Subgroups — #27381467

Order 13568.

2036 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287918	C₂	1	2	yes	maximal
2	C₂×#16810380	C₂	1	12	yes	maximal, complemented
3	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
4	#16811106	—	2	8	no	—
5	#16811101	—	2	2	no	—
6	#16811098	—	2	6	no	—
7	#41×Dic₅₃	C₄	1	16	yes	complemented
8	C₂×#13133176	C₂²	1	12	yes	—
9	#16810380	C₂²	1	16	yes	—
10	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#568	—	2	8	no	—
12	C₂×#131×C₅₃	C₄	1	8	yes	complemented
13	C₂²×M₁₆×C₅₃	C₂²	1	6	yes	—
14	Q₈×Dic₅₃	—	2	16	no	—
15	D₄×Dic₅₃	—	2	48	no	—
16	#13133176	—	2	12	no	—
17	#13133176	—	2	12	no	—
18	#13133176	C₂×C₄	1	24	yes	complemented
19	#13133176	C₂³	1	8	yes	—
20	C₂×C₄⋊Dic₅₃	—	2	12	no	—
21	C₂×C₂²⋊Dic₅₃	—	2	12	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	24	yes	complemented
23	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
24	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×#41×C₅₃	D₄	1	8	yes	—
26	C₂×#41×C₅₃	Q₈	1	4	yes	—
27	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×D₄×C₅₃	D₄	1	3	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
30	#131×C₅₃	—	2	32	no	—
31	C₂×M₁₆×C₅₃	—	2	24	no	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	12	yes	complemented
33	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	complemented
35	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
36	#41×C₅₃	C₄²	1	16	yes	—
37	#41×C₅₃	C₂²⋊C₄	1	16	yes	—
38	#41×C₅₃	C₄⋊C₄	1	32	yes	—
39	C₂×Q₈×C₅₃	C₂²⋊C₄	1	8	yes	—
40	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂×D₄×C₅₃	C₂²⋊C₄	1	24	yes	—
42	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
47	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	24	no	—
50	M₁₆×C₅₃	—	2	24	no	—
51	M₁₆×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	24	no	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
54	C₂²⋊Dic₅₃	—	2	24	no	—
55	C₂²⋊Dic₅₃	—	2	8	no	—
56	C₄⋊Dic₅₃	—	2	24	no	—
57	C₄⋊Dic₅₃	—	2	8	no	—
58	C₄×Dic₅₃	—	2	24	no	—
59	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
60	C₂³×C₅₃	—	8	3	no	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
62	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
63	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	Q₈×C₅₃	#95	1	16	yes	—
65	D₄×C₅₃	#95	1	48	yes	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
68	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
72	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	16	no	—
76	C₂×Dic₅₃	—	8	24	no	—
77	C₈×C₅₃	—	2	8	no	—
78	#20338	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	4	12	no	—
80	C₂²×C₅₃	—	4	1	no	—
81	C₂²×C₅₃	—	4	6	no	—
82	C₂²×C₅₃	C₂×#95	1	12	yes	—
83	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
84	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂×#95	1	12	yes	—
86	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
87	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
89	Dic₅₃	—	4	16	no	—
90	C₂×#568	D₅₃	1	1	yes	characteristic
91	#3428	—	53	2	no	—
92	C₂×H(3,ℤ/4)	—	53	12	no	—
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	—	2	1	no	—
95	C₂×C₅₃	—	2	12	no	—
96	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
97	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
98	#568	—	2	8	no	—
99	C₂×#131	Dic₅₃	1	8	yes	complemented
100	C₂²×M₁₆	C₂×D₅₃	1	6	yes	—
101	#520	—	106	8	no	—
102	#519	—	106	2	no	—
103	#518	—	106	6	no	—
104	C₄×#41	—	53	16	no	—
105	C₂×#117	—	53	12	no	—
106	H(3,ℤ/4)	—	53	16	no	—
