Subgroups — #28317078

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291620	C₂	1	2	yes	maximal
2	#20290362	C₂	1	2	yes	maximal, complemented
3	#20290355	C₂	1	2	yes	maximal, complemented
4	#20290308	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#88324	C₂	1	2	yes	maximal, complemented
6	#20290276	C₂	1	1	yes	maximal, characteristic
7	#20290078	C₂	1	2	yes	maximal
8	#20289919	C₂	1	1	yes	maximal, characteristic
9	#20285299	C₂	1	1	yes	maximal, characteristic
10	#16811675	—	2	4	no	—
11	#16811674	C₂²	1	4	yes	—
12	Q₈×#88324	C₂²	1	4	yes	—
13	#16811648	—	2	4	no	—
14	#16811645	C₂²	1	2	yes	—
15	#16811641	C₂²	1	1	yes	characteristic
16	#16811641	C₂²	1	1	yes	characteristic
17	#16811486	C₂²	1	2	yes	—
18	#16811484	C₂²	1	2	yes	—
19	Q₈⋊#88324	C₂²	1	4	yes	—
20	Q₈⋊#88324	C₂²	1	4	yes	—
21	#16811464	C₂²	1	1	yes	characteristic
22	#16811464	C₂²	1	1	yes	characteristic
23	C₈×#88324	C₂²	1	1	yes	characteristic
24	C₈×#88324	C₂²	1	1	yes	characteristic
25	#16811178	C₂²	1	2	yes	—
26	#16811028	C₄	1	1	yes	characteristic
27	#16810973	C₄	1	2	yes	—
28	#16810861	C₄	1	2	yes	—
29	#16810861	C₄	1	2	yes	—
30	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
31	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#494	C₄	1	1	yes	characteristic
34	#13133284	—	2	1	no	—
35	#13133284	—	2	4	no	—
36	#13133284	D₄	1	2	yes	—
37	C₄⋊#88324	—	2	4	no	—
38	C₄⋊#88324	C₂³	1	4	yes	—
39	#13133274	—	2	2	no	—
40	#13133274	—	2	1	no	—
41	#13133274	D₄	1	2	yes	—
42	#13133272	—	2	2	no	—
43	#13133272	—	2	4	no	—
44	C₄×#88324	—	2	4	no	—
45	C₄×#88324	C₂³	1	1	yes	characteristic
46	C₄×#88324	C₂³	1	1	yes	characteristic
47	C₄×#88324	C₂³	1	1	yes	characteristic
48	C₄×#88324	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	#13133217	C₂×C₄	1	2	yes	—
52	#13133208	—	2	2	no	—
53	#13133197	—	2	2	no	—
54	#13133193	C₂×C₄	1	4	yes	—
55	Q₈×Dic₅₃	C₂³	1	4	yes	—
56	#13133182	—	2	2	no	—
57	#13133175	—	2	2	no	—
58	C₄²×D₅₃	C₂³	1	1	yes	characteristic
59	#13133151	C₂×C₄	1	2	yes	—
60	#13133151	C₂×C₄	1	2	yes	—
61	#13133141	C₂×C₄	1	4	yes	—
62	#13133141	C₂×C₄	1	4	yes	—
63	C₄×#88322	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
67	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
68	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
74	C₂×Dic₁₀₆	—	2	2	no	—
75	C₂×#88324	—	2	4	no	—
76	C₂×#88324	—	2	1	no	—
77	C₂×#88324	—	2	1	no	—
78	C₂×#88324	C₂×D₄	1	2	yes	—
79	C₂×#88324	C₂×D₄	1	2	yes	—
80	Q₁₆×C₅₃	—	2	4	no	—
81	Q₁₆×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	#11719921	—	2	4	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
91	#11719919	—	2	4	no	—
92	C₄×Dic₅₃	—	2	4	no	—
93	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄×Dic₅₃	#41	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
96	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
97	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
98	C₈×D₅₃	—	2	2	no	—
99	C₈×D₅₃	—	2	2	no	—
100	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₅₃	—	2	4	no	—
102	Q₈×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	4	no	—
108	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
111	C₄×D₅₃	—	2	1	no	—
112	C₄×D₅₃	—	2	1	no	—
113	C₄×D₅₃	—	2	2	no	—
114	C₄×D₅₃	—	4	1	no	—
115	Dic₁₀₆	—	8	2	no	—
116	#88324	—	2	2	no	—
117	#88324	—	2	2	no	—
118	#88324	—	4	1	no	—
119	#88324	—	4	1	no	—
120	#88324	—	4	4	no	—
121	C₈×C₅₃	#131	1	2	yes	—
122	C₈×C₅₃	#131	1	2	yes	—
123	#88322	—	2	1	no	—
124	#88322	—	2	1	no	—
125	#17972	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×D₅₃	—	2	1	no	—
128	C₂×D₅₃	—	2	1	no	—
129	C₂×D₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
135	Dic₅₃	—	2	1	no	—
136	Dic₅₃	—	2	1	no	—
137	Dic₅₃	—	4	4	no	—
138	Dic₅₃	#578	1	2	yes	—
139	#3170	—	53	1	no	—
140	#2985	—	53	2	no	—
141	#1593	—	53	2	no	—
142	#1591	—	53	2	no	—
143	#1579	—	53	2	no	—
144	C₈×Q₁₆	—	53	2	no	—
145	#1568	—	53	1	no	—
146	#1497	—	53	2	no	—
