Subgroups — C₂×C₄₁×#423 #19106683

Order 5248.

576 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₄₁	C₂	1	2	yes	maximal
2	C₂²×C₄²×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#423	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₈×C₄₁	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₄₁	C₄	1	2	yes	—
6	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₄₁	—	2	8	no	—
8	C₂×M₁₆×C₄₁	C₂²	1	8	yes	—
9	C₂²×C₈×C₄₁	C₂²	1	4	yes	—
10	C₂×C₄²×C₄₁	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₄₁	C₄	1	4	yes	—
12	C₂×C₄²×C₄₁	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₄₁	C₂²	1	4	yes	—
14	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
15	C₄⋊C₈×C₄₁	C₂²	1	16	yes	complemented
16	C₂⁴×C₄₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₁	—	2	8	no	—
18	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₁	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₁	D₄	1	4	yes	—
24	C₂²×C₄×C₄₁	Q₈	1	4	yes	—
25	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
26	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₄₁	—	2	16	no	—
28	C₂×C₈×C₄₁	—	2	8	no	—
29	C₂×C₈×C₄₁	C₂³	1	8	yes	—
30	C₄²×C₄₁	C₂×C₄	1	4	yes	—
31	C₄²×C₄₁	C₂×C₄	1	4	yes	—
32	C₄²×C₄₁	C₂×C₄	1	4	yes	—
33	C₄²×C₄₁	C₂³	1	4	yes	—
34	C₂³×C₄₁	—	2	4	no	—
35	C₂³×C₄₁	C₄⋊C₄	1	4	yes	—
36	C₂³×C₄₁	C₂²×C₄	1	2	yes	—
37	C₂³×C₄₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₁	—	2	4	no	—
39	C₂×C₄×C₄₁	—	2	16	no	—
40	C₂×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
41	C₂×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₄₁	M₁₆	1	8	yes	complemented
43	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₁	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₄₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₄₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
52	C₈×C₄₁	—	2	8	no	—
53	C₂²×C₄₁	—	2	4	no	—
54	C₂²×C₄₁	—	2	8	no	—
55	C₂²×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
56	C₂²×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂²×C₄₁	C₂×M₁₆	1	4	yes	—
58	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₄₁	—	2	4	no	—
60	C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
61	C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
62	C₄×C₄₁	C₂×M₁₆	1	8	yes	complemented
63	C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
65	C₂×#423	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₄₁	—	2	4	no	—
67	C₂×C₄₁	#423	1	4	yes	direct product factor
68	C₂×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄₁	C₂²×M₁₆	1	2	yes	—
70	C₂²×M₁₆	C₂×C₄₁	1	2	yes	—
71	C₂²×C₄²	C₂×C₄₁	1	1	yes	characteristic
72	#423	C₂×C₄₁	1	8	yes	direct product factor
73	C₂×C₄⋊C₈	C₂×C₄₁	1	4	yes	complemented
74	C₄₁	C₂×#423	1	1	yes	characteristic, direct product factor, Sylow
75	C₂³×C₄	C₄×C₄₁	1	2	yes	—
76	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
77	C₂×M₁₆	—	2	8	no	—
78	C₂×M₁₆	C₂²×C₄₁	1	8	yes	—
79	C₂²×C₈	C₂²×C₄₁	1	4	yes	—
80	C₂×C₄²	C₄×C₄₁	1	1	yes	characteristic
81	C₂×C₄²	C₄×C₄₁	1	4	yes	—
82	C₂×C₄²	C₄×C₄₁	1	1	yes	characteristic
83	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
84	C₂×C₄²	C₂²×C₄₁	1	4	yes	—
85	C₄⋊C₈	C₂²×C₄₁	1	16	yes	complemented
86	C₂⁴	C₂×C₄×C₄₁	1	1	yes	characteristic
87	C₂²×C₄	—	2	8	no	—
88	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
89	C₂²×C₄	C₂×C₄×C₄₁	1	8	yes	—
90	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
91	C₂²×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄×C₄₁	1	2	yes	—
93	C₂²×C₄	D₄×C₄₁	1	4	yes	—
94	C₂²×C₄	Q₈×C₄₁	1	4	yes	—
95	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
97	M₁₆	—	2	16	no	—
98	C₂×C₈	—	2	8	no	—
99	C₂×C₈	C₂³×C₄₁	1	8	yes	—
100	C₄²	C₂×C₄×C₄₁	1	4	yes	—
101	C₄²	C₂×C₄×C₄₁	1	4	yes	—
102	C₄²	C₂×C₄×C₄₁	1	4	yes	—
103	C₄²	C₂³×C₄₁	1	4	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	C₄⋊C₄×C₄₁	1	4	yes	—
106	C₂³	C₂²×C₄×C₄₁	1	2	yes	—
107	C₂³	C₂²×C₄×C₄₁	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	16	no	—
110	C₂×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
111	C₂×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
112	C₂×C₄	M₁₆×C₄₁	1	8	yes	complemented
113	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
114	C₂×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
116	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
117	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
118	C₂×C₄	C₂×D₄×C₄₁	1	4	yes	—
119	C₂×C₄	C₂×Q₈×C₄₁	1	4	yes	—
120	C₂×C₄	C₂×Q₈×C₄₁	1	2	yes	—
121	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
122	C₈	—	2	8	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	8	no	—
125	C₂²	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
126	C₂²	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
127	C₂²	C₂×M₁₆×C₄₁	1	4	yes	—
128	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
131	C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
132	C₄	C₂×M₁₆×C₄₁	1	8	yes	complemented
133	C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
134	C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	C₄₁×#423	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
138	C₂	C₂²×M₁₆×C₄₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#423	C₂	direct (N × H)
4. C₂×C₄⋊C₈×C₄₁	C₂	semidirect
15. C₄⋊C₈×C₄₁	C₂²	semidirect
42. C₂×C₄×C₄₁	M₁₆	semidirect
62. C₄×C₄₁	C₂×M₁₆	semidirect
65. C₂×#423	C₄₁	direct (N × H)
67. C₂×C₄₁	#423	direct (N × H)
72. #423	C₂×C₄₁	direct (N × H)
73. C₂×C₄⋊C₈	C₂×C₄₁	semidirect
74. C₄₁	C₂×#423	direct (N × H)
85. C₄⋊C₈	C₂²×C₄₁	semidirect
112. C₂×C₄	M₁₆×C₄₁	semidirect
132. C₄	C₂×M₁₆×C₄₁	semidirect
136. C₂	C₄₁×#423	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₄₁	2
C₂²×C₄²×C₄₁	2
C₄₁×#423	2
C₂×C₄⋊C₈×C₄₁	2
C₂×#423	41

Subgroups — C2×C41×#423 #19106683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#423 #19106683