Subgroups — C₅₃×#8616 #27216128

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 238 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×#1567	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×#335	—	2	2	no	—
16	C₅₃×#335	—	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	D₈	1	2	yes	—
42	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	M₁₆×C₅₃	—	4	4	no	—
52	M₁₆×C₅₃	—	4	4	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	4	1	no	—
56	C₂×C₈×C₅₃	—	4	4	no	—
57	C₂×C₈×C₅₃	—	4	4	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	4	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	C₂×D₄	1	2	yes	—
66	C₄²×C₅₃	#41	1	2	yes	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	4	2	no	—
70	C₂³×C₅₃	—	4	2	no	—
71	C₂³×C₅₃	#107	1	4	yes	—
72	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	D₄×C₅₃	—	2	4	no	—
74	D₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	4	1	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#121	1	2	yes	—
86	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	2	yes	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	—	4	2	no	—
93	C₈×C₅₃	—	8	4	no	—
94	C₈×C₅₃	#100	1	2	yes	complemented
95	#8616	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	—	4	4	no	—
100	C₂²×C₅₃	—	4	4	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	#380	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
105	C₂²×C₅₃	#427	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
110	C₄×C₅₃	#413	1	1	yes	characteristic
111	C₄×C₅₃	#466	1	2	yes	—
112	C₄×C₅₃	#467	1	2	yes	—
113	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
114	#1800	C₂×C₅₃	1	2	yes	—
115	#1567	C₂×C₅₃	1	2	yes	complemented
116	#1292	C₂×C₅₃	1	2	yes	complemented
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	—	4	4	no	—
120	C₂×C₅₃	#1800	1	1	yes	characteristic
121	C₂×C₅₃	#1808	1	1	yes	characteristic
122	C₂×C₅₃	#1813	1	1	yes	characteristic
123	C₂³×C₈	C₄×C₅₃	1	2	yes	—
124	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
125	C₈×D₄	—	2	2	no	—
126	C₈×D₄	—	2	2	no	—
127	#423	C₂²×C₅₃	1	4	yes	—
128	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×#100	—	2	2	no	—
130	C₂×C₂²⋊C₈	C₄×C₅₃	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
132	#343	—	2	2	no	—
133	#335	—	2	2	no	—
134	#335	—	2	2	no	—
135	C₅₃	#8616	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
137	C₂³×C₄	C₂×C₄×C₅₃	1	2	yes	—
138	C₂×M₁₆	—	2	4	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	4	4	no	—
142	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	Q₈×C₅₃	1	2	yes	—
149	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	#100	—	4	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₄×C₈	D₄×C₅₃	1	2	yes	—
156	C₄×C₈	D₄×C₅₃	1	2	yes	—
157	C₂⁴	C₄⋊C₄×C₅₃	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	D₈×C₅₃	1	2	yes	—
161	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
170	M₁₆	—	4	4	no	—
171	M₁₆	—	4	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	—	4	4	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
180	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	C₂×D₄×C₅₃	1	2	yes	—
185	C₄²	#41×C₅₃	1	2	yes	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	4	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	#107×C₅₃	1	4	yes	—
191	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
192	D₄	—	2	4	no	—
193	D₄	—	2	4	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
203	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
204	C₂×C₄	#121×C₅₃	1	2	yes	—
205	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
206	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
208	C₂×C₄	#135×C₅₃	1	2	yes	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	8	4	no	—
213	C₈	#100×C₅₃	1	2	yes	complemented
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₅₃×#380	1	1	yes	characteristic
222	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
223	C₂²	C₅₃×#427	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×#100×C₅₃	1	1	yes	characteristic
228	C₄	C₅₃×#413	1	1	yes	characteristic
229	C₄	C₅₃×#466	1	2	yes	—
230	C₄	C₅₃×#467	1	2	yes	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	—	4	4	no	—
234	C₂	C₅₃×#1800	1	1	yes	characteristic
235	C₂	C₅₃×#1808	1	1	yes	characteristic
236	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×#1567	C₂	semidirect
4. C₅₃×#1292	C₂	semidirect
94. C₈×C₅₃	#100	semidirect
95. #8616	C₅₃	direct (N × H)
115. #1567	C₂×C₅₃	semidirect
116. #1292	C₂×C₅₃	semidirect
135. C₅₃	#8616	direct (N × H)
213. 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×#1567	2
C₅₃×#1292	2
#8616	53

Subgroups — C53×#8616 #27216128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8616 #27216128