Subgroups — C₂×C₈×D₅₂ #13123510

Order 1664.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88060	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₅₂	C₂	1	8	yes	maximal, direct product factor
7	C₂²×D₅₂	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
9	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#6103	C₄	1	2	yes	—
13	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
14	C₂²×#1055	C₂²	1	2	yes	—
15	C₂×C₈×D₁₃	—	2	8	no	—
16	C₂×C₈×D₁₃	C₂²	1	8	yes	complemented
17	C₄×D₅₂	C₄	1	4	yes	—
18	C₄×D₅₂	C₂²	1	4	yes	—
19	C₄×C₈×C₁₃	C₂²	1	4	yes	complemented
20	#88060	C₂²	1	8	yes	complemented
21	C₄⋊#1055	C₂²	1	4	yes	complemented
22	C₂³×D₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
27	C₂×D₅₂	C₈	1	8	yes	direct product factor
28	C₂×D₅₂	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₃	—	2	4	no	—
30	C₂×C₄×D₁₃	—	2	4	no	—
31	C₂×C₄×D₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₁₃	C₂³	1	4	yes	—
33	C₂×C₈×C₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	D₄	1	4	yes	complemented
35	C₂×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₈×C₁₃	C₂³	1	2	yes	—
37	C₄²×C₁₃	C₂×C₄	1	2	yes	—
38	C₄²×C₁₃	C₂³	1	2	yes	—
39	#6103	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
41	C₂×#1055	—	2	4	no	—
42	C₂×#1055	C₂³	1	4	yes	—
43	C₈×D₁₃	—	2	16	no	—
44	C₂×C₈×D₄	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₁₃	—	2	4	no	—
47	C₂²×D₁₃	—	2	4	no	—
48	C₂²×D₁₃	C₂×C₈	1	8	yes	complemented
49	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	C₂×C₈	1	4	yes	complemented
52	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₃	#41	1	2	yes	—
57	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₁₃	—	2	4	no	—
59	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
60	D₅₂	C₂×C₈	1	16	yes	direct product factor
61	C₄×D₁₃	—	2	8	no	—
62	C₄×D₁₃	—	2	4	no	—
63	C₄×D₁₃	—	2	4	no	—
64	C₈×C₁₃	—	2	2	no	—
65	C₈×C₁₃	C₂×D₄	1	4	yes	complemented
66	#1055	—	2	4	no	—
67	C₂³×C₈	—	13	2	no	—
68	C₂×C₄×D₄	—	13	1	no	—
69	C₈×D₄	—	13	8	no	—
70	C₂×C₄⋊C₈	—	13	1	no	—
71	C₂×C₂²⋊C₈	—	13	2	no	—
72	C₂×C₄×C₈	D₁₃	1	1	yes	characteristic, complemented
73	C₂²×C₁₃	C₄×D₄	1	2	yes	—
74	C₂²×C₁₃	C₂²×C₈	1	2	yes	—
75	C₂²×C₁₃	#131	1	2	yes	—
76	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₁₃	—	2	4	no	—
78	C₂×D₁₃	—	2	4	no	—
79	C₂×D₁₃	—	2	16	no	—
80	C₂×D₁₃	C₂²×C₈	1	8	yes	complemented
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	C₄×D₄	1	2	yes	—
83	C₄×C₁₃	C₂²×C₈	1	4	yes	complemented
84	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
85	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
86	Dic₁₃	—	2	4	no	—
87	C₂²×D₄	—	13	1	no	—
88	C₂³×C₄	—	13	2	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	—	13	8	no	—
91	C₂²×C₈	—	26	8	no	—
92	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₄×D₄	—	13	4	no	—
95	C₄×D₄	—	13	4	no	—
96	C₂×C₄⋊C₄	—	13	1	no	—
97	C₂×C₂²⋊C₄	—	13	2	no	—
98	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
99	C₄⋊C₈	—	13	4	no	—
100	C₂²⋊C₈	—	13	8	no	—
101	C₄×C₈	C₂×D₁₃	1	4	yes	complemented
102	C₂×C₁₃	C₈×D₄	1	4	yes	complemented
