Subgroups — C₅₃×#11322 #27218834

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 224 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	3	yes	maximal
2	C₂×C₅₃×#336	C₂	1	3	yes	maximal
3	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₅₃	C₂²	1	3	yes	—
5	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
7	C₂×C₄⋊C₈×C₅₃	—	2	12	no	—
8	C₂×C₄⋊C₈×C₅₃	—	2	6	no	—
9	C₂×#95×C₅₃	C₄	1	3	yes	—
10	C₂×#95×C₅₃	C₄	1	1	yes	characteristic
11	C₅₃×#336	—	2	6	no	—
12	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
13	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	—	2	3	no	—
16	C₂²×C₈×C₅₃	—	2	3	no	—
17	C₂²×C₈×C₅₃	—	2	6	no	—
18	C₂²×C₈×C₅₃	—	2	6	no	—
19	C₂²×C₈×C₅₃	—	2	3	no	—
20	C₂²×C₈×C₅₃	—	2	3	no	—
21	C₂²×C₈×C₅₃	D₄	1	6	yes	—
22	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₅₃	—	2	6	no	—
24	C₂×C₄²×C₅₃	—	2	6	no	—
25	C₂×C₄²×C₅₃	D₄	1	6	yes	—
26	C₂×C₄²×C₅₃	Q₈	1	6	yes	—
27	C₄⋊C₈×C₅₃	—	2	24	no	—
28	#95×C₅₃	—	2	6	no	—
29	#95×C₅₃	C₈	1	4	yes	complemented
30	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	6	no	—
32	C₂²×C₄×C₅₃	—	2	6	no	—
33	C₂²×C₄×C₅₃	—	2	6	no	—
34	C₂²×C₄×C₅₃	—	2	6	no	—
35	C₂²×C₄×C₅₃	—	2	3	no	—
36	C₂²×C₄×C₅₃	—	2	3	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	6	yes	—
39	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
40	C₂²×C₄×C₅₃	M₁₆	1	6	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	3	yes	—
47	C₂×C₈×C₅₃	—	2	12	no	—
48	C₂×C₈×C₅₃	—	2	6	no	—
49	C₂×C₈×C₅₃	—	2	6	no	—
50	C₂×C₈×C₅₃	—	4	3	no	—
51	C₂×C₈×C₅₃	—	4	6	no	—
52	C₂×C₈×C₅₃	—	4	3	no	—
53	C₂×C₈×C₅₃	—	4	3	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₄²×C₅₃	—	2	6	no	—
56	C₄²×C₅₃	—	2	12	no	—
57	C₄²×C₅₃	—	2	6	no	—
58	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
59	C₂³×C₅₃	C₄×D₄	1	3	yes	—
60	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
61	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
62	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
63	C₂³×C₅₃	#131	1	3	yes	—
64	C₂³×C₅₃	#131	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	6	no	—
66	C₂×C₄×C₅₃	—	2	6	no	—
67	C₂×C₄×C₅₃	—	2	12	no	—
68	C₂×C₄×C₅₃	—	2	12	no	—
69	C₂×C₄×C₅₃	—	2	6	no	—
70	C₂×C₄×C₅₃	—	2	6	no	—
71	C₂×C₄×C₅₃	—	2	6	no	—
72	C₂×C₄×C₅₃	—	4	6	no	—
73	C₂×C₄×C₅₃	—	4	6	no	—
74	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	complemented
75	C₂×C₄×C₅₃	#120	1	3	yes	—
76	C₂×C₄×C₅₃	#121	1	3	yes	—
77	C₂×C₄×C₅₃	C₂²⋊Q₈	1	6	yes	—
78	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
79	C₂×C₄×C₅₃	#123	1	3	yes	—
80	C₂×C₄×C₅₃	#125	1	3	yes	—
81	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
83	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
84	C₈×C₅₃	—	4	6	no	—
85	C₈×C₅₃	—	4	2	no	—
86	#11322	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	#384	1	1	yes	characteristic
88	C₂²×C₅₃	#385	1	3	yes	—
89	C₂²×C₅₃	#389	1	3	yes	—
90	C₂²×C₅₃	C₂×C₄⋊C₈	1	3	yes	—
91	C₂²×C₅₃	#423	1	3	yes	—
92	C₂²×C₅₃	#424	1	3	yes	—
93	C₂²×C₅₃	#424	1	3	yes	—
94	C₂²×C₅₃	C₈×D₄	1	3	yes	—
95	C₂²×C₅₃	#435	1	6	yes	—
96	C₂²×C₅₃	#436	1	3	yes	—
97	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
98	C₂²×C₅₃	#446	1	3	yes	—
99	C₄×C₅₃	—	2	12	no	—
100	C₄×C₅₃	—	4	6	no	—
101	C₄×C₅₃	#392	1	1	yes	characteristic
102	C₄×C₅₃	#395	1	3	yes	—
103	C₄×C₅₃	#398	1	3	yes	—
104	C₄×C₅₃	#399	1	1	yes	characteristic
105	C₂²×C₄⋊C₈	C₂×C₅₃	1	3	yes	—
106	C₂×#336	C₂×C₅₃	1	3	yes	—
107	C₄×#95	C₂×C₅₃	1	1	yes	characteristic
108	C₂×C₅₃	#1444	1	1	yes	characteristic
109	C₂×C₅₃	#1842	1	1	yes	characteristic
110	C₂×C₅₃	#1843	1	1	yes	characteristic
111	C₂×C₅₃	#1850	1	3	yes	—
112	C₂×C₅₃	#1851	1	3	yes	—
113	C₂×C₅₃	#1939	1	3	yes	—
114	C₂×C₅₃	#1940	1	3	yes	—
115	C₂³×C₈	C₂²×C₅₃	1	3	yes	—
