Subgroups — C₄×C₁₆×#6157 #28310182

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×#6157	C₂	1	4	yes	maximal
3	C₄×C₈×#6157	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
6	C₄×D₅₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×#6157	C₄	1	2	yes	—
8	C₂×C₈×#6157	C₂²	1	2	yes	—
9	C₄²×#6157	C₄	1	1	yes	characteristic
10	#16811509	C₄	1	2	yes	complemented
11	C₄×#11719910	C₄	1	2	yes	complemented
12	C₁₆×#6157	C₄	1	16	yes	direct product factor
13	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
14	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₄×#11719908	C₄	1	1	yes	characteristic, complemented
16	C₁₆×Dic₅₃	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×#6157	C₈	1	4	yes	—
20	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
21	D₅₃⋊C₁₆	C₂×C₄	1	8	yes	—
22	C₂×#11719910	C₂×C₄	1	4	yes	—
23	C₄×#88324	C₈	1	4	yes	—
24	C₈×#6157	C₈	1	8	yes	—
25	C₈×#6157	C₂×C₄	1	8	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
28	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#11719908	C₂×C₄	1	2	yes	—
30	C₁₆×D₅₃	C₂×C₄	1	8	yes	—
31	C₄×#88322	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×#6157	C₂×C₈	1	4	yes	—
37	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×#88324	C₂×C₈	1	4	yes	—
40	C₄×#6157	C₁₆	1	16	yes	direct product factor
41	C₄×#6157	C₄²	1	4	yes	—
42	C₄×#6157	C₂×C₈	1	4	yes	—
43	D₅₃⋊C₈	C₄²	1	4	yes	—
44	D₅₃⋊C₈	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₄²	1	1	yes	characteristic
48	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
49	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
50	C₂×#88322	C₄²	1	1	yes	characteristic
51	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
52	C₈×D₅₃	C₂×C₈	1	4	yes	—
53	C₈×D₅₃	C₂²×C₄	1	4	yes	—
54	#11719910	C₄²	1	8	yes	complemented
55	C₁₆×C₅₃	C₄²	1	4	yes	complemented
56	#11719908	C₄²	1	4	yes	complemented
57	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×#6157	C₄×C₈	1	8	yes	—
59	C₂×#6157	C₂×C₁₆	1	16	yes	—
60	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
63	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₄×D₅₃	C₂×C₁₆	1	8	yes	—
65	C₄×D₅₃	C₂×C₄²	1	2	yes	—
66	C₄×D₅₃	C₂²×C₈	1	2	yes	—
67	#88324	C₄×C₈	1	8	yes	—
68	C₈×C₅₃	C₄×C₈	1	2	yes	—
69	C₈×C₅₃	C₂×C₄²	1	2	yes	—
70	#88322	C₄×C₈	1	2	yes	—
71	#88322	C₂×C₄²	1	2	yes	—
72	C₄²×C₁₆	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₂×D₅₃	C₂×C₄×C₈	1	2	yes	—
75	C₂×D₅₃	C₂²×C₁₆	1	4	yes	—
76	#6157	C₄×C₁₆	1	16	yes	direct product factor
77	C₄×C₅₃	C₄×C₁₆	1	4	yes	complemented
78	C₄×C₅₃	C₄³	1	1	yes	characteristic
79	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
80	Dic₅₃	C₄×C₁₆	1	4	yes	complemented
81	Dic₅₃	C₂×C₄×C₈	1	2	yes	—
82	C₂×C₄×C₁₆	—	53	4	no	—
83	C₂×C₄×C₁₆	—	53	1	no	—
84	C₂×C₄×C₁₆	—	53	1	no	—
85	C₄²×C₈	—	53	1	no	—
86	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
87	C₂×C₅₃	C₂×C₄×C₁₆	1	2	yes	—
88	D₅₃	C₂×C₄×C₁₆	1	4	yes	—
89	C₂²×C₁₆	—	53	2	no	—
90	C₂²×C₁₆	—	53	2	no	—
91	C₂×C₄×C₈	—	53	2	no	—
92	C₂×C₄×C₈	—	53	2	no	—
