Subgroups — #27282660

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	#20275434	C₂	1	2	yes	maximal
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810374	—	2	2	no	—
12	#16810345	—	2	2	no	—
13	C₂×#107×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#340	C₄	1	1	yes	characteristic
17	C₅₃×#340	C₄	1	1	yes	characteristic
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	D₄	1	2	yes	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	4	4	no	—
24	C₄×#88322	—	2	1	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	Q₈	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#107×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	#95×C₅₃	—	2	2	no	—
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
51	C₄²×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	—	4	2	no	—
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	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	—	2	4	no	—
68	C₂×C₄×C₅₃	—	4	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	#131	1	2	yes	—
81	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×C₄×C₅₃	#137	1	2	yes	—
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	2	2	no	—
85	#88322	—	2	4	no	—
86	#88322	—	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	—	4	4	no	—
97	C₄×C₅₃	—	8	2	no	—
98	C₄×C₅₃	#423	1	1	yes	characteristic
99	C₄×C₅₃	#424	1	1	yes	characteristic
100	C₄×C₅₃	#446	1	2	yes	—
101	C₄×C₅₃	#468	1	2	yes	—
102	C₄×C₅₃	#501	1	2	yes	—
103	#1939	—	53	2	no	—
104	C₄×#107	D₅₃	1	1	yes	characteristic
105	C₂×#334	—	53	1	no	—
106	C₂×#322	—	53	1	no	—
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	Dic₅₃	1	2	yes	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄⋊C₈	—	106	2	no	—
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	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
120	#340	Dic₅₃	1	1	yes	characteristic
121	#340	Dic₅₃	1	1	yes	characteristic
122	#336	—	106	2	no	—
123	#334	—	106	2	no	—
124	#322	—	106	2	no	—
125	C₅₃	#9599	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	53	2	no	—
127	C₂²×C₈	—	106	1	no	—
128	C₂²×C₈	—	106	2	no	—
129	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
135	#107	—	2	4	no	—
136	C₄⋊C₈	—	106	2	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₄⋊C₈	—	212	4	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	106	1	no	—
141	C₄×C₈	—	106	1	no	—
142	C₄×C₈	—	106	1	no	—
143	C₄×C₈	Dic₁₀₆	1	2	yes	—
144	C₄×C₈	D₂₁₂	1	2	yes	—
145	#95	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
149	C₂²×C₄	#11719947	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	106	4	no	—
154	C₂×C₈	—	106	2	no	—
155	C₂×C₈	—	106	2	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	#11719917	1	4	yes	—
167	C₄²	—	4	2	no	—
168	C₄²	—	4	2	no	—
169	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
170	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
171	C₄²	Q₈×D₅₃	1	1	yes	characteristic
172	C₄²	#11719949	1	1	yes	characteristic
173	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	#13133192	1	1	yes	characteristic
184	C₂×C₄	#13133197	1	1	yes	characteristic
185	C₂×C₄	#13133208	1	1	yes	characteristic
186	C₂×C₄	#13133216	1	1	yes	characteristic
187	C₂×C₄	C₂×#11719917	1	2	yes	—
188	C₂×C₄	#13133224	1	2	yes	—
189	C₂×C₄	#13133241	1	2	yes	—
190	C₂×C₄	#13133249	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₈	—	106	4	no	—
194	C₈	—	424	2	no	—
195	C₈	—	212	2	no	—
196	C₂²	#16810522	1	1	yes	characteristic
197	C₂²	#16810695	1	2	yes	—
198	C₂²	#16810730	1	1	yes	characteristic
199	C₂²	#16810735	1	1	yes	characteristic
200	C₂²	#16810740	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	#16810647	1	1	yes	characteristic
206	C₄	#16810650	1	1	yes	characteristic
207	C₄	#16810754	1	2	yes	—
208	C₄	#16810870	1	2	yes	—
209	C₄	#16811031	1	2	yes	—
210	C₂	#20273580	1	2	yes	—
211	C₂	#20273587	1	2	yes	—
212	C₂	#20275439	1	1	yes	characteristic
213	C₂	#20275454	1	1	yes	characteristic
214	C₂	#20275464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#9599
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

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
#20275434	2
C₂×#16810374	2
C₂×#16810345	2
#20272556	2
C₄×#107×C₅₃	2
#9599	53