Subgroups — #27484055

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285239	C₂	1	1	yes	maximal, characteristic, complemented
2	#20275887	C₂	1	4	yes	maximal, complemented
3	#20275614	C₂	1	2	yes	maximal
4	#20274840	C₂	1	2	yes	maximal, complemented
5	#20274769	C₂	1	2	yes	maximal, complemented
6	#20274761	C₂	1	2	yes	maximal, complemented
7	#20274599	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#444	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811032	—	2	4	no	—
11	C₂×#13133153	C₂²	1	1	yes	characteristic
12	#16810661	C₂²	1	2	yes	—
13	#16810661	C₂²	1	2	yes	—
14	#16810661	C₂²	1	1	yes	characteristic
15	#16810640	C₂²	1	4	yes	—
16	#16810597	C₂²	1	4	yes	—
17	#16810578	C₂²	1	2	yes	—
18	#16810395	C₂²	1	4	yes	—
19	#16810383	C₂²	1	2	yes	—
20	#16810380	C₂²	1	4	yes	—
21	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#131×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#444	—	2	4	no	—
24	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#104×C₅₃	C₂²	1	2	yes	—
26	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#11719917	C₂³	1	2	yes	—
28	#13133217	—	2	2	no	—
29	#13133209	—	2	2	no	—
30	#13133196	—	4	2	no	—
31	C₄×Dic₁₀₆	—	4	2	no	—
32	#13133176	C₂³	1	2	yes	—
33	#13133176	C₂³	1	4	yes	—
34	#13133153	—	2	2	no	—
35	#13133151	—	2	2	no	—
36	#13133151	—	2	2	no	—
37	#13133151	—	2	1	no	—
38	#13133149	—	2	2	no	—
39	#13133149	—	2	2	no	—
40	#13133149	—	2	1	no	—
41	#13133144	—	4	4	no	—
42	#13133132	—	4	4	no	—
43	#13133127	—	2	2	no	—
44	C₈×Dic₅₃	—	2	2	no	—
45	C₂×#41×C₅₃	C₂³	1	2	yes	—
46	#135×C₅₃	—	2	2	no	—
47	#135×C₅₃	—	2	2	no	—
48	C₂×Q₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
49	C₂×SD₁₆×C₅₃	Q₈	1	2	yes	complemented
50	C₂×D₈×C₅₃	Q₈	1	1	yes	characteristic, complemented
51	#131×C₅₃	—	2	2	no	—
52	#131×C₅₃	—	2	2	no	—
53	#131×C₅₃	—	2	2	no	—
54	#131×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	—	2	2	no	—
56	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	—	2	2	no	—
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#108×C₅₃	—	2	2	no	—
62	#104×C₅₃	—	2	4	no	—
63	#97×C₅₃	D₄	1	4	yes	—
64	#41×C₅₃	—	2	2	no	—
65	#41×C₅₃	—	2	2	no	—
66	#41×C₅₃	—	2	2	no	—
67	#41×C₅₃	—	2	2	no	—
68	C₂×Q₈×C₅₃	C₂×Q₈	1	2	yes	—
69	C₂×D₄×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	2	2	no	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₁₀₆	—	4	2	no	—
75	Q₁₆×C₅₃	—	2	2	no	—
76	SD₁₆×C₅₃	—	2	4	no	—
77	D₈×C₅₃	—	2	2	no	—
78	M₁₆×C₅₃	—	2	2	no	—
79	M₁₆×C₅₃	—	2	2	no	—
80	M₁₆×C₅₃	—	4	2	no	—
81	M₁₆×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	4	2	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₃	#41	1	2	yes	—
90	C₂×C₈×C₅₃	#41	1	2	yes	—
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
93	C₄²×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	C₂×D₄	1	2	yes	—
95	C₂²⋊Dic₅₃	—	2	2	no	—
96	C₂²⋊Dic₅₃	—	2	4	no	—
97	C₄⋊Dic₅₃	—	2	4	no	—
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	—	4	2	no	—
100	#11719919	—	4	4	no	—
101	C₄×Dic₅₃	—	2	2	no	—
102	C₄×Dic₅₃	—	2	4	no	—
103	#11719917	—	2	2	no	—
104	#11719917	—	2	2	no	—
105	C₂×#88322	—	2	2	no	—
106	C₂³×C₅₃	—	2	2	no	—
107	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
108	Q₈×C₅₃	—	2	2	no	—
109	Q₈×C₅₃	—	2	2	no	—
110	D₄×C₅₃	—	2	2	no	—
111	D₄×C₅₃	—	2	2	no	—
112	D₄×C₅₃	—	4	2	no	—
113	D₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	1	no	—
119	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×Dic₅₃	—	4	4	no	—
126	C₂×Dic₅₃	—	4	2	no	—
127	Dic₁₀₆	—	8	4	no	—
128	C₈×C₅₃	—	4	2	no	—
129	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
131	C₈×C₅₃	#124	1	2	yes	—
132	C₈×C₅₃	#124	1	2	yes	—
133	#88322	—	4	2	no	—
134	#29155	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	4	1	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	8	2	no	—
138	C₂²×C₅₃	#535	1	1	yes	characteristic
139	C₂²×C₅₃	#554	1	1	yes	characteristic
140	C₂²×C₅₃	#555	1	1	yes	characteristic
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
145	C₄×C₅₃	#554	1	1	yes	characteristic
146	C₄×C₅₃	#555	1	1	yes	characteristic
147	Dic₅₃	—	8	4	no	—
148	Dic₅₃	—	8	2	no	—
149	#3162	—	53	1	no	—
150	C₂×#444	D₅₃	1	1	yes	characteristic
151	#2031	—	53	4	no	—
152	#1976	—	53	2	no	—
153	#1833	—	53	2	no	—
154	#1814	—	53	2	no	—
155	#1812	—	53	2	no	—
