Subgroups — C₄×C₅₃×#336 #27218204

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#336	C₂	1	4	yes	maximal
4	C₂³×C₈×C₅₃	C₂²	1	4	yes	—
5	C₂²×C₄²×C₅₃	C₄	1	4	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	—	2	4	no	—
9	C₂×C₄×C₈×C₅₃	—	2	4	no	—
10	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
11	C₄³×C₅₃	—	2	4	no	—
12	C₅₃×#336	C₄	1	16	yes	direct product factor
13	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	—	2	4	no	—
17	C₂²×C₈×C₅₃	—	2	4	no	—
18	C₂²×C₈×C₅₃	—	2	8	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	16	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
21	C₂×C₄²×C₅₃	—	2	8	no	—
22	C₂×C₄²×C₅₃	—	2	8	no	—
23	C₂×C₄²×C₅₃	—	2	4	no	—
24	C₂×C₄²×C₅₃	—	2	4	no	—
25	C₂×C₄²×C₅₃	C₈	1	16	yes	complemented
26	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	D₄	1	4	yes	—
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
31	C₄×C₈×C₅₃	—	2	16	no	—
32	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	—	2	4	no	—
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
37	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	4	yes	—
49	C₂×C₈×C₅₃	—	2	8	no	—
50	C₂×C₈×C₅₃	—	2	8	no	—
51	C₂×C₈×C₅₃	—	2	16	no	—
52	C₂×C₈×C₅₃	—	2	16	no	—
53	C₂×C₈×C₅₃	C₄²	1	16	yes	complemented
54	C₄²×C₅₃	—	2	32	no	—
55	C₄²×C₅₃	—	2	16	no	—
56	C₄²×C₅₃	C₂²⋊C₄	1	8	yes	—
57	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
58	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
60	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₅₃	#117	1	2	yes	—
63	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
64	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	—	2	8	no	—
67	C₂×C₄×C₅₃	—	2	16	no	—
68	C₂×C₄×C₅₃	—	2	16	no	—
69	C₂×C₄×C₅₃	#95	1	8	yes	—
70	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
71	C₂×C₄×C₅₃	C₄×C₈	1	16	yes	complemented
72	C₂×C₄×C₅₃	#97	1	8	yes	—
73	C₂×C₄×C₅₃	C₂²⋊C₈	1	16	yes	—
74	C₂×C₄×C₅₃	C₄⋊C₈	1	16	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	#117	1	4	yes	—
79	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
80	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
81	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
82	C₈×C₅₃	—	2	16	no	—
83	C₄×#336	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₅₃	C₄³	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
86	C₂²×C₅₃	#376	1	1	yes	characteristic
87	C₂²×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
88	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
89	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
90	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
91	C₂²×C₅₃	C₂×#97	1	2	yes	—
92	C₂²×C₅₃	C₄×M₁₆	1	4	yes	—
93	C₂²×C₅₃	C₂×C₂²⋊C₈	1	4	yes	—
94	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
95	C₂²×C₅₃	#431	1	8	yes	—
96	C₄×C₅₃	—	2	16	no	—
97	C₄×C₅₃	#336	1	16	yes	direct product factor
98	C₄×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
99	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
100	C₂²×C₄×C₈	C₂×C₅₃	1	2	yes	—
101	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic
102	C₂×#336	C₂×C₅₃	1	4	yes	—
103	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
104	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
105	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#336	1	2	yes	—
107	C₂×C₅₃	#1744	1	2	yes	—
108	C₂×C₅₃	C₄×C₂²⋊C₈	1	4	yes	—
109	C₂×C₅₃	C₄×C₄⋊C₈	1	4	yes	—
110	C₂³×C₈	C₂²×C₅₃	1	4	yes	—
111	C₂²×C₄²	C₄×C₅₃	1	4	yes	—
112	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
113	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	C₄×C₅₃	1	8	yes	complemented
117	C₄³	—	2	4	no	—
118	#336	C₄×C₅₃	1	16	yes	direct product factor
119	C₅₃	C₄×#336	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
121	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
122	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	—	2	8	no	—
