Subgroups — #27283333

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	#20275667	C₂	1	2	yes	maximal
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810370	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	—	2	1	no	—
7	C₂²×Dic₂₁₂	—	2	1	no	—
8	C₂×#13133217	C₂²	1	2	yes	—
9	#16811003	—	2	2	no	—
10	#16811003	C₄	1	4	yes	complemented
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	#16810385	—	2	4	no	—
13	#16810376	—	2	2	no	—
14	#16810370	—	2	4	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#336	—	2	1	no	—
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	4	no	—
22	C₂×Dic₂₁₂	—	2	4	no	—
23	C₂×Dic₂₁₂	—	4	2	no	—
24	C₂×Dic₂₁₂	—	4	2	no	—
25	C₂×Dic₂₁₂	—	4	2	no	—
26	C₂×Dic₂₁₂	—	4	2	no	—
27	C₂²×#88322	—	2	2	no	—
28	#13133217	—	2	2	no	—
29	#13133217	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊#88322	—	2	2	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	—	2	1	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₅₃	—	2	1	no	—
43	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	—	4	2	no	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	#95×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	4	2	no	—
53	C₂×Dic₁₀₆	—	2	4	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×Dic₁₀₆	—	4	2	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	4	1	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	D₈	1	2	yes	—
67	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
68	C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₄⋊C₄×C₅₃	Q₁₆	1	4	yes	complemented
70	C₄²×C₅₃	—	4	1	no	—
71	C₄²×C₅₃	—	4	1	no	—
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₃	—	2	4	no	—
77	C₄⋊Dic₅₃	—	2	4	no	—
78	C₄⋊Dic₅₃	—	4	2	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	4	no	—
82	Dic₂₁₂	—	4	8	no	—
83	Dic₂₁₂	—	4	8	no	—
84	C₂³×C₅₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	#135	1	2	yes	—
99	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×Dic₅₃	—	4	4	no	—
102	C₂×Dic₅₃	—	8	2	no	—
103	C₂×Dic₅₃	—	8	2	no	—
104	Dic₁₀₆	—	4	4	no	—
105	Dic₁₀₆	—	4	4	no	—
106	Dic₁₀₆	—	8	2	no	—
107	Dic₁₀₆	—	8	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	—	4	2	no	—
110	C₈×C₅₃	#102	1	4	yes	—
111	#88322	—	8	2	no	—
112	#9733	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	#390	1	1	yes	characteristic
114	C₂²×C₅₃	#459	1	1	yes	characteristic
115	C₂²×C₅₃	#460	1	1	yes	characteristic
116	C₂²×C₅₃	#467	1	2	yes	—
117	C₂²×C₅₃	#473	1	2	yes	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	4	4	no	—
121	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
122	C₄×C₅₃	#418	1	1	yes	characteristic
123	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
124	C₄×C₅₃	#459	1	1	yes	characteristic
125	C₄×C₅₃	#460	1	1	yes	characteristic
126	C₄×C₅₃	#494	1	1	yes	characteristic
127	C₄×C₅₃	#495	1	1	yes	characteristic
128	Dic₅₃	—	8	4	no	—
129	C₂×#494	—	53	1	no	—
130	#1988	—	53	2	no	—
131	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
132	C₂×#335	—	53	1	no	—
133	C₂×#332	—	53	2	no	—
134	C₂×C₅₃	#1669	1	2	yes	—
135	C₂×C₅₃	#1670	1	2	yes	—
136	C₂×C₅₃	#1969	1	1	yes	characteristic
137	C₂×C₅₃	#1979	1	1	yes	characteristic
138	C₂×C₅₃	#1982	1	1	yes	characteristic
139	C₂²×Q₁₆	—	106	1	no	—
140	C₂²×Q₁₆	—	106	1	no	—
141	C₂×C₄⋊Q₈	—	53	2	no	—
142	#494	—	53	4	no	—
143	#494	—	106	2	no	—
144	C₂×C₄⋊C₈	—	53	2	no	—
145	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂×C₄×C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
149	#340	—	106	4	no	—