116	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
117	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
118	C₂×C₄⋊C₈	—	2	12	no	—
119	C₂×C₄⋊C₈	—	2	6	no	—
120	C₂×#95	C₄×C₅₃	1	3	yes	—
121	C₂×#95	C₄×C₅₃	1	1	yes	characteristic
122	#336	—	2	6	no	—
123	C₅₃	#11322	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
125	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
126	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
127	C₂²×C₈	—	2	3	no	—
128	C₂²×C₈	—	2	3	no	—
129	C₂²×C₈	—	2	6	no	—
130	C₂²×C₈	—	2	6	no	—
131	C₂²×C₈	—	2	3	no	—
132	C₂²×C₈	—	2	3	no	—
133	C₂²×C₈	D₄×C₅₃	1	6	yes	—
134	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
135	C₂×C₄²	—	2	6	no	—
136	C₂×C₄²	—	2	6	no	—
137	C₂×C₄²	D₄×C₅₃	1	6	yes	—
138	C₂×C₄²	Q₈×C₅₃	1	6	yes	—
139	C₄⋊C₈	—	2	24	no	—
140	#95	—	2	6	no	—
141	#95	C₈×C₅₃	1	4	yes	complemented
142	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	6	no	—
144	C₂²×C₄	—	2	6	no	—
145	C₂²×C₄	—	2	6	no	—
146	C₂²×C₄	—	2	6	no	—
147	C₂²×C₄	—	2	3	no	—
148	C₂²×C₄	—	2	3	no	—
149	C₂²×C₄	C₄⋊C₄×C₅₃	1	6	yes	—
150	C₂²×C₄	C₄⋊C₄×C₅₃	1	6	yes	—
151	C₂²×C₄	C₂×C₈×C₅₃	1	6	yes	—
152	C₂²×C₄	M₁₆×C₅₃	1	6	yes	—
153	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
154	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
155	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
157	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
158	C₂²×C₄	#41×C₅₃	1	3	yes	—
159	C₂×C₈	—	2	12	no	—
160	C₂×C₈	—	2	6	no	—
161	C₂×C₈	—	2	6	no	—
162	C₂×C₈	—	4	3	no	—
163	C₂×C₈	—	4	6	no	—
164	C₂×C₈	—	4	3	no	—
165	C₂×C₈	—	4	3	no	—
166	C₂×C₈	—	4	1	no	—
167	C₄²	—	2	6	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₂³	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
171	C₂³	C₄×D₄×C₅₃	1	3	yes	—
172	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
173	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
174	C₂³	C₂×M₁₆×C₅₃	1	3	yes	—
175	C₂³	#131×C₅₃	1	3	yes	—
176	C₂³	#131×C₅₃	1	1	yes	characteristic
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	4	6	no	—
185	C₂×C₄	—	4	6	no	—
186	C₂×C₄	C₄⋊C₈×C₅₃	1	12	yes	complemented
187	C₂×C₄	#120×C₅₃	1	3	yes	—
188	C₂×C₄	#121×C₅₃	1	3	yes	—
189	C₂×C₄	C₂²⋊Q₈×C₅₃	1	6	yes	—
190	C₂×C₄	C₂²⋊Q₈×C₅₃	1	3	yes	—
191	C₂×C₄	#123×C₅₃	1	3	yes	—
192	C₂×C₄	#125×C₅₃	1	3	yes	—
193	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
194	C₂×C₄	C₄⋊Q₈×C₅₃	1	3	yes	—
195	C₂×C₄	C₄⋊Q₈×C₅₃	1	3	yes	—
196	C₈	—	4	6	no	—
197	C₈	—	4	2	no	—
198	C₂²	C₅₃×#384	1	1	yes	characteristic
199	C₂²	C₅₃×#385	1	3	yes	—
200	C₂²	C₅₃×#389	1	3	yes	—
201	C₂²	C₂×C₄⋊C₈×C₅₃	1	3	yes	—
202	C₂²	C₅₃×#423	1	3	yes	—
203	C₂²	C₅₃×#424	1	3	yes	—
204	C₂²	C₅₃×#424	1	3	yes	—
205	C₂²	C₈×D₄×C₅₃	1	3	yes	—
206	C₂²	C₅₃×#435	1	6	yes	—
207	C₂²	C₅₃×#436	1	3	yes	—
208	C₂²	C₈×Q₈×C₅₃	1	1	yes	characteristic
209	C₂²	C₅₃×#446	1	3	yes	—
210	C₄	—	2	12	no	—
211	C₄	—	4	6	no	—
212	C₄	C₅₃×#392	1	1	yes	characteristic
213	C₄	C₅₃×#395	1	3	yes	—
214	C₄	C₅₃×#398	1	3	yes	—
215	C₄	C₅₃×#399	1	1	yes	characteristic
216	C₂	C₅₃×#1444	1	1	yes	characteristic
217	C₂	C₅₃×#1842	1	1	yes	characteristic
218	C₂	C₅₃×#1843	1	1	yes	characteristic
219	C₂	C₅₃×#1850	1	3	yes	—
220	C₂	C₅₃×#1851	1	3	yes	—
221	C₂	C₅₃×#1939	1	3	yes	—
222	C₂	C₅₃×#1940	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
29. #95×C₅₃	C₈	semidirect
74. C₂×C₄×C₅₃	C₄⋊C₈	semidirect
86. #11322	C₅₃	direct (N × H)
123. C₅₃	#11322	direct (N × H)
141. #95	C₈×C₅₃	semidirect
186. C₂×C₄	C₄⋊C₈×C₅₃	semidirect

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×C₅₃×#336	2
C₄×#95×C₅₃	2
#11322	53

Subgroups — C53×#11322 #27218834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11322 #27218834