Subgroups — #27299191

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×#16811040	C₂	1	1	yes	maximal, characteristic
2	#20276314	C₂	1	2	yes	maximal, complemented
3	#20274515	C₂	1	1	yes	maximal, characteristic
4	#20272802	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2115	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	2	no	—
7	#16811040	—	2	2	no	—
8	C₂×#13133157	C₄	1	2	yes	complemented
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	2	yes	complemented
11	C₂²×#11719908	C₂²	1	1	yes	characteristic
12	#16810607	C₄	1	1	yes	characteristic
13	#16810597	C₄	1	1	yes	characteristic
14	#16810578	C₂²	1	2	yes	—
15	#16810398	—	2	2	no	—
16	C₁₆×Dic₅₃	—	2	2	no	—
17	#16810394	C₄	1	2	yes	—
18	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#349	C₄	1	4	yes	complemented
21	C₂×#11719917	C₂×C₄	1	2	yes	—
22	#13133176	C₂×C₄	1	2	yes	—
23	#13133157	—	2	2	no	—
24	#13133157	—	2	4	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₂×#11719908	C₂×C₄	1	2	yes	—
28	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
29	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	D₄	1	2	yes	—
31	C₂×#11719908	D₄	1	2	yes	—
32	#13133136	C₈	1	2	yes	—
33	#13133134	C₈	1	2	yes	—
34	#13133132	C₈	1	4	yes	—
35	#13133127	—	2	2	no	—
36	C₈×Dic₅₃	—	2	2	no	—
37	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#131×C₅₃	—	2	2	no	—
39	#131×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₅₃	—	2	2	no	—
45	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
46	C₂×C₁₆×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	4	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	C₄²	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	#41	1	2	yes	—
63	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄×Dic₅₃	—	2	2	no	—
66	#11719917	—	2	2	no	—
67	#11719917	C₂×C₈	1	4	yes	—
68	C₂×#88322	—	2	2	no	—
69	C₁₆×C₅₃	—	2	4	no	—
70	#11719908	—	2	2	no	—
71	#11719908	C₂²⋊C₄	1	1	yes	characteristic
72	#11719908	C₂²⋊C₄	1	1	yes	characteristic
73	#11719908	C₂²⋊C₄	1	2	yes	—
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	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#131	1	2	yes	—
86	C₂×Dic₅₃	—	2	4	no	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₅₃	#117	1	1	yes	characteristic
90	C₈×C₅₃	C₄×D₄	1	2	yes	—
91	#88322	—	2	4	no	—
92	#12613	—	53	1	no	maximal, 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	C₄×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₅₃	#405	1	1	yes	characteristic
101	C₄×C₅₃	C₈×D₄	1	2	yes	—
102	Dic₅₃	—	4	4	no	—
103	C₂×#504	—	53	1	no	—
104	#2115	D₅₃	1	1	yes	characteristic
105	#2108	—	53	2	no	—
106	#1757	—	53	1	no	—
107	#1378	—	53	2	no	—
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	C₂×D₅₃	1	1	yes	characteristic
113	#504	—	106	2	no	—
114	#504	—	106	2	no	—
115	C₂×M₃₂	—	53	2	no	—
116	C₂×M₃₂	—	53	1	no	—
117	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
118	C₂²×C₁₆	—	53	1	no	—
119	C₂²×C₁₆	—	53	2	no	—
120	#413	—	53	1	no	—
121	#410	—	53	1	no	—
122	#405	—	53	2	no	—
123	#349	Dic₅₃	1	4	yes	complemented
124	#346	—	106	2	no	—
125	C₄×C₁₆	—	106	2	no	—
126	#344	—	53	2	no	—
127	C₅₃	#12613	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
129	#131	—	2	2	no	—
130	#131	—	2	2	no	—
131	C₂×M₁₆	—	53	2	no	—
132	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
133	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
134	C₂²×C₈	C₄×D₅₃	1	2	yes	—
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	#117	—	53	2	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	106	4	no	—
139	M₃₂	—	106	2	no	—
140	M₃₂	C₂×Dic₅₃	1	4	yes	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	53	1	no	—
143	C₂×C₁₆	—	53	1	no	—
144	C₂×C₁₆	—	53	2	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	53	2	no	—
147	C₂×C₁₆	—	106	2	no	—
148	C₂×C₁₆	—	106	2	no	—
149	#101	—	53	2	no	—
150	#100	—	53	2	no	—
151	#99	—	53	4	no	—
152	#97	—	106	2	no	—
153	C₄×C₈	—	106	2	no	—
154	#41	—	2	2	no	—
155	#41	—	2	2	no	—
156	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
157	C₂×D₄	C₈×D₅₃	1	2	yes	—
158	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
160	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	53	4	no	—
164	M₁₆	—	106	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
169	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₈	D₄×D₅₃	1	2	yes	—
171	C₂×C₈	#11719947	1	2	yes	—
172	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₄⋊C₄	—	106	2	no	—
174	C₂²⋊C₄	—	53	4	no	—
175	C₄²	—	106	2	no	—
176	C₁₆	—	2	4	no	—
177	C₁₆	—	53	1	no	—
178	C₁₆	—	53	1	no	—
179	C₁₆	—	53	2	no	—
180	C₁₆	—	106	2	no	—
181	C₂³	C₈×Dic₅₃	1	2	yes	—
182	C₂³	C₂×C₈×D₅₃	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	—	106	4	no	—
189	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
191	C₂×C₄	#13133177	1	2	yes	—
192	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
193	C₂×C₄	#13133225	1	2	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	106	4	no	—
196	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
197	C₈	#13133173	1	1	yes	characteristic
198	C₈	D₄×Dic₅₃	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
203	C₂²	#16810686	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	212	4	no	—
206	C₄	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
207	C₄	#16810581	1	1	yes	characteristic
208	C₄	#16810686	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	2	1	no	—
211	C₂	—	4	2	no	—
212	C₂	#20274489	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2115	C₂
socle	C₂×C₅₃	C₈×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2115	D₅₃
hypocenter	C₅₃	#12613
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2115

Complements

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

N	H ≅ G/N	type
2. #20276314	C₂	semidirect
4. #20272802	C₂	semidirect
8. C₂×#13133157	C₄	semidirect
10. C₂²×#11719908	C₄	semidirect
20. C₅₃×#349	C₄	semidirect
123. #349	Dic₅₃	semidirect
127. C₅₃	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811040	2
#20276314	2
#20274515	2
#20272802	2
C₅₃×#2115	2
#12613	53