Subgroups — #27300015

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810441	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#20272792	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	1	yes	characteristic
7	C₂×#13133176	C₄	1	1	yes	characteristic
8	C₂²×#11719908	C₂²	1	2	yes	—
9	#16810578	—	2	4	no	—
10	C₂×#13133127	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810441	—	2	4	no	—
15	C₂²⋊#11719908	—	2	2	no	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	#13133176	—	2	2	no	—
25	#13133176	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
27	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	—	2	2	no	—
33	#13133127	—	2	2	no	—
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	D₄	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
57	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₂²⋊Dic₅₃	—	2	4	no	—
68	C₄⋊Dic₅₃	—	2	4	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	1	no	—
71	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄×Dic₅₃	M₁₆	1	2	yes	—
73	#11719917	—	2	8	no	—
74	C₂×#88322	—	2	1	no	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	C₂²⋊C₄	1	2	yes	—
79	C₂×#88322	M₁₆	1	2	yes	—
80	C₁₆×C₅₃	—	4	2	no	—
81	#11719908	—	4	2	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
87	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	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	2	no	—
94	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	2	2	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	#121	1	2	yes	—
109	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
110	#88322	—	2	2	no	—
111	#88322	—	2	2	no	—
112	#12784	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	#431	1	1	yes	characteristic
121	C₂²×C₅₃	#504	1	2	yes	—
122	C₂²×C₅₃	#504	1	2	yes	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	#380	1	1	yes	characteristic
125	C₄×C₅₃	#407	1	1	yes	characteristic
126	C₄×C₅₃	#423	1	1	yes	characteristic
127	C₄×C₅₃	#432	1	1	yes	characteristic
128	Dic₅₃	—	2	4	no	—
129	Dic₅₃	—	4	2	no	—
130	C₂×#405	—	53	1	no	—
131	C₂×C₄⋊C₁₆	—	53	2	no	—
132	C₂×C₂²⋊C₁₆	—	53	1	no	—
133	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
134	#1374	—	53	2	no	—
135	C₂×C₅₃	—	4	2	no	—
136	C₂×C₅₃	#1797	1	1	yes	characteristic
137	C₂×C₅₃	#2113	1	2	yes	—
138	C₂×C₅₃	#2151	1	1	yes	characteristic
139	C₂×C₅₃	#2151	1	1	yes	characteristic
140	C₂×C₅₃	#2164	1	2	yes	—
141	C₂²×M₁₆	—	53	1	no	—
142	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₂×#117	—	53	1	no	—
144	C₂²×C₁₆	—	53	2	no	—
145	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
146	#405	—	106	4	no	—
147	C₂×#97	—	53	1	no	—
148	C₂×#97	—	53	1	no	—
149	C₂×C₄×C₈	—	53	1	no	—
150	C₂×C₄×C₈	—	53	1	no	—
151	C₄⋊C₁₆	—	106	4	no	—
152	C₂²⋊C₁₆	—	2	2	no	—
153	C₂²⋊C₁₆	—	106	2	no	—
154	C₅₃	#12784	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
156	C₂×M₁₆	—	106	2	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	—	106	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	53	1	no	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	Dic₁₀₆	1	2	yes	—
164	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
166	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
167	#117	—	106	2	no	—
168	#117	—	106	2	no	—
169	C₂×C₄⋊C₄	—	53	2	no	—
170	C₂×C₂²⋊C₄	—	53	2	no	—
171	C₂×C₄²	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	106	2	no	—
177	C₂×C₁₆	—	106	2	no	—
178	C₂×C₁₆	—	106	2	no	—
179	#97	—	106	2	no	—
180	#97	—	106	2	no	—
181	C₄×C₈	—	53	4	no	—
182	C₄×C₈	—	106	2	no	—
183	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	53	2	no	—
189	C₂²×C₄	—	53	2	no	—
190	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	#11719912	1	2	yes	—
192	C₂²×C₄	#11719919	1	2	yes	—
193	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
194	M₁₆	—	106	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	53	2	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	106	1	no	—
203	C₂×C₈	—	106	1	no	—
204	C₂×C₈	—	106	1	no	—
205	C₂×C₈	#11719919	1	2	yes	—
206	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂×C₈	D₄×D₅₃	1	2	yes	—
208	C₂×C₈	#11719947	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄⋊C₄	—	106	4	no	—
211	C₂²⋊C₄	—	106	4	no	—
212	C₄²	—	53	2	no	—
213	C₄²	—	53	2	no	—
214	C₄²	—	106	1	no	—
215	C₄²	—	106	1	no	—
216	C₁₆	—	4	2	no	—
217	C₁₆	—	212	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	#13133146	1	2	yes	—
223	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	53	4	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	—	106	2	no	—
234	C₂×C₄	—	106	2	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	#13133146	1	2	yes	—
237	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
239	C₂×C₄	#13133173	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	106	2	no	—
245	C₈	—	106	2	no	—
246	C₈	#13133189	1	2	yes	—
247	C₈	#13133194	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
254	C₂²	C₂×#13133146	1	1	yes	characteristic
255	C₂²	#16810673	1	1	yes	characteristic
256	C₂²	#16811039	1	2	yes	—
257	C₂²	#16811041	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	106	4	no	—
260	C₄	—	212	2	no	—
261	C₄	#16810481	1	1	yes	characteristic
262	C₄	#16810585	1	1	yes	characteristic
263	C₄	#16810646	1	1	yes	characteristic
264	C₄	#16810677	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	#20274693	1	1	yes	characteristic
267	C₂	#20276327	1	2	yes	—
268	C₂	#20276473	1	1	yes	characteristic
269	C₂	#20276475	1	1	yes	characteristic
270	C₂	#20276519	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#12784
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810441	C₂	semidirect
4. #20272792	C₂	semidirect
154. C₅₃	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810578	2
C₂×#16810441	2
C₂×C₂²⋊#11719908	2
#20272792	2
C₂×C₅₃×C₂²⋊C₁₆	2
#12784	53