Subgroups — C₂×C₄₁×#427 #19106689

Order 5248.

512 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₄₁	C₂	1	2	yes	maximal
3	C₄₁×#427	C₂	1	8	yes	maximal, direct product factor
4	C₂×#107×C₄₁	C₂	1	2	yes	maximal, complemented
5	C₂×#106×C₄₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₄₁	C₄	1	8	yes	complemented
8	C₂²×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₄₁	C₂²	1	4	yes	—
11	#117×C₄₁	—	2	4	no	—
12	#117×C₄₁	C₂²	1	8	yes	—
13	C₂×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
15	C₂×C₄²×C₄₁	—	2	2	no	—
16	#107×C₄₁	C₂²	1	8	yes	complemented
17	#106×C₄₁	C₂²	1	8	yes	complemented
18	C₂⁴×C₄₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₁	—	2	4	no	—
20	C₂²×C₄×C₄₁	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₄₁	D₄	1	2	yes	—
22	C₂²×C₄×C₄₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₁	Q₈	1	4	yes	—
24	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
26	C₂×C₈×C₄₁	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₄₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₄₁	C₂×C₄	1	16	yes	complemented
29	C₂×C₈×C₄₁	C₂³	1	2	yes	—
30	C₂×C₈×C₄₁	C₂³	1	2	yes	—
31	C₄⋊C₄×C₄₁	—	2	4	no	—
32	C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
33	C₂²⋊C₄×C₄₁	—	2	8	no	—
34	C₄²×C₄₁	—	2	8	no	—
35	C₂³×C₄₁	—	2	4	no	—
36	C₂³×C₄₁	C₄⋊C₄	1	4	yes	—
37	C₂³×C₄₁	C₂×D₄	1	2	yes	—
38	C₂³×C₄₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₄₁	—	2	8	no	—
40	C₂×C₄×C₄₁	—	4	8	no	—
41	C₂×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₁	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₄₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
50	C₈×C₄₁	C₂²×C₄	1	4	yes	—
51	C₈×C₄₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄₁	—	2	4	no	—
53	C₂²×C₄₁	—	2	8	no	—
54	C₂²×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
55	C₂²×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
56	C₂²×C₄₁	#135	1	4	yes	—
57	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₄₁	—	4	8	no	—
59	C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
60	C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
61	C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
63	C₂×#427	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₄₁	—	2	4	no	—
65	C₂×C₄₁	#427	1	4	yes	direct product factor
66	C₂×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄₁	C₂×#135	1	2	yes	—
68	C₂³×C₈	C₂×C₄₁	1	1	yes	characteristic
69	C₂×#117	C₂×C₄₁	1	2	yes	—
70	#427	C₂×C₄₁	1	8	yes	direct product factor
71	C₂×#107	C₂×C₄₁	1	2	yes	complemented
72	C₂×#106	C₂×C₄₁	1	2	yes	complemented
73	C₄₁	C₂×#427	1	1	yes	characteristic, direct product factor, Sylow
74	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
75	C₂²×C₈	C₄×C₄₁	1	8	yes	complemented
76	C₂²×C₈	C₂²×C₄₁	1	1	yes	characteristic
77	C₂²×C₈	C₂²×C₄₁	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₄₁	1	4	yes	—
79	#117	—	2	4	no	—
80	#117	C₂²×C₄₁	1	8	yes	—
81	C₂×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
82	C₂×C₂²⋊C₄	—	2	2	no	—
83	C₂×C₄²	—	2	2	no	—
84	#107	C₂²×C₄₁	1	8	yes	complemented
85	#106	C₂²×C₄₁	1	8	yes	complemented
86	C₂⁴	D₄×C₄₁	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	C₂×C₄×C₄₁	1	4	yes	—
89	C₂²×C₄	D₄×C₄₁	1	1	yes	characteristic
90	C₂²×C₄	D₄×C₄₁	1	2	yes	—
91	C₂²×C₄	Q₈×C₄₁	1	4	yes	—
92	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
93	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
94	C₂×C₈	C₂×C₄×C₄₁	1	4	yes	—
95	C₂×C₈	C₂×C₄×C₄₁	1	4	yes	—
96	C₂×C₈	C₂×C₄×C₄₁	1	16	yes	complemented
97	C₂×C₈	C₂³×C₄₁	1	2	yes	—
98	C₂×C₈	C₂³×C₄₁	1	2	yes	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
101	C₂²⋊C₄	—	2	8	no	—
102	C₄²	—	2	8	no	—
103	C₂³	—	2	4	no	—
104	C₂³	C₄⋊C₄×C₄₁	1	4	yes	—
105	C₂³	C₂×D₄×C₄₁	1	2	yes	—
106	C₂³	C₂×D₄×C₄₁	1	1	yes	characteristic
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	4	8	no	—
109	C₂×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
110	C₂×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
111	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
112	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
113	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
114	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
115	C₂×C₄	C₂×Q₈×C₄₁	1	4	yes	—
116	C₂×C₄	C₂×Q₈×C₄₁	1	2	yes	—
117	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
118	C₈	C₂²×C₄×C₄₁	1	4	yes	—
119	C₈	C₂²×C₄×C₄₁	1	4	yes	—
120	C₂²	—	2	4	no	—
121	C₂²	—	2	8	no	—
122	C₂²	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
123	C₂²	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
124	C₂²	#135×C₄₁	1	4	yes	—
125	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
126	C₄	—	4	8	no	—
127	C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
128	C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
129	C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
130	C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
131	C₂	—	2	4	no	—
132	C₂	C₄₁×#427	1	4	yes	direct product factor
133	C₂	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
134	C₂	C₂×#135×C₄₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	D₄
commutator	C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂×D₄
prime radical	C₂⁴×C₄₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#427	C₂	direct (N × H)
4. C₂×#107×C₄₁	C₂	semidirect
5. C₂×#106×C₄₁	C₂	semidirect
7. C₂²×C₈×C₄₁	C₄	semidirect
16. #107×C₄₁	C₂²	semidirect
17. #106×C₄₁	C₂²	semidirect
28. C₂×C₈×C₄₁	C₂×C₄	semidirect
63. C₂×#427	C₄₁	direct (N × H)
65. C₂×C₄₁	#427	direct (N × H)
70. #427	C₂×C₄₁	direct (N × H)
71. C₂×#107	C₂×C₄₁	semidirect
72. C₂×#106	C₂×C₄₁	semidirect
73. C₄₁	C₂×#427	direct (N × H)
75. C₂²×C₈	C₄×C₄₁	semidirect
84. #107	C₂²×C₄₁	semidirect
85. #106	C₂²×C₄₁	semidirect
96. C₂×C₈	C₂×C₄×C₄₁	semidirect
132. C₂	C₄₁×#427	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₄₁	2
C₂×#117×C₄₁	2
C₄₁×#427	2
C₂×#107×C₄₁	2
C₂×#106×C₄₁	2
C₂×#427	41

Subgroups — C2×C41×#427 #19106689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#427 #19106689