Subgroups — #28302180

Order 13568.

334 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290262	C₂	1	1	yes	maximal, characteristic
2	#20289922	C₂	1	2	yes	maximal
3	C₂×#16811464	C₂	1	1	yes	maximal, characteristic
4	#20289548	C₂	1	2	yes	maximal
5	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	#16811486	C₄	1	2	yes	—
10	#16811484	C₄	1	2	yes	—
11	#88324⋊C₈	—	2	4	no	—
12	#16811464	C₄	1	2	yes	—
13	#16811464	C₄	1	2	yes	—
14	C₂×#13133153	C₄	1	1	yes	characteristic
15	M₁₆×Dic₅₃	C₂²	1	2	yes	—
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810387	C₄	1	2	yes	—
18	C₂×#108×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	—	2	1	no	—
20	C₂²×#88324	Q₈	1	2	yes	—
21	C₄⋊#88324	C₂×C₄	1	4	yes	—
22	C₂²⋊#88324	—	2	2	no	—
23	C₄×#88324	—	2	2	no	—
24	C₄×#88324	—	2	1	no	—
25	C₄×#88324	C₂×C₄	1	2	yes	—
26	C₄×#88324	C₂×C₄	1	2	yes	—
27	C₄×#88324	D₄	1	2	yes	—
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	#13133153	—	2	2	no	—
32	#13133127	—	2	2	no	—
33	C₈×Dic₅₃	—	2	2	no	—
34	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	#108×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×Dic₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	2	no	—
45	C₂×#88324	—	2	4	no	—
46	C₂×#88324	—	4	2	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×#88324	C₄⋊C₄	1	4	yes	—
49	M₁₆×C₅₃	—	2	2	no	—
50	M₁₆×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
56	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₅₃	#41	1	1	yes	characteristic
60	#11719917	—	2	2	no	—
61	#11719917	—	2	2	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	C₄²	1	1	yes	characteristic
66	C₂×#88322	C₄²	1	1	yes	characteristic
67	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
74	C₂×Dic₅₃	—	2	2	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	#117	1	1	yes	characteristic
78	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
80	#88324	—	2	4	no	—
81	#88324	—	4	2	no	—
82	#88324	—	4	4	no	—
83	C₈×C₅₃	—	4	2	no	—
84	C₈×C₅₃	#97	1	2	yes	—
85	C₈×C₅₃	#97	1	2	yes	—
86	#88322	—	2	1	no	—
87	#88322	—	2	1	no	—
88	#88322	—	4	2	no	—
89	#8295	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₅₃	#446	1	1	yes	characteristic
94	C₂²×C₅₃	#446	1	1	yes	characteristic
95	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₅₃	#405	1	1	yes	characteristic
97	C₄×C₅₃	#436	1	1	yes	characteristic
98	C₄×C₅₃	#436	1	1	yes	characteristic
99	Dic₅₃	—	2	1	no	—
100	Dic₅₃	—	2	1	no	—
101	Dic₅₃	#430	1	2	yes	—
102	Dic₅₃	#444	1	2	yes	—
103	C₄×#108	—	53	1	no	—
104	#1564	—	53	1	no	—
105	#1451	—	53	2	no	—
106	C₂×#322	—	53	1	no	—
107	#1278	—	53	2	no	—
108	C₂×C₅₃	—	2	1	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	#1770	1	1	yes	characteristic
111	C₂×C₅₃	#1773	1	1	yes	characteristic
112	C₂×C₅₃	#1778	1	1	yes	characteristic
113	#431	—	53	2	no	—
114	C₂×#108	—	53	1	no	—
115	C₂×#108	#6157	1	1	yes	characteristic
116	C₄×M₁₆	—	53	2	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	#341	—	53	2	no	—
121	#335	—	53	2	no	—
122	#334	—	53	2	no	—
123	#322	—	53	2	no	—
124	#322	—	53	2	no	—
125	#322	—	106	4	no	—
126	C₅₃	#8295	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	53	2	no	—
128	C₂×M₁₆	C₂×#6157	1	2	yes	—
129	C₂²×C₈	—	53	1	no	—
130	C₂²×C₈	—	53	2	no	—
131	C₂²×C₈	—	106	1	no	—
132	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
133	C₂×C₄²	—	53	1	no	—
134	#108	—	2	2	no	—
135	#108	—	106	2	no	—
136	C₄⋊C₈	—	53	4	no	—
137	C₂²⋊C₈	—	106	2	no	—
138	#97	—	106	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	53	1	no	—
142	C₄×C₈	—	53	1	no	—
143	C₄×C₈	—	53	1	no	—
144	C₄×C₈	—	53	1	no	—
145	C₄×C₈	—	53	2	no	—
146	C₄×C₈	—	106	2	no	—
147	C₄×C₈	—	106	2	no	—
148	C₄×C₈	—	106	1	no	—
149	C₂²×C₄	—	53	1	no	—
150	C₂²×C₄	—	53	1	no	—
151	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	106	2	no	—
155	M₁₆	—	106	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	53	1	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	106	1	no	—
161	C₂×C₈	—	106	1	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	4	no	—
164	C₂×C₈	—	106	2	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₂×C₈	#11719936	1	2	yes	—
168	C₂×C₈	C₄×#6157	1	1	yes	characteristic
169	C₂×C₈	C₄×#6157	1	1	yes	characteristic
170	C₂×C₈	#11719940	1	2	yes	—
171	C₄²	—	53	1	no	—
172	C₄²	—	53	1	no	—
173	C₄²	—	53	1	no	—
174	C₄²	—	53	1	no	—
175	C₂³	Q₈×#6157	1	1	yes	characteristic
176	C₂×C₄	—	53	1	no	—
177	C₂×C₄	—	53	1	no	—
178	C₂×C₄	—	53	1	no	—
179	C₂×C₄	—	53	1	no	—
180	C₂×C₄	—	106	2	no	—
181	C₂×C₄	—	106	2	no	—
182	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
183	C₂×C₄	D₄×#6157	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
186	C₂×C₄	#13133304	1	1	yes	characteristic
187	C₂×C₄	#13133305	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	—	106	4	no	—
190	C₈	—	106	1	no	—
191	C₈	—	106	1	no	—
192	C₈	—	212	4	no	—
193	C₈	—	212	2	no	—
194	C₈	—	212	2	no	—
195	C₈	#13133272	1	2	yes	—
196	C₈	#13133272	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
200	C₂²	#16811673	1	1	yes	characteristic
201	C₂²	#16811674	1	1	yes	characteristic
202	C₄	—	53	2	no	—
203	C₄	—	53	2	no	—
204	C₄	—	106	1	no	—
205	C₄	—	106	1	no	—
206	C₄	C₂×#13133272	1	1	yes	characteristic
207	C₄	#16811588	1	1	yes	characteristic
208	C₄	#16811657	1	1	yes	characteristic
209	C₄	#16811658	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#20290511	1	1	yes	characteristic
213	C₂	#20290524	1	1	yes	characteristic
214	C₂	#20290536	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#108×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂³×C₅₃	C₄×Q₈
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#108	#6157
hypocenter	C₅₃	#8295
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
126. C₅₃	#8295	semidirect

Maximal subgroups

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

subgroup	index
#20290262	2
#20289922	2
C₂×#16811464	2
#20289548	2
#108×Dic₅₃	2
#8295	53