Subgroups — #28310303

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊#108	C₂	1	2	yes	maximal
2	#20289785	C₂	1	4	yes	maximal, complemented
3	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×D₅₃⋊M₁₆	—	2	2	no	—
5	C₂×#13133288	—	2	2	no	—
6	D₅₃⋊#108	—	2	2	no	—
7	D₅₃⋊#108	—	2	2	no	—
8	D₅₃⋊#108	C₄	1	4	yes	—
9	D₅₃⋊#108	C₂²	1	4	yes	—
10	C₂×#13133287	C₄	1	4	yes	—
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	—
14	M₃₂×D₅₃	—	2	2	no	—
15	M₃₂×D₅₃	—	2	2	no	—
16	C₂×C₁₆×D₅₃	C₄	1	2	yes	—
17	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#11719936	—	2	4	no	—
19	D₅₃⋊M₁₆	—	2	4	no	—
20	D₅₃⋊M₁₆	—	2	4	no	—
21	D₅₃⋊M₁₆	—	4	2	no	—
22	D₅₃⋊M₁₆	—	4	2	no	—
23	C₂×#11719940	—	4	2	no	—
24	C₂×D₅₃⋊C₈	—	4	2	no	—
25	#13133288	—	2	4	no	—
26	#13133288	—	2	4	no	—
27	#13133287	—	2	4	no	—
28	#13133287	C₂×C₄	1	8	yes	—
29	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂²×#88322	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₅₃	D₄	1	2	yes	—
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
39	#13133157	—	2	2	no	—
40	#13133156	—	2	4	no	—
41	C₂×#11719908	C₂×C₄	1	2	yes	—
42	C₁₆×D₅₃	—	2	2	no	—
43	C₁₆×D₅₃	—	2	2	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
54	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
55	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
56	C₂×C₄×D₅₃	Q₁₆	1	1	yes	characteristic
57	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
58	#11719940	—	4	4	no	—
59	#11719940	—	4	4	no	—
60	C₂×#88324	—	4	4	no	—
61	#11719936	—	2	8	no	—
62	#11719936	—	4	4	no	—
63	D₅₃⋊C₈	—	4	4	no	—
64	D₅₃⋊C₈	—	4	4	no	—
65	C₂×C₈×C₅₃	C₄²	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
70	C₂×#88322	C₄⋊C₄	1	2	yes	—
71	C₂×#88322	C₄⋊C₄	1	2	yes	—
72	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
73	C₈×D₅₃	C₄²	1	1	yes	characteristic
74	C₈×D₅₃	C₄²	1	1	yes	characteristic
75	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
76	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
77	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
80	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
81	C₈×D₅₃	C₂²×C₄	1	2	yes	—
82	C₈×D₅₃	C₂×D₄	1	2	yes	—
83	C₁₆×C₅₃	—	2	2	no	—
84	#11719908	—	2	2	no	—
85	C₂³×C₅₃	#107	1	1	yes	characteristic
86	C₂²×D₅₃	—	4	2	no	—
87	C₂²×D₅₃	#103	1	2	yes	—
88	C₂²×D₅₃	#103	1	2	yes	—
89	C₂²×D₅₃	#107	1	1	yes	characteristic
90	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#95	1	2	yes	—
92	C₂×C₄×C₅₃	#106	1	2	yes	—
93	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×Dic₅₃	#102	1	2	yes	—
96	C₂×Dic₅₃	#102	1	2	yes	—
97	C₂×Dic₅₃	#107	1	1	yes	characteristic
98	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
99	C₄×D₅₃	#95	1	1	yes	characteristic
100	C₄×D₅₃	#95	1	1	yes	characteristic
101	C₄×D₅₃	#102	1	2	yes	—
102	C₄×D₅₃	#102	1	2	yes	—
103	C₄×D₅₃	#103	1	2	yes	—
104	C₄×D₅₃	#103	1	2	yes	—
105	C₄×D₅₃	#106	1	1	yes	characteristic
106	C₄×D₅₃	#106	1	1	yes	characteristic
107	C₄×D₅₃	C₂×C₂²⋊C₄	1	2	yes	—
108	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
109	#88324	—	4	8	no	—
110	C₈×C₅₃	#95	1	2	yes	—
111	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
112	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	#88322	#95	1	1	yes	characteristic
114	#88322	#95	1	1	yes	characteristic
115	#88322	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×#1385	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	4	1	no	—
118	C₂²×C₅₃	#340	1	2	yes	—
119	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
120	C₂×D₅₃	—	2	4	no	—
121	C₂×D₅₃	—	2	2	no	—
122	C₂×D₅₃	—	4	2	no	—
123	C₂×D₅₃	—	4	1	no	—
124	C₂×D₅₃	#340	1	1	yes	characteristic
125	C₂×D₅₃	#340	1	1	yes	characteristic
126	C₂×D₅₃	C₂×#103	1	2	yes	—
127	C₄×C₅₃	#340	1	2	yes	—
128	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
129	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
130	Dic₅₃	#340	1	1	yes	characteristic
131	Dic₅₃	#340	1	1	yes	characteristic
132	Dic₅₃	C₂×#102	1	2	yes	—
133	C₂²×M₃₂	—	53	1	no	—
134	C₂²×#108	—	53	2	no	—
