Subgroups — #27536664

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	4	yes	maximal
2	#20283078	C₂	1	2	yes	maximal
3	#20274879	C₂	1	4	yes	maximal
4	#20274875	C₂	1	2	yes	maximal
5	#20274872	C₂	1	2	yes	maximal
6	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133209	C₂²	1	4	yes	—
8	#16811025	—	2	16	no	—
9	C₂×#13133151	C₂²	1	8	yes	—
10	C₂×#13133149	C₂²	1	8	yes	—
11	#16810521	—	2	8	no	—
12	#16810493	C₂²	1	4	yes	—
13	#16810489	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	#13133209	—	2	16	no	—
20	C₂×C₄⋊Dic₅₃	—	2	8	no	—
21	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
22	C₂×#11719919	—	2	8	no	—
23	#13133151	—	2	16	no	—
24	#13133149	—	2	16	no	—
25	#13133124	—	2	8	no	—
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
28	C₂×C₄²×C₅₃	D₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	D₄	1	8	yes	—
33	C₄×C₈×C₅₃	Q₈	1	4	yes	—
34	C₄×C₈×C₅₃	Q₈	1	4	yes	—
35	C₄×C₈×C₅₃	Q₈	1	8	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	8	no	—
41	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
42	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
43	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
47	C₄²×C₅₃	—	2	8	no	—
48	C₄²×C₅₃	C₂×D₄	1	4	yes	—
49	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
50	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
51	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
52	C₄⋊Dic₅₃	—	2	16	no	—
53	C₄⋊Dic₅₃	—	4	16	no	—
54	#11719919	—	4	16	no	—
55	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	—	2	4	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	#127	1	4	yes	—
59	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
60	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
61	C₂×C₄×C₅₃	#135	1	8	yes	—
62	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
66	C₂×Dic₅₃	—	4	8	no	—
67	C₂×Dic₅₃	—	4	16	no	—
68	C₈×C₅₃	C₄⋊Q₈	1	8	yes	—
69	C₈×C₅₃	C₄⋊Q₈	1	8	yes	—
70	#33311	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×#135	1	4	yes	—
75	C₄×C₅₃	—	2	4	no	—
76	C₄×C₅₃	—	2	4	no	—
77	C₄×C₅₃	#499	1	8	yes	complemented
78	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
79	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
80	Dic₅₃	—	8	8	no	—
81	C₂×#499	—	53	4	no	—
82	#2859	—	53	2	no	—
83	#1846	—	53	4	no	—
84	#1845	—	53	2	no	—
85	#1844	—	53	2	no	—
86	C₄²×C₈	D₅₃	1	1	yes	characteristic
87	C₂×C₅₃	#2868	1	1	yes	characteristic
88	C₂×C₅₃	#3147	1	2	yes	—
89	C₂×C₅₃	C₂×#499	1	2	yes	—
90	C₂×C₅₃	#3160	1	2	yes	—
91	C₂×#125	—	53	4	no	—
92	#499	—	106	16	no	—
93	C₂×#107	—	53	8	no	—
94	C₂×#106	—	53	8	no	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
99	#389	—	106	8	no	—
100	#384	—	53	4	no	—
101	#382	—	53	4	no	—
102	C₄³	C₂×D₅₃	1	1	yes	characteristic
103	C₅₃	#33311	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
105	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
106	#125	—	106	16	no	—
107	C₂×C₄⋊C₄	—	53	8	no	—
108	C₂×C₄⋊C₄	—	106	8	no	—
109	C₂×C₄⋊C₄	—	106	8	no	—
110	C₂×C₄²	D₂₁₂	1	4	yes	—
111	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
112	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
114	#107	—	106	16	no	—
115	#106	—	106	16	no	—
116	C₄×C₈	Dic₁₀₆	1	4	yes	—
117	C₄×C₈	Dic₁₀₆	1	4	yes	—
118	C₄×C₈	Dic₁₀₆	1	8	yes	—
119	C₄×C₈	D₂₁₂	1	8	yes	—
120	#95	—	106	8	no	—
121	C₂²×C₄	—	106	8	no	—
122	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
123	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
124	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
125	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
126	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
127	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
128	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
129	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
130	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
131	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
132	C₄⋊C₄	—	106	16	no	—
133	C₄⋊C₄	—	212	16	no	—
134	C₄⋊C₄	—	212	16	no	—
135	C₄²	—	2	8	no	—
136	C₄²	C₂×Dic₁₀₆	1	4	yes	—
137	C₄²	C₂×Dic₁₀₆	1	2	yes	—
138	C₄²	C₂×Dic₁₀₆	1	2	yes	—
139	C₄²	C₂×D₂₁₂	1	4	yes	—
140	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	212	16	no	—
144	C₂×C₄	—	212	8	no	—
145	C₂×C₄	#13133214	1	4	yes	—
146	C₂×C₄	#13133217	1	4	yes	—
147	C₂×C₄	#13133217	1	4	yes	—
148	C₂×C₄	#13133239	1	8	yes	—
149	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
150	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
152	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
153	C₈	#13133217	1	8	yes	—
154	C₈	#13133217	1	8	yes	—
155	C₂²	C₂×#13133214	1	1	yes	characteristic
156	C₂²	C₂×#13133217	1	1	yes	characteristic
157	C₂²	C₂×#13133217	1	1	yes	characteristic
158	C₂²	C₂×#13133239	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	—	424	8	no	—
162	C₄	#16811025	1	8	yes	complemented
163	C₄	C₂×#13133217	1	2	yes	—
164	C₄	C₂×#13133217	1	2	yes	—
165	C₂	#20283113	1	1	yes	characteristic
166	C₂	#20285143	1	2	yes	—
167	C₂	C₂×#16811025	1	2	yes	—
168	C₂	#20285229	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#33311
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
77. C₄×C₅₃	#499	semidirect
103. C₅₃	#33311	semidirect
162. C₄	#16811025	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811025	2
#20283078	2
#20274879	2
#20274875	2
#20274872	2
C₄²×C₈×C₅₃	2
#33311	53