Subgroups — #27298972

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	2	yes	maximal
2	#20272807	C₂	1	4	yes	maximal, complemented
3	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133176	—	2	2	no	—
5	#16810661	—	2	4	no	—
6	#16810661	C₄	1	8	yes	complemented
7	C₂×#13133151	C₂²	1	4	yes	—
8	C₂×#13133149	—	2	2	no	—
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	—	2	4	no	—
11	C₂²×C₁₆×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
13	#13133176	—	2	8	no	—
14	#13133176	—	4	4	no	—
15	C₂×C₄⋊Dic₅₃	—	2	4	no	—
16	C₂×C₂²⋊Dic₅₃	—	4	2	no	—
17	C₂×C₄×Dic₅₃	—	4	2	no	—
18	#13133151	—	2	4	no	—
19	#13133151	C₂×C₄	1	8	yes	—
20	#13133149	—	2	8	no	—
21	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	4	yes	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	8	no	—
29	C₂×C₁₆×C₅₃	—	2	4	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
31	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	4	4	no	—
39	C₂×C₈×C₅₃	C₄²	1	4	yes	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²⋊Dic₅₃	—	4	8	no	—
49	C₄⋊Dic₅₃	—	2	8	no	—
50	C₄⋊Dic₅₃	—	4	4	no	—
51	C₄×Dic₅₃	—	4	8	no	—
52	C₁₆×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	—	2	4	no	—
54	C₂³×C₅₃	#102	1	4	yes	—
55	C₂³×C₅₃	#107	1	2	yes	—
56	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#95	1	4	yes	—
58	C₂×C₄×C₅₃	#102	1	4	yes	—
59	C₂×C₄×C₅₃	#103	1	8	yes	—
60	C₂×C₄×C₅₃	#106	1	4	yes	—
61	C₂×C₄×C₅₃	#107	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₅₃	—	4	8	no	—
67	C₂×Dic₅₃	—	8	8	no	—
68	C₈×C₅₃	#95	1	4	yes	—
69	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
72	#12556	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	2	8	no	—
74	C₂²×C₅₃	—	2	4	no	—
75	C₂²×C₅₃	#340	1	4	yes	—
76	C₂²×C₅₃	C₂×#102	1	2	yes	—
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₃	#508	1	4	yes	—
79	C₄×C₅₃	#340	1	4	yes	—
80	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
81	C₄×C₅₃	C₂×#103	1	2	yes	—
82	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
83	Dic₅₃	—	8	8	no	—
84	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
85	C₂×#427	—	53	2	no	—
86	#1380	—	53	4	no	—
87	C₂×C₅₃	—	2	4	no	—
88	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
89	C₂×C₅₃	#2138	1	4	yes	—
90	C₂×C₅₃	#2158	1	2	yes	—
91	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₂×#117	—	106	2	no	—
93	C₂²×C₁₆	—	2	4	no	—
94	C₂²×C₁₆	Dic₅₃	1	4	yes	complemented
95	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
96	#427	—	53	8	no	—
97	#427	—	106	4	no	—
98	C₂×#107	—	53	4	no	—
99	C₂×#106	—	106	2	no	—
100	C₅₃	#12556	1	1	yes	characteristic, complemented, Sylow
101	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
102	C₂²×C₈	Dic₁₀₆	1	2	yes	—
103	C₂²×C₈	C₄×D₅₃	1	4	yes	—
104	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
106	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
107	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
108	#117	—	106	8	no	—
109	#117	—	212	4	no	—
110	C₂×C₄⋊C₄	—	106	4	no	—
111	C₂×C₂²⋊C₄	—	212	2	no	—
112	C₂×C₄²	—	212	2	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	—	2	8	no	—
115	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
116	#107	—	53	8	no	—
117	#107	—	106	4	no	—
118	#106	—	106	8	no	—
119	C₂⁴	D₄₂₄	1	1	yes	characteristic
120	C₂²×C₄	—	212	4	no	—
121	C₂²×C₄	#11719913	1	4	yes	—
122	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
123	C₂²×C₄	Dic₂₁₂	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
125	C₂²×C₄	#11719921	1	4	yes	—
126	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
127	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
128	C₂×C₈	#11719919	1	8	yes	—
129	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
130	C₂×C₈	#11719921	1	4	yes	—
131	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
132	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
134	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
135	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
136	C₄⋊C₄	—	106	8	no	—
137	C₄⋊C₄	—	212	4	no	—
138	C₂²⋊C₄	—	212	8	no	—
139	C₄²	—	212	8	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	—	2	4	no	—
142	C₂³	#13133139	1	4	yes	—
143	C₂³	#13133151	1	2	yes	—
144	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
145	C₂×C₄	—	424	8	no	—
146	C₂×C₄	—	212	8	no	—
147	C₂×C₄	#13133124	1	4	yes	—
148	C₂×C₄	#13133139	1	4	yes	—
149	C₂×C₄	#13133142	1	8	yes	—
150	C₂×C₄	#13133149	1	4	yes	—
151	C₂×C₄	#13133151	1	2	yes	—
152	C₂×C₄	C₂×#11719921	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂×#11719913	1	2	yes	—
155	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
156	C₈	#13133124	1	4	yes	—
157	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
158	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
159	C₈	C₂×#11719919	1	2	yes	—
160	C₂²	—	2	4	no	—
161	C₂²	—	2	8	no	—
162	C₂²	#16810386	1	4	yes	—
163	C₂²	C₂×#13133139	1	2	yes	—
164	C₂²	C₂×#13133151	1	1	yes	characteristic
165	C₂²	#16811055	1	4	yes	—
166	C₄	—	424	8	no	—
167	C₄	#16810386	1	4	yes	—
168	C₄	C₂×#13133124	1	1	yes	characteristic
169	C₄	C₂×#13133142	1	2	yes	—
170	C₄	C₂×#13133149	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×#16810386	1	1	yes	characteristic
173	C₂	#20276423	1	4	yes	—
174	C₂	#20276499	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	#12556
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20272807	C₂	semidirect
6. #16810661	C₄	semidirect
11. C₂²×C₁₆×C₅₃	C₄	semidirect
94. C₂²×C₁₆	Dic₅₃	semidirect
100. C₅₃	#12556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
#20272807	2
C₂³×C₁₆×C₅₃	2
#12556	53