Subgroups — #27293145

Order 13568.

246 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276518	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
3	#20272637	C₂	1	2	yes	maximal, complemented
4	#20272635	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₄⋊#11719908	C₄	1	2	yes	—
9	C₂²⋊#11719908	C₄	1	2	yes	—
10	#16810399	C₂²	1	2	yes	—
11	#16810344	C₄	1	1	yes	characteristic
12	#16810344	C₄	1	1	yes	characteristic
13	#16810344	C₂²	1	1	yes	characteristic
14	#16810344	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
17	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
18	C₅₃×#346	C₂²	1	2	yes	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	—	2	1	no	—
21	C₂×#11719908	C₂×C₄	1	4	yes	—
22	C₄×#88322	—	2	1	no	—
23	C₄×#88322	—	2	1	no	—
24	C₄×#88322	C₂×C₄	1	1	yes	characteristic
25	C₄×#88322	C₂×C₄	1	1	yes	characteristic
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₄×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₄²	1	1	yes	characteristic
46	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	—	2	1	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₁₆×C₅₃	—	2	4	no	—
56	#11719908	—	2	4	no	—
57	C₂³×C₅₃	#97	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	#131	1	2	yes	—
71	C₈×C₅₃	#131	1	2	yes	—
72	#88322	—	4	1	no	—
73	#88322	—	4	1	no	—
74	#88322	—	4	1	no	—
75	#88322	—	4	1	no	—
76	#11415	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	#347	1	2	yes	—
80	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	—	2	1	no	—
85	C₄×C₅₃	#347	1	2	yes	—
86	C₄×C₅₃	#405	1	1	yes	characteristic
87	C₄×C₅₃	#405	1	1	yes	characteristic
88	#2163	—	53	1	no	—
89	#2163	D₅₃	1	1	yes	characteristic
90	C₂×#322	—	53	1	no	—
91	#1313	—	53	2	no	—
92	#1313	—	53	2	no	—
93	C₂×C₅₃	—	2	2	no	—
94	C₂×C₅₃	#1452	1	1	yes	characteristic
95	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#347	1	1	yes	characteristic
97	C₂×C₄×C₈	—	53	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₄⋊C₁₆	—	53	2	no	—
101	C₄⋊C₁₆	Dic₅₃	1	2	yes	—
102	C₂²⋊C₁₆	—	53	2	no	—
103	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
104	#346	—	53	2	no	—
105	#346	C₂×D₅₃	1	2	yes	—
106	#322	—	53	1	no	—
107	#322	—	53	1	no	—
108	#322	—	53	1	no	—
109	#322	—	53	1	no	—
110	C₅₃	#11415	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	106	1	no	—
112	C₂²×C₈	—	106	1	no	—
113	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
116	C₂×C₁₆	—	53	4	no	—
117	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	—	53	1	no	—
122	C₄×C₈	—	106	1	no	—
123	C₄×C₈	—	106	1	no	—
124	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
125	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
128	C₂²×C₄	#11719912	1	1	yes	characteristic
129	C₂²×C₄	#11719912	1	1	yes	characteristic
130	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	106	1	no	—
134	C₂×C₈	—	106	1	no	—
135	C₂×C₈	—	106	1	no	—
136	C₂×C₈	—	106	1	no	—
137	C₂×C₈	—	212	2	no	—
138	C₂×C₈	—	212	2	no	—
139	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
140	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₄²	#11719912	1	1	yes	characteristic
144	C₄²	#11719912	1	1	yes	characteristic
145	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₁₆	—	2	4	no	—
148	C₁₆	—	106	4	no	—
149	C₂³	#13133127	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	#13133127	1	1	yes	characteristic
155	C₂×C₄	#13133127	1	1	yes	characteristic
156	C₂×C₄	#13133127	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
159	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	212	1	no	—
163	C₈	—	212	1	no	—
164	C₈	—	212	1	no	—
165	C₈	—	212	1	no	—
166	C₈	#13133224	1	2	yes	—
167	C₈	#13133225	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#16810400	1	2	yes	—
171	C₂²	C₂×#13133127	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#16810400	1	2	yes	—
177	C₄	#16810579	1	1	yes	characteristic
178	C₄	#16810581	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#20273008	1	1	yes	characteristic
181	C₂	C₂×#16810400	1	1	yes	characteristic
182	C₂	C₂×#16810400	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2163	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#11415
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #20272637	C₂	semidirect
4. #20272635	C₂	semidirect
110. C₅₃	#11415	semidirect

Maximal subgroups

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

subgroup	index
#20276518	2
C₂×#16810344	2
#20272637	2
#20272635	2
C₅₃×#2163	2
#11415	53