Subgroups — #27300507

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810443	C₂	1	2	yes	maximal
3	#20276470	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	#16810444	—	2	2	no	—
14	#16810443	—	2	4	no	—
15	#16810441	—	2	4	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#364	—	2	2	no	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	#13133157	—	2	2	no	—
25	#13133157	—	2	2	no	—
26	C₂×#11719908	—	2	2	no	—
27	C₄×#88322	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	C₈	1	2	yes	—
30	C₈×Dic₅₃	C₈	1	2	yes	—
31	C₈×Dic₅₃	D₄	1	2	yes	—
32	C₈×Dic₅₃	D₄	1	2	yes	—
33	C₈×Dic₅₃	Q₈	1	2	yes	—
34	C₈×Dic₅₃	Q₈	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₅₃	—	2	2	no	—
39	M₃₂×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	Q₈	1	2	yes	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	—	2	1	no	—
46	C₂²×Dic₅₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₅₃	—	2	8	no	—
59	C₄×Dic₅₃	—	2	1	no	—
60	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
61	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
62	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
63	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	C₄⋊C₄	1	2	yes	—
67	C₂×#88322	C₄⋊C₄	1	2	yes	—
68	C₂×#88322	C₂×C₈	1	2	yes	—
69	C₂×#88322	M₁₆	1	2	yes	—
70	C₁₆×C₅₃	—	4	2	no	—
71	#11719908	—	4	2	no	—
72	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	4	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	4	no	—
84	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
86	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
87	C₈×C₅₃	#127	1	1	yes	characteristic
88	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
90	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
91	#88322	—	2	2	no	—
92	#88322	C₄⋊C₈	1	4	yes	—
93	#12900	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₅₃	#423	1	1	yes	characteristic
97	C₂²×C₅₃	#423	1	1	yes	characteristic
98	C₄×C₅₃	—	2	4	no	—
99	C₄×C₅₃	#384	1	1	yes	characteristic
100	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₅₃	#423	1	1	yes	characteristic
103	Dic₅₃	—	2	4	no	—
104	Dic₅₃	—	2	2	no	—
105	Dic₅₃	#364	1	4	yes	complemented
106	C₂×#364	—	53	2	no	—
107	C₂×#364	—	53	1	no	—
108	C₂×#364	D₅₃	1	1	yes	characteristic
109	#2150	—	53	2	no	—
110	C₄²×C₈	—	53	1	no	—
111	C₂×C₅₃	—	2	1	no	—
112	C₂×C₅₃	—	2	1	no	—
113	C₂×C₅₃	#1842	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#364	1	2	yes	—
115	C₂×M₃₂	—	53	2	no	—
116	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	C₄³	—	53	1	no	—
124	#364	—	2	2	no	—
125	#364	—	106	4	no	—
126	#364	—	106	2	no	—
127	C₄⋊C₁₆	—	106	4	no	—
128	C₅₃	#12900	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₈	—	53	1	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
133	C₂×C₄²	—	53	1	no	—
134	C₂×C₄²	—	53	1	no	—
135	C₂×C₄²	—	106	2	no	—
136	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	106	2	no	—
140	M₃₂	—	106	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	106	2	no	—
143	C₄×C₈	—	53	2	no	—
144	C₄×C₈	—	53	2	no	—
145	C₄×C₈	—	53	2	no	—
146	C₄×C₈	—	53	2	no	—
147	C₄×C₈	—	53	2	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	106	2	no	—
150	C₄×C₈	—	106	2	no	—
151	C₄×C₈	Dic₁₀₆	1	2	yes	—
152	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	53	2	no	—
155	C₂²×C₄	—	106	1	no	—
156	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	53	2	no	—
159	C₂×C₈	—	53	2	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	106	1	no	—
162	C₂×C₈	—	106	1	no	—
163	C₂×C₈	C₈×D₅₃	1	2	yes	—
164	C₂×C₈	#11719912	1	2	yes	—
165	C₂×C₈	#11719919	1	2	yes	—
166	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
167	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
168	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	—	53	2	no	—
174	C₄²	—	53	1	no	—
175	C₄²	—	53	1	no	—
176	C₄²	—	53	2	no	—
177	C₄²	—	106	8	no	—
178	C₄²	—	106	1	no	—
179	C₄²	#11719919	1	2	yes	—
180	C₁₆	—	4	2	no	—
181	C₁₆	—	212	2	no	—
182	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	53	2	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	106	4	no	—
188	C₂×C₄	—	106	4	no	—
189	C₂×C₄	—	106	2	no	—
190	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
191	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
192	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
193	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
194	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
195	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
196	C₈	—	53	4	no	—
197	C₈	—	106	2	no	—
198	C₈	#13133146	1	4	yes	—
199	C₈	#13133213	1	1	yes	characteristic
200	C₈	#13133215	1	1	yes	characteristic
201	C₈	#13133216	1	1	yes	characteristic
202	C₈	#13133216	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
205	C₂²	#16810645	1	1	yes	characteristic
206	C₂²	#16810646	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	53	4	no	—
209	C₄	—	106	4	no	—
210	C₄	—	106	2	no	—
211	C₄	#16810492	1	1	yes	characteristic
212	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
213	C₄	C₂×#13133146	1	1	yes	characteristic
214	C₄	#16810645	1	1	yes	characteristic
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#20274866	1	1	yes	characteristic
218	C₂	#364×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12900
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
105. Dic₅₃	#364	semidirect
128. C₅₃	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
C₂×#16810443	2
#20276470	2
C₄×C₈×Dic₅₃	2
C₂×C₅₃×#364	2
#12900	53