Subgroups — #28307755

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290921	C₂	1	1	yes	maximal, characteristic
2	#20289918	C₂	1	1	yes	maximal, characteristic
3	#20289626	C₂	1	2	yes	maximal, complemented
4	C₁₆×#88324	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₄	1	2	yes	—
7	#16811639	C₄	1	2	yes	—
8	#16811512	C₄	1	2	yes	—
9	#16811509	C₂²	1	2	yes	—
10	#16811464	C₄	1	2	yes	—
11	C₈×#88324	C₂²	1	2	yes	—
12	C₂²×#11719908	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
15	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
16	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
19	#13133291	C₈	1	8	yes	complemented
20	C₂×#11719910	C₂×C₄	1	4	yes	—
21	C₄⋊#88324	C₈	1	4	yes	—
22	C₂²⋊#88324	C₈	1	4	yes	—
23	C₄×#88324	C₂×C₄	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719908	C₈	1	4	yes	complemented
27	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
35	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
39	C₂×#88324	C₂×C₈	1	8	yes	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
46	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×#88322	C₄²	1	1	yes	characteristic
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
50	#11719910	C₂×C₈	1	8	yes	—
51	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
52	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
53	#11719908	C₂×C₈	1	2	yes	—
54	#11719908	C₂×C₈	1	2	yes	—
55	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
62	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
63	#88324	—	2	8	no	—
64	C₈×C₅₃	C₄×C₈	1	2	yes	—
65	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	#88322	C₄×C₈	1	2	yes	—
68	#88322	C₂²×C₈	1	2	yes	—
69	#11399	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	C₈²	1	2	yes	—
72	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₅₃	C₈²	1	2	yes	—
74	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
76	Dic₅₃	—	2	2	no	—
77	Dic₅₃	#504	1	4	yes	—
78	C₄×M₃₂	—	53	1	no	—
79	C₂×C₄×C₁₆	—	53	1	no	—
80	#1450	—	53	1	no	—
81	#1311	—	53	2	no	—
82	C₈×C₁₆	—	53	2	no	—
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
85	C₂×C₅₃	#2108	1	2	yes	—
86	C₂×M₃₂	—	53	2	no	—
87	C₂²×C₁₆	—	53	1	no	—
88	C₂²×C₁₆	#6157	1	1	yes	characteristic
89	#431	—	53	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	#346	—	53	2	no	—
92	C₄×C₁₆	—	53	2	no	—
93	C₄×C₁₆	—	53	1	no	—
94	C₄×C₁₆	—	53	1	no	—
95	C₄×C₁₆	—	53	1	no	—
96	C₄×C₁₆	—	53	1	no	—
97	#322	—	53	2	no	—
98	C₈²	—	53	2	no	—
99	C₅₃	#11399	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	53	1	no	—
101	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
102	C₂×C₄²	—	53	1	no	—
103	M₃₂	—	53	8	no	—
104	C₂×C₁₆	—	53	4	no	—
105	C₂×C₁₆	—	53	1	no	—
106	C₂×C₁₆	—	53	1	no	—
107	C₂×C₁₆	—	53	4	no	—
108	C₂×C₁₆	#88324	1	4	yes	complemented
109	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
110	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
111	C₄⋊C₈	—	53	4	no	—
112	C₂²⋊C₈	—	53	4	no	—
113	C₄×C₈	—	53	4	no	—
114	C₄×C₈	—	53	1	no	—
115	C₄×C₈	—	53	1	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₂²×C₄	—	53	2	no	—
119	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
120	C₂×C₈	—	53	4	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	8	no	—
124	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
125	C₂×C₈	C₄×#6157	1	1	yes	characteristic
126	C₂×C₈	C₂×#88324	1	2	yes	—
127	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
128	C₄²	—	53	2	no	—
129	C₄²	—	53	1	no	—
130	C₄²	—	53	1	no	—
131	C₁₆	—	53	8	no	—
132	C₁₆	—	53	2	no	—
133	C₁₆	—	53	2	no	—
134	C₁₆	C₂×#88324	1	2	yes	—
135	C₁₆	C₂×#88324	1	2	yes	—
136	C₂³	C₈×#6157	1	1	yes	characteristic
137	C₂×C₄	—	53	2	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	106	4	no	—
140	C₂×C₄	C₈×#6157	1	2	yes	—
141	C₂×C₄	C₈×#6157	1	1	yes	characteristic
142	C₂×C₄	C₄×#88324	1	2	yes	—
143	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
144	C₈	—	53	2	no	—
145	C₈	—	53	2	no	—
146	C₈	—	106	8	no	—
147	C₈	C₄×#88324	1	2	yes	—
148	C₈	C₂²×#88324	1	1	yes	characteristic
149	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	C₈×#88324	1	2	yes	—
152	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
153	C₄	—	53	4	no	—
154	C₄	—	106	2	no	—
155	C₄	C₈×#88324	1	2	yes	—
156	C₄	C₂×C₈×#6157	1	1	yes	characteristic
157	C₄	C₂×C₄×#88324	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	C₂×C₈×#88324	1	1	yes	characteristic
160	C₂	#20290923	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#11399
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #20289626	C₂	semidirect
4. C₁₆×#88324	C₂	semidirect
19. #13133291	C₈	semidirect
26. C₂×#11719908	C₈	semidirect
34. C₂×C₁₆×C₅₃	C₈	semidirect
99. C₅₃	#11399	semidirect
108. C₂×C₁₆	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290921	2
#20289918	2
#20289626	2
C₁₆×#88324	2
C₂×C₁₆×Dic₅₃	2
#11399	53