Subgroups — #27274665

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133136	C₂	1	1	yes	maximal, characteristic
2	#20273039	C₂	1	1	yes	maximal, characteristic
3	#20273004	C₂	1	2	yes	maximal
4	#20272565	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133136	C₄	1	2	yes	—
8	C₄×#11719917	C₂²	1	2	yes	—
9	#16810392	C₄	1	2	yes	—
10	#16810349	C₄	1	4	yes	—
11	#16810345	—	2	2	no	—
12	#16810344	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₃	—	2	2	no	—
15	C₈×Q₈×C₅₃	—	2	2	no	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#11719917	C₂×C₄	1	4	yes	—
21	C₄⋊#88322	—	2	2	no	—
22	#13133136	—	2	4	no	—
23	C₂²⋊#88322	C₂×C₄	1	4	yes	—
24	#13133128	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	—	2	2	no	—
27	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₅₃	—	2	2	no	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	4	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	C₂×D₄	1	2	yes	—
63	C₄²×C₅₃	#41	1	2	yes	—
64	#11719917	—	2	4	no	—
65	#11719917	—	2	4	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	2	4	no	—
68	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	Q₈×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	#97	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₅₃	#131	1	2	yes	—
84	C₈×C₅₃	—	2	1	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	4	2	no	—
87	C₈×C₅₃	#101	1	2	yes	complemented
88	#88322	—	4	4	no	—
89	#88322	—	4	4	no	—
90	#8165	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	—	2	1	no	—
93	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
95	C₂²×C₅₃	#405	1	1	yes	characteristic
96	C₄×C₅₃	—	2	4	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	4	2	no	—
100	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
101	C₄×C₅₃	#413	1	1	yes	characteristic
102	C₄×C₅₃	#435	1	4	yes	—
103	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
104	C₄×#101	—	53	1	no	—
105	#1461	—	53	1	no	—
106	#1451	—	53	2	no	—
107	#1283	—	53	2	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	#1752	1	1	yes	characteristic
111	C₂×C₅₃	C₄×#101	1	1	yes	characteristic
112	C₂×C₅₃	#1757	1	1	yes	characteristic
113	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
114	C₈×Q₈	—	2	2	no	—
115	C₈×Q₈	—	2	2	no	—
116	#431	—	53	2	no	—
117	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
118	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₂×#101	—	53	2	no	—
120	C₄×M₁₆	—	53	2	no	—
121	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	#343	—	53	2	no	—
124	#324	—	53	4	no	—
125	#322	—	106	2	no	—
126	#322	—	106	2	no	—
127	C₅₃	#8165	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
129	C₂×M₁₆	—	53	4	no	—
130	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
132	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
136	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
138	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	106	2	no	—
142	#101	—	106	4	no	—
143	C₂²⋊C₈	—	53	4	no	—
144	#97	—	106	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	106	2	no	—
148	C₄×C₈	—	106	2	no	—
149	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
150	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	#11719912	1	4	yes	—
154	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
155	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
159	M₁₆	—	106	4	no	—
160	M₁₆	—	106	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	106	4	no	—
166	C₂×C₈	—	106	4	no	—
167	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	#11719945	1	2	yes	—
175	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
176	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	#13133127	1	4	yes	—
186	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	#13133176	1	1	yes	characteristic
188	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
189	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
190	C₂×C₄	C₂×#11719912	1	2	yes	—
191	C₂×C₄	#13133223	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	212	4	no	—
196	C₈	—	212	4	no	—
197	C₈	#13133136	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂²	C₂×#13133127	1	1	yes	characteristic
202	C₂²	#16810578	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	C₂×#13133136	1	1	yes	characteristic
208	C₄	#16810607	1	1	yes	characteristic
209	C₄	#16810694	1	4	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#20274491	1	1	yes	characteristic
213	C₂	C₄×#13133136	1	1	yes	characteristic
214	C₂	#20274515	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator, prime radical	C₂³×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#8165
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
87. C₈×C₅₃	#101	semidirect
127. C₅₃	#8165	semidirect
197. C₈	#13133136	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133136	2
#20273039	2
#20273004	2
#20272565	2
C₂×C₈×Q₈×C₅₃	2
#8165	53