Subgroups — #27381471

Order 13568.

1956 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×#16811390	C₂	1	2	yes	maximal
2	C₂×#16810381	C₂	1	12	yes	maximal, complemented
3	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
4	#16811390	—	2	8	no	—
5	#16811390	—	2	8	no	—
6	C₂×#13133224	C₄	1	16	yes	—
7	C₂²×#11719917	C₂²	1	12	yes	—
8	#16810381	C₂²	1	16	yes	—
9	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#568	—	2	8	no	—
11	C₂×#131×C₅₃	C₄	1	8	yes	—
12	C₂²×M₁₆×C₅₃	C₂²	1	6	yes	—
13	#13133224	—	2	64	no	—
14	C₂×#11719917	—	2	24	no	—
15	C₂×#11719917	—	2	24	no	—
16	C₂×#11719917	C₂×C₄	1	24	yes	—
17	C₂×#11719917	C₂³	1	8	yes	—
18	C₂²×#88322	C₂×C₄	1	24	yes	—
19	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
20	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×#41×C₅₃	D₄	1	4	yes	—
22	C₂×#41×C₅₃	D₄	1	8	yes	—
23	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
24	C₂²×D₄×C₅₃	Q₈	1	3	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
26	#131×C₅₃	—	2	32	no	—
27	C₂×M₁₆×C₅₃	—	2	24	no	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	12	yes	—
29	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	12	yes	—
31	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
32	#41×C₅₃	C₄²	1	16	yes	—
33	#41×C₅₃	C₂²⋊C₄	1	32	yes	—
34	#41×C₅₃	C₂²⋊C₄	1	16	yes	—
35	C₂×Q₈×C₅₃	C₄⋊C₄	1	8	yes	—
36	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
37	C₂×D₄×C₅₃	C₄⋊C₄	1	24	yes	—
38	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	M₁₆×C₅₃	—	2	24	no	—
46	M₁₆×C₅₃	—	2	8	no	—
47	C₂×C₈×C₅₃	—	2	24	no	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
49	#11719917	—	2	48	no	—
50	#11719917	—	2	8	no	—
51	#11719917	—	2	8	no	—
52	C₂×#88322	—	2	48	no	—
53	C₂×#88322	C₂²×C₄	1	16	yes	—
54	C₂³×C₅₃	—	8	3	no	—
55	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
56	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
57	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
58	Q₈×C₅₃	#95	1	16	yes	—
59	D₄×C₅₃	#95	1	48	yes	—
60	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
61	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
62	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
65	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
69	C₈×C₅₃	—	2	8	no	—
70	#88322	—	2	16	no	—
71	#20340	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	—	4	12	no	—
74	C₂²×C₅₃	—	4	6	no	—
75	C₂²×C₅₃	C₂×#95	1	12	yes	—
76	C₂²×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
77	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₅₃	C₂×#95	1	12	yes	—
79	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
80	C₄×C₅₃	C₂²×C₂²⋊C₄	1	2	yes	—
81	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×#568	—	53	2	no	—
83	C₂×#568	D₅₃	1	1	yes	characteristic
84	C₂×#338	—	53	12	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	2	12	no	—
88	C₂×C₅₃	C₂²×#95	1	1	yes	characteristic
89	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
90	#568	—	2	8	no	—
91	#568	—	106	8	no	—
92	#568	—	106	8	no	—
93	C₂×#131	—	53	16	no	—
94	C₂×#131	Dic₅₃	1	8	yes	—
95	C₂²×M₁₆	—	53	12	no	—
96	C₂²×M₁₆	C₂×D₅₃	1	6	yes	—
97	#338	—	53	16	no	—
98	C₅₃	#20340	1	1	yes	characteristic, complemented, Sylow
99	C₂×#41	C₄×D₅₃	1	8	yes	—
100	C₂×#41	D₂₁₂	1	4	yes	—
101	C₂×#41	C₂×Dic₅₃	1	4	yes	—
102	C₂×#41	C₂²⋊D₅₃	1	8	yes	—
103	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
104	C₂²×D₄	Dic₁₀₆	1	3	yes	—
105	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
106	#131	—	2	32	no	—
107	#131	—	106	64	no	—
108	C₂×M₁₆	—	2	24	no	—
109	C₂×M₁₆	—	53	24	no	—
110	C₂×M₁₆	—	53	8	no	—
111	C₂×M₁₆	—	106	24	no	—
112	C₂×M₁₆	—	106	24	no	—
113	C₂×M₁₆	C₂×Dic₅₃	1	12	yes	—
114	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
115	C₂²×C₈	—	53	24	no	—
116	C₂²×C₈	C₂×Dic₅₃	1	12	yes	—
117	C₂⁴	C₂×Dic₁₀₆	1	3	yes	—
118	#41	C₄×Dic₅₃	1	16	yes	—
119	#41	#11719921	1	32	yes	—
120	#41	C₂²⋊Dic₅₃	1	16	yes	—
121	C₂×Q₈	#11719919	1	8	yes	—
122	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
123	C₂×D₄	#11719919	1	24	yes	—
124	C₂×D₄	C₄⋊Dic₅₃	1	12	yes	—
125	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
126	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
127	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
128	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
130	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
131	M₁₆	—	2	24	no	—
132	M₁₆	—	2	8	no	—
133	M₁₆	—	106	48	no	—
134	M₁₆	—	106	8	no	—
135	M₁₆	—	106	8	no	—
136	C₂×C₈	—	2	24	no	—
137	C₂×C₈	—	53	16	no	—
138	C₂×C₈	—	106	48	no	—
139	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
140	C₂³	—	8	3	no	—
141	C₂³	C₂×#11719919	1	12	yes	—
142	C₂³	C₂×C₄⋊Dic₅₃	1	6	yes	—
143	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
144	Q₈	#13133124	1	16	yes	—
145	D₄	#13133124	1	48	yes	—
146	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
147	C₂×C₄	C₂×#11719921	1	24	yes	—
148	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	12	yes	—
149	C₂×C₄	C₂×#11719919	1	12	yes	—
150	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
151	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
152	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₂²⋊D₅₃	1	2	yes	—
154	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
155	C₈	—	2	8	no	—
156	C₈	—	106	16	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	12	no	—
159	C₂²	—	4	6	no	—
160	C₂²	C₂×#13133124	1	12	yes	—
161	C₂²	C₂²×#11719919	1	2	yes	—
162	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
163	C₄	C₂×#13133124	1	12	yes	—
164	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
165	C₄	C₂²×#11719921	1	2	yes	—
166	C₄	C₂²×C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	—	2	1	no	—
169	C₂	—	2	12	no	—
170	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	#20340
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×#16810381	C₂	semidirect
98. C₅₃	#20340	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811390	2
C₂×#16810381	2
C₂×C₅₃×#568	2
#20340	53