Subgroups — #27283132

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₂×#16810372	C₂	1	2	yes	maximal
4	C₂×#16810369	C₂	1	2	yes	maximal
5	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	2	no	—
11	#16810372	—	2	4	no	—
12	#16810369	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	—	2	2	no	—
16	C₅₃×#391	C₄	1	4	yes	—
17	C₅₃×#382	—	2	1	no	—
18	C₅₃×#382	—	2	1	no	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	D₄	1	4	yes	—
26	C₄×#88322	—	4	2	no	—
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#95×C₅₃	—	2	4	no	—
43	#95×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₅₃	—	2	4	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	1	no	—
60	C₄⋊C₄×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	—	2	4	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	4	2	no	—
64	C₄²×C₅₃	SD₁₆	1	2	yes	—
65	C₄²×C₅₃	Q₁₆	1	2	yes	—
66	C₄²×C₅₃	C₂×D₄	1	2	yes	—
67	C₄²×C₅₃	#41	1	2	yes	—
68	C₂×#88322	—	2	4	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	4	no	—
73	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	Q₈×C₅₃	—	8	4	no	—
75	Q₈×C₅₃	—	8	2	no	—
76	Q₈×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	#103	1	4	yes	—
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	#121	1	2	yes	—
90	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#135	1	4	yes	—
95	C₂×C₄×C₅₃	#137	1	2	yes	—
96	#88322	—	4	4	no	—
97	#88322	—	8	2	no	—
98	#9693	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	#390	1	1	yes	characteristic
100	C₂²×C₅₃	#413	1	2	yes	—
101	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
102	C₂²×C₅₃	#416	1	2	yes	—
103	C₂²×C₅₃	#419	1	1	yes	characteristic
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	2	no	—
107	C₄×C₅₃	—	8	2	no	—
108	C₄×C₅₃	—	8	2	no	—
109	C₄×C₅₃	#461	1	2	yes	—
110	C₄×C₅₃	#462	1	2	yes	—
111	C₄×C₅₃	#488	1	2	yes	—
112	C₄×C₅₃	#495	1	2	yes	—
113	#2862	D₅₃	1	1	yes	characteristic
114	#1831	—	53	1	no	—
115	C₄×C₄⋊C₈	—	53	1	no	—
116	C₂×#333	—	53	2	no	—
117	C₂×#332	—	53	2	no	—
118	C₂×C₅₃	#1542	1	2	yes	—
119	C₂×C₅₃	#1554	1	2	yes	—
120	C₂×C₅₃	#1958	1	1	yes	characteristic
121	C₂×C₅₃	#1967	1	1	yes	characteristic
122	C₂×C₅₃	#1981	1	1	yes	characteristic
123	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	53	2	no	—
126	C₂×C₄⋊C₈	—	53	2	no	—
127	C₂×C₄×C₈	—	106	1	no	—
128	#391	—	2	2	no	—
129	#391	Dic₅₃	1	4	yes	—
130	#382	—	2	1	no	—
131	#382	—	2	1	no	—
132	C₄³	C₂×D₅₃	1	1	yes	characteristic
133	#336	—	106	2	no	—
134	#336	—	106	1	no	—
135	#333	—	106	4	no	—
136	#332	—	106	4	no	—
137	C₅₃	#9693	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
140	C₂²×C₈	—	106	2	no	—
141	C₂²×C₈	—	106	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
150	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
151	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	53	4	no	—
156	C₄⋊C₈	—	106	2	no	—
157	C₄⋊C₈	—	106	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄×C₈	—	212	2	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₈	—	106	4	no	—
175	C₂×C₈	—	212	2	no	—
176	C₂×C₈	—	212	4	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	—	212	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	#11719924	1	2	yes	—
188	C₄²	#11719925	1	2	yes	—
189	C₄²	D₄×D₅₃	1	2	yes	—
190	C₄²	#11719947	1	2	yes	—
191	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
192	Q₈	—	8	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	#13133141	1	4	yes	—
206	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
207	C₂×C₄	#13133188	1	2	yes	—
208	C₂×C₄	#13133204	1	1	yes	characteristic
209	C₂×C₄	#13133213	1	1	yes	characteristic
210	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
211	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
212	C₂×C₄	#13133237	1	4	yes	—
213	C₂×C₄	#13133248	1	2	yes	—
214	C₈	—	424	2	no	—
215	C₈	—	212	4	no	—
216	C₂²	#16810526	1	1	yes	characteristic
217	C₂²	#16810607	1	2	yes	—
218	C₂²	C₂×#13133141	1	1	yes	characteristic
219	C₂²	#16810619	1	2	yes	—
220	C₂²	#16810632	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#16810825	1	2	yes	—
227	C₄	#16810830	1	2	yes	—
228	C₄	#16810977	1	2	yes	—
229	C₄	#16811005	1	2	yes	—
230	C₂	#20273372	1	2	yes	—
231	C₂	#20273425	1	2	yes	—
232	C₂	#20275527	1	1	yes	characteristic
233	C₂	#20275565	1	1	yes	characteristic
234	C₂	#20275639	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2862	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2862	D₅₃
hypocenter	C₅₃	#9693
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
137. C₅₃	#9693	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
C₂×#16810372	2
C₂×#16810369	2
C₅₃×#2862	2
#9693	53