Subgroups — #27403041

Order 13568.

822 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20275473	C₂	1	3	yes	maximal
4	#20275471	C₂	1	3	yes	maximal
5	#20274842	C₂	1	3	yes	maximal
6	C₄×#13133150	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₃	—	2	1	no	—
9	Q₈×#88322	C₄	1	8	yes	complemented
10	C₂×#13133150	C₂²	1	3	yes	—
11	C₂×#13133150	C₂²	1	3	yes	—
12	C₂×C₄⋊#88322	C₂²	1	3	yes	—
13	C₂×C₄×#88322	C₂²	1	3	yes	—
14	#16810522	—	2	3	no	—
15	#16810493	C₂²	1	3	yes	—
16	C₄×C₄⋊Dic₅₃	C₂²	1	3	yes	—
17	#16810385	C₂²	1	6	yes	—
18	#16810385	C₂²	1	3	yes	—
19	#16810385	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#389	—	2	3	no	—
23	C₅₃×#384	C₂²	1	3	yes	—
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
25	C₂²×#88322	C₂³	1	1	yes	characteristic
26	C₂²×#88322	C₂³	1	3	yes	—
27	Q₈×Dic₅₃	—	4	4	no	—
28	C₂×C₄⋊Dic₅₃	—	2	3	no	—
29	C₂×C₄⋊Dic₅₃	—	2	3	no	—
30	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	—	2	3	no	—
33	#13133150	—	2	6	no	—
34	#13133150	—	2	6	no	—
35	C₄⋊#88322	C₂×C₄	1	12	yes	—
36	C₄×#88322	C₂×C₄	1	12	yes	—
37	#13133124	—	2	3	no	—
38	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×Q₈×C₅₃	—	4	4	no	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
44	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
46	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	—	2	3	no	—
48	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
49	#95×C₅₃	—	2	3	no	—
50	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂×Q₈×C₅₃	D₈	1	4	yes	—
52	C₂×Q₈×C₅₃	Q₁₆	1	4	yes	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	3	no	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	1	no	—
59	C₂²×Dic₅₃	—	2	3	no	—
60	C₄⋊C₄×C₅₃	—	2	1	no	—
61	C₄⋊C₄×C₅₃	—	2	3	no	—
62	C₄⋊C₄×C₅₃	—	2	3	no	—
63	C₄⋊C₄×C₅₃	—	4	6	no	—
64	C₄⋊C₄×C₅₃	—	8	3	no	—
65	C₄⋊C₄×C₅₃	C₄⋊C₄	1	12	yes	—
66	C₄⋊C₄×C₅₃	#41	1	2	yes	—
67	C₄²×C₅₃	—	4	6	no	—
68	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
69	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
70	C₄⋊Dic₅₃	—	2	1	no	—
71	C₄⋊Dic₅₃	—	2	3	no	—
72	C₄⋊Dic₅₃	—	2	3	no	—
73	C₄⋊Dic₅₃	—	4	6	no	—
74	C₄⋊Dic₅₃	—	8	3	no	—
75	C₄⋊Dic₅₃	#41	1	2	yes	—
76	C₄×Dic₅₃	—	4	6	no	—
77	C₂×#88322	—	2	3	no	—
78	C₂×#88322	—	2	3	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	C₂²×C₄	1	2	yes	—
82	C₂×#88322	C₂²×C₄	1	6	yes	—
83	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₅₃	#107	1	16	yes	complemented
85	C₂×C₄×C₅₃	—	2	3	no	—
86	C₂×C₄×C₅₃	—	2	3	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	3	no	—
91	C₂×C₄×C₅₃	—	4	3	no	—
92	C₂×C₄×C₅₃	—	4	3	no	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
95	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
96	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	4	1	no	—
105	C₂×Dic₅₃	—	4	1	no	—
106	C₂×Dic₅₃	—	4	3	no	—
107	C₂×Dic₅₃	—	4	3	no	—
108	C₂×Dic₅₃	—	4	3	no	—
109	#88322	—	2	6	no	—
110	#88322	—	2	2	no	—
111	#22740	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₅₃	#548	1	1	yes	characteristic
114	C₂²×C₅₃	#548	1	1	yes	characteristic
115	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
116	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂²×C₅₃	#577	1	1	yes	characteristic
118	C₂²×C₅₃	#577	1	1	yes	characteristic
119	C₄×C₅₃	—	2	6	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	—	8	3	no	—
122	C₄×C₅₃	C₂×#107	1	12	yes	—
123	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
124	C₄×C₅₃	#519	1	1	yes	characteristic
125	C₄×C₅₃	#557	1	1	yes	characteristic
126	C₄×C₅₃	#557	1	1	yes	characteristic
127	Dic₅₃	—	4	2	no	—
128	Dic₅₃	—	8	3	no	—
129	C₂×C₈×Q₈	—	53	1	no	—
130	Q₈×C₄⋊C₄	—	53	1	no	—
131	Q₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
132	#1947	—	53	3	no	—
133	#1947	—	53	3	no	—
134	#1834	—	53	3	no	—
135	C₄×#107	—	53	3	no	—
136	C₂×C₅₃	#2351	1	1	yes	characteristic
137	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
138	C₂×C₅₃	#2912	1	1	yes	characteristic
139	C₂×C₅₃	#3358	1	1	yes	characteristic
140	C₂×C₅₃	#3358	1	1	yes	characteristic
141	C₂×C₅₃	#3363	1	1	yes	characteristic
142	C₂×C₅₃	#3363	1	1	yes	characteristic
143	C₂×C₄×Q₈	—	2	1	no	—
144	C₂×C₄×Q₈	—	106	1	no	—
145	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
