Subgroups — C₅₃×#12551 #27220063

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1377	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₅₃	—	2	4	no	—
8	C₂×M₃₂×C₅₃	—	2	2	no	—
9	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
10	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₅₃	—	2	2	no	—
12	C₅₃×#405	—	2	2	no	—
13	C₅₃×#405	—	2	2	no	—
14	C₂×#97×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
16	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×M₁₆×C₅₃	—	2	2	no	—
18	C₂×M₁₆×C₅₃	—	2	2	no	—
19	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	C₂²×C₈×C₅₃	D₄	1	2	yes	—
26	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	#117×C₅₃	—	2	2	no	—
29	#117×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
31	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	M₃₂×C₅₃	—	2	8	no	—
34	M₃₂×C₅₃	—	2	4	no	—
35	M₃₂×C₅₃	C₂×C₄	1	8	yes	—
36	C₂×C₁₆×C₅₃	—	2	8	no	—
37	C₂×C₁₆×C₅₃	—	2	4	no	—
38	#97×C₅₃	—	2	4	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₄²	1	2	yes	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₂²⋊C₄×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₁₆×C₅₃	—	2	8	no	—
72	C₂³×C₅₃	—	4	2	no	—
73	C₂³×C₅₃	C₄×C₈	1	2	yes	—
74	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
75	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	#95	1	4	yes	—
80	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₅₃	#97	1	4	yes	—
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₈×C₅₃	—	2	4	no	—
93	C₈×C₅₃	#95	1	2	yes	—
94	C₈×C₅₃	#95	1	2	yes	—
95	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	#12551	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	4	no	—
100	C₂²×C₅₃	—	4	2	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	#336	1	4	yes	—
103	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₅₃	—	4	4	no	—
107	C₄×C₅₃	#336	1	4	yes	—
108	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
110	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×M₃₂	C₂×C₅₃	1	2	yes	—
113	C₂×#405	C₂×C₅₃	1	1	yes	characteristic
114	#1377	C₂×C₅₃	1	4	yes	complemented
115	C₂×C₅₃	—	2	4	no	—
116	C₂×C₅₃	—	2	1	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
119	C₂×C₅₃	#2115	1	2	yes	—
120	C₂²×M₁₆	C₄×C₅₃	1	1	yes	characteristic
121	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
122	C₂×#117	C₄×C₅₃	1	1	yes	characteristic
123	C₂×M₃₂	—	2	2	no	—
124	C₂×M₃₂	—	2	4	no	—
125	C₂×M₃₂	C₄×C₅₃	1	8	yes	complemented
126	C₂×M₃₂	C₂²×C₅₃	1	4	yes	—
127	C₂²×C₁₆	—	2	2	no	—
128	#405	—	2	2	no	—
129	#405	—	2	2	no	—
130	C₂×#97	C₂²×C₅₃	1	2	yes	—
131	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
132	C₅₃	#12551	1	1	yes	characteristic, direct product factor, Sylow
133	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	C₈×C₅₃	1	4	yes	—
137	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
139	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₅₃	1	2	yes	—
141	C₂²×C₈	D₄×C₅₃	1	2	yes	—
142	C₂²×C₈	D₄×C₅₃	1	2	yes	—
143	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
144	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
145	#117	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄⋊C₄	C₈×C₅₃	1	2	yes	—
148	C₂×C₂²⋊C₄	C₈×C₅₃	1	2	yes	—
149	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
150	M₃₂	—	2	4	no	—
151	M₃₂	—	2	8	no	—
152	M₃₂	C₂×C₄×C₅₃	1	8	yes	—
153	C₂×C₁₆	—	2	4	no	—
154	C₂×C₁₆	—	2	8	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₂⁴	C₂×C₈×C₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
160	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
161	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
162	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
163	C₂²×C₄	C₂×C₈×C₅₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
165	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
166	C₂²×C₄	M₁₆×C₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₄²×C₅₃	1	2	yes	—
172	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
175	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
176	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
177	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
178	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
179	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
180	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
181	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
183	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₁₆	—	2	8	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₄×C₈×C₅₃	1	2	yes	—
191	C₂³	C₂²⋊C₈×C₅₃	1	2	yes	—
192	C₂³	C₄⋊C₈×C₅₃	1	2	yes	—
193	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	#95×C₅₃	1	4	yes	—
197	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
198	C₂×C₄	#97×C₅₃	1	4	yes	—
199	C₂×C₄	C₂²⋊C₈×C₅₃	1	2	yes	—
200	C₂×C₄	C₂²⋊C₈×C₅₃	1	4	yes	—
201	C₂×C₄	C₄⋊C₈×C₅₃	1	2	yes	—
202	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
203	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
209	C₈	—	2	4	no	—
210	C₈	#95×C₅₃	1	2	yes	—
211	C₈	#95×C₅₃	1	2	yes	—
212	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
213	C₈	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₅₃×#336	1	4	yes	—
219	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
220	C₂²	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
221	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	C₅₃×#336	1	4	yes	—
224	C₄	C₂×#95×C₅₃	1	1	yes	characteristic
225	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
226	C₄	C₂×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
227	C₄	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
228	C₂	—	2	4	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	C₂×C₅₃×#336	1	1	yes	characteristic
232	C₂	C₅₃×#2115	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1377	C₂	semidirect
9. C₂×M₃₂×C₅₃	C₄	semidirect
97. #12551	C₅₃	direct (N × H)
114. #1377	C₂×C₅₃	semidirect
125. C₂×M₃₂	C₄×C₅₃	semidirect
132. C₅₃	#12551	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×M₃₂×C₅₃	2
C₂×C₅₃×#405	2
C₅₃×#1377	2
#12551	53

Subgroups — C53×#12551 #27220063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12551 #27220063