Subgroups — C₅₃×#8820 #27216332

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1845	C₂	1	2	yes	maximal
3	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1834	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₅₃	—	2	4	no	—
7	C₂×#107×C₅₃	—	2	2	no	—
8	C₂×#106×C₅₃	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#384	C₂²	1	1	yes	characteristic
12	C₅₃×#384	C₂²	1	2	yes	—
13	C₅₃×#384	C₂²	1	1	yes	characteristic
14	C₅₃×#340	—	2	4	no	—
15	C₅₃×#322	C₄	1	4	yes	complemented
16	C₂²×C₈×C₅₃	D₄	1	4	yes	—
17	C₂²×C₈×C₅₃	D₄	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	#107×C₅₃	—	4	8	no	—
27	#107×C₅₃	—	4	4	no	—
28	#106×C₅₃	—	4	4	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	4	yes	—
32	C₄×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₅₃	—	4	2	no	—
34	C₂²×C₄×C₅₃	—	4	1	no	—
35	C₂²×C₄×C₅₃	—	4	1	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₅₃	—	2	4	no	—
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
42	C₂×C₈×C₅₃	D₈	1	4	yes	—
43	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
44	C₄⋊C₄×C₅₃	—	4	2	no	—
45	C₄⋊C₄×C₅₃	—	4	2	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	—	4	4	no	—
49	C₄⋊C₄×C₅₃	—	4	4	no	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
54	C₂³×C₅₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	8	1	no	—
56	C₂×C₄×C₅₃	—	8	2	no	—
57	C₂×C₄×C₅₃	—	8	1	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	8	2	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₂×C₄×C₅₃	#137	1	4	yes	—
70	C₈×C₅₃	—	2	8	no	—
71	C₈×C₅₃	#107	1	8	yes	complemented
72	#8820	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	#384	1	1	yes	characteristic
74	C₂²×C₅₃	#493	1	1	yes	characteristic
75	C₂²×C₅₃	#494	1	1	yes	characteristic
76	C₂²×C₅₃	#496	1	2	yes	—
77	C₂²×C₅₃	#497	1	2	yes	—
78	C₄×C₅₃	—	16	1	no	—
79	C₄×C₅₃	—	16	2	no	—
80	C₄×C₅₃	—	16	1	no	—
81	C₄×C₅₃	C₂×#107	1	2	yes	—
82	C₄×C₅₃	#428	1	4	yes	—
83	C₄×C₅₃	#500	1	2	yes	—
84	C₄×C₅₃	#501	1	4	yes	—
85	#1845	C₂×C₅₃	1	1	yes	characteristic
86	#1845	C₂×C₅₃	1	2	yes	—
87	#1844	C₂×C₅₃	1	1	yes	characteristic
88	#1834	C₂×C₅₃	1	2	yes	—
89	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
90	C₂×C₅₃	#1721	1	1	yes	characteristic
91	C₂×C₅₃	#1722	1	1	yes	characteristic
92	C₂×C₅₃	#1723	1	2	yes	—
93	C₂×C₅₃	#1845	1	1	yes	characteristic
94	C₂×C₅₃	#1847	1	2	yes	—
95	C₂×#107	—	2	4	no	—
96	C₂×#107	—	2	2	no	—
97	C₂×#106	—	2	2	no	—
98	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
100	#384	C₂²×C₅₃	1	1	yes	characteristic
101	#384	C₂²×C₅₃	1	2	yes	—
102	#384	C₂²×C₅₃	1	1	yes	characteristic
103	#340	—	2	4	no	—
104	#322	C₄×C₅₃	1	4	yes	complemented
105	C₅₃	#8820	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×C₈	D₄×C₅₃	1	4	yes	—
107	C₂²×C₈	D₄×C₅₃	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
116	#107	—	4	8	no	—
117	#107	—	4	4	no	—
118	#106	—	4	4	no	—
119	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
120	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
121	C₄×C₈	Q₈×C₅₃	1	4	yes	—
122	C₄×C₈	Q₈×C₅₃	1	2	yes	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	—	4	1	no	—
126	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	C₄⋊C₄×C₅₃	1	8	yes	—
131	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
132	C₂×C₈	D₈×C₅₃	1	4	yes	—
133	C₂×C₈	Q₁₆×C₅₃	1	4	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
142	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
143	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
144	C₂³	#127×C₅₃	1	1	yes	characteristic
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
154	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
155	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
156	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
157	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
158	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
159	C₂×C₄	#137×C₅₃	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#107×C₅₃	1	8	yes	complemented
162	C₂²	C₅₃×#384	1	1	yes	characteristic
163	C₂²	C₅₃×#493	1	1	yes	characteristic
164	C₂²	C₅₃×#494	1	1	yes	characteristic
165	C₂²	C₅₃×#496	1	2	yes	—
166	C₂²	C₅₃×#497	1	2	yes	—
167	C₄	—	16	1	no	—
168	C₄	—	16	2	no	—
169	C₄	—	16	1	no	—
170	C₄	C₂×#107×C₅₃	1	2	yes	—
171	C₄	C₅₃×#428	1	4	yes	—
172	C₄	C₅₃×#500	1	2	yes	—
173	C₄	C₅₃×#501	1	4	yes	—
174	C₂	C₅₃×#1721	1	1	yes	characteristic
175	C₂	C₅₃×#1722	1	1	yes	characteristic
176	C₂	C₅₃×#1723	1	2	yes	—
177	C₂	C₅₃×#1845	1	1	yes	characteristic
178	C₂	C₅₃×#1847	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₅₃×#322	C₄	semidirect
71. C₈×C₅₃	#107	semidirect
72. #8820	C₅₃	direct (N × H)
104. #322	C₄×C₅₃	semidirect
105. C₅₃	#8820	direct (N × H)
161. C₈	#107×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#1845	2
C₅₃×#1844	2
C₅₃×#1834	2
C₂×C₅₃×#322	2
#8820	53

Subgroups — C53×#8820 #27216332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8820 #27216332