Subgroups — C₈×#41×C₄₁ #19106744

Order 5248.

456 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₈×D₄×C₄₁	C₂	1	6	yes	maximal, complemented
4	C₄₁×#431	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₈×C₄₁	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₄₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₄₁	C₂²	1	6	yes	—
8	C₄×Q₈×C₄₁	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₄₁	C₄	1	3	yes	—
11	C₄×D₄×C₄₁	C₂²	1	3	yes	—
12	#117×C₄₁	C₄	1	3	yes	—
13	C₂×C₄²×C₄₁	C₂²	1	3	yes	—
14	C₄⋊C₈×C₄₁	C₂²	1	6	yes	—
15	C₂²⋊C₈×C₄₁	C₂²	1	6	yes	—
16	C₄×C₈×C₄₁	C₂²	1	3	yes	—
17	C₄×C₈×C₄₁	C₂²	1	6	yes	—
18	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	#41×C₄₁	C₈	1	8	yes	direct product factor
20	C₂×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₄₁	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₄₁	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₄₁	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₄₁	C₂³	1	3	yes	—
25	C₂×C₈×C₄₁	—	2	12	no	—
26	C₂×C₈×C₄₁	C₂³	1	6	yes	—
27	C₂×C₈×C₄₁	C₂³	1	2	yes	—
28	C₄⋊C₄×C₄₁	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₄₁	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₄₁	C₂×C₄	1	6	yes	—
31	C₄²×C₄₁	C₂×C₄	1	3	yes	—
32	C₄²×C₄₁	C₂×C₄	1	3	yes	—
33	C₄²×C₄₁	C₂³	1	1	yes	characteristic
34	C₄²×C₄₁	C₂³	1	3	yes	—
35	C₂³×C₄₁	C₂²×C₄	1	3	yes	—
36	Q₈×C₄₁	C₂×C₈	1	4	yes	complemented
37	D₄×C₄₁	C₂×C₈	1	12	yes	complemented
38	C₂×C₄×C₄₁	—	2	6	no	—
39	C₂×C₄×C₄₁	—	2	6	no	—
40	C₂×C₄×C₄₁	C₂×C₈	1	12	yes	complemented
41	C₂×C₄×C₄₁	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₄₁	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
47	C₈×C₄₁	—	2	6	no	—
48	C₈×C₄₁	#41	1	4	yes	direct product factor
49	C₂²×C₄₁	—	2	3	no	—
50	C₂²×C₄₁	—	2	3	no	—
51	C₂²×C₄₁	C₂²×C₈	1	6	yes	—
52	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₄₁	—	2	3	no	—
54	C₄×C₄₁	—	2	3	no	—
55	C₄×C₄₁	C₂²×C₈	1	2	yes	—
56	C₄×C₄₁	C₂²×C₈	1	6	yes	—
57	C₄×C₄₁	#131	1	2	yes	—
58	C₄×C₄₁	C₂×#41	1	2	yes	—
59	C₈×#41	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₄₁	—	2	6	no	—
61	C₂×C₄₁	C₄×#41	1	1	yes	characteristic
62	C₂×C₄₁	C₂³×C₈	1	1	yes	characteristic
63	C₂×C₄₁	C₂×#131	1	1	yes	characteristic
64	C₄×#41	C₂×C₄₁	1	1	yes	characteristic
65	C₈×Q₈	C₂×C₄₁	1	2	yes	complemented
66	C₈×D₄	C₂×C₄₁	1	6	yes	complemented
67	#431	C₂×C₄₁	1	3	yes	complemented
68	C₂×C₄×C₈	C₂×C₄₁	1	3	yes	complemented
69	C₄₁	C₈×#41	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×#41	C₄×C₄₁	1	1	yes	characteristic
71	C₂²×C₈	C₂²×C₄₁	1	6	yes	—
72	C₄×Q₈	C₄×C₄₁	1	1	yes	characteristic
