Subgroups — #27414075

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	#20276295	C₂	1	8	yes	maximal, complemented
3	C₂×#16810393	C₂	1	2	yes	maximal
4	#20274442	C₂	1	2	yes	maximal
5	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#2905	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂²×#11719917	C₂²	1	2	yes	—
9	#16810672	—	2	2	no	—
10	#16810672	C₂²	1	4	yes	—
11	C₂×#13133153	—	2	2	no	—
12	#16810393	C₂²	1	8	yes	—
13	#16810388	C₂²	1	8	yes	—
14	#16810381	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#424	—	2	2	no	—
19	C₅₃×#424	C₂²	1	4	yes	—
20	C₅₃×#423	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	—	2	2	no	—
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂×#11719917	C₂³	1	4	yes	—
33	C₂²×#88322	—	2	2	no	—
34	C₂²×#88322	—	2	2	no	—
35	#13133153	—	2	8	no	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂×M₁₆×C₅₃	—	2	1	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
44	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	#117×C₅₃	—	2	1	no	—
48	#117×C₅₃	—	2	1	no	—
49	#117×C₅₃	Q₈	1	2	yes	—
50	#117×C₅₃	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₄⋊C₈×C₅₃	—	2	8	no	—
55	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	4	1	no	—
70	M₁₆×C₅₃	—	4	1	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₂²⋊C₄×C₅₃	—	2	2	no	—
80	C₂²⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
81	C₄²×C₅₃	—	2	2	no	—
82	C₄²×C₅₃	—	2	2	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	2	4	no	—
85	#11719917	—	2	4	no	—
86	#11719917	—	2	4	no	—
87	#11719917	—	4	2	no	—
88	#11719917	—	4	2	no	—
89	#11719917	—	4	2	no	—
90	#11719917	—	4	2	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	—	4	2	no	—
97	C₂³×C₅₃	—	8	1	no	—
98	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₅₃	#125	1	2	yes	—
101	C₂³×C₅₃	#125	1	2	yes	—
102	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	8	2	no	—
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#123	1	2	yes	—
110	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#124	1	2	yes	—
112	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#125	1	2	yes	—
114	C₂×C₄×C₅₃	#125	1	2	yes	—
115	C₂×C₄×C₅₃	#126	1	2	yes	—
116	C₂×C₄×C₅₃	#126	1	2	yes	—
117	C₂×C₄×C₅₃	#126	1	2	yes	—
118	C₂×C₄×C₅₃	#126	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
123	C₈×C₅₃	—	2	4	no	—
124	C₈×C₅₃	—	4	2	no	—
125	#88322	—	4	4	no	—
126	#88322	—	4	4	no	—
127	#23753	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	—	4	1	no	—
130	C₂²×C₅₃	—	4	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	1	no	—
133	C₂²×C₅₃	#400	1	1	yes	characteristic
134	C₂²×C₅₃	#400	1	1	yes	characteristic
135	C₂²×C₅₃	#400	1	2	yes	—
136	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×#125	1	2	yes	—
138	C₄×C₅₃	—	4	4	no	—
139	C₄×C₅₃	#400	1	2	yes	—
140	C₄×C₅₃	#400	1	1	yes	characteristic
141	C₄×C₅₃	#400	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
143	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
144	C₄×C₅₃	C₂×#126	1	2	yes	—
145	C₂×#430	—	53	1	no	—
146	#2905	D₅₃	1	1	yes	characteristic
147	#2103	—	53	8	no	—
148	C₂×#344	—	53	2	no	—
149	#1738	—	53	2	no	—
150	C₂×#338	—	53	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	2	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	C₂×#400	1	1	yes	characteristic
157	C₂²×M₁₆	—	53	2	no	—
158	C₂²×M₁₆	—	53	2	no	—
159	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
160	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
162	#430	—	53	4	no	—
163	#430	—	106	2	no	—
164	C₂×#108	—	106	2	no	—
165	#424	—	2	2	no	—
