Subgroups — #27299459

Order 13568.

326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276314	C₂	1	2	yes	maximal
2	C₂×#16810399	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	#16810399	C₄	1	4	yes	complemented
12	#16810398	—	2	2	no	—
13	C₁₆×Dic₅₃	—	2	2	no	—
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	4	yes	complemented
18	C₂×#11719917	C₂×C₄	1	2	yes	—
19	#13133176	C₂×C₄	1	2	yes	—
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	C₂×C₄	1	2	yes	—
23	C₂×#11719908	C₂×C₄	1	2	yes	—
24	C₂×#11719908	D₄	1	2	yes	—
25	C₂×#11719908	Q₈	1	2	yes	—
26	#13133153	C₈	1	4	yes	—
27	#13133151	C₈	1	2	yes	—
28	#13133149	C₈	1	2	yes	—
29	#13133127	—	2	2	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₅₃	—	2	2	no	—
35	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄²	1	2	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	C₄²	1	2	yes	—
53	C₂²⋊Dic₅₃	—	2	2	no	—
54	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
55	C₄×Dic₅₃	—	2	2	no	—
56	#11719917	—	2	2	no	—
57	#11719917	C₂×C₈	1	4	yes	—
58	C₂×#88322	—	2	2	no	—
59	C₁₆×C₅₃	—	2	4	no	—
60	#11719908	—	2	2	no	—
61	#11719908	C₄⋊C₄	1	4	yes	—
62	C₂³×C₅₃	#131	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
71	C₂×Dic₅₃	—	2	4	no	—
72	C₈×C₅₃	C₄×C₈	1	2	yes	—
73	C₈×C₅₃	C₄×C₈	1	2	yes	—
74	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₅₃	#117	1	1	yes	characteristic
76	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
77	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
78	#88322	—	2	4	no	—
79	#12665	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	2	no	—
81	C₂²×C₅₃	#405	1	1	yes	characteristic
82	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
83	C₂²×C₅₃	C₈×Q₈	1	1	yes	characteristic
84	C₄×C₅₃	—	4	2	no	—
85	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
88	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
89	Dic₅₃	—	4	4	no	—
90	C₂×#364	D₅₃	1	1	yes	characteristic
91	#2108	—	53	2	no	—
92	C₂×#346	—	53	1	no	—
93	#1772	—	53	1	no	—
94	#1379	—	53	2	no	—
95	C₂×C₅₃	—	2	1	no	—
96	C₂×C₅₃	—	2	1	no	—
97	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₅₃	#2171	1	2	yes	—
99	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂²×C₁₆	—	53	1	no	—
102	C₂×#108	—	53	1	no	—
103	#427	—	53	1	no	—
104	#405	—	53	2	no	—
105	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	#364	Dic₅₃	1	4	yes	complemented
107	#346	—	53	4	no	—
108	#346	—	106	2	no	—
109	C₄×C₁₆	—	106	2	no	—
110	#339	—	53	2	no	—
111	C₅₃	#12665	1	1	yes	characteristic, complemented, Sylow
112	C₂×M₁₆	—	53	2	no	—
113	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	#117	—	53	2	no	—
116	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	C₂×Dic₅₃	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	53	2	no	—
123	C₂×C₁₆	—	53	2	no	—
124	C₂×C₁₆	—	106	1	no	—
125	C₂×C₁₆	—	106	1	no	—
126	#108	—	53	4	no	—
127	#107	—	53	2	no	—
128	#106	—	53	2	no	—
129	#97	—	106	2	no	—
130	C₄×C₈	—	106	2	no	—
131	C₄×C₈	C₄×D₅₃	1	2	yes	—
132	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
135	M₁₆	—	53	4	no	—
136	M₁₆	—	106	2	no	—
137	C₂×C₈	—	106	2	no	—
138	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
139	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
140	C₂×C₈	C₈×D₅₃	1	2	yes	—
141	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
144	C₂×C₈	#11719947	1	1	yes	characteristic
145	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
146	C₂×C₈	#11719949	1	1	yes	characteristic
147	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	53	4	no	—
149	C₂²⋊C₄	—	106	2	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	106	2	no	—
152	C₄²	C₄×Dic₅₃	1	2	yes	—
153	C₁₆	—	2	4	no	—
154	C₁₆	—	53	4	no	—
155	C₁₆	—	106	2	no	—
156	C₂³	#13133225	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	106	4	no	—
160	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
161	C₂×C₄	#13133179	1	1	yes	characteristic
162	C₂×C₄	#13133182	1	1	yes	characteristic
163	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
165	C₂×C₄	#13133225	1	1	yes	characteristic
166	C₈	—	106	4	no	—
167	C₈	C₈×Dic₅₃	1	2	yes	—
168	C₈	C₈×Dic₅₃	1	2	yes	—
169	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
170	C₈	#13133175	1	1	yes	characteristic
171	C₈	D₄×Dic₅₃	1	1	yes	characteristic
172	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#16810581	1	1	yes	characteristic
175	C₂²	#16810688	1	1	yes	characteristic
176	C₂²	#16810747	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	212	4	no	—
179	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
180	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
181	C₄	#16810688	1	1	yes	characteristic
182	C₄	#16810747	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	#20274580	1	1	yes	characteristic
186	C₂	#20276556	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×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	#12665
LCS	G ▷ 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
11. #16810399	C₄	semidirect
17. C₅₃×#364	C₄	semidirect
106. #364	Dic₅₃	semidirect
111. C₅₃	#12665	semidirect

Maximal subgroups

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

subgroup	index
#20276314	2
C₂×#16810399	2
#20274594	2
#20272805	2
C₂×C₅₃×#364	2
#12665	53