135	#1385	—	53	4	no	—
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
139	D₅₃	—	2	1	no	—
140	D₅₃	—	2	2	no	—
141	D₅₃	#1385	1	2	yes	complemented
142	C₂²×M₁₆	—	106	2	no	—
143	C₂³×C₈	—	53	1	no	—
144	C₂×M₃₂	—	53	1	no	—
145	C₂×M₃₂	—	53	2	no	—
146	C₂×M₃₂	—	106	2	no	—
147	C₂×M₃₂	—	106	2	no	—
148	C₂×M₃₂	#6157	1	1	yes	characteristic
149	C₂²×C₁₆	—	53	2	no	—
150	C₂×#108	—	53	4	no	—
151	C₂×#108	—	53	4	no	—
152	C₂×#108	—	53	4	no	—
153	C₂×#108	—	106	2	no	—
154	C₂×#108	—	106	2	no	—
155	C₂×#108	—	106	2	no	—
156	C₅₃	C₂×#1385	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	—	53	1	no	—
158	C₂×M₁₆	—	106	4	no	—
159	C₂×M₁₆	—	106	4	no	—
160	C₂×M₁₆	—	106	4	no	—
161	C₂×M₁₆	—	212	2	no	—
162	C₂×M₁₆	—	212	2	no	—
163	C₂×M₁₆	—	212	2	no	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	—	53	2	no	—
167	C₂²×C₈	—	53	1	no	—
168	C₂²×C₈	—	53	1	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	—	53	2	no	—
171	C₂²×C₈	—	53	1	no	—
172	C₂²×C₈	—	53	1	no	—
173	C₂²×C₈	—	212	2	no	—
174	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	—	106	4	no	—
177	M₃₂	—	106	2	no	—
178	C₂×C₁₆	—	53	2	no	—
179	C₂×C₁₆	—	106	2	no	—
180	C₂×C₁₆	—	106	2	no	—
181	C₂×C₁₆	C₂×#6157	1	2	yes	—
182	#108	—	53	8	no	—
183	#108	—	106	4	no	—
184	#108	—	106	4	no	—
185	#108	—	106	4	no	—
186	C₂⁴	—	53	1	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	—	53	2	no	—
189	C₂²×C₄	—	53	2	no	—
190	C₂²×C₄	—	53	2	no	—
191	C₂²×C₄	—	53	2	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
197	M₁₆	—	106	8	no	—
198	M₁₆	—	212	4	no	—
199	M₁₆	—	212	4	no	—
200	M₁₆	—	212	4	no	—
201	C₂×C₈	—	53	2	no	—
202	C₂×C₈	—	53	2	no	—
203	C₂×C₈	—	53	1	no	—
204	C₂×C₈	—	53	1	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	53	1	no	—
211	C₂×C₈	—	53	1	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	1	no	—
214	C₂×C₈	—	53	1	no	—
215	C₂×C₈	—	212	4	no	—
216	C₂×C₈	—	212	4	no	—
217	C₂×C₈	—	212	4	no	—
218	C₂×C₈	C₄×#6157	1	2	yes	—
219	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊#6157	1	2	yes	—
221	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
222	C₁₆	—	2	2	no	—
223	C₁₆	—	106	2	no	—
224	C₂³	—	53	1	no	—
225	C₂³	—	53	2	no	—
226	C₂³	—	53	1	no	—
227	C₂³	—	53	2	no	—
228	C₂³	—	212	2	no	—
229	C₂³	#13133286	1	1	yes	characteristic
230	C₂×C₄	—	53	1	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	53	2	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	1	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	1	no	—
238	C₂×C₄	—	53	1	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	53	2	no	—
242	C₂×C₄	—	53	2	no	—
243	C₂×C₄	—	53	2	no	—
244	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
245	C₂×C₄	#13133285	1	2	yes	—
246	C₂×C₄	#13133286	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
248	C₈	—	53	1	no	—
249	C₈	—	53	1	no	—
250	C₈	—	53	2	no	—
251	C₈	—	212	8	no	—
252	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
253	C₈	C₂×C₄×#6157	1	1	yes	characteristic
254	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
255	C₂²	—	4	1	no	—
256	C₂²	—	53	1	no	—
257	C₂²	—	53	1	no	—
258	C₂²	—	53	2	no	—
259	C₂²	—	106	2	no	—
260	C₂²	—	106	4	no	—
261	C₂²	—	212	1	no	—
262	C₂²	—	212	2	no	—
263	C₂²	#16811497	1	2	yes	—
264	C₂²	C₂×#13133286	1	1	yes	characteristic
265	C₄	—	53	2	no	—
266	C₄	—	53	1	no	—
267	C₄	—	53	1	no	—
268	C₄	#16811497	1	2	yes	—
269	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
270	C₄	C₂×#13133285	1	1	yes	characteristic
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	53	2	no	—
274	C₂	—	106	1	no	—
275	C₂	—	106	2	no	—
276	C₂	C₂×#16811497	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133285
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂×C₅₃	C₂×#340
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	C₂×#1385
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20289785	C₂	semidirect
141. D₅₃	#1385	semidirect
156. C₅₃	C₂×#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊#108	2
#20289785	2
C₂×M₃₂×D₅₃	2
C₂×#1385	53