Subgroups — C₅₃×#10703 #27218215

Order 13568.

1142 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
3	C₂×C₅₃×#340	C₂	1	4	yes	maximal
4	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
5	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₅₃	—	2	2	no	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#97×C₅₃	—	2	4	no	—
9	C₂×#97×C₅₃	C₄	1	4	yes	complemented
10	C₄×C₄⋊C₄×C₅₃	—	2	4	no	—
11	C₄×C₄⋊C₄×C₅₃	C₄	1	8	yes	complemented
12	C₂×#95×C₅₃	—	2	2	no	—
13	C₅₃×#340	C₄	1	16	yes	complemented
14	C₂³×C₄×C₅₃	—	2	4	no	—
15	C₂³×C₄×C₅₃	D₄	1	2	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	—	2	4	no	—
18	C₂²×C₈×C₅₃	—	2	4	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
23	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₅₃	—	2	8	no	—
26	C₂×C₄²×C₅₃	—	4	8	no	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	4	yes	—
30	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
31	#97×C₅₃	—	2	8	no	—
32	#95×C₅₃	—	2	8	no	—
33	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	4	no	—
35	C₂²×C₄×C₅₃	—	2	4	no	—
36	C₂²×C₄×C₅₃	—	2	8	no	—
37	C₂²×C₄×C₅₃	—	2	16	no	—
38	C₂²×C₄×C₅₃	—	4	8	no	—
39	C₂²×C₄×C₅₃	—	4	4	no	—
40	C₂²×C₄×C₅₃	—	4	4	no	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	4	yes	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	—	2	16	no	—
53	C₂×C₈×C₅₃	C₄²	1	8	yes	complemented
54	C₄⋊C₄×C₅₃	—	2	8	no	—
55	C₄⋊C₄×C₅₃	—	2	16	no	—
56	C₄⋊C₄×C₅₃	C₄²	1	16	yes	complemented
57	C₄²×C₅₃	—	4	16	no	—
58	C₄²×C₅₃	C₄²	1	4	yes	—
59	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
60	C₄²×C₅₃	C₄⋊C₄	1	8	yes	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
62	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂³×C₅₃	C₄×D₄	1	4	yes	—
64	C₂³×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
65	C₂³×C₅₃	#137	1	4	yes	—
66	C₂×C₄×C₅₃	—	2	8	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	2	16	no	—
71	C₂×C₄×C₅₃	—	4	8	no	—
72	C₂×C₄×C₅₃	—	4	8	no	—
73	C₂×C₄×C₅₃	—	4	16	no	—
74	C₂×C₄×C₅₃	—	4	16	no	—
75	C₂×C₄×C₅₃	#95	1	8	yes	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₅₃	#117	1	2	yes	—
82	C₂×C₄×C₅₃	#117	1	4	yes	—
83	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
84	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
85	C₈×C₅₃	—	2	8	no	—
86	#10703	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
88	C₂²×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
89	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
90	C₂²×C₅₃	#417	1	4	yes	—
91	C₂²×C₅₃	#418	1	2	yes	—
92	C₂²×C₅₃	#419	1	2	yes	—
93	C₂²×C₅₃	#428	1	4	yes	—
94	C₂²×C₅₃	#440	1	8	yes	—
95	C₂²×C₅₃	#441	1	4	yes	—
96	C₂²×C₅₃	#442	1	4	yes	—
97	C₄×C₅₃	—	2	8	no	—
98	C₄×C₅₃	—	4	16	no	—
99	C₄×C₅₃	C₄³	1	1	yes	characteristic
100	C₄×C₅₃	#376	1	1	yes	characteristic
101	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
102	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
103	C₂²×#97	C₂×C₅₃	1	1	yes	characteristic
104	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
105	C₂×#340	C₂×C₅₃	1	4	yes	—
106	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
107	C₂×C₅₃	#1736	1	2	yes	—
108	C₂×C₅₃	#1762	1	4	yes	—
109	C₂×C₅₃	#1763	1	4	yes	—
110	C₂×C₅₃	#1776	1	4	yes	—
111	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
112	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
113	C₂²×C₄²	—	2	2	no	—
114	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
115	C₂×#97	—	2	4	no	—
116	C₂×#97	C₄×C₅₃	1	4	yes	complemented
117	C₄×C₄⋊C₄	—	2	4	no	—
118	C₄×C₄⋊C₄	C₄×C₅₃	1	8	yes	complemented
119	C₂×#95	—	2	2	no	—
120	#340	C₄×C₅₃	1	16	yes	complemented