150	#336	—	2	1	no	—
151	#335	—	106	2	no	—
152	#332	—	106	4	no	—
153	C₅₃	#9733	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	106	2	no	—
155	C₂×Q₁₆	—	106	4	no	—
156	C₂×Q₁₆	—	106	4	no	—
157	C₂×Q₁₆	—	212	2	no	—
158	C₂×Q₁₆	—	212	2	no	—
159	C₂×Q₁₆	—	212	2	no	—
160	C₂×Q₁₆	—	212	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	106	2	no	—
163	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	53	4	no	—
166	C₄⋊Q₈	—	106	2	no	—
167	C₄⋊Q₈	—	106	2	no	—
168	C₄⋊Q₈	—	106	2	no	—
169	C₂×C₄⋊C₄	—	106	2	no	—
170	C₂×C₄⋊C₄	—	106	2	no	—
171	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	106	2	no	—
176	C₄⋊C₈	—	106	2	no	—
177	C₄⋊C₈	D₂₁₂	1	2	yes	—
178	C₄×C₈	—	4	2	no	—
179	C₄×C₈	C₄×D₅₃	1	2	yes	—
180	C₄×C₈	D₂₁₂	1	2	yes	—
181	#95	—	2	1	no	—
182	C₂×Q₈	—	106	4	no	—
183	C₂×Q₈	—	212	2	no	—
184	C₂×Q₈	—	212	2	no	—
185	C₂×Q₈	—	212	2	no	—
186	C₂×Q₈	—	212	2	no	—
187	C₂×Q₈	—	212	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	212	2	no	—
190	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
191	C₂²×C₄	#11719945	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	Q₁₆	—	212	8	no	—
194	Q₁₆	—	212	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	—	212	4	no	—
202	C₂×C₈	#11719921	1	2	yes	—
203	C₂×C₈	#11719921	1	2	yes	—
204	C₂×C₈	#11719922	1	2	yes	—
205	C₂×C₈	#11719924	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	106	4	no	—
208	C₄⋊C₄	—	106	4	no	—
209	C₄⋊C₄	—	212	2	no	—
210	C₄⋊C₄	—	212	2	no	—
211	C₄⋊C₄	Dic₂₁₂	1	4	yes	complemented
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
216	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
217	C₄²	#11719945	1	1	yes	characteristic
218	C₂³	#13133191	1	1	yes	characteristic
219	Q₈	—	424	2	no	—
220	Q₈	—	424	2	no	—
221	Q₈	—	212	4	no	—
222	Q₈	—	212	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	424	2	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	—	212	4	no	—
230	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
231	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
232	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
233	C₂×C₄	#13133191	1	1	yes	characteristic
234	C₂×C₄	#13133205	1	1	yes	characteristic
235	C₂×C₄	#13133214	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
238	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
239	C₂×C₄	#13133239	1	2	yes	—
240	C₂×C₄	#13133242	1	1	yes	characteristic
241	C₂×C₄	#13133242	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	424	2	no	—
245	C₈	#13133137	1	4	yes	—
246	C₂²	#16810525	1	1	yes	characteristic
247	C₂²	#16810818	1	1	yes	characteristic
248	C₂²	#16810823	1	1	yes	characteristic
249	C₂²	#16810864	1	2	yes	—
250	C₂²	#16810898	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	424	4	no	—
255	C₄	C₂×#13133137	1	1	yes	characteristic
256	C₄	#16810627	1	1	yes	characteristic
257	C₄	C₄×Dic₂₁₂	1	2	yes	—
258	C₄	#16810818	1	1	yes	characteristic
259	C₄	#16810823	1	1	yes	characteristic
260	C₄	#16811003	1	1	yes	characteristic
261	C₄	#16811007	1	1	yes	characteristic
262	C₂	#20274080	1	2	yes	—
263	C₂	#20274085	1	2	yes	—
264	C₂	#20275576	1	1	yes	characteristic
265	C₂	#20275630	1	1	yes	characteristic
266	C₂	#20275642	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#9733
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #16811003	C₄	semidirect
69. C₄⋊C₄×C₅₃	Q₁₆	semidirect
153. C₅₃	#9733	semidirect
211. C₄⋊C₄	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811003	2
#20275667	2
C₂×#16810376	2
C₂×#16810370	2
C₈×C₄⋊C₄×C₅₃	2
#9733	53