Subgroups — #27403044

Order 13568.

822 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	Q₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	#20275474	C₂	1	6	yes	maximal
3	#20274843	C₂	1	3	yes	maximal
4	C₄×#13133151	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×#13133151	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	#13133151	—	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	D₈	1	4	yes	—
37	C₂×Q₈×C₅₃	Q₁₆	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	—	2	6	no	—
43	C₂×C₈×C₅₃	—	2	6	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
47	C₄⋊C₄×C₅₃	C₄⋊C₄	1	12	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
49	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
50	C₄⋊Dic₅₃	—	2	2	no	—
51	C₄⋊Dic₅₃	—	2	6	no	—
52	C₄⋊Dic₅₃	—	2	6	no	—
53	C₄⋊Dic₅₃	—	4	12	no	—
54	C₄⋊Dic₅₃	—	8	6	no	—
55	C₄⋊Dic₅₃	#41	1	4	yes	—
56	C₄×Dic₅₃	—	4	12	no	—
57	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
58	Q₈×C₅₃	#107	1	16	yes	complemented
59	C₂×C₄×C₅₃	—	2	6	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×D₈	1	6	yes	—
63	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
64	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
67	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
68	C₂×Dic₅₃	—	2	4	no	—
69	C₂×Dic₅₃	—	4	2	no	—
70	C₂×Dic₅₃	—	4	2	no	—
71	C₂×Dic₅₃	—	4	6	no	—
72	C₂×Dic₅₃	—	4	6	no	—
73	C₂×Dic₅₃	—	4	6	no	—
74	C₈×C₅₃	—	2	6	no	—
75	C₈×C₅₃	—	2	2	no	—
76	#22740	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₅₃	#548	1	2	yes	—
79	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
80	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
81	C₂²×C₅₃	#577	1	2	yes	—
82	C₄×C₅₃	—	2	6	no	—
83	C₄×C₅₃	C₂×#107	1	12	yes	—
84	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
85	C₄×C₅₃	#519	1	1	yes	characteristic
86	C₄×C₅₃	#557	1	2	yes	—
87	Dic₅₃	—	4	4	no	—
88	Dic₅₃	—	8	6	no	—
89	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
90	Q₈×C₄⋊C₄	—	53	2	no	—
91	#1947	—	53	6	no	—
92	#1834	—	53	3	no	—
93	C₄×#107	—	53	3	no	—
94	C₂×C₅₃	#2351	1	1	yes	characteristic
95	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
96	C₂×C₅₃	#2912	1	1	yes	characteristic
97	C₂×C₅₃	#3358	1	2	yes	—
98	C₂×C₅₃	#3363	1	2	yes	—
99	C₂×C₄×Q₈	—	106	2	no	—
100	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
101	C₈×Q₈	Dic₅₃	1	8	yes	complemented
102	C₂×#107	—	53	6	no	—
103	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
104	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
105	#389	—	106	6	no	—
106	#384	—	53	6	no	—
107	C₄×C₄⋊C₄	—	53	6	no	—
108	#340	—	53	3	no	—
109	#340	—	53	6	no	—
110	#340	—	53	1	no	—
111	C₅₃	#22740	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
115	C₄×Q₈	—	212	8	no	—
116	C₄×Q₈	Dic₁₀₆	1	4	yes	—
117	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
118	C₂×C₄⋊C₄	—	53	2	no	—
119	C₂×C₄⋊C₄	—	53	6	no	—
120	C₂×C₄⋊C₄	—	106	6	no	—
121	C₂×C₄⋊C₄	—	106	6	no	—
122	C₂×C₄⋊C₄	D₂₁₂	1	3	yes	—
123	C₂×C₄²	—	106	6	no	—
124	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
125	#107	—	106	12	no	—
126	C₄⋊C₈	C₂×Dic₅₃	1	12	yes	—
127	C₄×C₈	C₂×Dic₅₃	1	12	yes	—
128	#95	—	106	6	no	—
129	C₂×Q₈	D₄₂₄	1	4	yes	—
130	C₂×Q₈	Dic₂₁₂	1	4	yes	—
131	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
132	C₂²×C₄	—	106	2	no	—
133	C₂²×C₄	—	106	6	no	—
134	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
135	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
136	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
140	C₂×C₈	C₂²×Dic₅₃	1	6	yes	—
141	C₄⋊C₄	—	53	4	no	—
142	C₄⋊C₄	—	106	2	no	—
143	C₄⋊C₄	—	106	6	no	—
144	C₄⋊C₄	—	106	6	no	—
145	C₄⋊C₄	—	424	6	no	—
146	C₄⋊C₄	—	212	12	no	—
147	C₄⋊C₄	C₄⋊Dic₅₃	1	12	yes	—
148	C₄²	—	212	12	no	—
149	C₄²	C₂×Dic₁₀₆	1	6	yes	—
150	C₄²	C₂²×Dic₅₃	1	6	yes	—
151	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
152	Q₈	#13133151	1	16	yes	complemented
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	106	4	no	—
155	C₂×C₄	—	212	6	no	—
156	C₂×C₄	—	212	6	no	—
157	C₂×C₄	—	212	6	no	—
158	C₂×C₄	—	212	2	no	—
159	C₂×C₄	—	212	2	no	—
160	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
162	C₂×C₄	C₂×D₄₂₄	1	6	yes	—
163	C₂×C₄	C₂×Dic₂₁₂	1	6	yes	—
164	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
166	C₂×C₄	#41×D₅₃	1	2	yes	—
167	C₂×C₄	#13133265	1	2	yes	—
168	C₈	—	2	6	no	—
169	C₈	—	2	2	no	—
170	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂²	#16811285	1	2	yes	—
172	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
173	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
174	C₂²	#16811435	1	2	yes	—
175	C₄	—	2	6	no	—
176	C₄	—	424	6	no	—
177	C₄	—	212	4	no	—
178	C₄	C₂×#13133151	1	12	yes	—
179	C₄	#41×Dic₅₃	1	1	yes	characteristic
180	C₄	#16811101	1	1	yes	characteristic
181	C₄	#16811349	1	2	yes	—
182	C₂	#20277636	1	1	yes	characteristic
183	C₂	C₂²×#13133151	1	1	yes	characteristic
184	C₂	#20283310	1	1	yes	characteristic
185	C₂	#20287351	1	2	yes	—
186	C₂	#20287394	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	#22740
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
58. Q₈×C₅₃	#107	semidirect
101. C₈×Q₈	Dic₅₃	semidirect
111. C₅₃	#22740	semidirect
152. Q₈	#13133151	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
#20275474	2
#20274843	2
C₄×#13133151	2
C₂×C₈×Q₈×C₅₃	2
#22740	53