Subgroups — C₅₃×#8614 #27216126

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1800	C₂	1	2	yes	maximal
3	C₅₃×#1566	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1292	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₄	1	2	yes	—
6	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₈×D₄×C₅₃	—	2	2	no	—
8	C₈×D₄×C₅₃	—	2	2	no	—
9	C₅₃×#423	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#100×C₅₃	—	2	2	no	—
12	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#343	—	2	2	no	—
15	C₅₃×#334	—	2	2	no	—
16	C₅₃×#334	—	2	2	no	—
17	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₅₃	—	2	4	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	—	4	4	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×D₄×C₅₃	—	2	2	no	—
27	C₄×D₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄⋊C₈×C₅₃	—	2	4	no	—
33	C₄⋊C₈×C₅₃	—	2	4	no	—
34	#100×C₅₃	—	4	4	no	—
35	C₂²⋊C₈×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂×D₄×C₅₃	—	2	2	no	—
40	C₂×D₄×C₅₃	—	2	2	no	—
41	C₂×D₄×C₅₃	SD₁₆	1	4	yes	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	M₁₆×C₅₃	—	4	4	no	—
51	M₁₆×C₅₃	—	4	4	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	—	4	4	no	—
56	C₂×C₈×C₅₃	—	4	4	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	4	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	C₂×D₄	1	2	yes	—
65	C₄²×C₅₃	#41	1	2	yes	—
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	4	2	no	—
69	C₂³×C₅₃	—	4	2	no	—
70	C₂³×C₅₃	#106	1	4	yes	—
71	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	D₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#121	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₅₃	#135	1	2	yes	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	—	8	4	no	—
91	C₈×C₅₃	#100	1	2	yes	complemented
92	#8614	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	1	no	—
95	C₂²×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	—	4	4	no	—
97	C₂²×C₅₃	—	4	4	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	—	4	2	no	—
100	C₂²×C₅₃	#380	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
102	C₂²×C₅₃	#427	1	1	yes	characteristic
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
107	C₄×C₅₃	#413	1	1	yes	characteristic
108	C₄×C₅₃	#465	1	4	yes	—
109	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
110	#1800	C₂×C₅₃	1	2	yes	—
111	#1566	C₂×C₅₃	1	2	yes	complemented
112	#1292	C₂×C₅₃	1	2	yes	complemented
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	—	4	4	no	—
116	C₂×C₅₃	#1800	1	1	yes	characteristic
117	C₂×C₅₃	#1807	1	1	yes	characteristic
118	C₂×C₅₃	#1813	1	1	yes	characteristic
119	C₂³×C₈	C₄×C₅₃	1	2	yes	—
120	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
121	C₈×D₄	—	2	2	no	—
122	C₈×D₄	—	2	2	no	—
123	#423	C₂²×C₅₃	1	4	yes	—
124	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
125	C₂×#100	—	2	2	no	—
126	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
128	#343	—	2	2	no	—
129	#334	—	2	2	no	—
130	#334	—	2	2	no	—
131	C₅₃	#8614	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
133	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
134	C₂×M₁₆	—	2	4	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	4	4	no	—
138	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
140	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
141	C₄×D₄	—	2	2	no	—
142	C₄×D₄	—	2	2	no	—
143	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	Q₈×C₅₃	1	2	yes	—
145	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	2	4	no	—
149	#100	—	4	4	no	—
150	C₂²⋊C₈	—	2	4	no	—
151	C₄×C₈	D₄×C₅₃	1	2	yes	—
152	C₄×C₈	D₄×C₅₃	1	2	yes	—
153	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	SD₁₆×C₅₃	1	4	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
162	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
165	M₁₆	—	4	4	no	—
166	M₁₆	—	4	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	4	no	—
171	C₂×C₈	—	4	4	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	4	no	—
174	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
175	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₄²	C₂×D₄×C₅₃	1	2	yes	—
180	C₄²	#41×C₅₃	1	2	yes	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	#106×C₅₃	1	4	yes	—
186	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
187	D₄	—	2	8	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
197	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
198	C₂×C₄	#121×C₅₃	1	2	yes	—
199	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
200	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
201	C₂×C₄	#135×C₅₃	1	2	yes	—
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	8	4	no	—
206	C₈	#100×C₅₃	1	2	yes	complemented
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₅₃×#380	1	1	yes	characteristic
215	C₂²	C₂×#106×C₅₃	1	1	yes	characteristic
216	C₂²	C₅₃×#427	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	2	no	—
220	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
221	C₄	C₅₃×#413	1	1	yes	characteristic
222	C₄	C₅₃×#465	1	4	yes	—
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	4	4	no	—
226	C₂	C₅₃×#1800	1	1	yes	characteristic
227	C₂	C₅₃×#1807	1	1	yes	characteristic
228	C₂	C₅₃×#1813	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1566	C₂	semidirect
4. C₅₃×#1292	C₂	semidirect
91. C₈×C₅₃	#100	semidirect
92. #8614	C₅₃	direct (N × H)
111. #1566	C₂×C₅₃	semidirect
112. #1292	C₂×C₅₃	semidirect
131. C₅₃	#8614	direct (N × H)
206. C₈	#100×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₅₃	2
C₅₃×#1800	2
C₅₃×#1566	2
C₅₃×#1292	2
#8614	53

Subgroups — C53×#8614 #27216126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8614 #27216126