Subgroups — C₄×C₅₃×#334 #27215783

Order 13568.

470 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	C₄×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#334	C₂	1	2	yes	maximal
4	C₅₃×#1277	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	—	2	2	no	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	2	no	—
13	C₅₃×#334	C₄	1	8	yes	direct product factor
14	C₂²×C₈×C₅₃	—	2	4	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	—	2	4	no	—
24	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
25	C₄×C₈×C₅₃	—	2	4	no	—
26	C₄×C₈×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	—	4	4	no	—
28	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
29	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	8	no	—
39	C₂×C₈×C₅₃	—	4	4	no	—
40	C₂×C₈×C₅₃	—	4	4	no	—
41	C₂×C₈×C₅₃	C₄²	1	2	yes	—
42	C₂×C₈×C₅₃	C₄²	1	2	yes	—
43	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₄²×C₅₃	—	2	4	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	C₂×C₈	1	4	yes	—
48	C₄²×C₅₃	M₁₆	1	4	yes	—
49	C₄²×C₅₃	SD₁₆	1	4	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
51	C₄²×C₅₃	#41	1	2	yes	—
52	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
57	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₅₃	#106	1	4	yes	—
59	C₂×C₄×C₅₃	#108	1	4	yes	—
60	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	#135	1	2	yes	—
68	C₈×C₅₃	—	2	2	no	—
69	C₈×C₅₃	—	2	4	no	—
70	C₈×C₅₃	—	4	8	no	—
71	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
72	C₄×#334	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
75	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
76	C₂²×C₅₃	#427	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×#108	1	2	yes	—
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	#334	1	8	yes	direct product factor
81	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
83	C₄×C₅₃	#431	1	2	yes	—
84	C₄×C₅₃	C₄×SD₁₆	1	4	yes	—
85	C₄×C₄⋊C₈	C₂×C₅₃	1	2	yes	—
86	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
87	C₂×#334	C₂×C₅₃	1	2	yes	—
88	#1277	C₂×C₅₃	1	2	yes	—
89	C₂×C₅₃	C₂×#334	1	2	yes	—
90	C₂×C₅₃	#1564	1	2	yes	—
91	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
93	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
94	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
95	C₂×C₄×C₈	—	2	2	no	—
96	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
97	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
98	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
100	C₄³	C₂²×C₅₃	1	1	yes	characteristic
101	#336	—	2	2	no	—
102	#334	C₄×C₅₃	1	8	yes	direct product factor
103	C₅₃	C₄×#334	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
106	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
107	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
109	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
110	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
112	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	4	no	—
117	C₄×C₈	—	4	4	no	—
118	C₄×C₈	C₈×C₅₃	1	8	yes	complemented
119	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
120	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
121	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
122	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	8	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	C₄²×C₅₃	1	2	yes	—
132	C₂×C₈	C₄²×C₅₃	1	2	yes	—
133	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
134	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₂×C₈×C₅₃	1	4	yes	—
138	C₄²	M₁₆×C₅₃	1	4	yes	—
139	C₄²	SD₁₆×C₅₃	1	4	yes	—
140	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
141	C₄²	#41×C₅₃	1	2	yes	—
142	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
147	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
148	C₂×C₄	#106×C₅₃	1	4	yes	—
149	C₂×C₄	#108×C₅₃	1	4	yes	—
150	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
151	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
152	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
153	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
154	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
155	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
156	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
157	C₂×C₄	#135×C₅₃	1	2	yes	—
158	C₈	—	2	2	no	—
159	C₈	—	2	4	no	—
160	C₈	—	4	8	no	—
161	C₈	C₄×C₈×C₅₃	1	4	yes	complemented
162	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
163	C₂²	C₂×C₄⋊C₈×C₅₃	1	2	yes	—
164	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
165	C₂²	C₅₃×#427	1	1	yes	characteristic
166	C₂²	C₂×#108×C₅₃	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	C₅₃×#334	1	8	yes	direct product factor
170	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
171	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
172	C₄	C₅₃×#431	1	2	yes	—
173	C₄	C₄×SD₁₆×C₅₃	1	4	yes	—
174	C₂	C₂×C₅₃×#334	1	2	yes	—
175	C₂	C₅₃×#1564	1	2	yes	—
176	C₂	C₄×C₄⋊C₈×C₅₃	1	1	yes	characteristic
177	C₂	C₄×#106×C₅₃	1	1	yes	characteristic
178	C₂	C₄×#108×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₅₃	D₄
commutator	C₄	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#334	C₄	direct (N × H)
28. C₄×C₈×C₅₃	C₈	semidirect
71. C₈×C₅₃	C₄×C₈	semidirect
72. C₄×#334	C₅₃	direct (N × H)
80. C₄×C₅₃	#334	direct (N × H)
102. #334	C₄×C₅₃	direct (N × H)
103. C₅₃	C₄×#334	direct (N × H)
118. C₄×C₈	C₈×C₅₃	semidirect
161. C₈	C₄×C₈×C₅₃	semidirect
169. C₄	C₅₃×#334	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈×C₅₃	2
C₄²×C₈×C₅₃	2
C₂×C₅₃×#334	2
C₅₃×#1277	2
C₄×#334	53

Subgroups — C4×C53×#334 #27215783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#334 #27215783