Subgroups — #27273560

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	2	yes	—
6	#16810680	—	2	2	no	—
7	#16810680	C₄	1	4	yes	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₂²⋊#88322	—	2	1	no	—
10	C₂×#13133128	—	2	1	no	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	—	2	2	no	—
13	C₂×C₄×#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#432	—	2	2	no	—
17	C₅₃×#432	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
20	C₂×#97×C₅₃	—	2	1	no	—
21	C₂²×#88322	—	2	8	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	C₂×C₄	1	4	yes	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	C₂×C₄	1	4	yes	—
27	C₂²⋊#88322	—	2	4	no	—
28	#13133128	—	2	4	no	—
29	C₄×#88322	—	2	4	no	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₅₃	—	2	4	no	—
47	#97×C₅₃	—	2	4	no	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	4	no	—
66	C₄²×C₅₃	C₄²	1	2	yes	—
67	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×#88322	—	2	16	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	4	4	no	—
81	C₂×#88322	M₁₆	1	8	yes	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	#95	1	2	yes	—
85	C₂³×C₅₃	#108	1	4	yes	—
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	#95	1	2	yes	—
98	C₂×C₄×C₅₃	#95	1	2	yes	—
99	C₂×C₄×C₅₃	#95	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₅₃	#135	1	4	yes	—
108	C₈×C₅₃	—	4	4	no	—
109	#88322	—	2	8	no	—
110	#88322	—	4	4	no	—
111	#7875	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	8	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	#341	1	4	yes	—
116	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
117	C₂²×C₅₃	#423	1	2	yes	—
118	C₂²×C₅₃	#427	1	2	yes	—
119	C₂²×C₅₃	C₂×#108	1	2	yes	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
126	C₄×C₅₃	#407	1	2	yes	—
127	C₄×C₅₃	#416	1	2	yes	—
128	C₄×C₅₃	#416	1	2	yes	—
129	C₂×#432	—	53	1	no	—
130	C₂×#432	D₅₃	1	1	yes	characteristic
131	C₂²×C₄×C₈	—	53	1	no	—
132	#1277	—	53	4	no	—
133	C₂×C₅₃	—	2	4	no	—
134	C₂×C₅₃	#1568	1	4	yes	—
135	C₂×C₅₃	#1728	1	1	yes	characteristic
136	C₂×C₅₃	#1735	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
138	C₂³×C₈	—	53	2	no	—
139	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	#432	—	2	2	no	—
141	#432	—	53	4	no	—
142	#432	—	106	2	no	—
143	#432	Dic₅₃	1	4	yes	—
144	C₂×C₄⋊C₈	—	53	2	no	—
145	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₂²⋊C₈	—	106	1	no	—
148	C₂×#97	—	2	1	no	—
149	C₂×#97	—	106	1	no	—
150	C₂×C₄×C₈	—	53	2	no	—
151	C₂×C₄×C₈	—	53	2	no	—
152	C₂×C₄×C₈	—	106	2	no	—
153	C₂×C₄×C₈	—	106	2	no	—
154	C₅₃	#7875	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
156	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	53	4	no	—
160	C₂²×C₈	—	106	8	no	—
161	C₂²×C₈	—	106	4	no	—
162	C₂²×C₈	—	106	2	no	—
163	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₅₃	1	2	yes	—
165	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
166	C₂×C₄²	D₂₁₂	1	2	yes	—
167	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
169	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	53	4	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
175	C₂²⋊C₈	—	2	4	no	—
176	C₂²⋊C₈	—	106	4	no	—
177	#97	—	2	4	no	—
178	#97	—	106	4	no	—
179	C₄×C₈	—	106	2	no	—
180	C₄×C₈	—	106	2	no	—
181	C₄×C₈	—	106	4	no	—
182	C₂⁴	#11719919	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
187	C₂²×C₄	#11719919	1	1	yes	characteristic
188	C₂²×C₄	#11719919	1	1	yes	characteristic
189	C₂²×C₄	#11719919	1	1	yes	characteristic
190	C₂²×C₄	#11719919	1	2	yes	—
191	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
192	C₂²×C₄	#11719921	1	2	yes	—
193	C₂²×C₄	#11719921	1	2	yes	—
194	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
195	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	53	8	no	—
201	C₂×C₈	—	106	16	no	—
202	C₂×C₈	—	106	4	no	—
203	C₂×C₈	—	106	4	no	—
204	C₂×C₈	—	106	4	no	—
205	C₂×C₈	—	212	4	no	—
206	C₄²	C₄×Dic₅₃	1	2	yes	—
207	C₄²	#11719919	1	2	yes	—
208	C₄²	C₄⋊Dic₅₃	1	2	yes	—
209	C₄²	#11719921	1	2	yes	—
210	C₄²	#11719921	1	2	yes	—
211	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
212	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
214	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	#13133124	1	2	yes	—
219	C₂³	#13133152	1	4	yes	—
220	C₂³	C₂×#11719919	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	#13133124	1	2	yes	—
232	C₂×C₄	#13133124	1	2	yes	—
233	C₂×C₄	#13133124	1	2	yes	—
234	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
241	C₂×C₄	#13133237	1	4	yes	—
242	C₈	—	4	4	no	—
243	C₈	—	106	8	no	—
244	C₈	—	212	4	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	#16810387	1	4	yes	—
249	C₂²	C₂×#13133124	1	1	yes	characteristic
250	C₂²	#16810646	1	2	yes	—
251	C₂²	#16810659	1	2	yes	—
252	C₂²	C₂×#13133152	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	M₁₆×Dic₅₃	1	2	yes	—
259	C₄	#16810586	1	2	yes	—
260	C₄	#16810617	1	2	yes	—
261	C₄	#16810619	1	2	yes	—
262	C₂	—	2	4	no	—
263	C₂	#20273476	1	4	yes	—
264	C₂	#20274408	1	1	yes	characteristic
265	C₂	#20274431	1	1	yes	characteristic
266	C₂	C₂×#16810387	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#432	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#432	D₅₃
hypocenter	C₅₃	#7875
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20272549	C₂	semidirect
154. C₅₃	#7875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810680	2
C₂²×C₄×#88322	2
#20272549	2
C₂×C₅₃×#432	2
#7875	53