Subgroups — #27270636

Order 13568.

320 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	1	no	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄×#88322	C₄	1	2	yes	—
7	#16810378	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
11	C₂×#97×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	4	no	—
17	C₄⋊#88322	—	4	1	no	—
18	C₄⋊#88322	—	4	1	no	—
19	C₄×#88322	—	4	2	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	2	no	—
29	C₄⋊C₈×C₅₃	C₈	1	4	yes	complemented
30	#97×C₅₃	—	4	1	no	—
31	#97×C₅₃	—	4	1	no	—
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₄²×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	4	1	no	—
46	C₄²×C₅₃	—	4	1	no	—
47	C₄²×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	C₂×C₈	1	2	yes	—
49	C₄²×C₅₃	M₁₆	1	2	yes	—
50	C₄²×C₅₃	D₈	1	1	yes	characteristic
51	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂³×C₅₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
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	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#104	1	2	yes	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
87	C₈×C₅₃	—	4	4	no	—
88	#88322	—	4	4	no	—
89	#88322	—	8	2	no	—
90	#7022	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	#336	1	1	yes	characteristic
92	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
93	C₂²×C₅₃	#338	1	1	yes	characteristic
94	C₂²×C₅₃	#339	1	1	yes	characteristic
95	C₂²×C₅₃	#340	1	1	yes	characteristic
96	C₂²×C₅₃	#343	1	1	yes	characteristic
97	C₂²×C₅₃	#344	1	1	yes	characteristic
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	4	1	no	—
108	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
109	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
110	#1831	—	53	1	no	—
111	#1767	D₅₃	1	1	yes	characteristic
112	C₄×C₄⋊C₈	—	53	1	no	—
113	C₂×C₅₃	#1275	1	1	yes	characteristic
114	C₂×C₅₃	#1279	1	1	yes	characteristic
115	C₂×C₅₃	#1283	1	1	yes	characteristic
116	C₂×C₅₃	#1285	1	1	yes	characteristic
117	C₂×C₅₃	#1286	1	1	yes	characteristic
118	C₂×C₅₃	#1291	1	1	yes	characteristic
119	C₂×C₅₃	#1292	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	106	1	no	—
121	C₂×C₄⋊C₈	—	106	1	no	—
122	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
123	C₂×#97	—	2	1	no	—
124	C₂×C₄×C₈	—	53	2	no	—
125	C₄³	C₂×D₅₃	1	1	yes	characteristic
126	#336	—	2	1	no	—
127	#336	—	53	2	no	—
128	#336	—	106	1	no	—
129	#336	—	106	1	no	—
130	C₅₃	#7022	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	106	2	no	—
133	C₂²×C₈	—	106	2	no	—
134	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
135	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
138	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	106	4	no	—
143	C₄⋊C₈	—	212	1	no	—
144	C₄⋊C₈	—	212	1	no	—
145	C₄⋊C₈	#88322	1	4	yes	complemented
146	#97	—	4	1	no	—
147	#97	—	4	1	no	—
148	C₄×C₈	—	212	2	no	—
149	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
150	C₂²×C₄	#11719919	1	1	yes	characteristic
151	C₂²×C₄	#11719919	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₂²×C₄	#11719921	1	1	yes	characteristic
154	C₂²×C₄	#11719921	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	—	212	2	no	—
162	C₂×C₈	—	212	2	no	—
163	C₂×C₈	—	212	2	no	—
164	C₂×C₈	—	212	2	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	4	1	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×#88322	1	2	yes	—
172	C₄²	#11719917	1	2	yes	—
173	C₄²	#11719922	1	1	yes	characteristic
174	C₄²	#11719923	1	1	yes	characteristic
175	C₄²	#11719924	1	1	yes	characteristic
176	C₄²	#11719925	1	1	yes	characteristic
177	C₂³	#13133124	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	C₄×#88322	1	1	yes	characteristic
192	C₂×C₄	#13133128	1	1	yes	characteristic
193	C₂×C₄	C₂²⋊#88322	1	2	yes	—
194	C₂×C₄	#13133133	1	1	yes	characteristic
195	C₂×C₄	#13133135	1	1	yes	characteristic
196	C₂×C₄	#13133137	1	1	yes	characteristic
197	C₂×C₄	#13133138	1	1	yes	characteristic
198	C₂×C₄	#13133140	1	1	yes	characteristic
199	C₂×C₄	#13133141	1	1	yes	characteristic
200	C₂×C₄	#13133144	1	2	yes	—
201	C₂×C₄	C₄⋊#88322	1	2	yes	—
202	C₂×C₄	#13133148	1	1	yes	characteristic
203	C₂×C₄	#13133150	1	1	yes	characteristic
204	C₈	—	4	4	no	—
205	C₈	—	424	2	no	—
206	C₈	—	212	4	no	—
207	C₂²	#16810378	1	1	yes	characteristic
208	C₂²	#16810380	1	1	yes	characteristic
209	C₂²	#16810381	1	1	yes	characteristic
210	C₂²	#16810382	1	1	yes	characteristic
211	C₂²	#16810385	1	1	yes	characteristic
212	C₂²	#16810391	1	1	yes	characteristic
213	C₂²	#16810395	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	1	no	—
224	C₄	D₄⋊#88322	1	2	yes	—
225	C₄	Q₈⋊#88322	1	2	yes	—
226	C₂	#20272546	1	1	yes	characteristic
227	C₂	#20272556	1	1	yes	characteristic
228	C₂	#20272566	1	1	yes	characteristic
229	C₂	#20272572	1	1	yes	characteristic
230	C₂	#20272575	1	1	yes	characteristic
231	C₂	#20272586	1	1	yes	characteristic
232	C₂	#20272589	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1767	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1767	D₅₃
hypocenter	C₅₃	#7022
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
29. C₄⋊C₈×C₅₃	C₈	semidirect
130. C₅₃	#7022	semidirect
145. 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
#20274832	2
C₄×C₄⋊#88322	2
C₅₃×#1767	2
#7022	53