Subgroups — #27302100

Order 13568.

298 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	#20276475	C₂	1	2	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20274594	C₂	1	1	yes	maximal, characteristic
4	#20272805	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×#13133153	C₄	1	1	yes	characteristic
9	#16810661	C₄	1	1	yes	characteristic
10	#16810578	C₂²	1	2	yes	—
11	#16810441	—	2	4	no	—
12	C₄×#11719908	—	2	2	no	—
13	#16810383	C₄	1	2	yes	—
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#364	—	2	2	no	—
17	C₂×#11719917	C₂×C₄	1	2	yes	—
18	#13133176	C₂×C₄	1	2	yes	—
19	C₂×#11719908	—	2	1	no	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	—	2	1	no	—
23	C₂×#11719908	D₄	1	2	yes	—
24	C₂×#11719908	Q₈	1	2	yes	—
25	#13133153	—	2	2	no	—
26	#13133151	C₈	1	2	yes	—
27	#13133149	C₈	1	2	yes	—
28	#13133127	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₅₃	—	2	2	no	—
34	M₃₂×C₅₃	—	2	2	no	—
35	C₂×C₁₆×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²⋊Dic₅₃	—	2	2	no	—
53	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
54	C₄×Dic₅₃	—	2	2	no	—
55	#11719917	—	2	2	no	—
56	#11719917	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₁₆×C₅₃	—	4	2	no	—
59	#11719908	—	2	2	no	—
60	#11719908	—	2	2	no	—
61	C₂³×C₅₃	#131	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	4	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
70	C₂×Dic₅₃	—	2	4	no	—
71	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
72	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
73	C₈×C₅₃	#121	1	1	yes	characteristic
74	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₈×C₅₃	#125	1	1	yes	characteristic
76	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	#88322	—	4	2	no	—
78	#13161	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	2	no	—
80	C₂²×C₅₃	#424	1	1	yes	characteristic
81	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
82	C₂²×C₅₃	#446	1	1	yes	characteristic
83	C₄×C₅₃	—	2	4	no	—
84	C₄×C₅₃	#389	1	1	yes	characteristic
85	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₅₃	#436	1	1	yes	characteristic
87	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
88	Dic₅₃	—	4	4	no	—
89	C₂×#364	D₅₃	1	1	yes	characteristic
90	#2151	—	53	2	no	—
91	C₂×C₄×C₁₆	—	53	1	no	—
92	#1772	—	53	1	no	—
93	#1379	—	53	2	no	—
94	C₂×C₅₃	—	2	1	no	—
95	C₂×C₅₃	—	2	1	no	—
96	C₂×C₅₃	#1939	1	1	yes	characteristic
97	C₂×C₅₃	#2170	1	2	yes	—
98	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
99	C₂²×C₁₆	—	53	1	no	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂×#108	—	53	1	no	—
102	#427	—	53	1	no	—
103	#405	—	53	2	no	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	#364	—	2	2	no	—
106	C₄⋊C₁₆	—	106	4	no	—
107	C₄×C₁₆	—	106	2	no	—
108	#339	—	53	2	no	—
109	C₅₃	#13161	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	53	2	no	—
111	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	#117	—	53	2	no	—
114	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
115	M₃₂	—	2	2	no	—
116	M₃₂	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	53	2	no	—
119	C₂×C₁₆	—	53	2	no	—
120	C₂×C₁₆	—	106	1	no	—
121	C₂×C₁₆	—	106	1	no	—
122	C₂×C₁₆	—	106	1	no	—
123	C₂×C₁₆	—	106	1	no	—
124	#108	—	106	2	no	—
125	#107	—	53	2	no	—
126	#106	—	53	2	no	—
127	#97	—	106	2	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	Dic₁₀₆	1	2	yes	—
130	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
133	M₁₆	—	106	2	no	—
134	M₁₆	—	106	2	no	—
135	C₂×C₈	—	106	2	no	—
136	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
137	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
138	C₂×C₈	#11719912	1	2	yes	—
139	C₂×C₈	#11719919	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
142	C₂×C₈	#11719947	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
144	C₂×C₈	#11719949	1	1	yes	characteristic
145	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
146	C₄⋊C₄	—	53	4	no	—
147	C₂²⋊C₄	—	106	2	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	106	2	no	—
150	C₄²	#11719919	1	2	yes	—
151	C₁₆	—	4	2	no	—
152	C₁₆	—	106	2	no	—
153	C₁₆	—	106	2	no	—
154	C₂³	#13133225	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	106	4	no	—
158	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
159	C₂×C₄	#13133179	1	1	yes	characteristic
160	C₂×C₄	#13133182	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
162	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
163	C₂×C₄	#13133225	1	1	yes	characteristic
164	C₈	—	212	2	no	—
165	C₈	#13133146	1	2	yes	—
166	C₈	#13133146	1	2	yes	—
167	C₈	#13133188	1	1	yes	characteristic
168	C₈	#13133193	1	1	yes	characteristic
169	C₈	#13133208	1	1	yes	characteristic
170	C₈	#13133216	1	1	yes	characteristic
171	C₂²	—	2	2	no	—
172	C₂²	#16810651	1	1	yes	characteristic
173	C₂²	#16810688	1	1	yes	characteristic
174	C₂²	#16810750	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	212	4	no	—
177	C₄	#16810519	1	1	yes	characteristic
178	C₄	C₂×#13133146	1	1	yes	characteristic
179	C₄	#16810700	1	1	yes	characteristic
180	C₄	#16810747	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#20275431	1	1	yes	characteristic
184	C₂	#20276551	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	C₈×D₄
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#13161
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
109. C₅₃	#13161	semidirect

Maximal subgroups

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

subgroup	index
#20276475	2
C₂×C₄×#11719908	2
#20274594	2
#20272805	2
C₂×C₅₃×#364	2
#13161	53