103	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
104	C₂×C₁₃	C₂³×C₈	1	1	yes	characteristic
105	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
106	D₁₃	—	2	8	no	—
107	C₂⁴	—	13	2	no	—
108	C₂×D₄	—	13	8	no	—
109	C₂×D₄	—	13	4	no	—
110	C₂²×C₄	—	13	2	no	—
111	C₂²×C₄	—	13	4	no	—
112	C₂²×C₄	—	13	4	no	—
113	C₂²×C₄	—	26	4	no	—
114	C₂²×C₄	—	26	4	no	—
115	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
116	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	13	4	no	—
120	C₂×C₈	—	26	4	no	—
121	C₂×C₈	—	26	16	no	—
122	C₂×C₈	D₅₂	1	4	yes	direct product factor
123	C₂×C₈	C₂²×D₁₃	1	2	yes	—
124	C₂×C₈	C₂²×D₁₃	1	2	yes	—
125	C₄⋊C₄	—	13	4	no	—
126	C₂²⋊C₄	—	13	8	no	—
127	C₄²	C₄×D₁₃	1	2	yes	—
128	C₄²	C₂²×D₁₃	1	2	yes	—
129	C₁₃	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
130	C₂³	—	13	8	no	—
131	C₂³	—	13	4	no	—
132	C₂³	—	26	4	no	—
133	C₂³	—	26	4	no	—
134	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
135	D₄	—	13	16	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	13	4	no	—
138	C₂×C₄	—	26	4	no	—
139	C₂×C₄	—	26	8	no	—
140	C₂×C₄	—	26	4	no	—
141	C₂×C₄	—	26	4	no	—
142	C₂×C₄	C₈×D₁₃	1	4	yes	complemented
143	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
146	C₂×C₄	C₂×D₅₂	1	2	yes	—
147	C₂×C₄	#6127	1	2	yes	—
148	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	26	4	no	—
151	C₈	C₂×D₅₂	1	4	yes	direct product factor
152	C₂²	—	13	8	no	—
153	C₂²	—	26	4	no	—
154	C₂²	—	26	4	no	—
155	C₂²	—	26	16	no	—
156	C₂²	C₄×D₅₂	1	2	yes	—
157	C₂²	C₂×C₈×D₁₃	1	2	yes	—
158	C₂²	#88155	1	2	yes	—
159	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	26	4	no	—
162	C₄	C₄×D₅₂	1	2	yes	—
163	C₄	C₂×C₈×D₁₃	1	4	yes	complemented
164	C₄	C₂²×D₅₂	1	1	yes	characteristic
165	C₄	C₂×#6127	1	1	yes	characteristic
166	C₂	—	26	8	no	—
167	C₂	C₈×D₅₂	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₅₂	1	1	yes	characteristic
169	C₂	C₂²×C₈×D₁₃	1	1	yes	characteristic
170	C₂	C₂×#88155	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×C₄
prime radical	C₂²×D₅₂	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₁₃	C₂	semidirect
3. C₂×C₄×C₈×C₁₃	C₂	semidirect
4. C₂×#88060	C₂	semidirect
5. C₂×C₄⋊#1055	C₂	semidirect
6. C₈×D₅₂	C₂	direct (N × H)
16. C₂×C₈×D₁₃	C₂²	semidirect
19. C₄×C₈×C₁₃	C₂²	semidirect
20. #88060	C₂²	semidirect
21. C₄⋊#1055	C₂²	semidirect
27. C₂×D₅₂	C₈	direct (N × H)
34. C₂×C₈×C₁₃	D₄	semidirect
48. C₂²×D₁₃	C₂×C₈	semidirect
51. C₂×C₄×C₁₃	C₂×C₈	semidirect
60. D₅₂	C₂×C₈	direct (N × H)
65. C₈×C₁₃	C₂×D₄	semidirect
72. C₂×C₄×C₈	D₁₃	semidirect
80. C₂×D₁₃	C₂²×C₈	semidirect
83. C₄×C₁₃	C₂²×C₈	semidirect
101. C₄×C₈	C₂×D₁₃	semidirect
102. C₂×C₁₃	C₈×D₄	semidirect
122. C₂×C₈	D₅₂	direct (N × H)
129. C₁₃	C₂×C₈×D₄	semidirect
142. C₂×C₄	C₈×D₁₃	semidirect
151. C₈	C₂×D₅₂	direct (N × H)
163. C₄	C₂×C₈×D₁₃	semidirect
167. C₂	C₈×D₅₂	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×C₈×D₁₃	2
C₂×C₄×D₅₂	2
C₂×C₄×C₈×C₁₃	2
C₂×#88060	2
C₂×C₄⋊#1055	2
C₈×D₅₂	2
C₂×C₈×D₄	13

Subgroups — C2×C8×D52 #13123510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₅₂ #13123510