Subgroups — #27536754

Order 13568.

716 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	#20287818	C₂	1	2	yes	maximal
2	#20276320	C₂	1	6	yes	maximal, complemented
3	C₂×#16810401	C₂	1	6	yes	maximal, complemented
4	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic
5	#16811040	C₄	1	16	yes	complemented
6	C₂×#13133157	C₂²	1	12	yes	—
7	#16810401	C₂²	1	4	yes	—
8	#16810401	C₂²	1	12	yes	—
9	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
12	C₅₃×#405	C₄	1	3	yes	—
13	C₅₃×#405	C₂²	1	3	yes	—
14	C₄×M₁₆×C₅₃	C₄	1	3	yes	—
15	C₂×#97×C₅₃	C₂²	1	3	yes	—
16	#13133157	C₂×C₄	1	24	yes	complemented
17	#13133157	C₂³	1	8	yes	—
18	C₂×#11719908	C₂×C₄	1	24	yes	complemented
19	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
20	#131×C₅₃	—	2	4	no	—
21	#131×C₅₃	C₂×C₄	1	8	yes	—
22	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
23	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
26	C₄×Q₈×C₅₃	—	2	1	no	—
27	C₄×D₄×C₅₃	—	2	3	no	—
28	#117×C₅₃	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
30	#97×C₅₃	C₂×C₄	1	3	yes	—
31	#97×C₅₃	C₂×C₄	1	3	yes	—
32	#97×C₅₃	C₂³	1	1	yes	characteristic
33	#97×C₅₃	C₂³	1	3	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
35	#41×C₅₃	C₄²	1	4	yes	—
36	#41×C₅₃	M₁₆	1	4	yes	—
37	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
38	C₂×D₄×C₅₃	M₁₆	1	3	yes	—
39	C₂²×C₄×C₅₃	—	2	3	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
41	M₁₆×C₅₃	—	2	6	no	—
42	M₁₆×C₅₃	C₄²	1	12	yes	—
43	C₂×C₈×C₅₃	—	2	6	no	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
48	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₅₃	—	2	3	no	—
50	C₂²⋊C₄×C₅₃	—	2	3	no	—
51	C₄²×C₅₃	—	2	3	no	—
52	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	#11719908	C₂²×C₄	1	16	yes	—
55	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
56	Q₈×C₅₃	#97	1	4	yes	—
57	D₄×C₅₃	#97	1	12	yes	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	3	no	—
60	C₂×C₄×C₅₃	—	8	3	no	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
63	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	2	3	no	—
67	C₈×C₅₃	C₂×C₄²	1	6	yes	—
68	C₈×C₅₃	C₂³×C₄	1	2	yes	—
69	#33329	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	4	3	no	—
71	C₂²×C₅₃	C₂×#97	1	6	yes	—
72	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
73	C₄×C₅₃	—	4	3	no	—
74	C₄×C₅₃	—	4	1	no	—
75	C₄×C₅₃	C₂×#97	1	6	yes	—
76	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
77	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
78	#3407	—	53	2	no	—
79	#2875	D₅₃	1	1	yes	characteristic
80	#2110	—	53	6	no	—
81	C₂×#347	—	53	6	no	—
82	C₂×C₅₃	—	2	1	no	—
83	C₂×C₅₃	—	2	6	no	—
84	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
85	C₂×#131	Dic₅₃	1	1	yes	characteristic
86	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
87	C₄×#41	Dic₅₃	1	1	yes	characteristic
88	#504	—	53	16	no	—
89	C₂×M₃₂	—	53	12	no	—
90	#405	Dic₅₃	1	3	yes	—
91	#405	C₂×D₅₃	1	3	yes	—
92	C₄×M₁₆	Dic₅₃	1	3	yes	—
93	C₂×#97	C₂×D₅₃	1	3	yes	—
94	#347	—	53	4	no	—
95	#347	—	53	12	no	—
96	C₅₃	#33329	1	1	yes	characteristic, complemented, Sylow
97	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
98	#131	—	2	4	no	—
99	#131	C₄×D₅₃	1	8	yes	—
100	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
101	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
102	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
103	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×D₄	—	2	3	no	—
106	#117	C₂×Dic₅₃	1	3	yes	—
107	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
108	M₃₂	—	53	24	no	—
109	M₃₂	—	53	8	no	—
110	C₂×C₁₆	—	53	24	no	—
111	#97	C₂×Dic₅₃	1	3	yes	—
112	#97	C₂×Dic₅₃	1	3	yes	—
113	#97	C₂²×D₅₃	1	1	yes	characteristic
114	#97	C₂²×D₅₃	1	3	yes	—
115	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
116	#41	#11719917	1	4	yes	—
117	#41	C₄×Dic₅₃	1	4	yes	—
118	C₂×Q₈	#11719917	1	1	yes	characteristic
119	C₂×D₄	#11719917	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
122	M₁₆	—	2	6	no	—
123	M₁₆	C₄×Dic₅₃	1	12	yes	—
124	C₂×C₈	—	2	6	no	—
125	C₂×C₈	C₂×C₄×D₅₃	1	12	yes	—
126	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
127	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
129	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	3	no	—
131	C₂²⋊C₄	—	2	3	no	—
132	C₄²	—	2	3	no	—
133	C₄²	C₂²×Dic₅₃	1	3	yes	—
134	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
135	C₁₆	—	53	16	no	—
136	C₂³	C₂×#11719917	1	3	yes	—
137	Q₈	#13133128	1	4	yes	—
138	D₄	#13133128	1	12	yes	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	8	3	no	—
142	C₂×C₄	C₂×C₄×Dic₅₃	1	6	yes	—
143	C₂×C₄	C₂×#11719917	1	3	yes	—
144	C₂×C₄	C₂×#11719917	1	6	yes	—
145	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	3	no	—
148	C₈	C₂×C₄×Dic₅₃	1	6	yes	—
149	C₈	C₂²×C₄×D₅₃	1	2	yes	—
150	C₂²	—	4	3	no	—
151	C₂²	C₂×#13133128	1	6	yes	—
152	C₂²	C₂²×#11719917	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×#13133128	1	6	yes	—
156	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
157	C₄	C₂²×#11719917	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#13133128	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2875	C₂
socle	C₂×C₅₃	C₂²×#97
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2875	D₅₃
hypocenter	C₅₃	#33329
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #2875

Complements

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

N	H ≅ G/N	type
2. #20276320	C₂	semidirect
3. C₂×#16810401	C₂	semidirect
5. #16811040	C₄	semidirect
16. #13133157	C₂×C₄	semidirect
18. C₂×#11719908	C₂×C₄	semidirect
96. C₅₃	#33329	semidirect

Maximal subgroups

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

subgroup	index
#20287818	2
#20276320	2
C₂×#16810401	2
C₅₃×#2875	2
#33329	53