Subgroups — #28310488

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811522	C₂	1	2	yes	maximal
2	#20290921	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811512	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289769	C₂	1	2	yes	maximal, complemented
5	#20283137	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	2	yes	—
8	#16811522	C₄	1	8	yes	complemented
9	#16811512	—	2	1	no	—
10	#16811512	—	2	1	no	—
11	#16811512	—	2	1	no	—
12	#16811509	—	2	1	no	—
13	C₂×#13133225	C₂²	1	1	yes	characteristic
14	C₂×#13133223	C₄	1	1	yes	characteristic
15	#16811091	C₄	1	1	yes	characteristic
16	#16810580	—	2	1	no	—
17	#16810579	C₂²	1	1	yes	characteristic
18	#16810577	—	2	1	no	—
19	#16810575	—	2	1	no	—
20	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×D₅₃	—	2	1	no	—
23	C₅₃×#405	C₄	1	1	yes	characteristic
24	#13133291	—	2	2	no	—
25	#13133291	C₂×C₄	1	4	yes	complemented
26	C₂×#11719910	—	2	2	no	—
27	C₂×#11719910	—	2	2	no	—
28	C₂×#11719910	—	2	2	no	—
29	C₂×#11719910	C₂×C₄	1	4	yes	complemented
30	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
31	#13133225	—	2	2	no	—
32	#13133225	C₂×C₄	1	4	yes	—
33	#13133223	—	2	2	no	—
34	#13133223	C₈	1	4	yes	—
35	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
36	C₂×#11719912	—	2	1	no	—
37	C₂×#11719912	—	2	1	no	—
38	C₂²×#88322	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₅₃	—	2	1	no	—
40	C₂×C₈×D₅₃	—	2	1	no	—
41	#13133182	C₈	1	2	yes	—
42	#13133179	C₈	1	2	yes	—
43	#13133177	—	2	2	no	—
44	#13133174	—	2	1	no	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₅₃	—	2	1	no	—
47	#13133128	—	2	1	no	—
48	#13133127	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₈×Dic₅₃	—	2	1	no	—
51	C₈×Dic₅₃	D₄	1	1	yes	characteristic
52	C₈×Dic₅₃	D₄	1	1	yes	characteristic
53	C₈×Dic₅₃	D₄	1	1	yes	characteristic
54	C₈×Dic₅₃	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#97×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	—	2	1	no	—
62	C₂²×Dic₅₃	M₁₆	1	2	yes	—
63	#11719945	—	2	1	no	—
64	#11719945	—	2	1	no	—
65	#11719945	C₄²	1	2	yes	—
66	#11719945	C₂×C₈	1	2	yes	—
67	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
71	M₁₆×C₅₃	—	2	1	no	—
72	M₁₆×C₅₃	C₄²	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
78	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₄²×C₅₃	—	2	1	no	—
81	#11719921	—	2	2	no	—
82	#11719919	—	2	2	no	—
83	C₄×Dic₅₃	—	2	2	no	—
84	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
87	#11719917	—	2	1	no	—
88	#11719917	C₂×C₈	1	2	yes	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	C₂²×C₄	1	2	yes	—
91	C₂×#88322	C₂×D₄	1	1	yes	characteristic
92	C₂×#88322	C₂×D₄	1	1	yes	characteristic
93	C₂×#88322	#41	1	2	yes	—
94	#11719912	—	2	4	no	—
95	#11719912	—	2	4	no	—
96	C₈×D₅₃	—	2	4	no	—
97	C₈×D₅₃	—	2	4	no	—
98	#11719910	—	2	4	no	—
99	#11719910	—	2	4	no	—
100	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂²⋊D₅₃	—	2	2	no	—
110	C₂²⋊D₅₃	—	2	2	no	—
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	4	2	no	—
113	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
114	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
115	D₂₁₂	—	2	1	no	—
116	D₂₁₂	C₄×C₈	1	2	yes	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	4	4	no	—
120	Dic₁₀₆	—	2	1	no	—
121	Dic₁₀₆	C₄×C₈	1	2	yes	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
125	#88322	—	2	2	no	—
126	#88322	C₄×D₄	1	2	yes	—
127	#88322	C₄×D₄	1	2	yes	—
128	#12604	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂²×C₅₃	#431	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	—	4	2	no	—
135	C₄×C₅₃	—	4	2	no	—
136	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₅₃	#431	1	1	yes	characteristic
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	2	2	no	—
142	Dic₅₃	—	2	2	no	—
143	C₄×#131	—	53	1	no	—
144	C₂×#350	—	53	2	no	—
145	C₄×M₃₂	—	53	1	no	—
146	C₂×#346	—	53	1	no	—
147	#1379	—	53	2	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
151	C₂×C₅₃	#2117	1	2	yes	—
152	D₅₃	—	4	2	no	—
153	C₂×#131	—	53	1	no	—
154	C₂×#131	—	53	1	no	—
155	C₄×#41	—	53	1	no	—
156	C₂×M₃₂	—	53	2	no	—