147	#1450	—	53	1	no	—
148	C₂×C₅₃	#2931	1	1	yes	characteristic
149	C₂×C₅₃	#2950	1	1	yes	characteristic
150	C₂×C₅₃	#2957	1	1	yes	characteristic
151	D₅₃	—	4	1	no	—
152	#533	—	53	2	no	—
153	#500	—	53	1	no	—
154	#494	#6157	1	1	yes	characteristic
155	#487	—	53	2	no	—
156	#467	—	53	2	no	—
157	#467	—	53	2	no	—
158	#446	—	53	4	no	—
159	#446	—	106	4	no	—
160	C₈×Q₈	—	53	4	no	—
161	C₄×Q₁₆	—	53	2	no	—
162	C₄×Q₁₆	—	53	2	no	—
163	#433	—	106	4	no	—
164	#432	—	53	2	no	—
165	#431	—	53	1	no	—
166	#431	—	53	1	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	#335	—	53	2	no	—
169	#334	—	53	2	no	—
170	Q₈⋊C₈	—	53	4	no	—
171	Q₈⋊C₈	—	53	4	no	—
172	#322	—	53	1	no	—
173	#322	—	53	1	no	—
174	C₈²	—	53	1	no	—
175	C₈²	—	53	1	no	—
176	C₅₃	#17972	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	C₂×#6157	1	2	yes	—
178	C₂×Q₁₆	C₂×#6157	1	2	yes	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₄⋊Q₈	—	53	2	no	—
182	C₄⋊Q₈	C₂×#6157	1	2	yes	—
183	#125	—	106	2	no	—
184	C₂²⋊Q₈	—	53	4	no	—
185	C₂²⋊Q₈	—	106	2	no	—
186	C₄×Q₈	—	53	4	no	—
187	C₄×Q₈	—	106	2	no	—
188	#117	—	106	2	no	—
189	C₂×C₄²	—	53	1	no	—
190	#107	—	53	2	no	—
191	#107	—	53	2	no	—
192	C₄⋊C₈	—	53	2	no	—
193	C₄⋊C₈	—	53	4	no	—
194	C₄⋊C₈	—	106	1	no	—
195	C₄⋊C₈	—	106	4	no	—
196	C₄⋊C₈	—	106	4	no	—
197	#103	—	53	4	no	—
198	#103	—	53	4	no	—
199	C₂²⋊C₈	—	53	2	no	—
200	C₂²⋊C₈	—	106	1	no	—
201	C₂²⋊C₈	—	106	2	no	—
202	#97	—	106	2	no	—
203	#97	—	106	4	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	53	1	no	—
209	C₄×C₈	—	53	1	no	—
210	C₄×C₈	—	53	1	no	—
211	C₄×C₈	—	106	4	no	—
212	C₄×C₈	C₂×#6157	1	1	yes	characteristic
213	C₂×Q₈	—	106	2	no	—
214	C₂×Q₈	C₂²×#6157	1	4	yes	—
215	C₂²×C₄	—	53	1	no	—
216	C₂²×C₄	—	53	1	no	—
217	C₂²×C₄	—	53	1	no	—
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	4	no	—
220	C₂×C₈	—	53	2	no	—
221	C₂×C₈	—	53	2	no	—
222	C₂×C₈	—	53	1	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	106	2	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	1	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
230	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	53	4	no	—
234	C₄⋊C₄	—	106	4	no	—
235	C₄⋊C₄	—	106	4	no	—
236	C₄⋊C₄	—	106	2	no	—
237	C₂²⋊C₄	—	106	4	no	—
238	C₄²	—	53	1	no	—
239	C₄²	—	53	1	no	—
240	C₄²	—	53	1	no	—
241	C₄²	—	106	4	no	—
242	C₄²	C₂²×#6157	1	1	yes	characteristic
243	C₂³	—	53	1	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	424	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	53	1	no	—
249	C₂×C₄	—	53	1	no	—
250	C₂×C₄	—	53	1	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	106	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	4	no	—
255	C₂×C₄	—	212	1	no	—
256	C₂×C₄	D₄×#6157	1	1	yes	characteristic
257	C₂×C₄	D₄×#6157	1	1	yes	characteristic
258	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
259	C₈	—	106	2	no	—
260	C₈	—	106	2	no	—
261	C₈	—	106	1	no	—
262	C₈	—	106	1	no	—
263	C₈	—	212	4	no	—
264	C₈	—	212	1	no	—
265	C₈	—	212	1	no	—
266	C₈	#13133304	1	2	yes	—
267	C₈	#13133304	1	2	yes	—
268	C₂²	—	106	1	no	—
269	C₂²	—	106	1	no	—
270	C₂²	—	106	1	no	—
271	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	53	2	no	—
276	C₄	—	106	1	no	—
277	C₄	—	106	1	no	—
278	C₄	—	212	4	no	—
279	C₄	C₂×#13133304	1	1	yes	characteristic
280	C₄	C₂×#13133304	1	1	yes	characteristic
281	C₂	—	212	1	no	—
282	C₂	#20291437	1	1	yes	characteristic
283	C₂	#20291476	1	1	yes	characteristic
284	C₂	#20291495	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#494	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#494	#6157
hypocenter	C₅₃	#17972
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #20290362	C₂	semidirect
3. #20290355	C₂	semidirect
4. #20290308	C₂	semidirect
5. Q₁₆×#88324	C₂	semidirect
176. C₅₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#20291620	2
#20290362	2
#20290355	2
#20290308	2
Q₁₆×#88324	2
#20290276	2
#20290078	2
#20289919	2
#20285299	2
#17972	53