Subgroups — #27277021

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20274708	C₂	1	2	yes	maximal
3	#20273474	C₂	1	1	yes	maximal, characteristic
4	#20272589	C₂	1	2	yes	maximal
5	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₀₆	—	2	1	no	—
8	C₈×Dic₁₀₆	—	2	1	no	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	—	2	1	no	—
11	#16810647	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×#13133146	C₄	1	2	yes	—
14	C₂×#13133135	—	2	2	no	—
15	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
16	#16810391	—	2	2	no	—
17	#16810376	—	2	1	no	—
18	#16810376	—	2	1	no	—
19	C₅₃×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#335	—	2	1	no	—
22	C₅₃×#335	—	2	1	no	—
23	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
24	C₂×#11719917	—	2	2	no	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	C₄×Dic₁₀₆	—	2	1	no	—
29	C₄×Dic₁₀₆	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×#11719919	Q₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊#88322	—	2	1	no	—
34	C₄⋊#88322	—	2	1	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	#13133146	—	2	2	no	—
38	#13133146	—	2	2	no	—
39	#13133135	—	4	4	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₈×Dic₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₁₀₆	—	2	1	no	—
58	C₂×Dic₁₀₆	—	2	1	no	—
59	C₂×Dic₁₀₆	—	2	1	no	—
60	C₂×Dic₁₀₆	—	2	1	no	—
61	C₂×Dic₁₀₆	D₈	1	2	yes	—
62	C₂×Dic₁₀₆	Q₁₆	1	2	yes	—
63	M₁₆×C₅₃	—	4	2	no	—
64	M₁₆×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	4	1	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₃	—	2	1	no	—
77	C₄⋊Dic₅₃	—	2	1	no	—
78	#11719919	—	2	2	no	—
79	#11719919	—	2	2	no	—
80	C₄×Dic₅₃	—	2	2	no	—
81	C₄×Dic₅₃	—	2	2	no	—
82	#11719917	—	4	2	no	—
83	#11719917	—	4	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	4	2	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	2	no	—
102	C₂×Dic₅₃	—	2	2	no	—
103	C₂×Dic₅₃	—	2	2	no	—
104	C₂×Dic₅₃	—	4	2	no	—
105	C₂×Dic₅₃	#107	1	4	yes	—
106	Dic₁₀₆	—	2	2	no	—
107	Dic₁₀₆	—	2	2	no	—
108	Dic₁₀₆	—	2	2	no	—
109	Dic₁₀₆	—	2	2	no	—
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	—	2	1	no	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	8	2	no	—
114	C₈×C₅₃	#101	1	2	yes	complemented
115	#88322	—	4	2	no	—
116	#88322	—	8	2	no	—
117	#8617	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	#380	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
122	C₂²×C₅₃	#427	1	1	yes	characteristic
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
127	C₄×C₅₃	#413	1	1	yes	characteristic
128	C₄×C₅₃	#466	1	1	yes	characteristic
129	C₄×C₅₃	#466	1	1	yes	characteristic
130	C₄×C₅₃	#467	1	1	yes	characteristic
131	C₄×C₅₃	#467	1	1	yes	characteristic
132	Dic₅₃	—	2	4	no	—
133	Dic₅₃	—	4	2	no	—
134	C₂×C₈×Q₈	—	53	1	no	—
135	#1801	—	53	2	no	—
136	#1567	—	53	1	no	—
137	#1567	D₅₃	1	1	yes	characteristic
138	#1292	—	53	2	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	#1801	1	1	yes	characteristic
142	C₂×C₅₃	#1808	1	1	yes	characteristic
143	C₂×C₅₃	#1813	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	53	1	no	—
145	C₈×Q₈	—	106	1	no	—
146	C₈×Q₈	—	106	1	no	—
147	C₈×Q₈	—	106	1	no	—
148	C₈×Q₈	—	106	1	no	—
149	#423	—	53	2	no	—
150	#423	C₂×D₅₃	1	2	yes	—
151	C₂×C₄⋊C₈	—	53	2	no	—
152	C₂×C₄⋊C₈	—	53	1	no	—
153	C₂×#101	—	106	2	no	—
154	C₂×C₄×C₈	—	53	2	no	—
155	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
156	#343	—	106	2	no	—
157	#335	—	2	1	no	—
158	#335	—	2	1	no	—
159	#335	—	106	1	no	—
160	#335	—	106	1	no	—
