Subgroups — #27270619

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	C₄	1	2	yes	—
5	C₂×C₄×#88322	—	2	1	no	—
6	C₂×C₄×#88322	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	C₂×C₄×#88322	C₄	1	2	yes	—
9	#16810378	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	—	2	1	no	—
12	C₄³×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#336	—	2	1	no	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	2	2	no	—
17	C₄⋊#88322	C₈	1	4	yes	complemented
18	C₄×#88322	—	2	4	no	—
19	C₄×#88322	—	2	4	no	—
20	C₄×#88322	—	2	2	no	—
21	C₄×#88322	—	4	1	no	—
22	C₄×#88322	—	4	1	no	—
23	C₄×#88322	C₈	1	4	yes	complemented
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	2	no	—
33	C₄⋊C₈×C₅₃	C₈	1	4	yes	complemented
34	C₄×C₈×C₅₃	—	4	1	no	—
35	C₄×C₈×C₅₃	—	4	1	no	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₂×C₈	1	2	yes	—
50	C₄²×C₅₃	C₂×C₈	1	2	yes	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	M₁₆	1	2	yes	—
54	C₄²×C₅₃	M₁₆	1	2	yes	—
55	C₄²×C₅₃	D₈	1	1	yes	characteristic
56	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
57	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂³×C₅₃	#95	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#104	1	2	yes	—
86	C₂×C₄×C₅₃	#104	1	2	yes	—
87	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
90	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#108	1	2	yes	—
93	C₈×C₅₃	—	4	4	no	—
94	#88322	—	2	8	no	—
95	#88322	—	4	4	no	—
96	#7015	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	#336	1	1	yes	characteristic
98	C₂²×C₅₃	#336	1	1	yes	characteristic
99	C₂²×C₅₃	#336	1	1	yes	characteristic
100	C₂²×C₅₃	#339	1	1	yes	characteristic
101	C₂²×C₅₃	#339	1	1	yes	characteristic
102	C₂²×C₅₃	#340	1	1	yes	characteristic
103	C₂²×C₅₃	#341	1	1	yes	characteristic
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	C₈²	1	1	yes	characteristic
108	C₄×C₅₃	#322	1	1	yes	characteristic
109	C₄×C₅₃	#322	1	1	yes	characteristic
110	C₄×C₅₃	#322	1	1	yes	characteristic
111	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
112	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
113	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
114	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
115	C₄×C₅₃	#334	1	2	yes	—
116	C₄×C₅₃	#335	1	2	yes	—
117	C₄×C₄⋊C₈	—	53	1	no	—
118	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
119	C₄²×C₈	—	53	1	no	—
120	C₂×C₅₃	#1273	1	1	yes	characteristic
121	C₂×C₅₃	#1276	1	1	yes	characteristic
122	C₂×C₅₃	#1277	1	1	yes	characteristic
123	C₂×C₅₃	#1278	1	1	yes	characteristic
124	C₂×C₅₃	#1278	1	1	yes	characteristic
125	C₂×C₅₃	#1279	1	1	yes	characteristic
126	C₂×C₅₃	#1279	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	53	2	no	—
128	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
129	C₂×C₄×C₈	—	2	1	no	—
130	C₂×C₄×C₈	—	53	2	no	—
131	C₂×C₄×C₈	—	106	1	no	—
132	C₂×C₄×C₈	—	106	1	no	—
133	C₂×C₄×C₈	—	106	1	no	—
134	C₄³	C₂×D₅₃	1	1	yes	characteristic
135	#336	—	2	1	no	—
136	#336	—	106	1	no	—
137	C₅₃	#7015	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	106	2	no	—
140	C₂²×C₈	—	106	2	no	—
141	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	53	4	no	—
150	C₄⋊C₈	—	106	2	no	—
151	C₄⋊C₈	#88322	1	4	yes	complemented
152	C₄×C₈	—	4	1	no	—