146	C₈×Q₈	—	53	8	no	—
147	C₂×#107	—	53	3	no	—
148	C₂×#107	—	53	3	no	—
149	C₂×C₄⋊C₈	—	53	3	no	—
150	C₂×C₄×C₈	—	53	3	no	—
151	#389	—	2	3	no	—
152	#389	—	106	3	no	—
153	#384	—	53	3	no	—
154	#384	C₂×D₅₃	1	3	yes	—
155	C₄×C₄⋊C₄	—	53	3	no	—
156	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
157	#340	—	53	3	no	—
158	#340	—	53	6	no	—
159	#340	—	53	1	no	—
160	C₅₃	#22740	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	3	no	—
164	C₄×Q₈	—	4	4	no	—
165	C₄×Q₈	—	212	4	no	—
166	C₄×Q₈	Dic₁₀₆	1	4	yes	—
167	C₄×Q₈	C₄×D₅₃	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	3	no	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	C₂×C₄⋊C₄	—	53	1	no	—
171	C₂×C₄⋊C₄	—	53	3	no	—
172	C₂×C₄⋊C₄	—	106	3	no	—
173	C₂×C₄⋊C₄	—	106	3	no	—
174	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	3	yes	—
175	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
176	C₂×C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×C₄²	—	2	3	no	—
178	C₂×C₄²	—	106	3	no	—
179	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
180	#107	—	106	6	no	—
181	#107	—	106	6	no	—
182	C₄⋊C₈	—	53	12	no	—
183	C₄×C₈	—	53	12	no	—
184	#95	—	2	3	no	—
185	#95	—	106	3	no	—
186	C₂×Q₈	#11719919	1	4	yes	—
187	C₂×Q₈	#11719922	1	4	yes	—
188	C₂×Q₈	#11719925	1	4	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	3	no	—
191	C₂²×C₄	—	106	1	no	—
192	C₂²×C₄	—	106	3	no	—
193	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
195	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
196	C₂×C₈	—	53	2	no	—
197	C₂×C₈	—	53	6	no	—
198	C₂×C₈	—	106	3	no	—
199	C₂×C₈	—	106	3	no	—
200	C₂×C₈	—	106	1	no	—
201	C₂×C₈	—	106	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	3	no	—
204	C₄⋊C₄	—	2	3	no	—
205	C₄⋊C₄	—	4	6	no	—
206	C₄⋊C₄	—	8	3	no	—
207	C₄⋊C₄	—	53	2	no	—
208	C₄⋊C₄	—	106	1	no	—
209	C₄⋊C₄	—	106	3	no	—
210	C₄⋊C₄	—	106	3	no	—
211	C₄⋊C₄	—	424	3	no	—
212	C₄⋊C₄	—	212	6	no	—
213	C₄⋊C₄	#11719919	1	12	yes	—
214	C₄⋊C₄	#11719949	1	2	yes	—
215	C₄²	—	4	6	no	—
216	C₄²	—	212	6	no	—
217	C₄²	C₂×Dic₁₀₆	1	6	yes	—
218	C₄²	C₂×C₄×D₅₃	1	6	yes	—
219	C₂³	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
220	Q₈	#13133150	1	16	yes	complemented
221	C₂×C₄	—	2	3	no	—
222	C₂×C₄	—	2	3	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	3	no	—
227	C₂×C₄	—	4	3	no	—
228	C₂×C₄	—	4	3	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	212	3	no	—
231	C₂×C₄	—	212	3	no	—
232	C₂×C₄	—	212	3	no	—
233	C₂×C₄	—	212	1	no	—
234	C₂×C₄	—	212	1	no	—
235	C₂×C₄	C₂×#11719919	1	6	yes	—
236	C₂×C₄	C₂×#11719919	1	6	yes	—
237	C₂×C₄	C₂×#11719922	1	6	yes	—
238	C₂×C₄	C₂×#11719925	1	6	yes	—
239	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
241	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
243	C₂×C₄	#13133264	1	1	yes	characteristic
244	C₂×C₄	#13133265	1	1	yes	characteristic
245	C₈	—	106	6	no	—
246	C₈	—	106	2	no	—
247	C₂²	C₂²×#11719919	1	1	yes	characteristic
248	C₂²	#16811280	1	1	yes	characteristic
249	C₂²	#16811284	1	1	yes	characteristic
250	C₂²	C₂²×#11719922	1	1	yes	characteristic
251	C₂²	C₂²×#11719925	1	1	yes	characteristic
252	C₂²	#16811431	1	1	yes	characteristic
253	C₂²	#16811434	1	1	yes	characteristic
254	C₄	—	2	6	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	3	no	—
257	C₄	—	424	3	no	—
258	C₄	—	212	2	no	—
259	C₄	C₂×#13133150	1	12	yes	—
260	C₄	C₄×#11719949	1	1	yes	characteristic
261	C₄	#16811099	1	1	yes	characteristic
262	C₄	#16811349	1	1	yes	characteristic
263	C₄	#16811352	1	1	yes	characteristic
264	C₂	#20277633	1	1	yes	characteristic
265	C₂	C₂²×#13133150	1	1	yes	characteristic
266	C₂	#20283308	1	1	yes	characteristic
267	C₂	#20287346	1	1	yes	characteristic
268	C₂	#20287350	1	1	yes	characteristic
269	C₂	#20287389	1	1	yes	characteristic
270	C₂	#20287393	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	Q₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#22740
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×#88322	C₄	semidirect
84. Q₈×C₅₃	#107	semidirect
160. C₅₃	#22740	semidirect
220. Q₈	#13133150	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88322	2
Q₈×C₄⋊Dic₅₃	2
#20275473	2
#20275471	2
#20274842	2
C₄×#13133150	2
Q₈×C₄⋊C₄×C₅₃	2
#22740	53