—
46	M₁₆×C₄₁	C₄²	1	12	yes	—
47	M₁₆×C₄₁	C₂×C₈	1	12	yes	—
48	C₂×C₈×C₄₁	C₂×C₈	1	12	yes	—
49	C₂×C₈×C₄₁	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₄₁	C₂²×C₄	1	12	yes	—
51	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₁	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₄₁	C₂×C₈	1	6	yes	—
55	C₂²⋊C₄×C₄₁	C₂×C₈	1	6	yes	—
56	C₄²×C₄₁	C₂×C₈	1	6	yes	—
57	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₄₁	C₂²×C₄	1	3	yes	—
59	C₁₆×C₄₁	C₂²×C₄	1	4	yes	—
60	C₁₆×C₄₁	C₂²×C₄	1	12	yes	—
61	C₂³×C₄₁	C₂²×C₈	1	3	yes	—
62	Q₈×C₄₁	C₄×C₈	1	4	yes	—
63	D₄×C₄₁	C₄×C₈	1	12	yes	—
64	C₂×C₄×C₄₁	—	2	12	no	—
65	C₂×C₄×C₄₁	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₄₁	C₂²×C₈	1	6	yes	—
67	C₂×C₄×C₄₁	C₂²×C₈	1	6	yes	—
68	C₂×C₄×C₄₁	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
71	C₈×C₄₁	C₂×C₄²	1	6	yes	—
72	C₈×C₄₁	C₂²×C₈	1	2	yes	—
73	C₈×C₄₁	C₂²×C₈	1	6	yes	—
74	C₈×C₄₁	C₂³×C₄	1	2	yes	—
75	C₄×#504	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₄₁	—	2	6	no	—
77	C₂²×C₄₁	C₂×C₄×C₈	1	6	yes	—
78	C₂²×C₄₁	C₂³×C₈	1	1	yes	characteristic
79	C₄×C₄₁	—	2	6	no	—
80	C₄×C₄₁	C₂×C₄×C₈	1	6	yes	—
81	C₄×C₄₁	#504	1	4	yes	direct product factor
82	C₄×C₄₁	C₂²×C₄²	1	1	yes	characteristic
83	C₄×C₄₁	C₂³×C₈	1	1	yes	characteristic
84	C₂×#504	C₂×C₄₁	1	2	yes	—
85	C₄×#131	C₂×C₄₁	1	1	yes	characteristic
86	#2108	C₂×C₄₁	1	6	yes	complemented
87	C₄×M₃₂	C₂×C₄₁	1	3	yes	complemented
88	C₂×C₄×C₁₆	C₂×C₄₁	1	3	yes	complemented
89	C₂×C₄₁	—	2	6	no	—
90	C₂×C₄₁	C₂²×C₄×C₈	1	1	yes	characteristic
91	C₂×C₄₁	C₂×#504	1	2	yes	—
92	C₂×#131	C₄×C₄₁	1	1	yes	characteristic
93	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
94	C₄×#41	C₄×C₄₁	1	1	yes	characteristic
95	#504	C₄×C₄₁	1	16	yes	direct product factor
96	C₂×M₃₂	C₂²×C₄₁	1	6	yes	—
97	C₂²×C₁₆	C₂²×C₄₁	1	6	yes	—
98	#405	C₄×C₄₁	1	3	yes	—
99	#405	C₂²×C₄₁	1	3	yes	—
100	C₄×M₁₆	C₄×C₄₁	1	3	yes	—
101	C₂×C₄×C₈	C₂²×C₄₁	1	3	yes	—
102	#346	C₂²×C₄₁	1	6	yes	—
103	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
104	C₄×C₁₆	C₂²×C₄₁	1	3	yes	—
105	C₄×C₁₆	C₂²×C₄₁	1	6	yes	—
106	C₄₁	C₄×#504	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
108	#131	C₈×C₄₁	1	8	yes	—
109	#131	C₂×C₄×C₄₁	1	8	yes	—
110	C₂×M₁₆	C₂×C₄×C₄₁	1	3	yes	—
111	C₂×M₁₆	C₂×C₄×C₄₁	1	3	yes	—
112	C₂²×C₈	C₂×C₄×C₄₁	1	3	yes	—
113	C₂²×C₈	C₂³×C₄₁	1	3	yes	—
114	C₄×Q₈	C₈×C₄₁	1	2	yes	—
115	C₄×D₄	C₈×C₄₁	1	6	yes	—
116	#117	C₂×C₄×C₄₁	1	3	yes	—
117	C₂×C₄²	C₂×C₄×C₄₁	1	3	yes	—
118	M₃₂	C₂×C₄×C₄₁	1	24	yes	complemented
119	C₂×C₁₆	C₂×C₄×C₄₁	1	24	yes	complemented
120	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
121	C₂×C₁₆	C₂³×C₄₁	1	6	yes	—
122	#97	C₂×C₄×C₄₁	1	6	yes	—
123	C₄×C₈	C₂×C₄×C₄₁	1	3	yes	—