153	C₄×C₈	—	4	1	no	—
154	C₄×C₈	—	53	4	no	—
155	C₄×C₈	—	106	4	no	—
156	C₄×C₈	—	106	4	no	—
157	C₄×C₈	—	106	2	no	—
158	C₄×C₈	—	212	1	no	—
159	C₄×C₈	—	212	1	no	—
160	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
161	C₂²×C₄	#11719919	1	1	yes	characteristic
162	C₂²×C₄	#11719919	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
164	C₂²×C₄	#11719921	1	1	yes	characteristic
165	C₂²×C₄	#11719921	1	1	yes	characteristic
166	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	106	4	no	—
171	C₂×C₈	—	106	4	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	—	106	4	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	—	212	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	C₈×D₅₃	1	2	yes	—
180	C₄²	C₈×D₅₃	1	2	yes	—
181	C₄²	#11719912	1	2	yes	—
182	C₄²	#11719912	1	2	yes	—
183	C₄²	C₂×#88322	1	2	yes	—
184	C₄²	#11719917	1	2	yes	—
185	C₄²	#11719922	1	1	yes	characteristic
186	C₄²	#11719923	1	1	yes	characteristic
187	C₄²	#11719924	1	1	yes	characteristic
188	C₄²	#11719925	1	1	yes	characteristic
189	C₂³	#13133124	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
197	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
198	C₂×C₄	C₄×#88322	1	1	yes	characteristic
199	C₂×C₄	#13133127	1	1	yes	characteristic
200	C₂×C₄	#13133127	1	1	yes	characteristic
201	C₂×C₄	#13133128	1	1	yes	characteristic
202	C₂×C₄	#13133129	1	2	yes	—
203	C₂×C₄	#13133129	1	2	yes	—
204	C₂×C₄	C₂²⋊#88322	1	2	yes	—
205	C₂×C₄	#13133137	1	1	yes	characteristic
206	C₂×C₄	#13133138	1	1	yes	characteristic
207	C₂×C₄	#13133140	1	1	yes	characteristic
208	C₂×C₄	#13133141	1	1	yes	characteristic
209	C₂×C₄	#13133143	1	2	yes	—
210	C₂×C₄	#13133144	1	2	yes	—
211	C₂×C₄	#13133146	1	2	yes	—
212	C₂×C₄	#13133146	1	2	yes	—
213	C₂×C₄	C₄⋊#88322	1	2	yes	—
214	C₂×C₄	#13133148	1	1	yes	characteristic
215	C₂×C₄	#13133150	1	1	yes	characteristic
216	C₂×C₄	#13133152	1	2	yes	—
217	C₈	—	4	4	no	—
218	C₈	—	106	8	no	—
219	C₈	—	212	4	no	—
220	C₂²	#16810377	1	1	yes	characteristic
221	C₂²	#16810377	1	1	yes	characteristic
222	C₂²	#16810378	1	1	yes	characteristic
223	C₂²	#16810382	1	1	yes	characteristic
224	C₂²	#16810384	1	1	yes	characteristic
225	C₂²	#16810385	1	1	yes	characteristic
226	C₂²	#16810387	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	C₈×#88322	1	1	yes	characteristic
231	C₄	#16810344	1	1	yes	characteristic
232	C₄	#16810345	1	1	yes	characteristic
233	C₄	#16810345	1	1	yes	characteristic
234	C₄	#16810350	1	2	yes	—
235	C₄	D₄⋊#88322	1	2	yes	—
236	C₄	#16810353	1	2	yes	—
237	C₄	Q₈⋊#88322	1	2	yes	—
238	C₄	#16810373	1	2	yes	—
239	C₄	#16810375	1	2	yes	—
240	C₂	#20272542	1	1	yes	characteristic
241	C₂	#20272547	1	1	yes	characteristic
242	C₂	#20272549	1	1	yes	characteristic
243	C₂	#20272551	1	1	yes	characteristic
244	C₂	#20272553	1	1	yes	characteristic
245	C₂	#20272554	1	1	yes	characteristic
246	C₂	#20272556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₈²
Frattini	C₄³	C₂×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#7015
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. C₄⋊#88322	C₈	semidirect
23. C₄×#88322	C₈	semidirect
33. C₄⋊C₈×C₅₃	C₈	semidirect
137. C₅₃	#7015	semidirect
151. C₄⋊C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₄²×#88322	2
C₄×C₄⋊C₈×C₅₃	2
#7015	53