121	C₅₃	#10703	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³×C₄	—	2	4	no	—
123	C₂³×C₄	D₄×C₅₃	1	2	yes	—
124	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
128	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	—	2	8	no	—
131	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	16	yes	—
132	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
133	C₂×C₄²	—	2	8	no	—
134	C₂×C₄²	—	4	8	no	—
135	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
136	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
137	C₂×C₄²	D₄×C₅₃	1	4	yes	—
138	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
139	#97	—	2	8	no	—
140	#95	—	2	8	no	—
141	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	16	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	4	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
150	C₂²×C₄	C₂²⋊C₄×C₅₃	1	4	yes	—
151	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
152	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
155	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
156	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
157	C₂²×C₄	#41×C₅₃	1	4	yes	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	C₄²×C₅₃	1	8	yes	complemented
162	C₄⋊C₄	—	2	8	no	—
163	C₄⋊C₄	—	2	16	no	—
164	C₄⋊C₄	C₄²×C₅₃	1	16	yes	complemented
165	C₄²	—	4	16	no	—
166	C₄²	C₄²×C₅₃	1	4	yes	—
167	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
168	C₄²	C₄⋊C₄×C₅₃	1	8	yes	—
169	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
170	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
171	C₂³	C₄×D₄×C₅₃	1	4	yes	—
172	C₂³	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
173	C₂³	#137×C₅₃	1	4	yes	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	—	4	16	no	—
182	C₂×C₄	—	4	16	no	—
183	C₂×C₄	#95×C₅₃	1	8	yes	—
184	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
186	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
189	C₂×C₄	#117×C₅₃	1	2	yes	—
190	C₂×C₄	#117×C₅₃	1	4	yes	—
191	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
192	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
193	C₈	—	2	8	no	—
194	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
195	C₂²	C₄×C₂²⋊C₄×C₅₃	1	4	yes	—
196	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
197	C₂²	C₅₃×#417	1	4	yes	—
198	C₂²	C₅₃×#418	1	2	yes	—
199	C₂²	C₅₃×#419	1	2	yes	—
200	C₂²	C₅₃×#428	1	4	yes	—
201	C₂²	C₅₃×#440	1	8	yes	—
202	C₂²	C₅₃×#441	1	4	yes	—
203	C₂²	C₅₃×#442	1	4	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	4	16	no	—
206	C₄	C₄³×C₅₃	1	1	yes	characteristic
207	C₄	C₅₃×#376	1	1	yes	characteristic
208	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
209	C₄	C₄×C₄⋊C₄×C₅₃	1	4	yes	—
210	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
211	C₂	C₅₃×#1736	1	2	yes	—
212	C₂	C₅₃×#1762	1	4	yes	—
213	C₂	C₅₃×#1763	1	4	yes	—
214	C₂	C₅₃×#1776	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₃	C₂×D₄
commutator	C₄	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×#97×C₅₃	C₄	semidirect
11. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
13. C₅₃×#340	C₄	semidirect
53. C₂×C₈×C₅₃	C₄²	semidirect
56. C₄⋊C₄×C₅₃	C₄²	semidirect
86. #10703	C₅₃	direct (N × H)
116. C₂×#97	C₄×C₅₃	semidirect
118. C₄×C₄⋊C₄	C₄×C₅₃	semidirect
120. #340	C₄×C₅₃	semidirect
121. C₅₃	#10703	direct (N × H)
161. C₂×C₈	C₄²×C₅₃	semidirect
164. C₄⋊C₄	C₄²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₅₃	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₅₃×#340	2
#10703	53

Subgroups — C53×#10703 #27218215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10703 #27218215