124	C₄×C₈	C₂×C₄×C₄₁	1	3	yes	—
125	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
126	C₄×C₈	C₂³×C₄₁	1	3	yes	—
127	#41	C₄²×C₄₁	1	4	yes	—
128	#41	C₂×C₈×C₄₁	1	4	yes	—
129	C₂×Q₈	C₂×C₈×C₄₁	1	1	yes	characteristic
130	C₂×D₄	C₂×C₈×C₄₁	1	3	yes	—
131	C₂²×C₄	C₂×C₈×C₄₁	1	6	yes	—
132	C₂²×C₄	C₂²×C₄×C₄₁	1	3	yes	—
133	M₁₆	C₄²×C₄₁	1	12	yes	—
134	M₁₆	C₂×C₈×C₄₁	1	12	yes	—
135	C₂×C₈	C₂×C₈×C₄₁	1	12	yes	—
136	C₂×C₈	C₂²×C₄×C₄₁	1	3	yes	—
137	C₂×C₈	C₂²×C₄×C₄₁	1	12	yes	—
138	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₄₁	1	3	yes	—
140	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
141	C₄⋊C₄	C₂×C₈×C₄₁	1	6	yes	—
142	C₂²⋊C₄	C₂×C₈×C₄₁	1	6	yes	—
143	C₄²	C₂×C₈×C₄₁	1	6	yes	—
144	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
145	C₄²	C₂²×C₄×C₄₁	1	3	yes	—
146	C₁₆	C₂²×C₄×C₄₁	1	4	yes	—
147	C₁₆	C₂²×C₄×C₄₁	1	12	yes	—
148	C₂³	C₂²×C₈×C₄₁	1	3	yes	—
149	Q₈	C₄×C₈×C₄₁	1	4	yes	—
150	D₄	C₄×C₈×C₄₁	1	12	yes	—
151	C₂×C₄	—	2	12	no	—
152	C₂×C₄	C₂×C₄²×C₄₁	1	6	yes	—
153	C₂×C₄	C₂²×C₈×C₄₁	1	6	yes	—
154	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
155	C₂×C₄	C₂²×C₈×C₄₁	1	6	yes	—
156	C₂×C₄	C₂²×C₈×C₄₁	1	3	yes	—
157	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
158	C₈	C₂×C₄²×C₄₁	1	6	yes	—
159	C₈	C₂²×C₈×C₄₁	1	2	yes	—
160	C₈	C₂²×C₈×C₄₁	1	6	yes	—
161	C₈	C₂³×C₄×C₄₁	1	2	yes	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×C₈×C₄₁	1	6	yes	—
164	C₂²	C₂³×C₈×C₄₁	1	1	yes	characteristic
165	C₄	—	2	6	no	—
166	C₄	C₂×C₄×C₈×C₄₁	1	6	yes	—
167	C₄	C₄₁×#504	1	4	yes	direct product factor
168	C₄	C₂²×C₄²×C₄₁	1	1	yes	characteristic
169	C₄	C₂³×C₈×C₄₁	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×C₈×C₄₁	1	1	yes	characteristic
172	C₂	C₂×C₄₁×#504	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆×C₄₁	C₂²
commutator	C₂	C₂²×C₄×C₈×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂³×C₈
prime radical	C₂×#41×C₄₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#2108	C₂	semidirect
4. C₄×M₃₂×C₄₁	C₂	semidirect
5. C₂×C₄×C₁₆×C₄₁	C₂	semidirect
9. C₄₁×#504	C₄	direct (N × H)
31. M₃₂×C₄₁	C₂×C₄	semidirect
32. C₂×C₁₆×C₄₁	C₂×C₄	semidirect
75. C₄×#504	C₄₁	direct (N × H)
81. C₄×C₄₁	#504	direct (N × H)
86. #2108	C₂×C₄₁	semidirect
87. C₄×M₃₂	C₂×C₄₁	semidirect
88. C₂×C₄×C₁₆	C₂×C₄₁	semidirect
95. #504	C₄×C₄₁	direct (N × H)
106. C₄₁	C₄×#504	direct (N × H)
118. M₃₂	C₂×C₄×C₄₁	semidirect
119. C₂×C₁₆	C₂×C₄×C₄₁	semidirect
167. C₄	C₄₁×#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#504	2
C₄×#131×C₄₁	2
C₄₁×#2108	2
C₄×M₃₂×C₄₁	2
C₂×C₄×C₁₆×C₄₁	2
C₄×#504	41

Subgroups — C4×C41×#504 #23809526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×#504 #23809526