Subgroups — #27284103

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	#20272555	C₂	1	4	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	#16810521	C₄	1	2	yes	—
8	#16810507	—	2	2	no	—
9	#16810489	C₄	1	1	yes	characteristic
10	C₄×#11719919	—	2	2	no	—
11	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	#16810378	—	2	2	no	—
13	C₂×C₄×C₈×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	2	no	—
17	#13133209	—	2	2	no	—
18	#13133209	C₈	1	4	yes	complemented
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×#11719919	—	2	2	no	—
22	C₂×#11719919	—	2	2	no	—
23	C₂×C₄×Dic₅₃	—	2	2	no	—
24	C₄⋊#88322	—	2	4	no	—
25	#13133124	—	2	2	no	—
26	#13133124	—	2	2	no	—
27	#13133124	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₅₃	—	2	8	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	Q₈	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	8	no	—
45	C₂×C₈×C₅₃	—	2	4	no	—
46	C₄²×C₅₃	—	2	4	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	4	no	—
49	C₄²×C₅₃	C₂×C₈	1	2	yes	—
50	C₄²×C₅₃	M₁₆	1	2	yes	—
51	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
52	C₄²×C₅₃	#41	1	2	yes	—
53	C₄⋊Dic₅₃	—	2	2	no	—
54	C₄⋊Dic₅₃	—	2	4	no	—
55	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
56	#11719919	—	2	8	no	—
57	#11719919	—	4	4	no	—
58	C₄×Dic₅₃	—	4	4	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
68	C₂×C₄×C₅₃	#104	1	4	yes	—
69	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	2	yes	—
76	C₂×C₄×C₅₃	#135	1	4	yes	—
77	C₂×Dic₅₃	—	2	4	no	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₈×C₅₃	—	2	8	no	—
84	#88322	—	8	2	no	—
85	#9871	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#391	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
88	C₂²×C₅₃	#408	1	1	yes	characteristic
89	C₂²×C₅₃	#416	1	2	yes	—
90	C₂²×C₅₃	C₂×#104	1	2	yes	—
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	C₈×Q₈	1	2	yes	—
95	C₄×C₅₃	#446	1	2	yes	—
96	C₄×C₅₃	#488	1	2	yes	—
97	C₄×C₅₃	#499	1	2	yes	—
98	Dic₅₃	—	4	4	no	—
99	Dic₅₃	—	8	2	no	—
100	C₄×#125	—	53	1	no	—
101	#1831	—	53	1	no	—
102	C₄²×C₈	D₅₃	1	1	yes	characteristic
103	#1279	—	53	4	no	—
104	C₂×C₅₃	#1476	1	2	yes	—
105	C₂×C₅₃	#1494	1	2	yes	—
106	C₂×C₅₃	#1984	1	1	yes	characteristic
107	C₂×C₅₃	#1991	1	1	yes	characteristic
108	C₂×C₅₃	#1994	1	1	yes	characteristic
109	C₂×#125	—	53	1	no	—
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₂×C₄×C₈	—	2	2	no	—
112	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
113	#389	—	53	2	no	—
114	#387	—	106	2	no	—
115	#382	—	53	1	no	—
116	C₄×C₄⋊C₄	—	53	2	no	—
117	C₄×C₄⋊C₄	—	106	2	no	—
118	C₄³	C₂×D₅₃	1	1	yes	characteristic
119	#336	—	106	2	no	—
120	C₅₃	#9871	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	106	2	no	—
123	#125	—	53	4	no	—
124	#125	—	106	2	no	—
125	C₂×C₄⋊C₄	—	53	2	no	—
126	C₂×C₄⋊C₄	—	53	2	no	—
127	C₂×C₄⋊C₄	—	106	2	no	—
128	C₂×C₄⋊C₄	—	106	2	no	—
129	C₂×C₄²	—	106	2	no	—
130	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
132	C₂×C₄²	D₂₁₂	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	C₄⋊C₈	—	106	4	no	—
136	C₄×C₈	—	2	8	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	Dic₁₀₆	1	4	yes	—
139	#95	—	106	2	no	—
140	#95	—	106	2	no	—
141	#95	—	106	2	no	—
142	C₂²×C₄	—	106	2	no	—
143	C₂²×C₄	—	106	2	no	—
144	C₂²×C₄	#11719921	1	2	yes	—
145	C₂²×C₄	#11719921	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	212	2	no	—
152	C₂×C₈	—	212	4	no	—
153	C₄⋊C₄	—	53	4	no	—
154	C₄⋊C₄	—	106	8	no	—
155	C₄⋊C₄	—	106	2	no	—
156	C₄⋊C₄	—	106	4	no	—
157	C₄⋊C₄	—	212	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	212	4	no	—
162	C₄²	C₈×D₅₃	1	2	yes	—
163	C₄²	#11719912	1	2	yes	—
164	C₄²	C₂×Dic₁₀₆	1	2	yes	—
165	C₄²	#11719945	1	2	yes	—
166	C₂³	C₂×#11719921	1	1	yes	characteristic
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	106	4	no	—
173	C₂×C₄	—	212	2	no	—
174	C₂×C₄	—	212	2	no	—
175	C₂×C₄	—	212	2	no	—
176	C₂×C₄	—	212	2	no	—
177	C₂×C₄	—	212	2	no	—
178	C₂×C₄	#13133129	1	4	yes	—
179	C₂×C₄	#13133145	1	4	yes	—
180	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
181	C₂×C₄	#13133196	1	2	yes	—
182	C₂×C₄	#13133205	1	1	yes	characteristic
183	C₂×C₄	#13133217	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
186	C₂×C₄	#13133223	1	2	yes	—
187	C₂×C₄	#13133239	1	4	yes	—
188	C₈	—	2	8	no	—
189	C₈	—	424	2	no	—
190	C₂²	#16810529	1	1	yes	characteristic
191	C₂²	C₂×#13133129	1	1	yes	characteristic
192	C₂²	#16810589	1	1	yes	characteristic
193	C₂²	#16810620	1	2	yes	—
194	C₂²	C₂×#13133145	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	—	424	2	no	—
199	C₄	—	212	4	no	—
200	C₄	C₈×Dic₁₀₆	1	2	yes	—
201	C₄	#16810753	1	2	yes	—
202	C₄	#16810978	1	2	yes	—
203	C₄	#16811025	1	2	yes	—
204	C₂	#20273098	1	2	yes	—
205	C₂	#20273176	1	2	yes	—
206	C₂	#20275651	1	1	yes	characteristic
207	C₂	#20275680	1	1	yes	characteristic
208	C₂	#20275692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#9871
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
18. #13133209	C₈	semidirect
120. C₅₃	#9871	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133209	2
#20274832	2
#20272555	2
C₄²×C₈×C₅₃	2
#9871	53