Subgroups — #27282659

Order 13568.

378 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	#20275433	C₂	1	2	yes	maximal
2	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	C₄	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133146	—	2	2	no	—
9	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810386	C₄	1	1	yes	characteristic
11	#16810386	C₄	1	1	yes	characteristic
12	#16810377	—	2	2	no	—
13	#16810374	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	#13133151	—	2	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	#13133146	—	4	4	no	—
26	#13133124	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₅₃	#41	1	1	yes	characteristic
56	C₄⋊Dic₅₃	—	2	2	no	—
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	M₁₆	1	4	yes	—
60	C₄×Dic₅₃	—	4	2	no	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₅₃	#131	1	2	yes	—
75	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₅₃	#137	1	2	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	4	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	4	2	no	—
87	#88322	—	8	2	no	—
88	#9599	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#389	1	1	yes	characteristic
90	C₂²×C₅₃	#435	1	2	yes	—
91	C₂²×C₅₃	#441	1	1	yes	characteristic
92	C₂²×C₅₃	#442	1	1	yes	characteristic
93	C₂²×C₅₃	#443	1	2	yes	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	#423	1	1	yes	characteristic
97	C₄×C₅₃	#424	1	1	yes	characteristic
98	C₄×C₅₃	#446	1	2	yes	—
99	C₄×C₅₃	#468	1	2	yes	—
100	C₄×C₅₃	#501	1	2	yes	—
101	Dic₅₃	—	4	4	no	—
102	Dic₅₃	—	8	2	no	—
103	#1939	—	53	2	no	—
104	C₄×#107	—	53	1	no	—
105	C₂×#334	—	53	1	no	—
106	C₂×#322	D₅₃	1	1	yes	characteristic
107	#1279	—	53	2	no	—
108	C₂×C₅₃	#1586	1	2	yes	—
109	C₂×C₅₃	#1587	1	2	yes	—
110	C₂×C₅₃	#1940	1	1	yes	characteristic
111	C₂×C₅₃	#1943	1	1	yes	characteristic
112	C₂×C₅₃	#1945	1	1	yes	characteristic
113	C₂×#107	—	53	2	no	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄⋊C₈	—	106	2	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	53	2	no	—
120	#340	—	53	1	no	—
121	#340	—	53	1	no	—
122	#336	—	106	2	no	—
123	#334	—	106	2	no	—
124	#322	—	2	2	no	—
125	C₅₃	#9599	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	2	1	no	—
127	C₂²×C₈	—	106	2	no	—
128	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
130	C₂²×C₈	D₂₁₂	1	2	yes	—
131	C₂×C₄⋊C₄	—	53	2	no	—
132	C₂×C₄⋊C₄	—	53	2	no	—
133	C₂×C₄²	—	106	2	no	—
134	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	#107	—	106	4	no	—
136	C₄⋊C₈	—	106	2	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₄⋊C₈	—	212	4	no	—
139	C₄×C₈	—	2	1	no	—
140	C₄×C₈	—	2	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	Dic₁₀₆	1	2	yes	—
143	C₄×C₈	Dic₁₀₆	1	2	yes	—
144	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
145	#95	—	106	2	no	—
146	C₂²×C₄	—	106	2	no	—
147	C₂²×C₄	—	106	2	no	—
148	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂²×C₄	#11719945	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	#11719919	1	2	yes	—
162	C₂×C₈	#11719919	1	2	yes	—
163	C₄⋊C₄	—	53	4	no	—
164	C₄⋊C₄	—	106	2	no	—
165	C₄⋊C₄	—	106	2	no	—
166	C₄⋊C₄	—	106	2	no	—
167	C₄²	—	212	2	no	—
168	C₄²	—	212	2	no	—
169	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
170	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
171	C₄²	#11719945	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	106	4	no	—
176	C₂×C₄	—	212	2	no	—
177	C₂×C₄	—	212	2	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	—	212	2	no	—
181	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
182	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
183	C₂×C₄	#13133191	1	1	yes	characteristic
184	C₂×C₄	#13133196	1	1	yes	characteristic
185	C₂×C₄	#13133209	1	1	yes	characteristic
186	C₂×C₄	#13133217	1	1	yes	characteristic
187	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
188	C₂×C₄	#13133225	1	2	yes	—
189	C₂×C₄	#13133245	1	2	yes	—
190	C₂×C₄	#13133250	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	424	2	no	—
196	C₂²	#16810521	1	1	yes	characteristic
197	C₂²	#16810696	1	2	yes	—
198	C₂²	#16810731	1	1	yes	characteristic
199	C₂²	#16810736	1	1	yes	characteristic
200	C₂²	#16810741	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	424	2	no	—
204	C₄	—	212	4	no	—
205	C₄	#16810646	1	1	yes	characteristic
206	C₄	#16810651	1	1	yes	characteristic
207	C₄	#16810751	1	2	yes	—
208	C₄	#16810869	1	2	yes	—
209	C₄	#16811032	1	2	yes	—
210	C₂	#20273581	1	2	yes	—
211	C₂	#20273588	1	2	yes	—
212	C₂	#20275438	1	1	yes	characteristic
213	C₂	#20275453	1	1	yes	characteristic
214	C₂	#20275463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#9599
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
125. C₅₃	#9599	semidirect

Maximal subgroups

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

subgroup	index
#20275433	2
C₄×#13133151	2
C₂×#16810374	2
#20272555	2
C₂×C₅₃×#322	2
#9599	53