Subgroups — #27275212

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133146	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88322	C₂	1	2	yes	maximal
4	#20272542	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133146	C₄	1	4	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
11	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810377	C₄	1	4	yes	—
13	C₈×#88322	—	2	4	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂×C₄	1	4	yes	—
17	C₂²×#88322	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
22	#13133146	C₈	1	16	yes	complemented
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	D₄	1	2	yes	—
26	C₄×#88322	Q₈	1	2	yes	—
27	C₈×Dic₅₃	—	2	4	no	—
28	C₈×Dic₅₃	C₈	1	8	yes	complemented
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	2	no	—
33	C₄×C₈×C₅₃	—	2	2	no	—
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂²×Dic₅₃	C₄²	1	2	yes	—
38	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
46	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₃	#41	1	2	yes	—
49	C₄×Dic₅₃	—	2	2	no	—
50	C₄×Dic₅₃	—	2	4	no	—
51	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
52	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	C₄⋊C₄	1	4	yes	—
56	C₂×#88322	C₂×C₈	1	8	yes	—
57	C₂×#88322	C₂×C₈	1	4	yes	—
58	C₂×#88322	M₁₆	1	4	yes	—
59	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
66	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	#131	1	4	yes	—
68	C₂×Dic₅₃	—	2	2	no	—
69	C₂×Dic₅₃	—	2	2	no	—
70	C₂×Dic₅₃	—	2	2	no	—
71	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
72	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
73	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
74	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
75	C₈×C₅₃	—	2	4	no	—
76	C₈×C₅₃	—	2	8	no	—
77	#88322	—	2	8	no	—
78	#88322	C₄⋊C₈	1	8	yes	complemented
79	C₈×C₄⋊C₈	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
82	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
84	C₂²×C₅₃	#405	1	2	yes	—
85	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
86	C₄×C₅₃	#431	1	2	yes	—
87	C₄×C₅₃	C₈×D₄	1	4	yes	—
88	C₄×C₅₃	C₈×Q₈	1	4	yes	—
89	Dic₅₃	—	2	4	no	—
90	Dic₅₃	—	2	2	no	—
91	Dic₅₃	C₈²	1	4	yes	complemented
92	C₄×C₄⋊C₈	—	53	2	no	—
93	C₄²×C₈	—	53	1	no	—
94	C₂×C₈²	—	53	2	no	—
95	#1273	—	53	1	no	—
96	#1273	D₅₃	1	1	yes	characteristic
97	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
98	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
99	C₂×C₅₃	#1450	1	1	yes	characteristic
100	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₅₃	C₈×C₄⋊C₄	1	2	yes	—
102	C₂×C₄⋊C₈	—	53	4	no	—
103	C₂×C₄×C₈	—	53	2	no	—
104	C₂×C₄×C₈	—	53	1	no	—
105	C₂×C₄×C₈	—	53	2	no	—
106	C₂×C₄×C₈	—	53	2	no	—
107	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	C₄³	—	53	1	no	—
110	#336	—	53	4	no	—
111	C₈²	—	106	4	no	—
112	C₅₃	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	53	2	no	—
114	C₂²×C₈	—	53	4	no	—
115	C₂²×C₈	C₄×D₅₃	1	4	yes	—
116	C₂²×C₈	C₄×D₅₃	1	2	yes	—
117	C₂×C₄²	—	53	1	no	—
118	C₂×C₄²	—	53	2	no	—
119	C₂×C₄²	—	53	2	no	—
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₄⋊C₈	—	53	16	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	53	8	no	—
127	C₄×C₈	—	53	2	no	—
128	C₄×C₈	—	53	2	no	—
129	C₄×C₈	—	106	4	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	—	106	2	no	—
132	C₂²×C₄	—	53	1	no	—
133	C₂²×C₄	—	53	2	no	—
134	C₂²×C₄	—	53	1	no	—
135	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	—	53	4	no	—
143	C₂×C₈	—	53	4	no	—
144	C₂×C₈	—	53	8	no	—
145	C₂×C₈	—	106	4	no	—
146	C₂×C₈	—	106	4	no	—
147	C₂×C₈	C₈×D₅₃	1	4	yes	—
148	C₂×C₈	C₈×D₅₃	1	8	yes	—
149	C₄²	—	53	4	no	—
150	C₄²	—	53	8	no	—
151	C₄²	—	106	2	no	—
152	C₄²	—	106	4	no	—
153	C₄²	D₄×D₅₃	1	1	yes	characteristic
154	C₄²	#11719947	1	2	yes	—
155	C₄²	Q₈×D₅₃	1	1	yes	characteristic
156	C₂³	C₄²×D₅₃	1	1	yes	characteristic
157	C₂×C₄	—	53	4	no	—
158	C₂×C₄	—	53	2	no	—
159	C₂×C₄	—	53	4	no	—
160	C₂×C₄	—	53	2	no	—
161	C₂×C₄	—	106	2	no	—
162	C₂×C₄	—	106	2	no	—
163	C₂×C₄	—	106	2	no	—
164	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133173	1	1	yes	characteristic
166	C₂×C₄	#13133177	1	2	yes	—
167	C₂×C₄	#13133182	1	2	yes	—
168	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
169	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
170	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
171	C₂×C₄	#13133225	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	2	8	no	—
174	C₈	—	53	8	no	—
175	C₈	—	106	8	no	—
176	C₂²	#16810474	1	1	yes	characteristic
177	C₂²	C₄×C₈×D₅₃	1	2	yes	—
178	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
179	C₂²	#16810575	1	1	yes	characteristic
180	C₂²	#16810580	1	2	yes	—
181	C₄	—	53	4	no	—
182	C₄	—	106	4	no	—
183	C₄	—	106	2	no	—
184	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
185	C₄	#16810673	1	2	yes	—
186	C₄	#16810686	1	4	yes	—
187	C₄	#16810747	1	4	yes	—
188	C₂	C₈²×D₅₃	1	1	yes	characteristic
189	C₂	#20272994	1	2	yes	—
190	C₂	#20273000	1	1	yes	characteristic
191	C₂	#20274563	1	1	yes	characteristic
192	C₂	#20274576	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₈²
Fitting	C₅₃×#1273	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
22. #13133146	C₈	semidirect
28. C₈×Dic₅₃	C₈	semidirect
78. #88322	C₄⋊C₈	semidirect
91. Dic₅₃	C₈²	semidirect
112. C₅₃	C₈×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133146	2
C₄×C₈×Dic₅₃	2
C₂×C₈×#88322	2
#20272542	2
C₅₃×#1273	2
C₈×C₄⋊C₈	53