Subgroups — #27299138

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810581	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810405	C₂	1	1	yes	maximal, characteristic
3	M₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
4	#20272798	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₄	1	1	yes	characteristic
7	C₂×#13133176	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810581	—	2	2	no	—
10	#16810581	—	2	2	no	—
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810405	C₄	1	4	yes	complemented
16	#16810398	—	2	2	no	—
17	C₁₆×Dic₅₃	—	2	2	no	—
18	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#349	C₄	1	4	yes	complemented
21	C₂²×#88322	—	2	1	no	—
22	C₂²×#88322	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	C₈	1	4	yes	—
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
30	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	#13133157	—	2	2	no	—
34	#13133157	C₂×C₄	1	4	yes	—
35	C₂×#11719908	—	2	2	no	—
36	#13133127	—	2	2	no	—
37	#13133127	—	2	2	no	—
38	C₈×Dic₅₃	—	2	1	no	—
39	C₈×Dic₅₃	—	2	1	no	—
40	C₈×Dic₅₃	D₄	1	2	yes	—
41	C₈×Dic₅₃	D₄	1	2	yes	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆×C₅₃	—	2	2	no	—
44	C₂×M₁₆×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	2	no	—
50	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
51	C₂×C₁₆×C₅₃	—	2	2	no	—
52	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	1	no	—
55	C₂²×C₄×C₅₃	—	2	1	no	—
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
62	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
63	M₁₆×C₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	C₄²	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₂²⋊Dic₅₃	—	2	4	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	C₄×Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	—	2	1	no	—
76	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄×Dic₅₃	M₁₆	1	2	yes	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×#88322	C₂×C₈	1	2	yes	—
85	C₂×#88322	C₂×C₈	1	2	yes	—
86	C₂×#88322	M₁₆	1	2	yes	—
87	C₁₆×C₅₃	—	2	4	no	—
88	#11719908	—	2	4	no	—
89	C₂³×C₅₃	—	2	1	no	—
90	C₂³×C₅₃	—	2	1	no	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	C₄×C₈	1	2	yes	—
93	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	C₂×Dic₅₃	—	4	2	no	—
110	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	C₄×D₄	1	2	yes	—
113	C₈×C₅₃	C₄×D₄	1	2	yes	—
114	#88322	—	2	2	no	—
115	#88322	C₂²⋊C₈	1	2	yes	—
116	#88322	C₂²⋊C₈	1	2	yes	—
117	#12602	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	4	2	no	—
121	C₂²×C₅₃	—	4	2	no	—
122	C₂²×C₅₃	—	4	2	no	—
123	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂²×C₅₃	#431	1	1	yes	characteristic
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
129	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₅₃	#431	1	1	yes	characteristic
131	Dic₅₃	—	2	4	no	—
132	Dic₅₃	—	4	2	no	—
133	C₂×#405	—	53	1	no	—
134	C₂×#349	—	53	1	no	—
135	C₂×#349	D₅₃	1	1	yes	characteristic
136	C₄×M₃₂	—	53	2	no	—
137	#1377	—	53	2	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	4	2	no	—
141	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
142	C₂×C₅₃	#2115	1	2	yes	—
143	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
144	C₂³×C₈	—	53	1	no	—
145	C₂×#117	—	53	1	no	—
146	C₂×M₃₂	—	53	2	no	—
147	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
148	#405	—	106	2	no	—
149	#405	—	106	2	no	—
150	C₂×#97	—	53	1	no	—
151	C₂×#97	—	53	1	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	#349	—	53	4	no	—
155	#349	Dic₅₃	1	4	yes	complemented
156	#346	—	106	2	no	—
157	C₄×C₁₆	—	106	2	no	—
158	C₅₃	#12602	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
163	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
164	C₂²×C₈	—	53	1	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	4	no	—
167	C₂²×C₈	—	106	1	no	—
168	C₂²×C₈	—	106	1	no	—
169	C₂²×C₈	—	106	2	no	—
170	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	#117	—	106	2	no	—
173	#117	—	106	2	no	—
174	C₂×C₄⋊C₄	—	53	2	no	—
175	C₂×C₂²⋊C₄	—	53	2	no	—
176	C₂×C₄²	—	53	1	no	—
177	C₂×C₄²	—	53	1	no	—
178	M₃₂	—	2	2	no	—
179	M₃₂	—	53	4	no	—
180	M₃₂	—	106	2	no	—
181	M₃₂	C₂×Dic₅₃	1	4	yes	—
182	C₂×C₁₆	—	2	2	no	—
183	C₂×C₁₆	—	106	2	no	—
184	#97	—	106	2	no	—
185	#97	—	106	2	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	106	1	no	—
189	C₄×C₈	—	106	1	no	—
190	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	53	2	no	—
196	C₂²×C₄	—	53	2	no	—
197	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₈×D₅₃	1	2	yes	—
199	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	53	1	no	—
206	C₂×C₈	—	53	1	no	—
207	C₂×C₈	—	53	2	no	—
208	C₂×C₈	—	53	2	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	1	no	—
213	C₂×C₈	—	106	1	no	—
214	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₂×C₈	D₄×D₅₃	1	2	yes	—
217	C₂×C₈	#11719947	1	2	yes	—
218	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
219	C₄⋊C₄	—	106	4	no	—
220	C₂²⋊C₄	—	106	4	no	—
221	C₄²	—	53	2	no	—
222	C₄²	—	53	2	no	—
223	C₄²	—	106	1	no	—
224	C₄²	—	106	1	no	—
225	C₁₆	—	2	4	no	—
226	C₁₆	—	106	4	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₈×Dic₅₃	1	2	yes	—
231	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	53	4	no	—
239	C₂×C₄	—	106	2	no	—
240	C₂×C₄	—	212	2	no	—
241	C₂×C₄	—	212	2	no	—
242	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₄	#13133173	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
248	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	53	2	no	—
251	C₈	—	53	2	no	—
252	C₈	—	106	2	no	—
253	C₈	#13133177	1	2	yes	—
254	C₈	D₄×Dic₅₃	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
261	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
262	C₂²	#16810673	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	106	4	no	—
265	C₄	—	212	2	no	—
266	C₄	C₂²⋊C₄×Dic₅₃	1	1	yes	characteristic
267	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
268	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
269	C₄	#16810673	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	2	no	—
273	C₂	C₂²⋊C₈×Dic₅₃	1	1	yes	characteristic
274	C₂	#20276334	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	#12602
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₅₃	C₂	semidirect
4. #20272798	C₂	semidirect
15. #16810405	C₄	semidirect
20. C₅₃×#349	C₄	semidirect
155. #349	Dic₅₃	semidirect
158. C₅₃	#12602	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810581	2
C₂×#16810405	2
M₃₂×Dic₅₃	2
#20272798	2
C₂×C₅₃×#349	2
#12602	53