Subgroups — #28303529

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290648	C₂	1	2	yes	maximal
2	#20289545	C₂	1	2	yes	maximal
3	#20289545	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133284	—	2	2	no	—
6	C₂×#13133284	—	2	2	no	—
7	C₂×C₄⋊#88324	C₂²	1	4	yes	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	2	4	no	—
16	#13133284	—	4	4	no	—
17	#13133284	—	4	4	no	—
18	C₄⋊#88324	—	2	4	no	—
19	C₄⋊#88324	—	2	4	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₄×#88322	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	—	2	2	no	—
31	C₈×Dic₅₃	—	2	2	no	—
32	C₈×Dic₅₃	C₈	1	4	yes	complemented
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×#88324	—	4	4	no	—
45	C₂×#88324	—	4	4	no	—
46	C₂×#88324	—	4	4	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	1	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	—	2	1	no	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	2	2	no	—
57	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
58	C₄×Dic₅₃	M₁₆	1	2	yes	—
59	C₄×Dic₅₃	D₈	1	2	yes	—
60	C₄×Dic₅₃	SD₁₆	1	4	yes	—
61	C₄×Dic₅₃	Q₁₆	1	2	yes	—
62	C₄×Dic₅₃	#41	1	2	yes	—
63	C₄×Dic₅₃	#41	1	2	yes	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	C₂×C₈	1	2	yes	—
69	C₂×#88322	M₁₆	1	2	yes	—
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	#106	1	2	yes	—
76	C₂×C₄×C₅₃	#107	1	2	yes	—
77	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
86	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
87	C₂×Dic₅₃	#106	1	2	yes	—
88	C₂×Dic₅₃	#107	1	2	yes	—
89	C₂×Dic₅₃	#108	1	2	yes	—
90	C₂×Dic₅₃	#108	1	2	yes	—
91	C₂×Dic₅₃	#124	1	2	yes	—
92	C₂×Dic₅₃	#125	1	2	yes	—
93	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
94	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
95	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
96	#88324	—	8	4	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	2	2	no	—
99	#88322	—	2	2	no	—
100	#88322	—	2	2	no	—
101	#8784	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	#382	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₅₃	#423	1	1	yes	characteristic
105	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
106	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
107	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	#431	1	1	yes	characteristic
112	C₄×C₅₃	#431	1	1	yes	characteristic
113	C₄×C₅₃	#432	1	1	yes	characteristic
114	C₄×C₅₃	#432	1	1	yes	characteristic
115	Dic₅₃	—	2	2	no	—
116	Dic₅₃	#334	1	4	yes	—
117	Dic₅₃	#335	1	4	yes	—
118	Dic₅₃	#486	1	2	yes	—
119	Dic₅₃	#487	1	2	yes	—
120	#1842	—	53	2	no	—
121	C₄²×C₈	—	53	1	no	—
122	#1277	—	53	2	no	—
123	#1277	—	53	2	no	—
124	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
126	C₂×C₅₃	#1566	1	1	yes	characteristic
127	C₂×C₅₃	#1567	1	1	yes	characteristic
128	C₂×C₅₃	#1831	1	1	yes	characteristic
129	C₂×C₅₃	#1834	1	1	yes	characteristic
130	C₂×C₅₃	#1837	1	1	yes	characteristic
131	C₂×C₄⋊C₈	—	53	4	no	—
132	C₂×C₄⋊C₈	—	106	2	no	—
133	C₂×C₄⋊C₈	—	106	2	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	1	no	—
136	C₂×C₄×C₈	—	53	1	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	#6157	1	1	yes	characteristic
140	C₄³	—	53	1	no	—
141	C₅₃	#8784	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	1	no	—
144	C₂²×C₈	—	106	4	no	—
145	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
146	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	—	53	1	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
154	C₄⋊C₈	—	106	4	no	—
155	C₄⋊C₈	—	106	4	no	—
156	C₄⋊C₈	—	212	4	no	—
157	C₄⋊C₈	—	212	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	53	4	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	106	2	no	—
162	C₄×C₈	—	106	2	no	—
163	C₄×C₈	—	106	2	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
170	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	1	no	—
180	C₂×C₈	—	106	1	no	—
181	C₂×C₈	—	212	4	no	—
182	C₂×C₈	—	212	4	no	—
183	C₂×C₈	—	212	4	no	—
184	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
185	C₂×C₈	#11719940	1	2	yes	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	53	4	no	—
189	C₄²	—	53	2	no	—
190	C₄²	—	53	2	no	—
191	C₄²	—	53	2	no	—
192	C₄²	—	53	2	no	—
193	C₄²	—	53	2	no	—
194	C₄²	—	53	2	no	—
195	C₄²	—	106	2	no	—
196	C₄²	—	106	2	no	—
197	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	53	2	no	—
203	C₂×C₄	—	53	2	no	—
204	C₂×C₄	—	53	2	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	53	2	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	1	no	—
209	C₂×C₄	—	53	1	no	—
210	C₂×C₄	—	53	2	no	—
211	C₂×C₄	—	53	2	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	106	1	no	—
214	C₂×C₄	—	106	1	no	—
215	C₂×C₄	#13133285	1	2	yes	—
216	C₂×C₄	#13133286	1	2	yes	—
217	C₂×C₄	#13133296	1	1	yes	characteristic
218	C₂×C₄	#13133296	1	1	yes	characteristic
219	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
221	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
222	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	106	2	no	—
226	C₈	—	106	2	no	—
227	C₈	—	424	4	no	—
228	C₂²	#16811567	1	1	yes	characteristic
229	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
230	C₂²	#16811626	1	1	yes	characteristic
231	C₂²	C₂×#13133285	1	1	yes	characteristic
232	C₂²	C₂×#13133286	1	1	yes	characteristic
233	C₂²	D₅₃⋊#108	1	1	yes	characteristic
234	C₂²	D₅₃⋊#108	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	53	2	no	—
238	C₄	—	53	2	no	—
239	C₄	—	53	4	no	—
240	C₄	—	53	4	no	—
241	C₄	—	106	2	no	—
242	C₄	#16811640	1	1	yes	characteristic
243	C₄	#16811641	1	1	yes	characteristic
244	C₄	#16811643	1	1	yes	characteristic
245	C₄	#16811646	1	1	yes	characteristic
246	C₂	D₅₃⋊#334	1	1	yes	characteristic
247	C₂	D₅₃⋊#335	1	1	yes	characteristic
248	C₂	#20290269	1	1	yes	characteristic
249	C₂	#20290272	1	1	yes	characteristic
250	C₂	#20290631	1	1	yes	characteristic
251	C₂	#20290636	1	1	yes	characteristic
252	C₂	#20290640	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8784
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
32. C₈×Dic₅₃	C₈	semidirect
141. C₅₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#20290648	2
#20289545	2
C₄×C₈×Dic₅₃	2
#8784	53