Subgroups — C₂²×C₁₆×#6157 #28327881

Order 13568.

1022 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×#6157	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#6157	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₅₃⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×#6157	C₄	1	1	yes	characteristic
7	C₂×D₅₃⋊C₁₆	C₄	1	6	yes	complemented
8	C₂×D₅₃⋊C₁₆	C₂²	1	6	yes	—
9	C₂²×#11719910	C₄	1	2	yes	complemented
10	C₂×C₈×#6157	C₄	1	6	yes	—
11	C₂×C₈×#6157	C₂²	1	6	yes	—
12	C₁₆×#6157	C₂²	1	16	yes	direct product factor
13	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×D₅₃	C₄	1	6	yes	complemented
16	C₂×C₁₆×D₅₃	C₂²	1	6	yes	—
17	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
18	C₂³×#6157	C₈	1	2	yes	—
19	C₂×D₅₃⋊C₈	C₈	1	6	yes	—
20	C₂×D₅₃⋊C₈	C₂×C₄	1	6	yes	—
21	C₂²×#88324	C₈	1	2	yes	—
22	C₂×C₄×#6157	C₈	1	6	yes	—
23	C₂×C₄×#6157	C₂×C₄	1	6	yes	—
24	D₅₃⋊C₁₆	C₂×C₄	1	12	yes	complemented
25	D₅₃⋊C₁₆	C₂³	1	4	yes	—
26	C₂×#11719910	C₂×C₄	1	12	yes	complemented
27	C₈×#6157	C₂×C₄	1	12	yes	—
28	C₈×#6157	C₂³	1	4	yes	—
29	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
34	C₂×C₈×D₅₃	C₂³	1	3	yes	—
35	C₂×#11719908	C₂×C₄	1	6	yes	complemented
36	C₁₆×D₅₃	C₂×C₄	1	12	yes	complemented
37	C₁₆×D₅₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	C₂×C₄	1	6	yes	complemented
40	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×#6157	C₁₆	1	16	yes	direct product factor
42	C₂²×#6157	C₂×C₈	1	12	yes	—
43	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₄²	1	3	yes	—
46	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
47	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
49	C₂×#88324	C₂×C₈	1	12	yes	—
50	C₄×#6157	C₂×C₈	1	12	yes	—
51	C₄×#6157	C₂²×C₄	1	4	yes	—
52	D₅₃⋊C₈	C₂×C₈	1	12	yes	—
53	D₅₃⋊C₈	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₅₃	C₄²	1	3	yes	—
55	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
56	C₂×#88322	C₄²	1	3	yes	—
57	C₂×#88322	C₂²×C₄	1	3	yes	—
58	C₈×D₅₃	C₄²	1	6	yes	—
59	C₈×D₅₃	C₂²×C₄	1	3	yes	—
60	C₈×D₅₃	C₂²×C₄	1	3	yes	—
61	C₈×D₅₃	C₂²×C₄	1	3	yes	—
62	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
63	#11719910	C₂²×C₄	1	8	yes	complemented
64	C₁₆×C₅₃	C₂²×C₄	1	4	yes	complemented
65	#11719908	C₂²×C₄	1	4	yes	complemented
66	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
67	C₂²×D₅₃	C₄×C₈	1	3	yes	—
68	C₂²×D₅₃	C₂×C₁₆	1	8	yes	—
69	C₂²×D₅₃	C₂²×C₈	1	3	yes	—
70	C₂×#6157	C₂×C₁₆	1	48	yes	direct product factor
71	C₂×#6157	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₅₃	C₄×C₈	1	3	yes	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
74	C₂×Dic₅₃	C₄×C₈	1	3	yes	—
