Subgroups — #27384631

Order 13568.

982 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	#20277122	C₂	1	2	yes	maximal
2	#20274875	C₂	1	2	yes	maximal, complemented
3	#20274843	C₂	1	2	yes	maximal, complemented
4	#20274744	C₂	1	4	yes	maximal
5	C₄×#13133151	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133176	—	2	2	no	—
8	C₂×#13133151	C₂²	1	8	yes	—
9	#16810493	C₂²	1	4	yes	—
10	#16810489	—	2	2	no	—
11	#16810483	—	2	2	no	—
12	#16810483	C₂²	1	4	yes	—
13	C₄×C₄⋊Dic₅₃	—	2	2	no	—
14	C₄×C₄⋊Dic₅₃	C₂²	1	4	yes	—
15	#16810386	C₂²	1	8	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#13133176	—	4	8	no	—
23	C₂×C₄⋊Dic₅₃	—	2	4	no	—
24	C₂×C₄⋊Dic₅₃	—	2	4	no	—
25	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
26	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
27	C₂×C₄×Dic₅₃	—	2	4	no	—
28	#13133151	—	2	16	no	—
29	#13133124	—	2	4	no	—
30	#13133124	—	2	4	no	—
31	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	—	2	8	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	D₈	1	4	yes	—
53	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	2	8	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	16	no	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₅₃	#41	1	8	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
73	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂²⋊Dic₅₃	—	4	8	no	—
75	C₄⋊Dic₅₃	—	2	16	no	—
76	C₄⋊Dic₅₃	—	4	8	no	—
77	C₄⋊Dic₅₃	—	4	8	no	—
78	C₄×Dic₅₃	—	4	8	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₅₃	C₂×D₈	1	2	yes	—
83	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
84	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	8	no	—
90	C₂×C₄×C₅₃	#107	1	16	yes	complemented
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
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	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₅₃	#135	1	4	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
104	C₂×Dic₅₃	—	4	8	no	—
105	C₂×Dic₅₃	—	4	8	no	—
106	C₂×Dic₅₃	—	4	8	no	—
107	C₈×C₅₃	—	2	4	no	—
108	C₈×C₅₃	#117	1	8	yes	complemented
109	#20817	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	C₂×#107	1	4	yes	—
115	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₅₃	#532	1	2	yes	—
117	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
118	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#135	1	2	yes	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	C₂×#107	1	4	yes	—
123	C₄×C₅₃	C₂×#107	1	4	yes	—
124	C₄×C₅₃	#427	1	4	yes	—
125	C₄×C₅₃	C₂×#117	1	2	yes	—
126	C₄×C₅₃	#533	1	2	yes	—
127	Dic₅₃	—	8	8	no	—
128	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
129	#2285	—	53	2	no	—
130	#1845	—	53	2	no	—
131	#1834	—	53	2	no	—
132	#1808	—	53	4	no	—
133	C₄×#107	—	53	4	no	—
134	C₂×C₅₃	—	2	4	no	—
135	C₂×C₅₃	#2285	1	1	yes	characteristic
136	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
138	C₂×C₅₃	#3169	1	2	yes	—
139	C₂×C₅₃	#3170	1	2	yes	—
140	C₂³×C₈	C₂×D₅₃	1	2	yes	—
141	C₂×#117	—	106	2	no	—
142	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
143	C₂×#107	—	53	8	no	—
144	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
148	#384	—	53	4	no	—
149	#382	—	106	2	no	—
150	#380	—	53	4	no	—
151	#380	—	106	2	no	—
152	C₄×C₄⋊C₄	—	53	4	no	—
153	C₄×C₄⋊C₄	—	106	2	no	—
154	#340	—	53	8	no	—
155	C₅₃	#20817	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
157	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
158	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
161	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
162	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
163	#117	—	212	8	no	—
164	C₂×C₄⋊C₄	—	53	8	no	—
165	C₂×C₄⋊C₄	—	106	4	no	—
166	C₂×C₄⋊C₄	—	106	4	no	—
167	C₂×C₂²⋊C₄	—	106	4	no	—
168	C₂×C₄²	—	106	4	no	—
169	C₂×C₄²	Dic₁₀₆	1	4	yes	—
170	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
171	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	#107	—	106	16	no	—
176	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
177	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
178	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
179	#95	—	106	4	no	—
180	#95	—	106	4	no	—
181	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	106	8	no	—
185	C₂²×C₄	D₄₂₄	1	4	yes	—
186	C₂²×C₄	Dic₂₁₂	1	4	yes	—
187	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
188	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
189	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
190	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
196	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	16	no	—
199	C₂×C₈	#11719945	1	8	yes	—
200	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
201	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
202	C₄⋊C₄	—	106	16	no	—
203	C₄⋊C₄	—	212	8	no	—
204	C₄⋊C₄	—	212	8	no	—
205	C₂²⋊C₄	—	212	8	no	—
206	C₄²	—	212	8	no	—
207	C₄²	C₄⋊Dic₅₃	1	4	yes	—
208	C₄²	C₄⋊Dic₅₃	1	4	yes	—
209	C₄²	C₂×Dic₁₀₆	1	2	yes	—
210	C₄²	C₂×Dic₁₀₆	1	2	yes	—
211	C₄²	C₂²×Dic₅₃	1	2	yes	—
212	C₄²	C₂²×Dic₅₃	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
216	C₂³	C₂×D₄₂₄	1	2	yes	—
217	C₂³	C₂×Dic₂₁₂	1	2	yes	—
218	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	212	8	no	—
225	C₂×C₄	—	212	8	no	—
226	C₂×C₄	—	212	8	no	—
227	C₂×C₄	#13133151	1	16	yes	complemented
228	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
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	2	yes	—
233	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
234	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
235	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
236	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
237	C₂×C₄	#13133239	1	4	yes	—
238	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
240	C₂×C₄	C₂×#11719945	1	4	yes	—
241	C₈	—	2	4	no	—
242	C₈	#13133176	1	8	yes	complemented
243	C₂²	—	2	4	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	C₂×#13133151	1	4	yes	—
248	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
249	C₂²	#16811173	1	2	yes	—
250	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
251	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
252	C₂²	C₂×#13133239	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	424	8	no	—
256	C₄	C₂×#13133151	1	4	yes	—
257	C₄	C₂×#13133151	1	4	yes	—
258	C₄	#16810661	1	4	yes	—
259	C₄	C₂×#13133176	1	2	yes	—
260	C₄	#16811180	1	2	yes	—
261	C₂	—	2	4	no	—
262	C₂	#20277122	1	1	yes	characteristic
263	C₂	C₂²×#13133151	1	1	yes	characteristic
264	C₂	C₂×#16810661	1	1	yes	characteristic
265	C₂	#20285294	1	2	yes	—
266	C₂	#20285301	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#20817
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20274875	C₂	semidirect
3. #20274843	C₂	semidirect
5. C₄×#13133151	C₂	semidirect
18. C₂×C₄×C₈×C₅₃	C₄	semidirect
90. C₂×C₄×C₅₃	#107	semidirect
108. C₈×C₅₃	#117	semidirect
144. C₂×C₄×C₈	Dic₅₃	semidirect
155. C₅₃	#20817	semidirect
227. C₂×C₄	#13133151	semidirect
242. C₈	#13133176	semidirect

Maximal subgroups

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

subgroup	index
#20277122	2
#20274875	2
#20274843	2
#20274744	2
C₄×#13133151	2
C₂²×C₄×C₈×C₅₃	2
#20817	53