Subgroups — #27353237

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	#20273486	C₂	1	8	yes	maximal, complemented
3	#20273471	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	2	no	—
6	#16810647	—	2	2	no	—
7	#16810647	—	2	2	no	—
8	#16810647	C₂²	1	4	yes	—
9	#16810647	C₂²	1	4	yes	—
10	C₂×C₄⋊#88322	—	2	2	no	—
11	C₂×C₄⋊#88322	C₂²	1	4	yes	—
12	#16810374	C₂²	1	16	yes	complemented
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#431	C₄	1	4	yes	—
15	C₅₃×#431	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	4	no	—
21	C₂×#11719917	—	4	8	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	—	2	4	no	—
26	C₄⋊#88322	C₂³	1	8	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
40	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂³	1	4	yes	—
43	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₃	—	4	4	no	—
60	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
61	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
66	C₄²×C₅₃	C₂×D₄	1	2	yes	—
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
69	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
71	#11719917	—	4	8	no	—
72	#11719917	—	4	8	no	—
73	C₂×#88322	—	2	8	no	—
74	C₂×#88322	—	4	8	no	—
75	C₂³×C₅₃	—	4	1	no	—
76	C₂³×C₅₃	—	4	1	no	—
77	C₂³×C₅₃	#106	1	4	yes	—
78	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	1	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	#106	1	4	yes	—
89	C₂×C₄×C₅₃	#106	1	8	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₈×C₅₃	—	2	8	no	—
103	#88322	—	4	8	no	—
104	#17813	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	4	4	no	—
110	C₂²×C₅₃	C₂×#106	1	4	yes	—
111	C₂²×C₅₃	C₂×#106	1	2	yes	—
112	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	2	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	C₂×#106	1	4	yes	—
118	C₄×C₅₃	C₂×#106	1	2	yes	—
119	C₄×C₅₃	#430	1	2	yes	—
120	C₄×C₅₃	#568	1	2	yes	—
121	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
122	C₂×#431	D₅₃	1	1	yes	characteristic
123	C₂×#423	—	53	2	no	—
124	#1570	—	53	8	no	—
125	#1566	—	53	4	no	—
126	C₂×C₅₃	—	2	4	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	#2905	1	1	yes	characteristic
129	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
131	C₂²×M₁₆	—	106	2	no	—
132	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
133	#431	Dic₅₃	1	4	yes	—
134	#431	C₂×D₅₃	1	4	yes	—
135	#423	—	53	4	no	—
136	#423	—	53	4	no	—
137	#423	—	106	2	no	—
138	#423	—	106	2	no	—
139	C₂×C₄⋊C₈	—	53	4	no	—
140	C₂×C₄⋊C₈	—	106	2	no	—
141	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
142	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
143	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
144	#334	—	53	16	no	—
145	C₅₃	#17813	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
147	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
148	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂×M₁₆	—	106	4	no	—
150	C₂×M₁₆	—	106	4	no	—
151	C₂×M₁₆	—	212	8	no	—
152	C₂²×C₈	—	106	4	no	—
153	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
154	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
155	C₂×C₄²	Dic₁₀₆	1	2	yes	—
156	C₂×C₄²	D₂₁₂	1	2	yes	—
157	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
162	C₄⋊C₈	—	53	8	no	—
163	C₄⋊C₈	—	106	4	no	—
164	C₄⋊C₈	—	106	4	no	—
165	C₄⋊C₈	—	106	4	no	—
166	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
167	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
168	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
169	C₄×C₈	C₂²×D₅₃	1	4	yes	—
170	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	#11719913	1	8	yes	—
174	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
175	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
176	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
180	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
182	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
183	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
186	M₁₆	—	212	8	no	—
187	M₁₆	—	212	8	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	106	8	no	—
190	C₂×C₈	—	212	8	no	—
191	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
192	C₄²	C₄⋊Dic₅₃	1	2	yes	—
193	C₄²	C₄⋊Dic₅₃	1	2	yes	—
194	C₄²	C₄⋊Dic₅₃	1	2	yes	—
195	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
196	C₄²	C₂×Dic₁₀₆	1	2	yes	—
197	C₄²	C₂×D₂₁₂	1	2	yes	—
198	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
199	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
200	C₄²	C₂²×Dic₅₃	1	2	yes	—
201	C₄²	C₂³×D₅₃	1	1	yes	characteristic
202	C₂³	—	4	1	no	—
203	C₂³	—	4	1	no	—
204	C₂³	#13133149	1	4	yes	—
205	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	#13133149	1	4	yes	—
216	C₂×C₄	#13133149	1	8	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719913	1	4	yes	—
224	C₂×C₄	C₂×#11719913	1	4	yes	—
225	C₂×C₄	C₂×#11719913	1	4	yes	—
226	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
228	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
229	C₈	—	2	8	no	—
230	C₈	—	212	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	4	no	—
236	C₂²	C₂×#13133149	1	4	yes	—
237	C₂²	C₂×#13133149	1	2	yes	—
238	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	C₂×#13133149	1	4	yes	—
244	C₄	C₂×#13133149	1	2	yes	—
245	C₄	#16810672	1	2	yes	—
246	C₄	#16811390	1	2	yes	—
247	C₄	C₂²×#11719913	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	4	no	—
250	C₂	#20283279	1	1	yes	characteristic
251	C₂	C₂²×#13133149	1	1	yes	characteristic
252	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #20273486	C₂	semidirect
3. #20273471	C₂	semidirect
12. #16810374	C₂²	semidirect
145. C₅₃	#17813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
#20273486	2
#20273471	2
C₂×C₅₃×#431	2
#17813	53