156	#1773	—	53	1	no	—
157	C₂×C₅₃	—	2	1	no	—
158	C₂×C₅₃	—	2	1	no	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	—	8	2	no	—
161	C₂×C₅₃	#2563	1	1	yes	characteristic
162	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#131	C₂×D₅₃	1	2	yes	—
164	#533	—	53	2	no	—
165	#501	—	106	4	no	—
166	#444	—	2	4	no	—
167	C₂×#108	—	53	1	no	—
168	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
169	#427	—	53	1	no	—
170	#427	—	53	2	no	—
171	#427	—	53	2	no	—
172	#421	—	53	4	no	—
173	C₂×#104	C₂×D₅₃	1	2	yes	—
174	#410	—	53	4	no	—
175	#405	—	53	2	no	—
176	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
177	#344	—	53	4	no	—
178	#339	—	53	2	no	—
179	H(3,ℤ/4)	—	53	4	no	—
180	C₅₃	#29155	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	C₂²×D₅₃	1	2	yes	—
182	#135	—	2	2	no	—
183	#135	—	2	2	no	—
184	C₂×Q₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
185	C₂×SD₁₆	Dic₁₀₆	1	2	yes	complemented
186	C₂×D₈	Dic₁₀₆	1	1	yes	characteristic, complemented
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	53	2	no	—
193	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	106	2	no	—
198	#125	—	106	2	no	—
199	C₂²⋊Q₈	—	212	2	no	—
200	C₄×Q₈	—	212	2	no	—
201	#117	—	53	4	no	—
202	#117	—	53	2	no	—
203	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
204	#108	—	2	2	no	—
205	#108	—	106	2	no	—
206	#107	—	106	1	no	—
207	#107	—	106	2	no	—
208	#107	—	106	2	no	—
209	#106	—	106	1	no	—
210	#106	—	106	2	no	—
211	#106	—	106	2	no	—
212	#104	—	2	4	no	—
213	#104	—	212	4	no	—
214	#99	—	212	4	no	—
215	#97	—	106	2	no	—
216	#97	C₂²⋊D₅₃	1	4	yes	—
217	C₄×C₈	—	106	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	—	212	2	no	—
223	C₂×Q₈	C₂×Dic₁₀₆	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×Dic₁₀₆	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	2	4	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	106	2	no	—
237	M₁₆	—	106	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
243	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
244	C₂×C₈	#11719945	1	2	yes	—
245	C₂×C₈	#11719947	1	1	yes	characteristic
246	C₂×C₈	#11719947	1	1	yes	characteristic
247	C₂×C₈	#11719947	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
250	C₄⋊C₄	—	106	4	no	—
251	C₄⋊C₄	—	106	2	no	—
252	C₄⋊C₄	—	212	4	no	—
253	C₄⋊C₄	—	212	2	no	—
254	C₂²⋊C₄	—	106	2	no	—
255	C₂²⋊C₄	—	106	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	106	2	no	—
258	C₄²	—	106	4	no	—
259	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	#13133263	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	424	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	212	2	no	—
275	C₂×C₄	—	212	4	no	—
276	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
278	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
279	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
281	C₂×C₄	#13133263	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	212	2	no	—
284	C₈	#13133196	1	2	yes	—
285	C₈	#13133196	1	2	yes	—
286	C₈	#13133204	1	2	yes	—
287	C₈	#13133204	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	#16811187	1	1	yes	characteristic
292	C₂²	#16811330	1	1	yes	characteristic
293	C₂²	#16811338	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	424	2	no	—
297	C₄	—	424	4	no	—
298	C₄	C₂×#13133196	1	1	yes	characteristic
299	C₄	C₂×#13133204	1	1	yes	characteristic
300	C₄	#16811330	1	1	yes	characteristic
301	C₄	#16811338	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	#20279888	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811338
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#444	C₂
socle	C₂×C₅₃	#2563
prime radical	C₂×#135×C₅₃	C₂²
hypercenter	C₂×#444	D₅₃
hypocenter	C₅₃	#29155
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
1. #20285239	C₂	semidirect
2. #20275887	C₂	semidirect
4. #20274840	C₂	semidirect
5. #20274769	C₂	semidirect
6. #20274761	C₂	semidirect
7. #20274599	C₂	semidirect
48. C₂×Q₁₆×C₅₃	Q₈	semidirect
49. C₂×SD₁₆×C₅₃	Q₈	semidirect
50. C₂×D₈×C₅₃	Q₈	semidirect
180. C₅₃	#29155	semidirect
184. C₂×Q₁₆	Dic₁₀₆	semidirect
185. C₂×SD₁₆	Dic₁₀₆	semidirect
186. C₂×D₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285239	2
#20275887	2
#20275614	2
#20274840	2
#20274769	2
#20274761	2
#20274599	2
C₂×C₅₃×#444	2
#29155	53