Subgroups — #27277216

Order 13568.

482 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	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273140	C₂	1	2	yes	maximal
4	#20272612	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	—	2	1	no	—
9	#16810647	C₂²	1	2	yes	—
10	#16810376	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	D₄⋊#88322	—	2	4	no	—
13	C₂³×C₈×C₅₃	—	2	1	no	—
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₅₃	—	2	1	no	—
16	C₈×D₄×C₅₃	—	2	1	no	—
17	C₈×D₄×C₅₃	C₄	1	2	yes	—
18	C₈×D₄×C₅₃	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#11719917	—	2	2	no	—
23	#13133153	—	4	2	no	—
24	#13133153	—	4	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₂²⋊#88322	—	2	2	no	—
29	C₄×#88322	—	2	2	no	—
30	C₂²×D₄×C₅₃	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	—	4	2	no	—
36	C₂²×C₈×C₅₃	—	4	2	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×D₄×C₅₃	—	2	1	no	—
40	C₄×D₄×C₅₃	—	2	1	no	—
41	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₅₃	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₅₃	—	2	1	no	—
47	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²⋊C₈×C₅₃	—	2	2	no	—
49	C₂²⋊C₈×C₅₃	—	2	2	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂⁴×C₅₃	—	2	1	no	—
55	C₂×D₄×C₅₃	—	2	4	no	—
56	C₂×D₄×C₅₃	—	2	1	no	—
57	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	—	4	1	no	—
64	C₂²×C₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	4	no	—
74	C₂×C₈×C₅₃	—	4	4	no	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	D₈	1	2	yes	—
80	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
81	C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₂²⋊C₄×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₃	#41	1	1	yes	characteristic
89	#11719917	—	4	2	no	—
90	#11719917	—	4	2	no	—
91	C₂×#88322	—	4	2	no	—
92	C₂×#88322	—	4	2	no	—
93	C₂³×C₅₃	—	2	4	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	4	1	no	—
97	C₂³×C₅₃	—	4	1	no	—
98	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
99	D₄×C₅₃	—	2	4	no	—
100	D₄×C₅₃	—	2	2	no	—
101	D₄×C₅₃	#108	1	4	yes	complemented
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	2	1	no	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	#102	1	2	yes	—
125	C₈×C₅₃	#102	1	2	yes	—
126	#88322	—	8	2	no	—
127	#88322	—	8	2	no	—
128	#8658	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	—	4	4	no	—
133	C₂²×C₅₃	—	4	4	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	#380	1	1	yes	characteristic
137	C₂²×C₅₃	#474	1	1	yes	characteristic
138	C₂²×C₅₃	#476	1	1	yes	characteristic
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	4	2	no	—
143	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
144	C₄×C₅₃	#417	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#108	1	2	yes	—
146	C₄×C₅₃	#459	1	1	yes	characteristic
147	C₄×C₅₃	#461	1	1	yes	characteristic
148	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
149	#1848	—	53	1	no	—
150	#1564	—	53	1	no	—
151	#1486	—	53	2	no	—
152	#1301	—	53	2	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	4	4	no	—
156	C₂×C₅₃	#1802	1	1	yes	characteristic
157	C₂×C₅₃	#1810	1	1	yes	characteristic
158	C₂×C₅₃	#1814	1	1	yes	characteristic
159	C₂³×C₈	—	2	1	no	—
160	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	—	2	1	no	—
163	C₈×D₄	Dic₅₃	1	2	yes	—
164	C₈×D₄	Dic₅₃	1	2	yes	—
165	#431	—	53	2	no	—
166	C₂×#108	—	106	1	no	—
167	C₂×#108	—	106	1	no	—
168	#423	—	53	2	no	—
169	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
172	#335	—	106	1	no	—
173	#334	—	106	1	no	—
174	D₄⋊C₈	—	106	4	no	—
175	C₅₃	#8658	1	1	yes	characteristic, complemented, Sylow
176	C₂²×D₄	Dic₁₀₆	1	1	yes	characteristic
177	C₂³×C₄	—	2	1	no	—
178	C₂×M₁₆	—	106	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	4	2	no	—
183	C₂²×C₈	—	4	2	no	—
184	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	D₂₁₂	1	2	yes	—
189	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
190	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
193	#108	—	212	2	no	—
194	#108	—	212	2	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	106	2	no	—
197	C₄⋊C₈	—	106	2	no	—
198	C₄⋊C₈	—	106	2	no	—
199	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	106	2	no	—
203	C₄×C₈	—	106	2	no	—
204	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂⁴	—	2	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
220	C₂²×C₄	#11719945	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
222	M₁₆	—	212	2	no	—
223	M₁₆	—	212	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	212	2	no	—
232	C₂×C₈	—	212	2	no	—
233	C₂×C₈	#11719922	1	2	yes	—
234	C₂×C₈	#11719923	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
244	C₄²	#11719945	1	1	yes	characteristic
245	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
247	C₂³	—	2	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	#13133196	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	#13133153	1	4	yes	complemented
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
267	C₂×C₄	#13133176	1	1	yes	characteristic
268	C₂×C₄	#13133191	1	1	yes	characteristic
269	C₂×C₄	#13133191	1	1	yes	characteristic
270	C₂×C₄	#13133196	1	1	yes	characteristic
271	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
272	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
273	C₂×C₄	#13133242	1	1	yes	characteristic
274	C₂×C₄	#13133247	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	2	no	—
278	C₈	—	424	2	no	—
279	C₈	—	424	2	no	—
280	C₈	#13133138	1	2	yes	—
281	C₈	#13133138	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#16810483	1	1	yes	characteristic
290	C₂²	#16810903	1	1	yes	characteristic
291	C₂²	#16810913	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	C₂×#13133138	1	1	yes	characteristic
297	C₄	#16810624	1	1	yes	characteristic
298	C₄	C₂×#13133153	1	2	yes	—
299	C₄	#16810818	1	1	yes	characteristic
300	C₄	#16810828	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	4	no	—
304	C₂	#20274715	1	1	yes	characteristic
305	C₂	#20274753	1	1	yes	characteristic
306	C₂	#20274769	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×D₄×C₅₃	Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#8658
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20274887	C₂	semidirect
2. #20273463	C₂	semidirect
4. #20272612	C₂	semidirect
101. D₄×C₅₃	#108	semidirect
175. C₅₃	#8658	semidirect
255. D₄	#13133153	semidirect

Maximal subgroups

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

subgroup	index
#20274887	2
#20273463	2
#20273140	2
#20272612	2
C₂×C₈×D₄×C₅₃	2
#8658	53