157	C₂²×C₁₆	—	53	2	no	—
158	#405	—	53	1	no	—
159	#405	—	106	1	no	—
160	#405	—	106	1	no	—
161	#405	#6157	1	1	yes	characteristic
162	C₄×M₁₆	—	53	1	no	—
163	C₄×M₁₆	—	106	1	no	—
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	106	1	no	—
166	#350	—	53	8	no	—
167	#346	—	106	1	no	—
168	#346	—	106	1	no	—
169	#346	—	106	1	no	—
170	C₄×C₁₆	—	106	1	no	—
171	C₅₃	#12604	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	53	1	no	—
173	#131	—	53	4	no	—
174	#131	—	53	4	no	—
175	#131	—	106	2	no	—
176	#131	—	106	2	no	—
177	C₂×M₁₆	—	53	1	no	—
178	C₂×M₁₆	—	106	1	no	—
179	C₂×M₁₆	—	106	1	no	—
180	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	106	1	no	—
183	C₂²×C₈	—	106	1	no	—
184	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
185	C₄×Q₈	—	53	2	no	—
186	C₄×D₄	—	53	2	no	—
187	C₄×D₄	—	106	2	no	—
188	#117	—	106	1	no	—
189	#117	C₂×#6157	1	1	yes	characteristic
190	C₂×C₄²	—	53	1	no	—
191	C₂×C₄²	—	106	1	no	—
192	M₃₂	—	53	4	no	—
193	M₃₂	—	106	2	no	—
194	C₂×C₁₆	—	53	4	no	—
195	C₂×C₁₆	—	106	2	no	—
196	C₂×C₁₆	—	106	2	no	—
197	C₂×C₁₆	—	106	2	no	—
198	#97	—	2	1	no	—
199	#97	—	106	1	no	—
200	#97	—	106	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	53	1	no	—
203	C₄×C₈	—	53	1	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	106	1	no	—
207	C₄×C₈	—	106	1	no	—
208	#41	—	53	2	no	—
209	#41	—	53	2	no	—
210	#41	—	106	1	no	—
211	#41	—	106	1	no	—
212	C₂×Q₈	—	53	1	no	—
213	C₂×D₄	—	53	1	no	—
214	C₂×D₄	—	106	1	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	106	2	no	—
217	C₂²×C₄	—	106	1	no	—
218	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	53	2	no	—
221	M₁₆	—	106	4	no	—
222	M₁₆	—	106	1	no	—
223	M₁₆	—	106	4	no	—
224	M₁₆	C₄×#6157	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	2	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	1	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
234	C₂×C₈	#11719936	1	2	yes	—
235	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
237	C₄⋊C₄	—	106	2	no	—
238	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	106	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	53	1	no	—
243	C₄²	—	53	1	no	—
244	C₄²	—	53	2	no	—
245	C₄²	—	106	2	no	—
246	C₁₆	—	106	4	no	—
247	C₁₆	—	106	4	no	—
248	C₂³	—	106	1	no	—
249	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
250	Q₈	—	53	2	no	—
251	Q₈	—	106	1	no	—
252	D₄	—	53	2	no	—
253	D₄	—	106	2	no	—
254	D₄	—	106	2	no	—
255	D₄	—	106	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	53	2	no	—
258	C₂×C₄	—	53	2	no	—
259	C₂×C₄	—	106	1	no	—
260	C₂×C₄	—	106	2	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	—	212	4	no	—
263	C₂×C₄	—	212	2	no	—
264	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
266	C₂×C₄	#13133296	1	1	yes	characteristic
267	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
268	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	53	2	no	—
272	C₈	—	53	2	no	—
273	C₈	—	106	2	no	—
274	C₈	#13133274	1	2	yes	—
275	C₈	#13133274	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	106	2	no	—
278	C₂²	—	212	2	no	—
279	C₂²	C₄×#11719936	1	1	yes	characteristic
280	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
281	C₂²	#16811641	1	1	yes	characteristic
282	C₄	—	4	2	no	—
283	C₄	—	106	2	no	—
284	C₄	—	106	2	no	—
285	C₄	—	106	2	no	—
286	C₄	C₄×C₂²⋊#6157	1	1	yes	characteristic
287	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
288	C₄	C₂×#13133274	1	1	yes	characteristic
289	C₄	#16811642	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	212	2	no	—
293	C₂	C₄×#13133274	1	1	yes	characteristic
294	C₂	#20290958	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×#88322	C₂³
hypercenter	#405	#6157
hypocenter	C₅₃	#12604
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20290921	C₂	semidirect
3. C₂×#16811512	C₂	semidirect
4. #20289769	C₂	semidirect
8. #16811522	C₄	semidirect
25. #13133291	C₂×C₄	semidirect
29. C₂×#11719910	C₂×C₄	semidirect
171. C₅₃	#12604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811522	2
#20290921	2
C₂×#16811512	2
#20289769	2
#20283137	2
#12604	53