107	C₅₃	#20338	1	1	yes	characteristic, complemented, Sylow
108	C₂×#41	Dic₁₀₆	1	4	yes	—
109	C₂×#41	C₄×D₅₃	1	8	yes	—
110	C₂×#41	C₂×Dic₅₃	1	4	yes	—
111	C₂×#41	C₂²⋊D₅₃	1	8	yes	—
112	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
113	C₂²×D₄	D₂₁₂	1	3	yes	—
114	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
115	#131	—	2	32	no	—
116	C₂×M₁₆	—	2	24	no	—
117	C₂×M₁₆	C₂×Dic₅₃	1	12	yes	complemented
118	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
119	C₂²×C₈	C₂×Dic₅₃	1	12	yes	complemented
120	C₄×Q₈	—	106	16	no	—
121	C₄×D₄	—	106	48	no	—
122	#117	—	53	24	no	—
123	#117	—	53	8	no	—
124	#117	—	106	12	no	—
125	#117	—	106	12	no	—
126	C₂×C₄⋊C₄	—	106	12	no	—
127	C₂×C₂²⋊C₄	—	106	12	no	—
128	C₂×C₄²	—	53	24	no	—
129	C₂⁴	C₂×D₂₁₂	1	3	yes	—
130	#41	C₄×Dic₅₃	1	16	yes	—
131	#41	#11719919	1	32	yes	—
132	#41	C₂²⋊Dic₅₃	1	16	yes	—
133	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
134	C₂×Q₈	#11719921	1	8	yes	—
135	C₂×D₄	C₄⋊Dic₅₃	1	12	yes	—
136	C₂×D₄	#11719921	1	24	yes	—
137	C₂²×C₄	—	106	24	no	—
138	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
139	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
140	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
141	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
143	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
144	M₁₆	—	2	24	no	—
145	M₁₆	—	2	8	no	—
146	C₂×C₈	—	2	24	no	—
147	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
148	C₄⋊C₄	—	106	24	no	—
149	C₄⋊C₄	—	106	8	no	—
150	C₂²⋊C₄	—	106	24	no	—
151	C₂²⋊C₄	—	106	8	no	—
152	C₄²	—	53	16	no	—
153	C₄²	—	106	24	no	—
154	C₂³	—	8	3	no	—
155	C₂³	C₂×#11719921	1	12	yes	—
156	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
157	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
158	Q₈	#13133124	1	16	yes	—
159	D₄	#13133124	1	48	yes	—
160	C₂×C₄	—	106	16	no	—
161	C₂×C₄	—	424	24	no	—
162	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
163	C₂×C₄	C₂×#11719921	1	12	yes	—
164	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	12	yes	—
165	C₂×C₄	C₂×#11719919	1	24	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
167	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
168	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
170	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
171	C₈	—	2	8	no	—
172	C₂²	—	4	12	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	6	no	—
175	C₂²	C₂×#13133124	1	12	yes	—
176	C₂²	C₂²×#11719921	1	2	yes	—
177	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
178	C₄	—	212	16	no	—
179	C₄	C₂×#13133124	1	12	yes	—
180	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
181	C₄	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
182	C₄	C₂²×#11719919	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	12	no	—
186	C₂	C₂²×#13133124	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#568	C₂
socle	C₂×C₅₃	C₂²×#95
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#568	D₅₃
hypocenter	C₅₃	#20338
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
2. C₂×#16810380	C₂	semidirect
7. #41×Dic₅₃	C₄	semidirect
12. C₂×#131×C₅₃	C₄	semidirect
18. #13133176	C₂×C₄	semidirect
22. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
32. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
34. C₂²×C₈×C₅₃	C₂×C₄	semidirect
99. C₂×#131	Dic₅₃	semidirect
107. C₅₃	#20338	semidirect
117. C₂×M₁₆	C₂×Dic₅₃	semidirect
119. C₂²×C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20287918	2
C₂×#16810380	2
C₂×C₅₃×#568	2
#20338	53