Subgroups — #27353246

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20273495	C₂	1	8	yes	maximal, complemented
3	#20273474	C₂	1	2	yes	maximal, complemented
4	#20273471	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
6	#16810680	—	2	2	no	—
7	#16810680	C₂²	1	4	yes	—
8	#16810676	—	2	2	no	—
9	#16810676	C₂²	1	4	yes	—
10	#16810647	C₂²	1	4	yes	—
11	#16810587	—	2	2	no	—
12	C₄×#11719917	—	2	2	no	—
13	#16810376	C₂²	1	8	yes	complemented
14	#16810374	C₂²	1	8	yes	complemented
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#431	C₄	1	4	yes	—
17	C₅₃×#431	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×#11719917	—	2	4	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	C₂³	1	8	yes	—
25	C₂²⋊#88322	—	2	4	no	—
26	C₂²⋊#88322	—	2	4	no	—
27	#13133128	—	2	4	no	—
28	C₄×#88322	—	2	4	no	—
29	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
42	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
43	C₂²⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄×C₈×C₅₃	C₂³	1	2	yes	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	2	yes	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
78	#11719917	—	4	8	no	—
79	C₂×#88322	—	2	8	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	4	1	no	—
82	C₂³×C₅₃	—	4	1	no	—
83	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#135	1	4	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	—	2	4	no	—
108	#88322	—	4	8	no	—
109	#17815	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	#427	1	2	yes	—
117	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	#427	1	2	yes	—
124	C₄×C₅₃	#430	1	2	yes	—
125	C₄×C₅₃	#568	1	2	yes	—
126	C₄×C₅₃	C₂×#135	1	2	yes	—
127	#2920	—	53	2	no	—
128	C₂×#431	D₅₃	1	1	yes	characteristic
129	#1572	—	53	8	no	—
130	#1567	—	53	2	no	—
131	#1566	—	53	2	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	#2905	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
136	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
137	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
138	#432	—	53	4	no	—
139	#432	—	106	2	no	—
140	#431	—	53	4	no	—
141	#431	—	106	2	no	—
142	#431	Dic₅₃	1	4	yes	—
143	#431	C₂×D₅₃	1	4	yes	—
144	#423	—	53	4	no	—
145	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
146	#407	—	106	2	no	—
147	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
148	C₄×M₁₆	—	106	2	no	—
149	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	#335	—	53	8	no	—
152	#334	—	53	8	no	—
153	C₅₃	#17815	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
155	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂×M₁₆	—	106	4	no	—
158	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
160	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
161	C₂×C₄²	Dic₁₀₆	1	2	yes	—
162	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
163	C₂×C₄²	D₂₁₂	1	2	yes	—
164	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
168	C₄⋊C₈	—	53	8	no	—
169	C₄⋊C₈	—	106	4	no	—
170	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
171	C₂²⋊C₈	—	106	4	no	—
172	C₂²⋊C₈	—	106	4	no	—
173	C₂²⋊C₈	C₂×Dic₅₃	1	8	yes	—
174	#97	—	106	4	no	—
175	C₄×C₈	—	106	4	no	—
176	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
177	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
178	C₄×C₈	C₂²×D₅₃	1	2	yes	—
179	C₄×C₈	C₂²×D₅₃	1	2	yes	—
180	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
188	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
192	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
193	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
196	M₁₆	—	212	8	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	106	8	no	—
200	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
201	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
202	C₄²	C₄⋊Dic₅₃	1	2	yes	—
203	C₄²	C₄⋊Dic₅₃	1	2	yes	—
204	C₄²	C₄⋊Dic₅₃	1	2	yes	—
205	C₄²	C₂×Dic₁₀₆	1	2	yes	—
206	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₄²	C₂×D₂₁₂	1	2	yes	—
208	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
209	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₄²	C₂²×Dic₅₃	1	2	yes	—
211	C₄²	C₂³×D₅₃	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	#13133239	1	4	yes	—
235	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₈	—	212	8	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#16810661	1	2	yes	—
248	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	#16810661	1	2	yes	—
255	C₄	#16810672	1	2	yes	—
256	C₄	#16811390	1	2	yes	—
257	C₄	C₂×#13133239	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	#20283279	1	1	yes	characteristic
261	C₂	C₂×#16810661	1	1	yes	characteristic
262	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	#17815
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. #20273495	C₂	semidirect
3. #20273474	C₂	semidirect
4. #20273471	C₂	semidirect
13. #16810376	C₂²	semidirect
14. #16810374	C₂²	semidirect
153. C₅₃	#17815	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20273495	2
#20273474	2
#20273471	2
C₂×C₅₃×#431	2
#17815	53