Subgroups — #27304270

Order 13568.

238 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810456	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276310	C₂	1	1	yes	maximal, characteristic
4	#20272849	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133156	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
10	#16810577	C₄	1	1	yes	characteristic
11	#16810456	C₄	1	4	yes	complemented
12	C₂×#13133123	C₂²	1	1	yes	characteristic
13	C₂×#13133123	C₂²	1	1	yes	characteristic
14	C₃₂×D₅₃	C₄	1	4	yes	complemented
15	#16810399	C₄	1	1	yes	characteristic
16	#16810398	C₄	1	1	yes	characteristic
17	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#11719912	C₈	1	1	yes	characteristic
22	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
23	#13133174	C₈	1	1	yes	characteristic
24	#13133156	C₈	1	4	yes	—
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
27	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
28	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
29	#13133128	C₈	1	1	yes	characteristic
30	#13133127	C₈	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	#13133123	C₂×C₄	1	2	yes	—
33	#13133123	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
38	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	#11719921	C₁₆	1	2	yes	—
44	#11719919	C₁₆	1	2	yes	—
45	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×#88322	C₂×C₈	1	1	yes	characteristic
47	C₂×#88322	C₂×C₈	1	1	yes	characteristic
48	#11719912	C₁₆	1	4	yes	—
49	C₈×D₅₃	C₄²	1	2	yes	—
50	C₈×D₅₃	C₂×C₈	1	2	yes	—
51	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
52	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	#11719908	C₄²	1	2	yes	—
55	#11719908	C₂×C₈	1	2	yes	—
56	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
60	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
61	C₄×D₅₃	C₄×C₈	1	2	yes	—
62	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
63	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
64	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
65	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	#88322	C₄×C₈	1	2	yes	—
67	#88322	C₂×C₁₆	1	2	yes	—
68	#13549	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₂×D₅₃	—	2	2	no	—
71	C₂×D₅₃	C₄×C₁₆	1	2	yes	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
75	Dic₅₃	—	2	2	no	—
76	Dic₅₃	C₄×C₁₆	1	2	yes	—
77	C₂×M₆₄	—	53	1	no	—
78	C₂²×C₃₂	—	53	1	no	—
79	#2108	—	53	1	no	—
80	#1397	—	53	1	no	—
81	#1397	—	53	1	no	—
82	C₄×C₃₂	—	53	1	no	—
83	C₄×C₃₂	D₅₃	1	1	yes	characteristic, complemented
84	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
85	C₂×C₅₃	#2258	1	2	yes	—
86	D₅₃	—	2	2	no	—
87	C₂×M₃₂	—	53	1	no	—
88	C₂²×C₁₆	—	53	1	no	—
89	#405	—	53	1	no	—
90	M₆₄	—	53	4	no	—
91	C₂×C₃₂	—	53	1	no	—
92	C₂×C₃₂	—	53	1	no	—
93	C₂×C₃₂	—	53	4	no	—
94	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
95	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
96	#346	—	53	1	no	—
97	#346	—	53	1	no	—
98	C₄×C₁₆	—	53	1	no	—
99	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
100	C₅₃	#13549	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	53	1	no	—
102	C₂²×C₈	—	53	1	no	—
103	#117	—	53	1	no	—
104	M₃₂	—	53	4	no	—
105	C₂×C₁₆	—	53	1	no	—
106	C₂×C₁₆	—	53	1	no	—
107	C₂×C₁₆	—	53	2	no	—
108	C₂×C₁₆	—	53	2	no	—
109	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
111	#97	—	53	1	no	—
112	#97	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₃₂	—	53	2	no	—
116	C₃₂	—	53	2	no	—
117	C₃₂	C₄×D₅₃	1	2	yes	—
118	C₃₂	C₄×D₅₃	1	2	yes	—
119	C₂²×C₄	—	53	1	no	—
120	M₁₆	—	53	4	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	2	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	53	2	no	—
128	C₂²⋊C₄	—	53	2	no	—
129	C₄²	—	53	1	no	—
130	C₄²	C₈×D₅₃	1	1	yes	characteristic
131	C₁₆	—	53	2	no	—
132	C₁₆	—	53	2	no	—
133	C₁₆	C₈×D₅₃	1	2	yes	—
134	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
135	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂³	—	53	1	no	—
137	C₂×C₄	—	53	1	no	—
138	C₂×C₄	—	53	2	no	—
139	C₂×C₄	—	53	2	no	—
140	C₂×C₄	—	53	2	no	—
141	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
143	C₈	—	53	2	no	—
144	C₈	—	53	2	no	—
145	C₈	C₁₆×D₅₃	1	2	yes	—
146	C₈	C₄²×D₅₃	1	1	yes	characteristic
147	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
148	C₂²	—	53	2	no	—
149	C₂²	—	106	2	no	—
150	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	53	2	no	—
153	C₄	—	106	2	no	—
154	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
155	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
156	C₂	—	106	2	no	—
157	C₂	C₄×C₁₆×D₅₃	1	1	yes	characteristic
158	C₂	#20277008	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂²×D₅₃	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#13549
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
4. #20272849	C₂	semidirect
5. #20272848	C₂	semidirect
6. C₃₂×Dic₅₃	C₂	semidirect
7. C₄×C₃₂×C₅₃	C₂	semidirect
11. #16810456	C₄	semidirect
14. C₃₂×D₅₃	C₄	semidirect
83. C₄×C₃₂	D₅₃	semidirect
100. C₅₃	#13549	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810456	2
C₂×C₃₂×D₅₃	2
#20276310	2
#20272849	2
#20272848	2
C₃₂×Dic₅₃	2
C₄×C₃₂×C₅₃	2
#13549	53