75	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
76	C₄×D₅₃	C₄×C₈	1	6	yes	—
77	C₄×D₅₃	C₂×C₄²	1	3	yes	—
78	C₄×D₅₃	C₂²×C₈	1	3	yes	—
79	C₄×D₅₃	C₂²×C₈	1	3	yes	—
80	C₄×D₅₃	C₂³×C₄	1	1	yes	characteristic
81	#88324	C₂²×C₈	1	8	yes	—
82	C₈×C₅₃	C₂×C₄²	1	3	yes	—
83	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
84	#88322	C₂×C₄²	1	3	yes	—
85	#88322	C₂³×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₆	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	C₄×C₁₆	1	4	yes	complemented
88	C₂²×C₅₃	C₂×C₄×C₈	1	3	yes	—
89	C₂×D₅₃	C₄×C₁₆	1	12	yes	complemented
90	C₂×D₅₃	C₂×C₄×C₈	1	3	yes	—
91	C₂×D₅₃	C₂²×C₁₆	1	12	yes	—
92	C₂×D₅₃	C₂³×C₈	1	1	yes	characteristic
93	#6157	C₂²×C₁₆	1	16	yes	direct product factor
94	C₄×C₅₃	C₂×C₄×C₈	1	3	yes	—
95	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
96	Dic₅₃	C₂×C₄×C₈	1	3	yes	—
97	Dic₅₃	C₂³×C₈	1	1	yes	characteristic
98	C₂³×C₁₆	—	53	1	no	—
99	C₂³×C₁₆	—	53	1	no	—
100	C₂²×C₄×C₈	—	53	1	no	—
101	C₂×C₄×C₁₆	—	53	12	no	—
102	C₂×C₅₃	C₂×C₄×C₁₆	1	6	yes	complemented
103	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
104	D₅₃	C₂×C₄×C₁₆	1	6	yes	complemented
105	D₅₃	C₂³×C₁₆	1	2	yes	—
106	C₂³×C₈	—	53	1	no	—
107	C₂³×C₈	—	53	1	no	—
108	C₂²×C₄²	—	53	1	no	—
109	C₂²×C₁₆	—	53	6	no	—
110	C₂²×C₁₆	—	53	2	no	—
111	C₂²×C₁₆	—	53	6	no	—
112	C₂²×C₁₆	—	53	1	no	—
113	C₂²×C₁₆	—	53	6	no	—
114	C₂²×C₁₆	—	53	6	no	—
115	C₂²×C₁₆	#6157	1	1	yes	characteristic, direct product factor
116	C₂×C₄×C₈	—	53	6	no	—
117	C₂×C₄×C₈	—	53	6	no	—
118	C₄×C₁₆	—	53	16	no	—
119	C₅₃	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	—	53	2	no	—
121	C₂³×C₄	—	53	1	no	—
122	C₂²×C₈	—	53	6	no	—
123	C₂²×C₈	—	53	2	no	—
124	C₂²×C₈	—	53	6	no	—
125	C₂²×C₈	—	53	1	no	—
126	C₂²×C₈	—	53	3	no	—
127	C₂²×C₈	—	53	3	no	—
128	C₂²×C₈	—	53	3	no	—
129	C₂²×C₈	—	53	3	no	—
130	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
131	C₂×C₄²	—	53	6	no	—
132	C₂×C₄²	—	53	6	no	—
133	C₂×C₁₆	—	53	4	no	—
134	C₂×C₁₆	—	53	12	no	—
135	C₂×C₁₆	—	53	12	no	—
136	C₂×C₁₆	—	53	6	no	—
137	C₂×C₁₆	—	53	4	no	—
138	C₂×C₁₆	—	53	12	no	—
139	C₂×C₁₆	C₂×#6157	1	6	yes	direct product factor
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	53	12	no	—
142	C₂⁴	—	53	1	no	—
143	C₂²×C₄	—	53	16	no	—
144	C₂²×C₄	—	53	12	no	—
145	C₂²×C₄	—	53	1	no	—
146	C₂²×C₄	—	53	3	no	—
147	C₂²×C₄	—	53	3	no	—
148	C₂²×C₄	—	53	3	no	—
149	C₂²×C₄	—	53	3	no	—
150	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
151	C₂×C₈	—	53	12	no	—
152	C₂×C₈	—	53	4	no	—
153	C₂×C₈	—	53	12	no	—
154	C₂×C₈	—	53	3	no	—
155	C₂×C₈	—	53	3	no	—
156	C₂×C₈	—	53	3	no	—
157	C₂×C₈	—	53	3	no	—
158	C₂×C₈	—	53	6	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	3	no	—
