Subgroups — C₅₃×#20762 #27228274

Order 13568.

910 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2285	C₂	1	2	yes	maximal
3	C₅₃×#1836	C₂	1	8	yes	maximal, complemented
4	C₅₃×#1735	C₂	1	4	yes	maximal
5	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂×#117×C₅₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#97×C₅₃	C₄	1	8	yes	complemented
9	C₂×#97×C₅₃	C₂²	1	4	yes	—
10	C₅₃×#384	C₂²	1	4	yes	—
11	C₅₃×#382	C₂²	1	4	yes	—
12	C₅₃×#380	—	2	4	no	—
13	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
14	C₅₃×#340	C₂²	1	16	yes	—
15	C₂³×C₄×C₅₃	D₄	1	2	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	—	2	8	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
20	#117×C₅₃	—	2	16	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
22	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
23	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
24	C₂×C₄²×C₅₃	—	2	4	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₅₃	D₄	1	2	yes	—
27	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
29	#97×C₅₃	C₂×C₄	1	16	yes	—
30	#95×C₅₃	—	2	8	no	—
31	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	—	2	8	no	—
33	C₂²×C₄×C₅₃	—	4	4	no	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₄×C₅₃	#41	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	16	no	—
42	C₂×C₈×C₅₃	—	2	8	no	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
44	C₄⋊C₄×C₅₃	—	2	16	no	—
45	C₄⋊C₄×C₅₃	—	4	16	no	—
46	C₂²⋊C₄×C₅₃	—	4	8	no	—
47	C₄²×C₅₃	—	4	8	no	—
48	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
49	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
50	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
51	C₂³×C₅₃	—	2	4	no	—
52	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂³×C₅₃	#124	1	4	yes	—
54	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	8	no	—
56	C₂×C₄×C₅₃	—	4	8	no	—
57	C₂×C₄×C₅₃	—	4	8	no	—
58	C₂×C₄×C₅₃	—	4	16	no	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₅₃	#117	1	8	yes	—
62	C₂×C₄×C₅₃	#124	1	4	yes	—
63	C₂×C₄×C₅₃	#125	1	8	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	4	yes	—
67	C₈×C₅₃	—	2	8	no	—
68	#20762	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂²×C₅₃	—	2	8	no	—
70	C₂²×C₅₃	—	2	4	no	—
71	C₂²×C₅₃	#382	1	4	yes	—
72	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₅₃	C₂×#124	1	2	yes	—
74	C₂²×C₅₃	#575	1	4	yes	—
75	C₄×C₅₃	—	2	8	no	—
76	C₄×C₅₃	—	8	8	no	—
77	C₄×C₅₃	#382	1	4	yes	—
78	C₄×C₅₃	C₂×#117	1	2	yes	—
79	C₄×C₅₃	C₂×#125	1	2	yes	—
80	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
81	#2285	C₂×C₅₃	1	2	yes	—
82	#1836	C₂×C₅₃	1	8	yes	complemented
83	#1735	C₂×C₅₃	1	4	yes	—
84	C₂×C₅₃	—	2	4	no	—
85	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
86	C₂×C₅₃	#2911	1	2	yes	—
87	C₂×C₅₃	#3135	1	4	yes	—
88	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
89	C₂×#117	C₂²×C₅₃	1	4	yes	—
90	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
91	C₂×#97	C₄×C₅₃	1	8	yes	complemented
92	C₂×#97	C₂²×C₅₃	1	4	yes	—
93	#384	C₂²×C₅₃	1	4	yes	—
94	#382	C₂²×C₅₃	1	4	yes	—
95	#380	—	2	4	no	—
96	C₄×C₄⋊C₄	—	2	4	no	—
97	#340	C₂²×C₅₃	1	16	yes	—
98	C₅₃	#20762	1	1	yes	characteristic, direct product factor, Sylow
99	C₂³×C₄	D₄×C₅₃	1	2	yes	—
100	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
101	C₂²×C₈	—	2	8	no	—
102	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
103	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
104	#117	—	2	16	no	—
105	C₂×C₄⋊C₄	—	2	8	no	—
106	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
107	C₂×C₂²⋊C₄	—	2	4	no	—
108	C₂×C₄²	—	2	4	no	—
109	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
110	C₂×C₄²	D₄×C₅₃	1	2	yes	—
111	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
112	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
113	#97	C₂×C₄×C₅₃	1	16	yes	—
114	#95	—	2	8	no	—
115	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	4	4	no	—
118	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
119	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
120	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
121	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
123	C₂²×C₄	#41×C₅₃	1	8	yes	—
124	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
125	C₂×C₈	—	2	16	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
128	C₄⋊C₄	—	2	16	no	—
129	C₄⋊C₄	—	4	16	no	—
130	C₂²⋊C₄	—	4	8	no	—
131	C₄²	—	4	8	no	—
132	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
133	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
134	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
135	C₂³	—	2	4	no	—
136	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
137	C₂³	#124×C₅₃	1	4	yes	—
138	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	4	8	no	—
141	C₂×C₄	—	4	8	no	—
142	C₂×C₄	—	4	16	no	—
143	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	8	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
145	C₂×C₄	#117×C₅₃	1	8	yes	—
146	C₂×C₄	#124×C₅₃	1	4	yes	—
147	C₂×C₄	#125×C₅₃	1	8	yes	—
148	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
151	C₈	—	2	8	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	C₅₃×#382	1	4	yes	—
155	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
156	C₂²	C₂×#124×C₅₃	1	2	yes	—
157	C₂²	C₅₃×#575	1	4	yes	—
158	C₄	—	2	8	no	—
159	C₄	—	8	8	no	—
160	C₄	C₅₃×#382	1	4	yes	—
161	C₄	C₂×#117×C₅₃	1	2	yes	—
162	C₄	C₂×#125×C₅₃	1	2	yes	—
163	C₂	—	2	4	no	—
164	C₂	C₂×C₅₃×#382	1	1	yes	characteristic
165	C₂	C₅₃×#2911	1	2	yes	—
166	C₂	C₅₃×#3135	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1836	C₂	semidirect
8. C₂×#97×C₅₃	C₄	semidirect
68. #20762	C₅₃	direct (N × H)
82. #1836	C₂×C₅₃	semidirect
91. C₂×#97	C₄×C₅₃	semidirect
98. C₅₃	#20762	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#97×C₅₃	2
C₅₃×#2285	2
C₅₃×#1836	2
C₅₃×#1735	2
#20762	53

Subgroups — C53×#20762 #27228274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#20762 #27228274