Subgroups — #27403031

Order 13568.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	#20275479	C₂	1	6	yes	maximal
3	#20274843	C₂	1	3	yes	maximal
4	C₄×#13133149	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₅₃	—	2	2	no	—
7	C₂×#13133149	C₂²	1	6	yes	—
8	#16810522	—	2	6	no	—
9	#16810493	C₂²	1	6	yes	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	6	yes	—
11	#16810386	C₂²	1	6	yes	—
12	#16810386	C₂²	1	3	yes	—
13	#16810386	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₅₃	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
18	Q₈×Dic₅₃	—	4	8	no	—
19	C₂×C₄⋊Dic₅₃	—	2	6	no	—
20	C₂×C₄⋊Dic₅₃	—	2	6	no	—
21	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
23	C₂×C₄×Dic₅₃	—	2	6	no	—
24	#13133149	—	2	12	no	—
25	#13133124	—	2	6	no	—
26	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
29	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
33	C₄⋊C₈×C₅₃	C₂×C₄	1	12	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
35	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×Q₈×C₅₃	SD₁₆	1	8	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	2	2	no	—
41	C₂²×Dic₅₃	—	2	6	no	—
42	C₂×C₈×C₅₃	—	2	6	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
46	C₄⋊C₄×C₅₃	C₄⋊C₄	1	12	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
48	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
49	C₄⋊Dic₅₃	—	2	2	no	—
50	C₄⋊Dic₅₃	—	2	6	no	—
51	C₄⋊Dic₅₃	—	2	6	no	—
52	C₄⋊Dic₅₃	—	4	12	no	—
53	C₄⋊Dic₅₃	—	8	6	no	—
54	C₄⋊Dic₅₃	#41	1	4	yes	—
55	C₄×Dic₅₃	—	4	12	no	—
56	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
57	Q₈×C₅₃	#106	1	16	yes	complemented
58	C₂×C₄×C₅₃	—	2	3	no	—
59	C₂×C₄×C₅₃	—	2	3	no	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
62	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
63	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
66	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
67	C₂×Dic₅₃	—	2	4	no	—
68	C₂×Dic₅₃	—	4	2	no	—
69	C₂×Dic₅₃	—	4	2	no	—
70	C₂×Dic₅₃	—	4	6	no	—
71	C₂×Dic₅₃	—	4	6	no	—
72	C₂×Dic₅₃	—	4	6	no	—
73	C₈×C₅₃	—	2	6	no	—
74	C₈×C₅₃	—	2	2	no	—
75	#22738	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₅₃	#548	1	2	yes	—
78	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
79	C₂²×C₅₃	#576	1	1	yes	characteristic
80	C₂²×C₅₃	#578	1	1	yes	characteristic
81	C₄×C₅₃	—	2	6	no	—
82	C₄×C₅₃	C₂×#106	1	12	yes	—
83	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
84	C₄×C₅₃	#519	1	1	yes	characteristic
85	C₄×C₅₃	#557	1	2	yes	—
86	Dic₅₃	—	4	4	no	—
87	Dic₅₃	—	8	6	no	—
88	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
89	Q₈×C₄⋊C₄	—	53	2	no	—
90	#1948	—	53	6	no	—
91	#1834	—	53	3	no	—
92	C₄×#106	—	53	3	no	—
93	C₂×C₅₃	#2351	1	1	yes	characteristic
94	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
95	C₂×C₅₃	#2913	1	1	yes	characteristic
96	C₂×C₅₃	#3359	1	2	yes	—
97	C₂×C₅₃	#3362	1	2	yes	—
98	C₂×C₄×Q₈	—	106	2	no	—
99	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
100	C₈×Q₈	Dic₅₃	1	8	yes	complemented
101	C₂×#106	—	53	6	no	—
102	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
103	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
104	#389	—	106	6	no	—
105	#384	—	53	6	no	—
106	C₄×C₄⋊C₄	—	53	6	no	—
107	#340	—	53	3	no	—
108	#340	—	53	6	no	—
109	#340	—	53	1	no	—
110	C₅₃	#22738	1	1	yes	characteristic, complemented, Sylow
111	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
114	C₄×Q₈	—	212	8	no	—
115	C₄×Q₈	Dic₁₀₆	1	4	yes	—
116	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
117	C₂×C₄⋊C₄	—	53	2	no	—
118	C₂×C₄⋊C₄	—	53	6	no	—
119	C₂×C₄⋊C₄	—	106	6	no	—
120	C₂×C₄⋊C₄	—	106	6	no	—
121	C₂×C₄⋊C₄	D₂₁₂	1	3	yes	—
122	C₂×C₄²	—	106	6	no	—
123	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
124	#106	—	106	12	no	—
125	C₄⋊C₈	C₂×Dic₅₃	1	12	yes	—
126	C₄×C₈	C₂×Dic₅₃	1	12	yes	—
127	#95	—	106	6	no	—
128	C₂×Q₈	#11719913	1	8	yes	—
129	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
130	C₂²×C₄	—	106	2	no	—
131	C₂²×C₄	—	106	6	no	—
132	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
133	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
134	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
138	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
139	C₄⋊C₄	—	53	4	no	—
140	C₄⋊C₄	—	106	2	no	—
141	C₄⋊C₄	—	106	6	no	—
142	C₄⋊C₄	—	106	6	no	—
143	C₄⋊C₄	—	424	6	no	—
144	C₄⋊C₄	—	212	12	no	—
145	C₄⋊C₄	C₄⋊Dic₅₃	1	12	yes	—
146	C₄²	—	212	12	no	—
147	C₄²	C₂×Dic₁₀₆	1	6	yes	—
148	C₄²	C₂²×Dic₅₃	1	6	yes	—
149	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
150	Q₈	#13133149	1	16	yes	complemented
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	106	4	no	—
154	C₂×C₄	—	212	6	no	—
155	C₂×C₄	—	212	6	no	—
156	C₂×C₄	—	212	6	no	—
157	C₂×C₄	—	212	2	no	—
158	C₂×C₄	—	212	2	no	—
159	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
160	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
161	C₂×C₄	C₂×#11719913	1	12	yes	—
162	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
163	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
164	C₂×C₄	#41×D₅₃	1	2	yes	—
165	C₂×C₄	#13133265	1	2	yes	—
166	C₈	—	2	6	no	—
167	C₈	—	2	2	no	—
168	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
169	C₂²	#16811285	1	2	yes	—
170	C₂²	C₂²×#11719913	1	2	yes	—
171	C₂²	#16811428	1	1	yes	characteristic
172	C₂²	#16811440	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	424	6	no	—
175	C₄	—	212	4	no	—
176	C₄	C₂×#13133149	1	12	yes	—
177	C₄	#41×Dic₅₃	1	1	yes	characteristic
178	C₄	#16811101	1	1	yes	characteristic
179	C₄	#16811349	1	2	yes	—
180	C₂	#20277636	1	1	yes	characteristic
181	C₂	C₂²×#13133149	1	1	yes	characteristic
182	C₂	#20283315	1	1	yes	characteristic
183	C₂	#20287358	1	2	yes	—
184	C₂	#20287387	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#22738
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₅₃	C₄	semidirect
57. Q₈×C₅₃	#106	semidirect
100. C₈×Q₈	Dic₅₃	semidirect
110. C₅₃	#22738	semidirect
150. Q₈	#13133149	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₅₃	2
#20275479	2
#20274843	2
C₄×#13133149	2
C₂×C₈×Q₈×C₅₃	2
#22738	53