Subgroups — C₅₃×#12613 #27220125

Order 13568.

410 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2115	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2108	C₂	1	2	yes	maximal, complemented
4	C₅₃×#1757	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#504	—	2	2	no	—
8	C₅₃×#504	—	2	2	no	—
9	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
10	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#413	C₄	1	1	yes	characteristic
15	C₅₃×#410	C₄	1	1	yes	characteristic
16	C₅₃×#405	C₂²	1	2	yes	—
17	C₅₃×#349	C₄	1	4	yes	complemented
18	C₅₃×#346	—	2	2	no	—
19	C₄×C₁₆×C₅₃	—	2	2	no	—
20	C₅₃×#344	C₄	1	2	yes	—
21	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
22	#131×C₅₃	—	2	2	no	—
23	#131×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	#117×C₅₃	C₂×C₄	1	2	yes	—
30	M₃₂×C₅₃	—	2	2	no	—
31	M₃₂×C₅₃	—	2	2	no	—
32	M₃₂×C₅₃	—	2	4	no	—
33	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₁₆×C₅₃	—	2	2	no	—
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
41	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
42	#101×C₅₃	C₈	1	2	yes	—
43	#100×C₅₃	C₈	1	2	yes	—
44	#99×C₅₃	C₈	1	4	yes	—
45	#97×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	#41×C₅₃	—	2	2	no	—
48	#41×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
51	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₅₃	—	2	2	no	—
55	M₁₆×C₅₃	—	2	2	no	—
56	M₁₆×C₅₃	—	2	4	no	—
57	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	C₄²	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	#41	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₁₆×C₅₃	—	2	2	no	—
70	C₁₆×C₅₃	—	2	4	no	—
71	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₅₃	C₄×C₈	1	2	yes	—
75	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	Q₈×C₅₃	—	2	2	no	—
77	D₄×C₅₃	—	2	4	no	—
78	D₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#131	1	2	yes	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	2	4	no	—
89	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₈×C₅₃	#117	1	1	yes	characteristic
91	C₈×C₅₃	C₄×D₄	1	2	yes	—
92	#12613	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	1	no	—
95	C₂²×C₅₃	—	4	2	no	—
96	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×D₄	1	2	yes	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	4	4	no	—
100	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	#405	1	1	yes	characteristic
102	C₄×C₅₃	C₈×D₄	1	2	yes	—
103	C₂×#504	C₂×C₅₃	1	1	yes	characteristic
104	#2115	C₂×C₅₃	1	1	yes	characteristic
105	#2108	C₂×C₅₃	1	2	yes	complemented
106	#1757	C₂×C₅₃	1	1	yes	characteristic
107	#1378	C₂×C₅₃	1	2	yes	complemented
108	C₂×C₅₃	—	2	2	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	4	2	no	—
111	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
113	#504	—	2	2	no	—
114	#504	—	2	2	no	—
115	C₂×M₃₂	C₄×C₅₃	1	2	yes	complemented
116	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
117	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
118	C₂²×C₁₆	C₄×C₅₃	1	2	yes	complemented
119	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
120	#413	C₄×C₅₃	1	1	yes	characteristic
121	#410	C₄×C₅₃	1	1	yes	characteristic
122	#405	C₂²×C₅₃	1	2	yes	—
123	#349	C₄×C₅₃	1	4	yes	complemented
124	#346	—	2	2	no	—
125	C₄×C₁₆	—	2	2	no	—
126	#344	C₄×C₅₃	1	2	yes	—
127	C₅₃	#12613	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
132	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
133	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
135	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
136	#117	C₂×C₄×C₅₃	1	2	yes	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	2	4	no	—
140	M₃₂	C₂×C₄×C₅₃	1	4	yes	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
145	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
146	C₂×C₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
147	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
148	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
149	#101	C₈×C₅₃	1	2	yes	—
150	#100	C₈×C₅₃	1	2	yes	—
151	#99	C₈×C₅₃	1	4	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₂×C₈×C₅₃	1	1	yes	characteristic
157	C₂×D₄	C₂×C₈×C₅₃	1	2	yes	—
158	C₂×D₄	C₂×C₈×C₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	4	no	—
164	M₁₆	C₂×C₈×C₅₃	1	4	yes	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₄²×C₅₃	1	2	yes	—
169	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
172	C₂×C₈	#41×C₅₃	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	C₂×C₈×C₅₃	1	4	yes	—
175	C₄²	—	2	2	no	—
176	C₁₆	—	2	2	no	—
177	C₁₆	—	2	4	no	—
178	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
179	C₁₆	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
180	C₁₆	C₂²⋊C₄×C₅₃	1	2	yes	—
181	C₂³	C₄×C₈×C₅₃	1	2	yes	—
182	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
183	Q₈	—	2	2	no	—
184	D₄	—	2	4	no	—
185	D₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
190	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
191	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
192	C₂×C₄	C₂²×C₈×C₅₃	1	1	yes	characteristic
193	C₂×C₄	#131×C₅₃	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	4	no	—
196	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
197	C₈	#117×C₅₃	1	1	yes	characteristic
198	C₈	C₄×D₄×C₅₃	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
203	C₂²	C₈×D₄×C₅₃	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	4	4	no	—
206	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
207	C₄	C₅₃×#405	1	1	yes	characteristic
208	C₄	C₈×D₄×C₅₃	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	C₈×C₂²⋊C₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₅₃	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₅₃ ◁ C₂²×C₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2108	C₂	semidirect
5. C₅₃×#1378	C₂	semidirect
9. C₂×M₃₂×C₅₃	C₄	semidirect
12. C₂²×C₁₆×C₅₃	C₄	semidirect
17. C₅₃×#349	C₄	semidirect
92. #12613	C₅₃	direct (N × H)
105. #2108	C₂×C₅₃	semidirect
107. #1378	C₂×C₅₃	semidirect
115. C₂×M₃₂	C₄×C₅₃	semidirect
118. C₂²×C₁₆	C₄×C₅₃	semidirect
123. #349	C₄×C₅₃	semidirect
127. C₅₃	#12613	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#504	2
C₅₃×#2115	2
C₅₃×#2108	2
C₅₃×#1757	2
C₅₃×#1378	2
#12613	53

Subgroups — C53×#12613 #27220125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12613 #27220125