161	C₅₃	#8617	1	1	yes	characteristic, complemented, Sylow
162	C₂²×Q₈	—	53	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	106	2	no	—
165	C₂²×C₈	—	53	2	no	—
166	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
168	C₄×Q₈	—	106	1	no	—
169	C₄×Q₈	—	106	1	no	—
170	C₄×Q₈	—	106	1	no	—
171	C₄×Q₈	—	106	1	no	—
172	C₂×C₄⋊C₄	—	53	1	no	—
173	C₂×C₄⋊C₄	—	53	2	no	—
174	C₂×C₄²	—	53	2	no	—
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	106	2	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	—	106	1	no	—
181	C₄⋊C₈	—	106	2	no	—
182	C₄⋊C₈	—	106	2	no	—
183	C₄⋊C₈	—	106	1	no	—
184	#101	—	212	4	no	—
185	C₄×C₈	—	106	2	no	—
186	C₄×C₈	—	106	2	no	—
187	C₄×C₈	D₂₁₂	1	1	yes	characteristic
188	C₄×C₈	D₂₁₂	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×Q₈	—	53	2	no	—
192	C₂×Q₈	—	53	2	no	—
193	C₂×Q₈	—	106	1	no	—
194	C₂×Q₈	—	106	1	no	—
195	C₂×Q₈	—	106	1	no	—
196	C₂×Q₈	—	106	1	no	—
197	C₂²×C₄	—	53	2	no	—
198	C₂²×C₄	—	53	2	no	—
199	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	212	2	no	—
205	M₁₆	—	212	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	106	2	no	—
210	C₂×C₈	—	212	2	no	—
211	C₂×C₈	—	212	2	no	—
212	C₂×C₈	#11719921	1	1	yes	characteristic
213	C₂×C₈	#11719921	1	1	yes	characteristic
214	C₂×C₈	#11719921	1	1	yes	characteristic
215	C₂×C₈	#11719921	1	1	yes	characteristic
216	C₄⋊C₄	—	106	1	no	—
217	C₄⋊C₄	—	106	1	no	—
218	C₄⋊C₄	—	106	2	no	—
219	C₄⋊C₄	—	106	2	no	—
220	C₄²	—	106	2	no	—
221	C₄²	—	106	2	no	—
222	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
223	C₄²	#11719947	1	1	yes	characteristic
224	C₄²	#11719949	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
227	Q₈	—	106	2	no	—
228	Q₈	—	106	2	no	—
229	Q₈	—	106	2	no	—
230	Q₈	—	106	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	53	4	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	106	2	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	212	2	no	—
239	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
240	C₂×C₄	#13133175	1	1	yes	characteristic
241	C₂×C₄	#13133188	1	1	yes	characteristic
242	C₂×C₄	#13133192	1	1	yes	characteristic
243	C₂×C₄	#13133193	1	1	yes	characteristic
244	C₂×C₄	#13133197	1	1	yes	characteristic
245	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
246	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
247	C₂×C₄	#13133236	1	1	yes	characteristic
248	C₂×C₄	#13133240	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	8	2	no	—
253	C₈	—	424	2	no	—
254	C₈	—	212	2	no	—
255	C₈	#13133135	1	2	yes	complemented
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	#16810480	1	1	yes	characteristic
259	C₂²	#107×D₅₃	1	1	yes	characteristic
260	C₂²	#16810660	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	106	4	no	—
265	C₄	—	212	2	no	—
266	C₄	C₂×#13133135	1	1	yes	characteristic
267	C₄	#16810605	1	1	yes	characteristic
268	C₄	#16810856	1	1	yes	characteristic
269	C₄	#16810860	1	1	yes	characteristic
270	C₄	#16810861	1	1	yes	characteristic
271	C₄	#16810865	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	#20274708	1	1	yes	characteristic
275	C₂	#20274741	1	1	yes	characteristic
276	C₂	#20274762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810480
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1567	D₅₃
hypocenter	C₅₃	#8617
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
114. C₈×C₅₃	#101	semidirect
161. C₅₃	#8617	semidirect
255. C₈	#13133135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
#20274708	2
#20273474	2
#20272589	2
C₅₃×#1567	2
#8617	53