Subgroups — #27283922

Order 13568.

526 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	#20275677	C₂	1	2	yes	maximal
2	#20275677	C₂	1	2	yes	maximal
3	#20274843	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	#16810493	C₂²	1	2	yes	—
9	#16810386	—	2	2	no	—
10	#16810385	—	2	4	no	—
11	#16810385	—	2	4	no	—
12	#16810345	—	2	2	no	—
13	C₂²×Q₁₆×C₅₃	—	2	2	no	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#494	—	2	2	no	—
16	C₅₃×#494	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×#88322	D₄	1	2	yes	—
19	C₂²×#88322	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	2	2	no	—
23	C₄×#88322	—	2	1	no	—
24	C₄×#88322	—	2	1	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₂×Q₁₆×C₅₃	—	2	8	no	—
29	C₂×Q₁₆×C₅₃	—	4	8	no	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	2	2	no	—
34	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂×Q₈×C₅₃	—	2	4	no	—
41	C₂×Q₈×C₅₃	—	4	2	no	—
42	C₂×Q₈×C₅₃	—	4	8	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	Q₁₆×C₅₃	—	4	16	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₄⋊C₄×C₅₃	—	2	4	no	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	1	yes	characteristic
58	C₄²×C₅₃	#41	1	1	yes	characteristic
59	C₄⋊Dic₅₃	—	4	4	no	—
60	C₄⋊Dic₅₃	—	4	4	no	—
61	C₄⋊Dic₅₃	—	4	4	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	4	no	—
66	C₂×#88322	Q₁₆	1	4	yes	complemented
67	C₂³×C₅₃	#127	1	1	yes	characteristic
68	Q₈×C₅₃	—	4	8	no	—
69	Q₈×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	8	4	no	—
72	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	#135	1	2	yes	—
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
81	C₂×C₄×C₅₃	#137	1	4	yes	—
82	C₂×Dic₅₃	—	8	2	no	—
83	C₂×Dic₅₃	—	8	2	no	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₈×C₅₃	—	2	4	no	—
86	#88322	—	2	4	no	—
87	#88322	—	4	2	no	—
88	#9835	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#390	1	1	yes	characteristic
90	C₂²×C₅₃	#494	1	1	yes	characteristic
91	C₂²×C₅₃	#495	1	1	yes	characteristic
92	C₂²×C₅₃	#496	1	2	yes	—
93	C₂²×C₅₃	#497	1	2	yes	—
94	C₄×C₅₃	—	8	4	no	—
95	C₄×C₅₃	#418	1	2	yes	—
96	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
97	C₄×C₅₃	#441	1	2	yes	—
98	C₄×C₅₃	#467	1	2	yes	—
99	C₄×C₅₃	#470	1	2	yes	—
100	C₄×C₅₃	#490	1	2	yes	—
101	Dic₅₃	—	16	2	no	—
102	C₂×#494	D₅₃	1	1	yes	characteristic
103	#1990	—	53	2	no	—
104	#1990	—	53	2	no	—
105	#1834	—	53	1	no	—
106	C₂×#322	—	53	1	no	—
107	C₂×C₅₃	#1720	1	2	yes	—
108	C₂×C₅₃	#1723	1	2	yes	—
109	C₂×C₅₃	#1961	1	1	yes	characteristic
110	C₂×C₅₃	#1969	1	1	yes	characteristic
111	C₂×C₅₃	#1971	1	1	yes	characteristic
112	C₂²×Q₁₆	—	2	2	no	—
113	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
114	#494	—	2	2	no	—
115	#494	Dic₅₃	1	4	yes	complemented
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	#384	—	53	2	no	—
120	#340	—	106	4	no	—
121	#340	—	106	4	no	—
122	#340	—	106	2	no	—
123	#322	—	106	2	no	—
124	C₅₃	#9835	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	2	2	no	—
126	C₂×Q₁₆	—	2	8	no	—
127	C₂×Q₁₆	—	4	8	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂×Dic₅₃	1	4	yes	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	106	2	no	—
138	C₂×C₄⋊C₄	—	106	2	no	—
139	C₂×C₄⋊C₄	—	106	2	no	—
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	C₄×C₈	—	106	1	no	—
142	C₄×C₈	—	106	1	no	—
143	C₄×C₈	—	106	1	no	—
144	C₄×C₈	—	106	1	no	—
145	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
146	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	8	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	212	2	no	—
152	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
155	Q₁₆	—	4	16	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	212	4	no	—
168	C₄⋊C₄	—	212	4	no	—
169	C₄⋊C₄	—	212	4	no	—
170	C₄²	#11719947	1	1	yes	characteristic
171	C₄²	#11719947	1	1	yes	characteristic
172	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂³	#13133213	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	424	2	no	—
180	C₂×C₄	—	424	2	no	—
181	C₂×C₄	—	424	2	no	—
182	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133188	1	1	yes	characteristic
186	C₂×C₄	#13133188	1	1	yes	characteristic
187	C₂×C₄	#13133204	1	1	yes	characteristic
188	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
189	C₂×C₄	#13133240	1	2	yes	—
190	C₂×C₄	#13133243	1	4	yes	—
191	C₂×C₄	#13133249	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	106	4	no	—
194	C₈	—	212	2	no	—
195	C₂²	#16810526	1	1	yes	characteristic
196	C₂²	#16811001	1	1	yes	characteristic
197	C₂²	#16811008	1	1	yes	characteristic
198	C₂²	#16811010	1	2	yes	—
199	C₂²	#16811016	1	2	yes	—
200	C₄	—	8	4	no	—
201	C₄	—	848	2	no	—
202	C₄	#16810628	1	2	yes	—
203	C₄	Q₁₆×Dic₅₃	1	2	yes	—
204	C₄	#16810730	1	2	yes	—
205	C₄	#16810861	1	2	yes	—
206	C₄	#16810878	1	2	yes	—
207	C₄	#16810987	1	2	yes	—
208	C₂	#20274378	1	2	yes	—
209	C₂	#20274390	1	2	yes	—
210	C₂	#20275540	1	1	yes	characteristic
211	C₂	#20275577	1	1	yes	characteristic
212	C₂	#20275587	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#9835
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
16. C₅₃×#494	C₄	semidirect
66. C₂×#88322	Q₁₆	semidirect
115. #494	Dic₅₃	semidirect
124. C₅₃	#9835	semidirect

Maximal subgroups

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

subgroup	index
#20275677	2
#20274843	2
C₂×#16810345	2
C₂×C₅₃×#494	2
#9835	53