126	C₂²×C₈	C₂×C₄×C₅₃	1	16	yes	—
127	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
128	C₂×C₄²	—	2	4	no	—
129	C₂×C₄²	—	2	4	no	—
130	C₂×C₄²	—	2	8	no	—
131	C₂×C₄²	—	2	8	no	—
132	C₂×C₄²	C₈×C₅₃	1	16	yes	complemented
133	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
134	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
135	C₂×C₄²	D₄×C₅₃	1	2	yes	—
136	C₂×C₄²	D₄×C₅₃	1	4	yes	—
137	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
138	C₄×C₈	—	2	16	no	—
139	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	C₄²×C₅₃	1	8	yes	—
144	C₂²×C₄	C₄²×C₅₃	1	4	yes	—
145	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
146	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
147	C₂²×C₄	C₂×C₈×C₅₃	1	16	yes	—
148	C₂²×C₄	C₂×C₈×C₅₃	1	8	yes	—
149	C₂²×C₄	M₁₆×C₅₃	1	8	yes	—
150	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
152	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
154	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
155	C₂²×C₄	#41×C₅₃	1	4	yes	—
156	C₂×C₈	—	2	8	no	—
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	2	16	no	—
159	C₂×C₈	—	2	16	no	—
160	C₂×C₈	C₄²×C₅₃	1	16	yes	complemented
161	C₄²	—	2	32	no	—
162	C₄²	—	2	16	no	—
163	C₄²	C₂²⋊C₄×C₅₃	1	8	yes	—
164	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
165	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
166	C₂³	C₂×C₄²×C₅₃	1	2	yes	—
167	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
168	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
169	C₂³	#117×C₅₃	1	2	yes	—
170	C₂³	C₂²×C₈×C₅₃	1	4	yes	—
171	C₂³	C₂×M₁₆×C₅₃	1	4	yes	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	16	no	—
175	C₂×C₄	—	2	16	no	—
176	C₂×C₄	#95×C₅₃	1	8	yes	—
177	C₂×C₄	C₄×C₈×C₅₃	1	8	yes	—
178	C₂×C₄	C₄×C₈×C₅₃	1	16	yes	complemented
179	C₂×C₄	#97×C₅₃	1	8	yes	—
180	C₂×C₄	C₂²⋊C₈×C₅₃	1	16	yes	—
181	C₂×C₄	C₄⋊C₈×C₅₃	1	16	yes	—
182	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
183	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
185	C₂×C₄	#117×C₅₃	1	4	yes	—
186	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
187	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
188	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
189	C₈	—	2	16	no	—
190	C₂²	C₄³×C₅₃	1	1	yes	characteristic
191	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
192	C₂²	C₅₃×#376	1	1	yes	characteristic
193	C₂²	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
194	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
195	C₂²	C₂×C₄×C₈×C₅₃	1	2	yes	—
196	C₂²	C₂×C₄×C₈×C₅₃	1	4	yes	—
197	C₂²	C₂×#97×C₅₃	1	2	yes	—
198	C₂²	C₄×M₁₆×C₅₃	1	4	yes	—
199	C₂²	C₂×C₂²⋊C₈×C₅₃	1	4	yes	—
200	C₂²	C₂×C₄⋊C₈×C₅₃	1	4	yes	—
201	C₂²	C₅₃×#431	1	8	yes	—
202	C₄	—	2	16	no	—
203	C₄	C₅₃×#336	1	16	yes	direct product factor
204	C₄	C₄×C₂²⋊C₄×C₅₃	1	4	yes	—
205	C₄	C₄×C₄⋊C₄×C₅₃	1	4	yes	—
206	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
207	C₂	C₄²×C₈×C₅₃	1	1	yes	characteristic
208	C₂	C₄×#97×C₅₃	1	1	yes	characteristic
209	C₂	C₂×C₅₃×#336	1	2	yes	—
210	C₂	C₅₃×#1744	1	2	yes	—
211	C₂	C₄×C₂²⋊C₈×C₅₃	1	4	yes	—
212	C₂	C₄×C₄⋊C₈×C₅₃	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₅₃	C₂²
commutator	C₂	C₄²×C₈×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₈×C₅₃	C₄	semidirect
12. C₅₃×#336	C₄	direct (N × H)
25. C₂×C₄²×C₅₃	C₈	semidirect
53. C₂×C₈×C₅₃	C₄²	semidirect
71. C₂×C₄×C₅₃	C₄×C₈	semidirect
83. C₄×#336	C₅₃	direct (N × H)
97. C₄×C₅₃	#336	direct (N × H)
116. C₂×C₄×C₈	C₄×C₅₃	semidirect
118. #336	C₄×C₅₃	direct (N × H)
119. C₅₃	C₄×#336	direct (N × H)
132. C₂×C₄²	C₈×C₅₃	semidirect
160. C₂×C₈	C₄²×C₅₃	semidirect
178. C₂×C₄	C₄×C₈×C₅₃	semidirect
203. C₄	C₅₃×#336	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×C₄×C₈×C₅₃	2
C₂×C₄³×C₅₃	2
C₂×C₅₃×#336	2
C₄×#336	53

Subgroups — C4×C53×#336 #27218204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#336 #27218204