Subgroups — C₄×C₅₃×#335 #27215784

Order 13568.

470 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	C₄×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	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×#335	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	D₈	1	2	yes	—
50	C₄²×C₅₃	Q₁₆	1	2	yes	—
51	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
52	C₄²×C₅₃	#41	1	2	yes	—
53	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
59	C₂×C₄×C₅₃	#107	1	4	yes	—
60	C₂×C₄×C₅₃	#108	1	4	yes	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	2	yes	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	4	no	—
72	C₈×C₅₃	—	4	8	no	—
73	C₈×C₅₃	C₄×C₈	1	4	yes	complemented
74	C₄×#335	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₃	#427	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×#108	1	2	yes	—
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	—	2	2	no	—
82	C₄×C₅₃	#335	1	8	yes	direct product factor
83	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
84	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₅₃	#431	1	2	yes	—
86	C₄×C₅₃	C₄×D₈	1	2	yes	—
87	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
88	C₄×C₄⋊C₈	C₂×C₅₃	1	2	yes	—
89	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
90	C₂×#335	C₂×C₅₃	1	2	yes	—
91	#1277	C₂×C₅₃	1	2	yes	—
92	C₂×C₅₃	C₂×#335	1	2	yes	—
93	C₂×C₅₃	#1564	1	2	yes	—
94	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
96	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
97	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
98	C₂×C₄×C₈	—	2	2	no	—
99	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
100	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
103	C₄³	C₂²×C₅₃	1	1	yes	characteristic
104	#336	—	2	2	no	—
105	#335	C₄×C₅₃	1	8	yes	direct product factor
106	C₅₃	C₄×#335	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₈	—	2	4	no	—
108	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
112	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
113	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
114	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
115	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	—	4	4	no	—
121	C₄×C₈	C₈×C₅₃	1	8	yes	complemented
122	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
123	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
124	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	C₄²×C₅₃	1	2	yes	—
135	C₂×C₈	C₄²×C₅₃	1	2	yes	—
136	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
137	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	C₂×C₈×C₅₃	1	4	yes	—
141	C₄²	M₁₆×C₅₃	1	4	yes	—
142	C₄²	D₈×C₅₃	1	2	yes	—
143	C₄²	Q₁₆×C₅₃	1	2	yes	—
144	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
145	C₄²	#41×C₅₃	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
151	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
152	C₂×C₄	#107×C₅₃	1	4	yes	—
153	C₂×C₄	#108×C₅₃	1	4	yes	—
154	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
155	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
156	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
157	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
158	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
159	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
160	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
162	C₂×C₄	#135×C₅₃	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	4	8	no	—
166	C₈	C₄×C₈×C₅₃	1	4	yes	complemented
167	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
168	C₂²	C₂×C₄⋊C₈×C₅₃	1	2	yes	—
169	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
170	C₂²	C₅₃×#427	1	1	yes	characteristic
171	C₂²	C₂×#108×C₅₃	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	C₅₃×#335	1	8	yes	direct product factor
175	C₄	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
176	C₄	C₄×M₁₆×C₅₃	1	1	yes	characteristic
177	C₄	C₅₃×#431	1	2	yes	—
178	C₄	C₄×D₈×C₅₃	1	2	yes	—
179	C₄	C₄×Q₁₆×C₅₃	1	2	yes	—
180	C₂	C₂×C₅₃×#335	1	2	yes	—
181	C₂	C₅₃×#1564	1	2	yes	—
182	C₂	C₄×C₄⋊C₈×C₅₃	1	1	yes	characteristic
183	C₂	C₄×#107×C₅₃	1	1	yes	characteristic
184	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×#335	C₄	direct (N × H)
28. C₄×C₈×C₅₃	C₈	semidirect
73. C₈×C₅₃	C₄×C₈	semidirect
74. C₄×#335	C₅₃	direct (N × H)
82. C₄×C₅₃	#335	direct (N × H)
105. #335	C₄×C₅₃	direct (N × H)
106. C₅₃	C₄×#335	direct (N × H)
121. C₄×C₈	C₈×C₅₃	semidirect
166. C₈	C₄×C₈×C₅₃	semidirect
174. C₄	C₅₃×#335	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×#335	2
C₅₃×#1277	2
C₄×#335	53

Subgroups — C4×C53×#335 #27215784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#335 #27215784