Subgroups — #27275287

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274570	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272549	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×#13133127	—	2	2	no	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
11	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810377	—	2	2	no	—
13	#16810376	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#335	C₄	1	4	yes	complemented
17	C₂²×#88322	—	2	2	no	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	C₂×C₄	1	4	yes	—
25	#13133127	—	4	4	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₈×Dic₅₃	—	2	4	no	—
29	C₈×Dic₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₂×C₈×C₅₃	—	4	2	no	—
47	C₂×C₈×C₅₃	C₄²	1	2	yes	—
48	C₂×C₈×C₅₃	C₄²	1	2	yes	—
49	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	#41	1	1	yes	characteristic
53	C₄²×C₅₃	#41	1	1	yes	characteristic
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	2	1	no	—
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	—	2	4	no	—
59	C₄×Dic₅₃	D₈	1	2	yes	—
60	C₄×Dic₅₃	Q₁₆	1	2	yes	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	M₁₆	1	4	yes	—
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	2	2	no	—
84	C₂×Dic₅₃	—	2	2	no	—
85	C₂×Dic₅₃	#107	1	4	yes	—
86	C₂×Dic₅₃	#108	1	4	yes	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	4	4	no	—
89	C₈×C₅₃	#97	1	4	yes	complemented
90	#88322	—	2	4	no	—
91	#88322	—	4	4	no	—
92	#8274	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₅₃	#423	1	2	yes	—
95	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
96	C₂²×C₅₃	#427	1	1	yes	characteristic
97	C₂²×C₅₃	C₂×#108	1	2	yes	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
101	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
102	C₄×C₅₃	#432	1	2	yes	—
103	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
104	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
105	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
106	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
107	Dic₅₃	—	2	4	no	—
108	Dic₅₃	—	2	4	no	—
109	#1767	—	53	2	no	—
110	C₄²×C₈	—	53	1	no	—
111	C₂×#335	—	53	1	no	—
112	C₂×#335	D₅₃	1	1	yes	characteristic
113	#1277	—	53	2	no	—
114	C₂×C₅₃	#1567	1	2	yes	—
115	C₂×C₅₃	#1568	1	2	yes	—
116	C₂×C₅₃	#1767	1	1	yes	characteristic
117	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
118	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
119	C₂×C₄⋊C₈	—	53	2	no	—
120	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
121	C₂×#97	—	106	2	no	—
122	C₂×C₄×C₈	—	53	2	no	—
123	C₂×C₄×C₈	—	53	2	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
126	C₄³	—	53	1	no	—
127	#336	—	106	2	no	—
128	#335	—	53	4	no	—
129	#335	Dic₅₃	1	4	yes	complemented
130	C₅₃	#8274	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	53	2	no	—
133	C₂²×C₈	—	106	2	no	—
134	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄²	—	53	1	no	—
137	C₂×C₄²	—	53	2	no	—
138	C₂×C₄²	—	53	1	no	—
139	C₂×C₄²	—	53	2	no	—
140	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	53	4	no	—
143	C₄⋊C₈	—	106	2	no	—
144	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
145	#97	—	212	4	no	—
146	C₄×C₈	—	106	4	no	—
147	C₄×C₈	—	106	4	no	—
148	C₄×C₈	—	106	1	no	—
149	C₄×C₈	—	106	1	no	—
150	C₄×C₈	C₄×D₅₃	1	2	yes	—
151	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
152	C₂²×C₄	—	53	2	no	—
153	C₂²×C₄	—	53	2	no	—
154	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	53	4	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	2	no	—
164	C₂×C₈	—	106	4	no	—
165	C₂×C₈	—	212	2	no	—
166	C₂×C₈	—	212	2	no	—
167	C₂×C₈	#11719912	1	4	yes	—
168	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
169	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
170	C₄²	—	53	2	no	—
171	C₄²	—	53	2	no	—
172	C₄²	—	106	4	no	—
173	C₄²	—	106	2	no	—
174	C₄²	—	106	2	no	—
175	C₄²	—	106	1	no	—
176	C₄²	—	106	1	no	—
177	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₄²	#11719947	1	1	yes	characteristic
179	C₄²	#11719949	1	1	yes	characteristic
180	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
181	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	53	4	no	—
184	C₂×C₄	—	53	4	no	—
185	C₂×C₄	—	106	2	no	—
186	C₂×C₄	—	106	2	no	—
187	C₂×C₄	—	106	2	no	—
188	C₂×C₄	—	106	2	no	—
189	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
190	C₂×C₄	#13133175	1	1	yes	characteristic
191	C₂×C₄	#13133179	1	1	yes	characteristic
192	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
193	C₂×C₄	#13133182	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
195	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
196	C₂×C₄	C₂×#11719912	1	2	yes	—
197	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
198	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
199	C₂×C₄	#13133236	1	1	yes	characteristic
200	C₂×C₄	#13133240	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	4	4	no	—
203	C₈	—	106	4	no	—
204	C₈	—	212	4	no	—
205	C₈	#13133127	1	4	yes	complemented
206	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
207	C₂²	#16810645	1	2	yes	—
208	C₂²	#107×D₅₃	1	1	yes	characteristic
209	C₂²	#16810660	1	1	yes	characteristic
210	C₂²	#108×D₅₃	1	2	yes	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	106	4	no	—
214	C₄	—	106	4	no	—
215	C₄	C₂×#13133127	1	1	yes	characteristic
216	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
217	C₄	#16810679	1	2	yes	—
218	C₄	#16810705	1	1	yes	characteristic
219	C₄	D₈×Dic₅₃	1	1	yes	characteristic
220	C₄	#16810717	1	1	yes	characteristic
221	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
222	C₂	#20273472	1	2	yes	—
223	C₂	#20273477	1	2	yes	—
224	C₂	#20274570	1	1	yes	characteristic
225	C₂	#107×Dic₅₃	1	1	yes	characteristic
226	C₂	#108×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#8274
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
13. #16810376	C₄	semidirect
16. C₅₃×#335	C₄	semidirect
89. C₈×C₅₃	#97	semidirect
129. #335	Dic₅₃	semidirect
130. C₅₃	#8274	semidirect
205. C₈	#13133127	semidirect

Maximal subgroups

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

subgroup	index
#20274570	2
C₄×C₈×Dic₅₃	2
C₂×#16810376	2
#20272549	2
C₂×C₅₃×#335	2
#8274	53