Subgroups — C₅₃×#21715 #27229227

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₅₃	C₂	1	2	yes	maximal
2	C₅₃×#2399	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1985	C₂	1	2	yes	maximal
4	C₅₃×#1940	C₂	1	4	yes	maximal
5	C₅₃×#1843	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊C₄×C₅₃	C₂	1	4	yes	maximal
7	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
9	C₈×Q₈×C₅₃	—	2	8	no	—
10	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
13	C₂×#97×C₅₃	C₂²	1	2	yes	—
14	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#398	C₄	1	4	yes	—
19	C₅₃×#395	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₅₃×#376	C₂²	1	1	yes	characteristic
22	C₅₃×#336	C₂²	1	8	yes	—
23	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₅₃	—	2	4	no	—
29	C₄×Q₈×C₅₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
36	C₄⋊C₈×C₅₃	—	2	8	no	—
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	—	2	8	no	—
39	#97×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	4	no	—
41	C₄×C₈×C₅₃	—	2	8	no	—
42	C₄×C₈×C₅₃	D₄	1	4	yes	—
43	C₄×C₈×C₅₃	Q₈	1	8	yes	complemented
44	#95×C₅₃	C₂×C₄	1	4	yes	—
45	#95×C₅₃	C₂×C₄	1	4	yes	—
46	#95×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	—	2	8	no	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	8	no	—
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
63	C₄⋊C₄×C₅₃	—	2	8	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	16	no	—
66	C₄²×C₅₃	—	2	4	no	—
67	C₄²×C₅₃	—	2	4	no	—
68	C₄²×C₅₃	C₂×D₄	1	2	yes	—
69	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
70	C₄²×C₅₃	#41	1	2	yes	—
71	C₄²×C₅₃	#41	1	4	yes	—
72	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	Q₈×C₅₃	—	2	16	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	8	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	2	8	no	—
80	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₅₃	C₄×Q₈	1	8	yes	—
82	C₂×C₄×C₅₃	#131	1	8	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	2	4	no	—
89	C₈×C₅₃	—	4	4	no	—
90	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
91	#21715	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
94	C₂²×C₅₃	C₂×#131	1	4	yes	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	8	no	—
97	C₄×C₅₃	—	2	8	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₅₃	#532	1	1	yes	characteristic
101	C₄×C₅₃	#533	1	2	yes	—
102	C₂×C₈×Q₈	C₂×C₅₃	1	2	yes	—
103	#2399	C₂×C₅₃	1	1	yes	characteristic
104	#1985	C₂×C₅₃	1	2	yes	—
105	#1940	C₂×C₅₃	1	4	yes	—
106	#1843	C₂×C₅₃	1	1	yes	characteristic
107	C₈×C₄⋊C₄	C₂×C₅₃	1	4	yes	—
108	#1726	C₂×C₅₃	1	1	yes	characteristic
109	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₅₃	#2929	1	2	yes	—
111	C₂×C₅₃	#2960	1	4	yes	—
112	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
113	C₈×Q₈	—	2	8	no	—
114	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
116	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
117	C₂×#97	C₂²×C₅₃	1	2	yes	—
118	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
120	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
122	#398	C₄×C₅₃	1	4	yes	—
123	#395	C₄×C₅₃	1	4	yes	—
124	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
125	#376	C₂²×C₅₃	1	1	yes	characteristic
126	#336	C₂²×C₅₃	1	8	yes	—
127	C₅₃	#21715	1	1	yes	characteristic, direct product factor, Sylow
128	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
129	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
130	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
132	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
133	C₄×Q₈	—	2	4	no	—
134	C₄×Q₈	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
138	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
139	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
141	C₄⋊C₈	—	2	8	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	2	8	no	—
144	#97	—	2	2	no	—
145	#97	—	2	4	no	—
146	C₄×C₈	—	2	8	no	—
147	C₄×C₈	D₄×C₅₃	1	4	yes	—
148	C₄×C₈	Q₈×C₅₃	1	8	yes	complemented
149	#95	C₂×C₄×C₅₃	1	4	yes	—
150	#95	C₂×C₄×C₅₃	1	4	yes	—
151	#95	C₂×C₄×C₅₃	1	4	yes	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
157	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
158	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	8	no	—
164	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
166	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
167	C₂×C₈	C₂×Q₈×C₅₃	1	4	yes	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄²	—	2	4	no	—
172	C₄²	—	2	4	no	—
173	C₄²	C₂×D₄×C₅₃	1	2	yes	—
174	C₄²	C₂×Q₈×C₅₃	1	4	yes	—
175	C₄²	#41×C₅₃	1	2	yes	—
176	C₄²	#41×C₅₃	1	4	yes	—
177	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
178	Q₈	—	2	16	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
186	C₂×C₄	C₄×Q₈×C₅₃	1	8	yes	—
187	C₂×C₄	#131×C₅₃	1	8	yes	—
188	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Q₈×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
191	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	4	no	—
194	C₈	—	4	4	no	—
195	C₈	C₄⋊Q₈×C₅₃	1	4	yes	—
196	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
197	C₂²	C₂×C₄×Q₈×C₅₃	1	2	yes	—
198	C₂²	C₂×#131×C₅₃	1	4	yes	—
199	C₄	—	2	8	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	8	no	—
202	C₄	—	4	4	no	—
203	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
204	C₄	C₅₃×#532	1	1	yes	characteristic
205	C₄	C₅₃×#533	1	2	yes	—
206	C₂	C₄×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
207	C₂	C₅₃×#2929	1	2	yes	—
208	C₂	C₅₃×#2960	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
43. C₄×C₈×C₅₃	Q₈	semidirect
91. #21715	C₅₃	direct (N × H)
127. C₅₃	#21715	direct (N × H)
148. C₄×C₈	Q₈×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×Q₈×C₅₃	2
C₅₃×#2399	2
C₅₃×#1985	2
C₅₃×#1940	2
C₅₃×#1843	2
C₈×C₄⋊C₄×C₅₃	2
C₅₃×#1726	2
#21715	53

Subgroups — C53×#21715 #27229227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21715 #27229227