Subgroups — C₅₃×#11971 #27219483

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1364	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
6	C₅₃×#500	—	2	1	no	—
7	C₅₃×#500	—	2	1	no	—
8	C₅₃×#500	C₄	1	2	yes	—
9	C₅₃×#500	C₂²	1	2	yes	—
10	C₂×#107×C₅₃	—	2	1	no	—
11	C₂×#107×C₅₃	C₄	1	2	yes	—
12	C₅₃×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₅₃	—	2	1	no	—
17	C₅₃×#335	—	2	1	no	—
18	C₅₃×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₅₃	—	4	1	no	—
20	C₂×M₁₆×C₅₃	—	4	1	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	4	1	no	—
26	C₄⋊Q₈×C₅₃	—	4	1	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	—	2	2	no	—
31	#107×C₅₃	—	2	2	no	—
32	#107×C₅₃	—	2	2	no	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	C₂×C₄	1	4	yes	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	4	1	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	—	4	2	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	8	2	no	—
44	C₂×Q₈×C₅₃	—	8	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
49	M₁₆×C₅₃	—	8	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	4	2	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	D₈	1	1	yes	characteristic
63	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₁₆×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	#102	1	1	yes	characteristic
67	Q₈×C₅₃	—	8	4	no	—
68	Q₈×C₅₃	—	8	2	no	—
69	Q₈×C₅₃	—	8	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	4	4	no	—
74	C₂×C₄×C₅₃	—	8	4	no	—
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
78	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	2	1	no	—
85	C₈×C₅₃	—	8	2	no	—
86	C₈×C₅₃	#101	1	2	yes	—
87	C₈×C₅₃	#137	1	2	yes	—
88	#11971	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₅₃	—	2	1	no	—
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	#357	1	2	yes	—
92	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	8	4	no	—
96	C₄×C₅₃	#357	1	2	yes	—
97	C₄×C₅₃	#362	1	2	yes	—
98	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
99	C₄×C₅₃	#419	1	1	yes	characteristic
100	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
101	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
102	C₂×#500	C₂×C₅₃	1	1	yes	characteristic
103	#2162	C₂×C₅₃	1	1	yes	characteristic
104	#1567	C₂×C₅₃	1	1	yes	characteristic
105	#1364	C₂×C₅₃	1	4	yes	complemented
106	C₂×C₅₃	—	2	2	no	—
107	C₂×C₅₃	#1546	1	1	yes	characteristic
108	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	C₄×C₅₃	1	2	yes	—
114	#500	C₂²×C₅₃	1	2	yes	—
115	C₂×#107	—	2	1	no	—
116	C₂×#107	C₄×C₅₃	1	2	yes	—
117	#423	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
119	C₄⋊C₁₆	C₂²×C₅₃	1	2	yes	—
120	C₂²⋊C₁₆	—	2	1	no	—
121	C₄×C₁₆	—	2	1	no	—
122	#335	—	2	1	no	—
123	#335	C₂²×C₅₃	1	2	yes	—
124	C₅₃	#11971	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×Q₈	—	4	1	no	—
126	C₂×M₁₆	—	4	1	no	—
127	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
128	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	1	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	—	2	2	no	—
139	#107	—	2	2	no	—
140	#107	C₂×C₄×C₅₃	1	4	yes	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	4	1	no	—
143	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
144	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
145	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
146	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	8	2	no	—
150	C₂×Q₈	—	8	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
154	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
155	M₁₆	—	8	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
168	C₄²	D₈×C₅₃	1	1	yes	characteristic
169	C₄²	SD₁₆×C₅₃	1	1	yes	characteristic
170	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
171	C₁₆	—	4	2	no	—
172	C₂³	#102×C₅₃	1	1	yes	characteristic
173	Q₈	—	8	2	no	—
174	Q₈	—	8	2	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
182	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
183	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
184	C₂×C₄	D₁₆×C₅₃	1	2	yes	—
185	C₂×C₄	SD₃₂×C₅₃	1	2	yes	—
186	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	8	2	no	—
192	C₈	#101×C₅₃	1	2	yes	—
193	C₈	#137×C₅₃	1	2	yes	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	C₅₃×#357	1	2	yes	—
197	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	C₅₃×#357	1	2	yes	—
202	C₄	C₅₃×#362	1	2	yes	—
203	C₄	C₂×#101×C₅₃	1	1	yes	characteristic
204	C₄	C₅₃×#419	1	1	yes	characteristic
205	C₄	C₂×D₁₆×C₅₃	1	1	yes	characteristic
206	C₄	C₂×SD₃₂×C₅₃	1	1	yes	characteristic
207	C₂	—	2	2	no	—
208	C₂	C₅₃×#1546	1	1	yes	characteristic
209	C₂	C₂×C₅₃×#357	1	1	yes	characteristic
210	C₂	C₂×C₅₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₂³×C₅₃	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅₃×#1364	C₂	semidirect
88. #11971	C₅₃	direct (N × H)
105. #1364	C₂×C₅₃	semidirect
124. C₅₃	#11971	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×#500	2
C₅₃×#2162	2
C₅₃×#1567	2
C₅₃×#1364	2
#11971	53

Subgroups — C53×#11971 #27219483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11971 #27219483