93	C₂×C₄×C₈	—	53	1	no	—
94	C₂×C₄×C₈	—	53	1	no	—
95	C₄³	—	53	1	no	—
96	C₄×C₁₆	—	53	16	no	—
97	C₄×C₁₆	—	53	2	no	—
98	C₄×C₁₆	—	53	2	no	—
99	C₄×C₁₆	—	53	2	no	—
100	C₄×C₁₆	—	53	1	no	—
101	C₄×C₁₆	#6157	1	1	yes	characteristic, direct product factor
102	C₅₃	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	—	53	2	no	—
104	C₂²×C₈	—	53	1	no	—
105	C₂²×C₈	—	53	1	no	—
106	C₂×C₄²	—	53	2	no	—
107	C₂×C₄²	—	53	4	no	—
108	C₂×C₄²	—	53	1	no	—
109	C₂×C₁₆	—	53	4	no	—
110	C₂×C₁₆	—	53	8	no	—
111	C₂×C₁₆	—	53	2	no	—
112	C₂×C₁₆	—	53	8	no	—
113	C₂×C₁₆	C₂×#6157	1	2	yes	—
114	C₄×C₈	—	53	8	no	—
115	C₄×C₈	—	53	8	no	—
116	C₄×C₈	—	53	4	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	C₂×#6157	1	1	yes	characteristic
121	C₂²×C₄	—	53	4	no	—
122	C₂²×C₄	—	53	2	no	—
123	C₂²×C₄	—	53	1	no	—
124	C₂×C₈	—	53	4	no	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	53	4	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	—	53	4	no	—
131	C₂×C₈	C₄×#6157	1	1	yes	characteristic
132	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
133	C₄²	—	53	16	no	—
134	C₄²	—	53	4	no	—
135	C₄²	—	53	4	no	—
136	C₄²	—	53	2	no	—
137	C₄²	—	53	1	no	—
138	C₄²	C₄×#6157	1	1	yes	characteristic
139	C₁₆	—	53	8	no	—
140	C₁₆	—	53	4	no	—
141	C₁₆	C₄×#6157	1	4	yes	direct product factor
142	C₂³	—	53	1	no	—
143	C₂×C₄	—	53	8	no	—
144	C₂×C₄	—	53	16	no	—
145	C₂×C₄	—	53	2	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	2	no	—
148	C₂×C₄	—	53	2	no	—
149	C₂×C₄	—	53	8	no	—
150	C₂×C₄	C₈×#6157	1	2	yes	—
151	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
152	C₈	—	53	8	no	—
153	C₈	—	53	2	no	—
154	C₈	—	53	2	no	—
155	C₈	C₈×#6157	1	2	yes	—
156	C₈	C₂×C₄×#6157	1	2	yes	—
157	C₂²	—	53	2	no	—
158	C₂²	—	53	4	no	—
159	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
160	C₄	—	53	16	no	—
161	C₄	—	53	2	no	—
162	C₄	—	53	4	no	—
163	C₄	C₁₆×#6157	1	4	yes	direct product factor
164	C₄	C₄²×#6157	1	1	yes	characteristic
165	C₄	C₂×C₈×#6157	1	1	yes	characteristic
166	C₂	—	53	4	no	—
167	C₂	C₄×C₈×#6157	1	1	yes	characteristic
168	C₂	C₂×C₁₆×#6157	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄×C₁₆	#6157
commutator, hypocenter	C₅₃	C₄²×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×D₅₃	C₂³
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #16811509	C₄	semidirect
11. C₄×#11719910	C₄	semidirect
12. C₁₆×#6157	C₄	direct (N × H)
15. C₄×#11719908	C₄	semidirect
16. C₁₆×Dic₅₃	C₄	semidirect
17. C₄×C₁₆×C₅₃	C₄	semidirect
40. C₄×#6157	C₁₆	direct (N × H)
54. #11719910	C₄²	semidirect
55. C₁₆×C₅₃	C₄²	semidirect
56. #11719908	C₄²	semidirect
76. #6157	C₄×C₁₆	direct (N × H)
77. C₄×C₅₃	C₄×C₁₆	semidirect
80. Dic₅₃	C₄×C₁₆	semidirect
101. C₄×C₁₆	#6157	direct (N × H)
102. C₅₃	C₄²×C₁₆	semidirect
141. C₁₆	C₄×#6157	direct (N × H)
163. C₄	C₁₆×#6157	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×D₅₃⋊C₁₆	2
C₂×C₁₆×#6157	2
C₄×C₈×#6157	2
C₄×C₁₆×D₅₃	2
C₄²×C₁₆	53

Subgroups — C4×C16×#6157 #28310182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₆×#6157 #28310182