Subgroups — #27271036

Order 13568.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274594	C₂	1	1	yes	maximal, characteristic
2	#20272998	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
4	#16810676	—	2	1	no	—
5	C₂×#13133153	C₄	1	1	yes	characteristic
6	#16810661	C₄	1	1	yes	characteristic
7	#16810578	—	2	1	no	—
8	#16810383	—	2	1	no	—
9	#16810344	C₄	1	2	yes	complemented
10	C₈×#88322	—	2	1	no	—
11	C₂×M₃₂×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#364	—	2	1	no	—
14	C₅₃×#364	C₄	1	2	yes	complemented
15	C₂×#11719917	—	2	1	no	—
16	#13133176	—	2	1	no	—
17	#13133153	—	2	1	no	—
18	#13133153	C₈	1	2	yes	complemented
19	#13133151	—	2	1	no	—
20	#13133149	C₈	1	2	yes	complemented
21	C₄⋊#88322	—	2	2	no	—
22	C₂²⋊#88322	—	2	2	no	—
23	#13133127	—	4	1	no	—
24	C₄×#88322	—	2	2	no	—
25	C₈×Dic₅₃	—	4	1	no	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	M₃₂×C₅₃	—	2	2	no	—
30	M₃₂×C₅₃	—	4	1	no	—
31	C₂×C₁₆×C₅₃	—	4	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²⋊Dic₅₃	—	4	1	no	—
54	C₄⋊Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	4	1	no	—
56	#11719917	—	2	2	no	—
57	#11719917	—	4	1	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	4	1	no	—
60	C₁₆×C₅₃	—	4	2	no	—
61	C₂³×C₅₃	#104	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
71	C₂×Dic₅₃	—	4	2	no	—
72	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
73	C₈×C₅₃	#97	1	1	yes	characteristic
74	C₈×C₅₃	#102	1	1	yes	characteristic
75	C₈×C₅₃	#103	1	1	yes	characteristic
76	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
77	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₅₃	#106	1	1	yes	characteristic
79	C₈×C₅₃	#107	1	1	yes	characteristic
80	#88322	—	4	4	no	—
81	#88322	—	4	2	no	—
82	#7181	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	4	1	no	—
84	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
86	C₂²×C₅₃	#339	1	1	yes	characteristic
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	4	1	no	—
89	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
90	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
91	C₄×C₅₃	#336	1	1	yes	characteristic
92	C₄×C₅₃	#340	1	1	yes	characteristic
93	Dic₅₃	—	8	2	no	—
94	C₂×#364	D₅₃	1	1	yes	characteristic
95	#1772	—	53	1	no	—
96	#1450	—	53	1	no	—
97	C₂×C₅₃	—	2	1	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	#1279	1	1	yes	characteristic
101	C₂×M₃₂	—	2	1	no	—
102	#431	—	106	1	no	—
103	C₂×#108	—	53	1	no	—
104	#427	—	53	1	no	—
105	#405	—	106	1	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	#364	—	2	1	no	—
108	#364	Dic₅₃	1	2	yes	complemented
109	#339	—	106	1	no	—
110	#322	—	53	2	no	—
111	C₈²	—	106	1	no	—
112	C₅₃	#7181	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₁₆	—	106	1	no	—
114	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	#117	—	106	1	no	—
117	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	—	4	1	no	—
120	C₂×C₁₆	—	4	1	no	—
121	#108	—	53	2	no	—
122	#108	—	106	1	no	—
123	#107	—	106	1	no	—
124	#106	—	53	2	no	—
125	C₄⋊C₈	—	106	2	no	—
126	C₂²⋊C₈	—	106	2	no	—
127	#97	—	212	1	no	—
128	C₄×C₈	—	106	2	no	—
129	C₄×C₈	—	212	1	no	—
130	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
131	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	#11719921	1	1	yes	characteristic
136	M₁₆	—	106	2	no	—
137	M₁₆	—	212	1	no	—
138	C₂×C₈	—	106	4	no	—
139	C₂×C₈	—	212	1	no	—
140	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
141	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
142	C₂×C₈	#11719912	1	1	yes	characteristic
143	C₂×C₈	#11719912	1	1	yes	characteristic
144	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
145	C₂×C₈	#11719919	1	1	yes	characteristic
146	C₂×C₈	#11719919	1	1	yes	characteristic
147	C₂×C₈	#11719922	1	1	yes	characteristic
148	C₂×C₈	#11719923	1	1	yes	characteristic
149	C₂×C₈	#11719924	1	1	yes	characteristic
150	C₂×C₈	#11719925	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
152	C₄⋊C₄	—	106	2	no	—
153	C₂²⋊C₄	—	212	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	212	1	no	—
156	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
157	C₄²	#11719921	1	1	yes	characteristic
158	C₁₆	—	4	2	no	—
159	C₂³	#13133143	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	212	2	no	—
164	C₂×C₄	#13133124	1	1	yes	characteristic
165	C₂×C₄	#13133129	1	1	yes	characteristic
166	C₂×C₄	#13133129	1	1	yes	characteristic
167	C₂×C₄	#13133137	1	1	yes	characteristic
168	C₂×C₄	#13133140	1	1	yes	characteristic
169	C₂×C₄	#13133143	1	1	yes	characteristic
170	C₈	—	212	4	no	—
171	C₈	—	212	2	no	—
172	C₈	C₈×Dic₅₃	1	1	yes	characteristic
173	C₈	#13133127	1	1	yes	characteristic
174	C₈	#13133138	1	1	yes	characteristic
175	C₈	#13133141	1	1	yes	characteristic
176	C₈	#13133146	1	1	yes	characteristic
177	C₈	#13133146	1	1	yes	characteristic
178	C₈	#13133148	1	1	yes	characteristic
179	C₈	#13133150	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	#16810350	1	1	yes	characteristic
182	C₂²	#16810353	1	1	yes	characteristic
183	C₂²	#16810384	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	424	2	no	—
187	C₄	#16810350	1	1	yes	characteristic
188	C₄	#16810353	1	1	yes	characteristic
189	C₄	#16810377	1	1	yes	characteristic
190	C₄	#16810385	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#20272554	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133138
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	#1279
prime radical	C₂³×C₅₃	#104
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#7181
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #16810344	C₄	semidirect
14. C₅₃×#364	C₄	semidirect
18. #13133153	C₈	semidirect
20. #13133149	C₈	semidirect
108. #364	Dic₅₃	semidirect
112. C₅₃	#7181	semidirect

Maximal subgroups

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

subgroup	index
#20274594	2
#20272998	2
C₂×C₅₃×#364	2
#7181	53