Subgroups — #27300523

Order 13568.

318 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810447	C₂	1	4	yes	maximal
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	#20274879	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133149	C₂²	1	2	yes	—
8	#16810489	—	2	1	no	—
9	#16810447	—	2	8	no	—
10	#16810399	C₄	1	2	yes	complemented
11	C₄×#11719908	C₄	1	2	yes	complemented
12	C₂×#108×C₅₃	—	2	1	no	—
13	C₂×#108×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	#13133157	D₄	1	4	yes	—
18	#13133157	Q₈	1	4	yes	—
19	C₂×#11719908	C₂×C₄	1	4	yes	—
20	#13133151	—	4	2	no	—
21	#13133149	—	2	4	no	—
22	#13133124	—	4	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	#108×C₅₃	—	2	4	no	—
28	#108×C₅₃	—	4	2	no	—
29	C₄⋊C₈×C₅₃	—	4	2	no	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	M₁₆×C₅₃	—	4	2	no	—
38	M₁₆×C₅₃	—	4	2	no	—
39	C₂×C₈×C₅₃	—	4	2	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₅₃	SD₁₆	1	2	yes	—
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	#11719908	C₄⋊C₄	1	8	yes	—
54	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	4	2	no	—
56	C₂×C₄×C₅₃	#106	1	2	yes	—
57	C₂×C₄×C₅₃	#106	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×Dic₅₃	—	8	4	no	—
65	C₂×Dic₅₃	—	8	2	no	—
66	C₈×C₅₃	—	8	2	no	—
67	C₈×C₅₃	#106	1	2	yes	—
68	C₈×C₅₃	#106	1	2	yes	—
69	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
70	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
71	#12904	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
74	C₂²×C₅₃	#492	1	1	yes	characteristic
75	C₂²×C₅₃	#498	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	#384	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
80	C₄×C₅₃	#492	1	1	yes	characteristic
81	C₄×C₅₃	#498	1	1	yes	characteristic
82	Dic₅₃	—	16	2	no	—
83	C₂×#365	—	53	4	no	—
84	C₄×M₃₂	—	53	1	no	—
85	#1848	D₅₃	1	1	yes	characteristic
86	#1846	—	53	1	no	—
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	#1844	1	1	yes	characteristic
90	C₂×C₅₃	#2155	1	2	yes	—
91	C₂×M₃₂	—	53	2	no	—
92	C₂×#108	—	2	1	no	—
93	C₂×#108	C₂×D₅₃	1	2	yes	—
94	C₂×#107	—	106	1	no	—
95	C₂×#106	—	53	2	no	—
96	#423	—	2	1	no	—
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	#382	—	106	1	no	—
99	#365	—	106	8	no	—
100	#346	—	53	2	no	—
101	C₄×C₁₆	—	53	2	no	—
102	C₅₃	#12904	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	2	2	no	—
104	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	106	2	no	—
107	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
108	M₃₂	—	53	4	no	—
109	M₃₂	—	53	4	no	—
110	C₂×C₁₆	—	53	4	no	—
111	#108	—	2	4	no	—
112	#108	—	4	2	no	—
113	#107	—	212	2	no	—
114	#106	—	106	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
117	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
118	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
120	#95	—	212	2	no	—
121	C₂²×C₄	—	212	2	no	—
122	C₂²×C₄	#11719924	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	#11719919	1	1	yes	characteristic
128	C₂×C₈	#11719919	1	1	yes	characteristic
129	C₂×C₈	#11719923	1	2	yes	—
130	C₂×C₈	#11719924	1	2	yes	—
131	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₈	D₄×D₅₃	1	2	yes	—
134	C₂×C₈	Q₈×D₅₃	1	2	yes	—
135	C₄⋊C₄	—	212	2	no	—
136	C₄⋊C₄	—	212	2	no	—
137	C₄²	#11719919	1	1	yes	characteristic
138	C₄²	#11719919	1	1	yes	characteristic
139	C₄²	#11719923	1	2	yes	—
140	C₁₆	—	53	8	no	—
141	C₂³	C₂×#11719924	1	1	yes	characteristic
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	424	4	no	—
144	C₂×C₄	—	424	2	no	—
145	C₂×C₄	#13133148	1	2	yes	—
146	C₂×C₄	#13133148	1	2	yes	—
147	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
148	C₂×C₄	#13133213	1	1	yes	characteristic
149	C₂×C₄	#13133215	1	1	yes	characteristic
150	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
151	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
152	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	#13133148	1	2	yes	—
155	C₈	#13133148	1	2	yes	—
156	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
157	C₈	#13133216	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×#13133148	1	1	yes	characteristic
160	C₂²	#16810997	1	1	yes	characteristic
161	C₂²	#16811019	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	848	2	no	—
165	C₄	#16810492	1	1	yes	characteristic
166	C₄	C₂×#13133148	1	1	yes	characteristic
167	C₄	#16810995	1	1	yes	characteristic
168	C₄	#16811019	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#20274871	1	1	yes	characteristic
172	C₂	#20276489	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133148
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#12904
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
10. #16810399	C₄	semidirect
11. C₄×#11719908	C₄	semidirect
102. C₅₃	#12904	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810447	2
C₄×#13133157	2
#20274879	2
C₅₃×#1848	2
#12904	53