Subgroups — #27277849

Order 13568.

362 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274831	C₂	1	2	yes	maximal
2	#20274460	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	2	yes	maximal
4	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810377	—	2	4	no	—
10	#16810377	C₄	1	2	yes	—
11	#16810377	C₄	1	2	yes	—
12	#16810375	—	2	4	no	—
13	#16810374	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#334	—	2	2	no	—
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄×#88322	D₄	1	2	yes	—
25	C₄×#88322	Q₈	1	2	yes	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	2	no	—
31	C₄⋊C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₄×C₈×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	4	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×Dic₅₃	—	2	2	no	—
49	C₄×Dic₅₃	—	2	2	no	—
50	C₄×Dic₅₃	—	2	4	no	—
51	C₄×Dic₅₃	—	4	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	—	2	2	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	4	no	—
56	C₂×#88322	C₄⋊C₄	1	4	yes	—
57	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#131	1	2	yes	—
66	C₂×C₄×C₅₃	#131	1	2	yes	—
67	C₂×C₄×C₅₃	#135	1	4	yes	—
68	C₂×Dic₅₃	—	2	4	no	—
69	C₂×Dic₅₃	—	2	4	no	—
70	C₂×Dic₅₃	—	2	4	no	—
71	C₈×C₅₃	—	2	2	no	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	—	8	2	no	—
74	#88322	—	2	4	no	—
75	#88322	—	8	2	no	—
76	#8811	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	#384	1	1	yes	characteristic
78	C₂²×C₅₃	#424	1	2	yes	—
79	C₂²×C₅₃	#427	1	2	yes	—
80	C₂²×C₅₃	#430	1	2	yes	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	#424	1	2	yes	—
84	C₄×C₅₃	#424	1	1	yes	characteristic
85	C₄×C₅₃	#424	1	1	yes	characteristic
86	C₄×C₅₃	#495	1	2	yes	—
87	C₄×C₅₃	#499	1	2	yes	—
88	Dic₅₃	—	4	4	no	—
89	#1831	—	53	2	no	—
90	#1744	—	53	1	no	—
91	C₂×#335	—	53	2	no	—
92	C₂×#334	—	53	1	no	—
93	C₂×#334	D₅₃	1	1	yes	characteristic
94	C₂×C₅₃	#1572	1	4	yes	—
95	C₂×C₅₃	#1843	1	1	yes	characteristic
96	C₂×C₅₃	#1846	1	1	yes	characteristic
97	C₂×C₅₃	#1849	1	1	yes	characteristic
98	C₂×C₄⋊C₈	—	53	2	no	—
99	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
100	C₂×C₄×C₈	—	53	1	no	—
101	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₄³	—	53	1	no	—
103	#336	—	53	2	no	—
104	#336	—	53	2	no	—
105	#336	—	106	4	no	—
106	#335	—	106	4	no	—
107	#334	—	2	2	no	—
108	#334	—	106	2	no	—
109	C₅₃	#8811	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	53	2	no	—
112	C₂²×C₈	—	106	2	no	—
113	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₄²	—	53	2	no	—
116	C₂×C₄²	—	53	2	no	—
117	C₂×C₄²	—	53	2	no	—
118	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	106	2	no	—
122	C₄⋊C₈	—	106	2	no	—
123	C₄×C₈	—	53	2	no	—
124	C₄×C₈	—	53	2	no	—
125	C₄×C₈	Dic₁₀₆	1	2	yes	—
126	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
127	C₂²×C₄	—	53	4	no	—
128	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	53	4	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	—	106	2	no	—
138	C₂×C₈	—	212	2	no	—
139	C₂×C₈	—	212	4	no	—
140	C₂×C₈	#11719919	1	2	yes	—
141	C₂×C₈	#11719919	1	2	yes	—
142	C₄²	—	106	2	no	—
143	C₄²	—	106	2	no	—
144	C₄²	—	106	4	no	—
145	C₄²	—	212	2	no	—
146	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₄²	D₄×D₅₃	1	1	yes	characteristic
148	C₄²	Q₈×D₅₃	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
150	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	106	4	no	—
153	C₂×C₄	—	106	4	no	—
154	C₂×C₄	—	106	4	no	—
155	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
156	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
157	C₂×C₄	#13133213	1	1	yes	characteristic
158	C₂×C₄	#13133215	1	1	yes	characteristic
159	C₂×C₄	#13133216	1	1	yes	characteristic
160	C₂×C₄	#13133216	1	1	yes	characteristic
161	C₂×C₄	#13133223	1	2	yes	—
162	C₂×C₄	#13133225	1	2	yes	—
163	C₂×C₄	#13133238	1	4	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	8	2	no	—
167	C₈	—	106	4	no	—
168	C₈	—	424	2	no	—
169	C₂²	#16810492	1	1	yes	characteristic
170	C₂²	#16810648	1	2	yes	—
171	C₂²	#16810658	1	2	yes	—
172	C₂²	#16810670	1	2	yes	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	212	4	no	—
176	C₄	#16810648	1	2	yes	—
177	C₄	#16810649	1	1	yes	characteristic
178	C₄	#16810651	1	1	yes	characteristic
179	C₄	#16811004	1	2	yes	—
180	C₄	#16811023	1	2	yes	—
181	C₂	#20273492	1	4	yes	—
182	C₂	#20274868	1	1	yes	characteristic
183	C₂	#20274876	1	1	yes	characteristic
184	C₂	#20274888	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#8811
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
109. C₅₃	#8811	semidirect

Maximal subgroups

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

subgroup	index
#20274831	2
#20274460	2
C₂×#16810375	2
C₂×#16810374	2
C₂×C₅₃×#334	2
#8811	53