73	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
74	C₄×D₄	C₄×C₄₁	1	3	yes	—
75	C₄×D₄	C₂²×C₄₁	1	3	yes	—
76	#117	C₄×C₄₁	1	3	yes	—
77	C₂×C₄²	C₂²×C₄₁	1	3	yes	—
78	C₄⋊C₈	C₂²×C₄₁	1	6	yes	—
79	C₂²⋊C₈	C₂²×C₄₁	1	6	yes	—
80	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
81	C₄×C₈	C₂²×C₄₁	1	3	yes	—
82	C₄×C₈	C₂²×C₄₁	1	6	yes	—
83	#41	C₈×C₄₁	1	8	yes	direct product factor
84	C₂×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
85	C₂×D₄	C₂×C₄×C₄₁	1	3	yes	—
86	C₂²×C₄	C₂×C₄×C₄₁	1	3	yes	—
87	C₂²×C₄	C₂×C₄×C₄₁	1	3	yes	—
88	C₂²×C₄	C₂³×C₄₁	1	3	yes	—
89	C₂×C₈	—	2	12	no	—
90	C₂×C₈	C₂³×C₄₁	1	6	yes	—
91	C₂×C₈	C₂³×C₄₁	1	2	yes	—
92	C₄⋊C₄	C₂×C₄×C₄₁	1	3	yes	—
93	C₄⋊C₄	C₂×C₄×C₄₁	1	3	yes	—
94	C₂²⋊C₄	C₂×C₄×C₄₁	1	6	yes	—
95	C₄²	C₂×C₄×C₄₁	1	3	yes	—
96	C₄²	C₂×C₄×C₄₁	1	3	yes	—
97	C₄²	C₂³×C₄₁	1	1	yes	characteristic
98	C₄²	C₂³×C₄₁	1	3	yes	—
99	C₂³	C₂²×C₄×C₄₁	1	3	yes	—
100	Q₈	C₂×C₈×C₄₁	1	4	yes	complemented
101	D₄	C₂×C₈×C₄₁	1	12	yes	complemented
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	C₂×C₈×C₄₁	1	12	yes	complemented
105	C₂×C₄	C₂²×C₄×C₄₁	1	3	yes	—
106	C₂×C₄	C₂²×C₄×C₄₁	1	3	yes	—
107	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
108	C₂×C₄	C₂²×C₄×C₄₁	1	3	yes	—
109	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
110	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
111	C₈	—	2	6	no	—
112	C₈	#41×C₄₁	1	4	yes	direct product factor
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	C₂²×C₈×C₄₁	1	6	yes	—
116	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂²×C₈×C₄₁	1	2	yes	—
120	C₄	C₂²×C₈×C₄₁	1	6	yes	—
121	C₄	#131×C₄₁	1	2	yes	—
122	C₄	C₂×#41×C₄₁	1	2	yes	—
123	C₂	—	2	6	no	—
124	C₂	C₄×#41×C₄₁	1	1	yes	characteristic
125	C₂	C₂³×C₈×C₄₁	1	1	yes	characteristic
126	C₂	C₂×#131×C₄₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₄₁	C₂	semidirect
3. C₈×D₄×C₄₁	C₂	semidirect
4. C₄₁×#431	C₂	semidirect
5. C₂×C₄×C₈×C₄₁	C₂	semidirect
19. #41×C₄₁	C₈	direct (N × H)
36. Q₈×C₄₁	C₂×C₈	semidirect
37. D₄×C₄₁	C₂×C₈	semidirect
40. C₂×C₄×C₄₁	C₂×C₈	semidirect
48. C₈×C₄₁	#41	direct (N × H)
59. C₈×#41	C₄₁	direct (N × H)
65. C₈×Q₈	C₂×C₄₁	semidirect
66. C₈×D₄	C₂×C₄₁	semidirect
67. #431	C₂×C₄₁	semidirect
68. C₂×C₄×C₈	C₂×C₄₁	semidirect
69. C₄₁	C₈×#41	direct (N × H)
83. #41	C₈×C₄₁	direct (N × H)
100. Q₈	C₂×C₈×C₄₁	semidirect
101. D₄	C₂×C₈×C₄₁	semidirect
104. C₂×C₄	C₂×C₈×C₄₁	semidirect
112. C₈	#41×C₄₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#41×C₄₁	2
C₈×Q₈×C₄₁	2
C₈×D₄×C₄₁	2
C₄₁×#431	2
C₂×C₄×C₈×C₄₁	2
C₈×#41	41

Subgroups — C8×#41×C41 #19106744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#41×C₄₁ #19106744