Subgroups — #27302256

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810407	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	2	no	—
8	#16810407	—	2	8	no	—
9	C₄×#11719908	—	2	2	no	—
10	C₂×#131×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#532	C₄	1	1	yes	characteristic
12	C₅₃×#436	—	2	2	no	—
13	C₈×D₄×C₅₃	—	2	2	no	—
14	C₅₃×#423	C₄	1	1	yes	characteristic
15	C₅₃×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	#13133157	—	2	4	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	D₄	1	4	yes	complemented
22	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
23	#131×C₅₃	—	2	4	no	—
24	#131×C₅₃	—	2	4	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
31	#127×C₅₃	C₈	1	1	yes	characteristic
32	#124×C₅₃	C₈	1	2	yes	—
33	#121×C₅₃	—	2	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	#41×C₅₃	—	2	4	no	—
41	#41×C₅₃	—	2	4	no	—
42	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×D₄×C₅₃	—	2	2	no	—
44	C₂×D₄×C₅₃	—	2	2	no	—
45	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₅₃	—	2	4	no	—
50	M₁₆×C₅₃	—	4	4	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	4	no	—
62	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
65	#11719908	—	2	4	no	—
66	#11719908	—	4	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	Q₈×C₅₃	—	2	4	no	—
70	D₄×C₅₃	—	2	4	no	—
71	D₄×C₅₃	—	4	4	no	—
72	D₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#131	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	#121	1	2	yes	—
86	C₈×C₅₃	#124	1	1	yes	characteristic
87	C₈×C₅₃	#127	1	1	yes	characteristic
88	#13191	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	—	4	4	no	—
91	C₂²×C₅₃	—	4	2	no	—
92	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂²×C₅₃	#436	1	2	yes	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	4	2	no	—
96	C₄×C₅₃	#350	1	4	yes	complemented
97	C₄×C₅₃	#390	1	1	yes	characteristic
98	C₄×C₅₃	#408	1	1	yes	characteristic
99	C₄×C₅₃	C₈×D₄	1	2	yes	—
100	#2931	D₅₃	1	1	yes	characteristic
101	#2150	—	53	1	no	—
102	C₂×#350	—	53	4	no	—
103	C₂×C₄×C₁₆	—	53	1	no	—
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	—	8	2	no	—
106	C₂×C₅₃	#1954	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#350	1	2	yes	—
108	C₂×#131	C₂×D₅₃	1	2	yes	—
109	#532	Dic₅₃	1	1	yes	characteristic
110	C₂×M₃₂	—	53	2	no	—
111	C₂²×C₁₆	—	53	2	no	—
112	#436	—	2	2	no	—
113	C₈×D₄	—	2	2	no	—
114	#423	Dic₅₃	1	1	yes	characteristic
115	#408	Dic₅₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	106	2	no	—
118	#350	—	106	8	no	—
119	C₄×C₁₆	—	106	2	no	—
120	C₅₃	#13191	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	C₂×Dic₅₃	1	2	yes	—
122	#131	—	2	4	no	—
123	#131	—	2	4	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
129	C₄⋊Q₈	#88322	1	1	yes	characteristic
130	#127	#88322	1	1	yes	characteristic
131	#124	#88322	1	2	yes	—
132	#121	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
135	M₃₂	—	106	4	no	—
136	C₂×C₁₆	—	53	4	no	—
137	C₂×C₁₆	—	106	2	no	—
138	C₂×C₁₆	—	106	2	no	—
139	C₂×C₁₆	—	106	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	4	no	—
142	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
143	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
144	#41	—	2	4	no	—
145	#41	—	2	4	no	—
146	C₂×Q₈	C₂×#88322	1	2	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	C₂×#88322	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	#11719917	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
153	M₁₆	—	2	4	no	—
154	M₁₆	—	4	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
160	C₂×C₈	D₄×D₅₃	1	2	yes	—
161	C₂×C₈	#11719947	1	2	yes	—
162	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₄²	C₂×#88322	1	1	yes	characteristic
167	C₄²	C₂×#88322	1	1	yes	characteristic
168	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
169	C₁₆	—	106	4	no	—
170	C₁₆	—	212	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂×#11719917	1	1	yes	characteristic
173	Q₈	—	2	4	no	—
174	D₄	—	2	4	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²⋊#88322	1	2	yes	—
182	C₂×C₄	C₂²⋊#88322	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
185	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
186	C₂×C₄	#13133224	1	2	yes	—
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#13133188	1	2	yes	—
190	C₈	#13133204	1	1	yes	characteristic
191	C₈	#13133213	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
196	C₂²	#16810701	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	#16810407	1	4	yes	complemented
200	C₄	#16810526	1	1	yes	characteristic
201	C₄	#16810590	1	1	yes	characteristic
202	C₄	D₄×#88322	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	8	2	no	—
205	C₂	#20275510	1	1	yes	characteristic
206	C₂	C₂×#16810407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2931	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₅₃×#532	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2931	D₅₃
hypocenter	C₅₃	#13191
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. C₂×#16810407	C₂	semidirect
3. C₂×C₄×#11719908	C₂	semidirect
21. C₂×#11719908	D₄	semidirect
96. C₄×C₅₃	#350	semidirect
120. C₅₃	#13191	semidirect
199. C₄	#16810407	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×#16810407	2
C₂×C₄×#11719908	2
C₅₃×#2931	2
#13191	53