161	C₂×C₈	C₄×#6157	1	3	yes	—
162	C₂×C₈	C₂²×#6157	1	3	yes	—
163	C₄²	—	53	12	no	—
164	C₄²	—	53	4	no	—
165	C₁₆	—	53	8	no	—
166	C₁₆	—	53	4	no	—
167	C₁₆	C₂²×#6157	1	4	yes	direct product factor
168	C₂³	—	53	3	no	—
169	C₂³	—	53	3	no	—
170	C₂³	—	53	8	no	—
171	C₂³	C₈×#6157	1	1	yes	characteristic
172	C₂×C₄	—	53	48	no	—
173	C₂×C₄	—	53	8	no	—
174	C₂×C₄	—	53	3	no	—
175	C₂×C₄	—	53	3	no	—
176	C₂×C₄	—	53	6	no	—
177	C₂×C₄	—	53	3	no	—
178	C₂×C₄	—	53	3	no	—
179	C₂×C₄	—	53	3	no	—
180	C₂×C₄	—	53	1	no	—
181	C₂×C₄	C₈×#6157	1	3	yes	—
182	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
183	C₈	—	53	8	no	—
184	C₈	—	53	3	no	—
185	C₈	—	53	1	no	—
186	C₈	C₂×C₄×#6157	1	3	yes	—
187	C₈	C₂³×#6157	1	1	yes	characteristic
188	C₂²	—	53	12	no	—
189	C₂²	—	53	12	no	—
190	C₂²	—	53	1	no	—
191	C₂²	—	53	3	no	—
192	C₂²	C₁₆×#6157	1	4	yes	direct product factor
193	C₂²	C₂×C₈×#6157	1	3	yes	—
194	C₄	—	53	16	no	—
195	C₄	—	53	1	no	—
196	C₄	—	53	3	no	—
197	C₄	C₂×C₈×#6157	1	3	yes	—
198	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
199	C₂	—	53	2	no	—
200	C₂	—	53	6	no	—
201	C₂	C₂×C₁₆×#6157	1	6	yes	direct product factor
202	C₂	C₂²×C₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₁₆	#6157
commutator, hypocenter	C₅₃	C₂²×C₄×C₁₆
Frattini	C₈	C₂³×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂³×C₅₃	C₄×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₈×D₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×#6157	C₂	direct (N × H)
7. C₂×D₅₃⋊C₁₆	C₄	semidirect
9. C₂²×#11719910	C₄	semidirect
12. C₁₆×#6157	C₂²	direct (N × H)
14. C₂²×#11719908	C₄	semidirect
15. C₂×C₁₆×D₅₃	C₄	semidirect
17. C₂²×C₁₆×C₅₃	C₄	semidirect
24. D₅₃⋊C₁₆	C₂×C₄	semidirect
26. C₂×#11719910	C₂×C₄	semidirect
35. C₂×#11719908	C₂×C₄	semidirect
36. C₁₆×D₅₃	C₂×C₄	semidirect
39. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
41. C₂²×#6157	C₁₆	direct (N × H)
63. #11719910	C₂²×C₄	semidirect
64. C₁₆×C₅₃	C₂²×C₄	semidirect
65. #11719908	C₂²×C₄	semidirect
70. C₂×#6157	C₂×C₁₆	direct (N × H)
87. C₂²×C₅₃	C₄×C₁₆	semidirect
89. C₂×D₅₃	C₄×C₁₆	semidirect
93. #6157	C₂²×C₁₆	direct (N × H)
102. C₂×C₅₃	C₂×C₄×C₁₆	semidirect
104. D₅₃	C₂×C₄×C₁₆	semidirect
115. C₂²×C₁₆	#6157	direct (N × H)
119. C₅₃	C₂²×C₄×C₁₆	semidirect
139. C₂×C₁₆	C₂×#6157	direct (N × H)
167. C₁₆	C₂²×#6157	direct (N × H)
192. C₂²	C₁₆×#6157	direct (N × H)
201. C₂	C₂×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
C₂²×D₅₃⋊C₁₆	2
C₂²×C₈×#6157	2
C₂×C₁₆×#6157	2
C₂²×C₁₆×D₅₃	2
C₂²×C₄×C₁₆	53

Subgroups — C22×C16×#6157 #28327881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₆×#6157 #28327881