Subgroups — #27299469

Order 13568.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276319	C₂	1	2	yes	maximal
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	#20274594	C₂	1	1	yes	maximal, characteristic
4	#20272802	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	1	yes	characteristic
7	C₂×#13133157	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	#16810400	—	2	4	no	—
12	#16810399	C₄	1	2	yes	complemented
13	C₄×#11719908	C₄	1	2	yes	complemented
14	#16810383	C₄	1	2	yes	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#364	C₄	1	2	yes	complemented
18	C₅₃×#364	C₄	1	2	yes	complemented
19	C₂×#11719917	C₂×C₄	1	2	yes	—
20	#13133176	C₂×C₄	1	2	yes	—
21	#13133157	—	2	1	no	—
22	#13133157	—	2	1	no	—
23	#13133157	D₄	1	2	yes	—
24	#13133157	Q₈	1	2	yes	—
25	C₂×#11719908	C₂×C₄	1	2	yes	—
26	C₂×#11719908	C₂×C₄	1	2	yes	—
27	#13133153	—	2	1	no	—
28	#13133153	—	2	1	no	—
29	#13133151	—	2	1	no	—
30	#13133149	—	2	1	no	—
31	#13133127	—	2	2	no	—
32	C₈×Dic₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₅₃	—	2	2	no	—
37	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
57	C₄²×C₅₃	—	2	1	no	—
58	C₄²×C₅₃	C₄²	1	1	yes	characteristic
59	C₄²×C₅₃	C₄²	1	1	yes	characteristic
60	C₂²⋊Dic₅₃	—	2	2	no	—
61	C₄⋊Dic₅₃	—	2	2	no	—
62	C₄×Dic₅₃	—	2	2	no	—
63	#11719917	—	2	2	no	—
64	#11719917	—	2	2	no	—
65	C₂×#88322	—	2	2	no	—
66	C₁₆×C₅₃	—	2	4	no	—
67	#11719908	—	2	2	no	—
68	#11719908	C₄⋊C₄	1	4	yes	—
69	C₂³×C₅₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
79	C₂×Dic₅₃	—	4	2	no	—
80	C₈×C₅₃	#97	1	1	yes	characteristic
81	C₈×C₅₃	#97	1	1	yes	characteristic
82	C₈×C₅₃	#97	1	1	yes	characteristic
83	C₈×C₅₃	#97	1	1	yes	characteristic
84	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₅₃	#117	1	1	yes	characteristic
86	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
88	#88322	—	4	2	no	—
89	#12667	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	4	1	no	—
91	C₂²×C₅₃	#405	1	1	yes	characteristic
92	C₂²×C₅₃	#436	1	1	yes	characteristic
93	C₂²×C₅₃	#446	1	1	yes	characteristic
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	4	1	no	—
96	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
98	C₄×C₅₃	#436	1	1	yes	characteristic
99	C₄×C₅₃	#446	1	1	yes	characteristic
100	Dic₅₃	—	8	2	no	—
101	C₂×#364	D₅₃	1	1	yes	characteristic
102	#2110	—	53	2	no	—
103	C₄×M₃₂	—	53	1	no	—
104	#1772	—	53	1	no	—
105	#1378	—	53	2	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	#1770	1	1	yes	characteristic
110	C₂×M₃₂	—	53	1	no	—
111	C₂×M₃₂	—	53	1	no	—
112	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
113	C₂×#108	—	53	1	no	—
114	#427	—	53	1	no	—
115	#405	—	53	2	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	#364	Dic₅₃	1	2	yes	complemented
118	#364	Dic₅₃	1	2	yes	complemented
119	#347	—	106	4	no	—
120	#346	—	53	2	no	—
121	C₄×C₁₆	—	53	2	no	—
122	#339	—	53	2	no	—
123	C₅₃	#12667	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	—	53	2	no	—
125	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	#117	—	53	2	no	—
128	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	53	2	no	—
131	M₃₂	—	53	2	no	—
132	M₃₂	—	106	1	no	—
133	M₃₂	—	106	1	no	—
134	M₃₂	C₂×Dic₅₃	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	53	2	no	—
137	C₂×C₁₆	—	53	2	no	—
138	#108	—	106	1	no	—
139	#108	—	106	1	no	—
140	#107	—	106	1	no	—
141	#106	—	106	1	no	—
142	#97	—	106	2	no	—
143	C₄×C₈	—	106	2	no	—
144	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
145	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
150	M₁₆	—	106	2	no	—
151	M₁₆	—	106	2	no	—
152	C₂×C₈	—	106	2	no	—
153	C₂×C₈	#11719912	1	1	yes	characteristic
154	C₂×C₈	#11719912	1	1	yes	characteristic
155	C₂×C₈	#11719912	1	1	yes	characteristic
156	C₂×C₈	#11719912	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
158	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
160	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
161	C₂×C₈	#11719947	1	1	yes	characteristic
162	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
163	C₂×C₈	#11719949	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
165	C₄⋊C₄	—	106	2	no	—
166	C₂²⋊C₄	—	106	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	106	2	no	—
169	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
170	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
171	C₁₆	—	2	4	no	—
172	C₁₆	—	53	4	no	—
173	C₁₆	—	106	2	no	—
174	C₂³	#13133225	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	212	2	no	—
179	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
180	C₂×C₄	#13133179	1	1	yes	characteristic
181	C₂×C₄	#13133182	1	1	yes	characteristic
182	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
184	C₂×C₄	#13133225	1	1	yes	characteristic
185	C₈	—	212	2	no	—
186	C₈	#13133127	1	1	yes	characteristic
187	C₈	#13133127	1	1	yes	characteristic
188	C₈	#13133127	1	1	yes	characteristic
189	C₈	#13133127	1	1	yes	characteristic
190	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
191	C₈	#13133175	1	1	yes	characteristic
192	C₈	D₄×Dic₅₃	1	1	yes	characteristic
193	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#16810581	1	1	yes	characteristic
196	C₂²	#16810700	1	1	yes	characteristic
197	C₂²	#16810750	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	424	2	no	—
201	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
202	C₄	C₂×#13133127	1	1	yes	characteristic
203	C₄	#16810700	1	1	yes	characteristic
204	C₄	#16810750	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#20274585	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133175
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂×C₅₃	#1770
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12667
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
12. #16810399	C₄	semidirect
13. C₄×#11719908	C₄	semidirect
17. C₅₃×#364	C₄	semidirect
18. C₅₃×#364	C₄	semidirect
117. #364	Dic₅₃	semidirect
118. #364	Dic₅₃	semidirect
123. C₅₃	#12667	semidirect

Maximal subgroups

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

subgroup	index
#20276319	2
C₄×#13133157	2
#20274594	2
#20272802	2
C₂×C₅₃×#364	2
#12667	53