Subgroups — #27300528

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810450	C₂	1	4	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20274875	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810493	—	2	1	no	—
9	#16810450	—	2	8	no	—
10	C₄×#11719908	C₄	1	4	yes	complemented
11	C₂×#107×C₅₃	—	2	1	no	—
12	C₂×#107×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#384	—	2	1	no	—
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	C₂×C₄⋊Dic₅₃	—	4	2	no	—
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	C₂×#11719908	D₄	1	4	yes	—
19	C₂×#11719908	Q₈	1	4	yes	—
20	#13133151	—	2	4	no	—
21	#13133151	—	4	2	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	#107×C₅₃	—	2	4	no	—
28	#107×C₅₃	—	4	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₃	—	4	2	no	—
32	C₂²×C₄×C₅₃	D₈	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₃	D₈	1	2	yes	—
37	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	4	2	no	—
43	C₄⋊C₄×C₅₃	—	4	2	no	—
44	C₄⋊C₄×C₅₃	—	8	4	no	—
45	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₅₃	Q₁₆	1	2	yes	—
47	C₄⋊Dic₅₃	—	4	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	8	4	no	—
50	#11719908	C₄⋊C₄	1	8	yes	complemented
51	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	8	2	no	—
53	C₂×C₄×C₅₃	—	8	4	no	—
54	C₂×C₄×C₅₃	#107	1	4	yes	—
55	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
56	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₅₃	—	8	2	no	—
64	C₂×Dic₅₃	—	8	4	no	—
65	C₈×C₅₃	#107	1	4	yes	—
66	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
67	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
68	#12906	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
70	C₂²×C₅₃	#493	1	1	yes	characteristic
71	C₂²×C₅₃	#500	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
73	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
74	C₄×C₅₃	—	16	2	no	—
75	C₄×C₅₃	#367	1	8	yes	complemented
76	C₄×C₅₃	#384	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₄×C₅₃	#494	1	1	yes	characteristic
79	C₄×C₅₃	#500	1	1	yes	characteristic
80	Dic₅₃	—	16	2	no	—
81	C₂×#367	—	53	4	no	—
82	C₂×C₄×C₁₆	—	53	1	no	—
83	#1845	—	53	1	no	—
84	#1845	D₅₃	1	1	yes	characteristic
85	C₂×C₅₃	#1845	1	1	yes	characteristic
86	C₂×C₅₃	C₂×#367	1	2	yes	—
87	C₂×C₅₃	#2248	1	1	yes	characteristic
88	C₂×C₅₃	#2248	1	1	yes	characteristic
89	C₂×C₅₃	#2254	1	2	yes	—
90	C₂²×C₁₆	—	53	2	no	—
91	C₂×#107	—	2	1	no	—
92	C₂×#107	—	53	2	no	—
93	C₂×#107	—	106	1	no	—
94	C₂×#107	C₂×D₅₃	1	2	yes	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	#384	—	2	1	no	—
97	#384	—	106	1	no	—
98	#367	—	106	8	no	—
99	C₄×C₁₆	—	53	4	no	—
100	C₅₃	#12906	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	—	4	2	no	—
105	C₂×C₄⋊C₄	—	106	2	no	—
106	C₂×C₄⋊C₄	—	212	2	no	—
107	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂×C₁₆	—	53	4	no	—
109	C₂×C₁₆	—	53	4	no	—
110	C₂×C₁₆	—	53	4	no	—
111	#107	—	2	4	no	—
112	#107	—	4	2	no	—
113	#107	—	106	4	no	—
114	#107	—	212	2	no	—
115	C₄×C₈	Dic₁₀₆	1	2	yes	—
116	C₄×C₈	C₄×D₅₃	1	2	yes	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	212	2	no	—
119	C₂²×C₄	#11719922	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂×C₈	#11719919	1	2	yes	—
122	C₂×C₈	#11719922	1	2	yes	—
123	C₂×C₈	#11719925	1	2	yes	—
124	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
126	C₂×C₈	D₄×D₅₃	1	2	yes	—
127	C₂×C₈	Q₈×D₅₃	1	2	yes	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄⋊C₄	—	424	4	no	—
132	C₄⋊C₄	—	212	2	no	—
133	C₄⋊C₄	—	212	2	no	—
134	C₄²	#11719919	1	2	yes	—
135	C₄²	#11719925	1	2	yes	—
136	C₁₆	—	53	8	no	—
137	C₂³	C₂×#11719922	1	1	yes	characteristic
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	424	4	no	—
141	C₂×C₄	—	424	2	no	—
142	C₂×C₄	#13133150	1	4	yes	—
143	C₂×C₄	#13133160	1	4	yes	—
144	C₂×C₄	#13133161	1	4	yes	—
145	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
146	C₂×C₄	#13133213	1	1	yes	characteristic
147	C₂×C₄	#13133215	1	1	yes	characteristic
148	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
149	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
151	C₈	#13133150	1	4	yes	—
152	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
153	C₈	#13133216	1	2	yes	—
154	C₂²	C₂×#13133150	1	1	yes	characteristic
155	C₂²	#16810999	1	1	yes	characteristic
156	C₂²	#16811026	1	1	yes	characteristic
157	C₂²	C₂×#13133160	1	2	yes	—
158	C₂²	C₂×#13133161	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	848	2	no	—
161	C₄	#16810450	1	8	yes	complemented
162	C₄	#16810492	1	1	yes	characteristic
163	C₄	C₂×#13133150	1	1	yes	characteristic
164	C₄	#16811002	1	1	yes	characteristic
165	C₄	#16811026	1	1	yes	characteristic
166	C₂	#20274874	1	1	yes	characteristic
167	C₂	C₂×#16810450	1	2	yes	—
168	C₂	#20276970	1	1	yes	characteristic
169	C₂	#20276972	1	1	yes	characteristic
170	C₂	#20276995	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1845	D₅₃
hypocenter	C₅₃	#12906
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×#11719908	C₄	semidirect
50. #11719908	C₄⋊C₄	semidirect
75. C₄×C₅₃	#367	semidirect
100. C₅₃	#12906	semidirect
161. C₄	#16810450	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810450	2
C₂×C₄×#11719908	2
#20274875	2
C₅₃×#1845	2
#12906	53