166	#424	C₂×D₅₃	1	4	yes	—
167	#423	—	2	2	no	—
168	#344	—	53	8	no	—
169	#341	—	53	8	no	—
170	#338	—	53	4	no	—
171	C₅₃	#23753	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	53	4	no	—
180	C₂×M₁₆	—	53	4	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	—	106	2	no	—
186	C₂×M₁₆	—	106	2	no	—
187	C₂×M₁₆	—	106	2	no	—
188	C₂×M₁₆	—	106	2	no	—
189	C₂×M₁₆	—	106	2	no	—
190	C₂×M₁₆	—	106	2	no	—
191	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
192	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
193	C₂²×C₈	—	106	2	no	—
194	C₂²×C₈	—	106	2	no	—
195	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
196	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	Dic₁₀₆	1	2	yes	—
200	#117	C₂²×D₅₃	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
203	C₂×C₄²	—	2	1	no	—
204	#108	—	106	8	no	—
205	C₄⋊C₈	—	2	8	no	—
206	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
209	C₂²×C₄	#11719945	1	1	yes	characteristic
210	C₂²×C₄	#11719945	1	1	yes	characteristic
211	C₂²×C₄	#11719947	1	2	yes	—
212	C₂²×C₄	#11719947	1	2	yes	—
213	C₂²×C₄	#11719949	1	2	yes	—
214	C₂²×C₄	#11719949	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	—	106	4	no	—
225	M₁₆	—	106	4	no	—
226	M₁₆	—	106	4	no	—
227	M₁₆	—	212	2	no	—
228	M₁₆	—	212	2	no	—
229	M₁₆	—	212	2	no	—
230	M₁₆	—	212	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	106	4	no	—
237	C₂×C₈	—	212	2	no	—
238	C₂×C₈	—	212	2	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	—	212	2	no	—
241	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	C₂×Dic₁₀₆	1	4	yes	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₂³	—	8	1	no	—
249	C₂³	#13133196	1	1	yes	characteristic
250	C₂³	#13133196	1	1	yes	characteristic
251	C₂³	#13133208	1	2	yes	—
252	C₂³	#13133208	1	2	yes	—
253	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	#13133196	1	1	yes	characteristic
257	C₂×C₄	#13133196	1	1	yes	characteristic
258	C₂×C₄	#13133198	1	1	yes	characteristic
259	C₂×C₄	#13133198	1	1	yes	characteristic
260	C₂×C₄	#13133198	1	2	yes	—
261	C₂×C₄	#13133206	1	1	yes	characteristic
262	C₂×C₄	#13133206	1	2	yes	—
263	C₂×C₄	#13133206	1	1	yes	characteristic
264	C₂×C₄	#13133208	1	2	yes	—
265	C₂×C₄	#13133208	1	2	yes	—
266	C₂×C₄	#13133212	1	2	yes	—
267	C₂×C₄	#13133212	1	2	yes	—
268	C₂×C₄	#13133212	1	2	yes	—
269	C₂×C₄	#13133212	1	2	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#11719947	1	2	yes	—
272	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719949	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	212	4	no	—
277	C₈	—	212	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#16810562	1	1	yes	characteristic
284	C₂²	#16810562	1	1	yes	characteristic
285	C₂²	#16810562	1	2	yes	—
286	C₂²	C₂×#13133196	1	1	yes	characteristic
287	C₂²	C₂×#13133208	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	#16810562	1	2	yes	—
290	C₄	#16810562	1	1	yes	characteristic
291	C₄	#16810562	1	1	yes	characteristic
292	C₄	C₂×#13133198	1	1	yes	characteristic
293	C₄	C₂×#13133206	1	1	yes	characteristic
294	C₄	C₂×#13133212	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₂×#16810562	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133206
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2905	C₂
socle	C₂×C₅₃	C₂×#400
prime radical	C₂⁴×C₅₃	C₂×Q₈
hypercenter	#2905	D₅₃
hypocenter	C₅₃	#23753
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #20276295	C₂	semidirect
171. C₅₃	#23753	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20276295	2
C₂×#16810393	2
#20274442	2
C₂×#16810381	2
C₅₃×#2905	2
#23753	53