Subgroups — #27402915

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 294 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	#20277627	C₂	1	2	yes	maximal
3	#20275476	C₂	1	2	yes	maximal
4	#20275475	C₂	1	4	yes	maximal
5	#20274870	C₂	1	2	yes	maximal
6	#106×Dic₅₃	C₂	1	2	yes	maximal
7	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1844	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	C₂×#13133182	—	2	2	no	—
11	C₈×Dic₁₀₆	C₄	1	8	yes	complemented
12	C₂×#13133149	C₂²	1	1	yes	characteristic
13	C₂×#13133149	C₂²	1	1	yes	characteristic
14	C₂×#13133148	C₂²	1	4	yes	—
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133146	C₂²	1	2	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
18	#16810519	—	2	2	no	—
19	#16810518	—	2	4	no	—
20	#16810492	C₂²	1	4	yes	—
21	C₄⋊C₄×Dic₅₃	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
23	#16810386	C₂²	1	2	yes	—
24	#16810385	C₂²	1	4	yes	—
25	C₂×#106×C₅₃	C₂²	1	2	yes	—
26	C₂×#106×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#384	C₂²	1	2	yes	—
29	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	2	yes	—
31	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₀₆	Q₈	1	4	yes	—
33	#13133182	—	4	8	no	—
34	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
37	C₂×#11719919	—	2	4	no	—
38	C₂×#11719919	—	2	4	no	—
39	C₂×#11719919	D₄	1	2	yes	—
40	C₂×C₄×Dic₅₃	—	2	2	no	—
41	C₂×C₄×Dic₅₃	—	2	4	no	—
42	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
43	#13133149	—	2	2	no	—
44	#13133149	Q₈	1	4	yes	complemented
45	#13133148	—	4	4	no	—
46	C₄⋊#88322	C₂×C₄	1	4	yes	—
47	#13133146	C₂×C₄	1	8	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
49	#13133124	—	2	2	no	—
50	#13133124	—	2	4	no	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#106×C₅₃	—	2	4	no	—
58	#106×C₅₃	—	4	2	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
60	C₂²×C₄×C₅₃	—	2	4	no	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅₃	—	2	2	no	—
65	C₂²×Dic₅₃	—	2	2	no	—
66	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
67	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
68	C₂×Dic₁₀₆	C₄⋊C₄	1	4	yes	—
69	C₂×Dic₁₀₆	SD₁₆	1	8	yes	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₅₃	#41	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	4	no	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	4	4	no	—
79	C₄⋊C₄×C₅₃	—	8	2	no	—
80	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
82	C₄⋊Dic₅₃	—	2	2	no	—
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
86	C₄⋊Dic₅₃	C₂×Q₈	1	4	yes	—
87	#11719919	—	4	8	no	—
88	#11719919	—	8	4	no	—
89	#11719919	C₄⋊C₄	1	8	yes	—
90	C₄×Dic₅₃	—	4	4	no	—
91	C₄×Dic₅₃	—	4	8	no	—
92	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
93	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	C₂²×C₄	1	4	yes	—
97	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	4	no	—
112	C₂×Dic₅₃	—	2	4	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	C₂×Dic₅₃	—	4	2	no	—
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×Dic₅₃	C₂×SD₁₆	1	8	yes	—
121	Dic₁₀₆	#106	1	16	yes	complemented
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	C₄×Q₈	1	4	yes	complemented
124	#88322	—	2	4	no	—
125	Q₈×#106	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
128	C₂²×C₅₃	#550	1	1	yes	characteristic
129	C₂²×C₅₃	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₅₃	#577	1	2	yes	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	8	4	no	—
133	C₄×C₅₃	C₂×#106	1	4	yes	—
134	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₅₃	#520	1	1	yes	characteristic
136	C₄×C₅₃	#557	1	1	yes	characteristic
137	C₄×C₅₃	Q₈²	1	1	yes	characteristic
138	Dic₅₃	—	2	4	no	—
139	Dic₅₃	—	4	4	no	—
140	Dic₅₃	—	8	2	no	—
141	Dic₅₃	C₂×#106	1	8	yes	—
142	C₂×C₈×Q₈	—	53	1	no	—
143	Q₈×C₄⋊C₄	—	53	2	no	—
144	#1948	—	53	4	no	—
145	#1948	—	53	2	no	—
146	#1844	—	53	2	no	—
147	#1844	D₅₃	1	1	yes	characteristic
148	C₄×#106	—	53	2	no	—
149	C₄×#106	—	53	1	no	—
150	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₅₃	C₂²×#106	1	1	yes	characteristic
152	C₂×C₅₃	#2912	1	1	yes	characteristic
153	C₂×C₅₃	Q₈×SD₁₆	1	2	yes	—
154	C₂×C₅₃	#3392	1	2	yes	—
155	C₂×C₄×Q₈	—	53	1	no	—
156	C₂×C₄×Q₈	—	106	2	no	—
157	C₈×Q₈	—	53	8	no	—
158	C₂×#106	—	53	1	no	—
159	C₂×#106	—	53	1	no	—
160	C₂×#106	—	53	4	no	—
161	C₂×#106	C₂×D₅₃	1	2	yes	—
162	C₂×#106	C₂×D₅₃	1	2	yes	—
163	C₂×C₄⋊C₈	—	53	2	no	—
164	C₂×C₄⋊C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	53	2	no	—
166	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
167	#389	—	106	4	no	—
168	#389	—	106	2	no	—
169	#384	—	53	4	no	—
170	#384	C₂×D₅₃	1	2	yes	—
171	C₄×C₄⋊C₄	—	53	4	no	—
172	C₄×C₄⋊C₄	—	53	2	no	—
173	#340	—	53	2	no	—
174	#340	—	53	4	no	—
175	C₅₃	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	53	1	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
180	C₄×Q₈	—	53	4	no	—
181	C₄×Q₈	—	53	4	no	—
182	C₄×Q₈	—	212	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	53	2	no	—
186	C₂×C₄⋊C₄	—	53	1	no	—
187	C₂×C₄⋊C₄	—	53	2	no	—
188	C₂×C₄⋊C₄	—	53	2	no	—
189	C₂×C₄⋊C₄	—	106	4	no	—
190	C₂×C₄⋊C₄	—	106	4	no	—
191	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
192	C₂×C₄²	—	53	2	no	—
193	C₂×C₄²	—	106	4	no	—
194	C₂×C₄²	—	106	2	no	—
195	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	#106	—	2	4	no	—
197	#106	—	4	2	no	—
198	#106	—	53	4	no	—
199	#106	—	106	2	no	—
200	#106	—	212	4	no	—
201	C₄⋊C₈	—	53	8	no	—
202	C₄⋊C₈	—	53	4	no	—
203	C₄×C₈	—	53	8	no	—
204	C₄×C₈	C₄×D₅₃	1	4	yes	—
205	#95	—	106	4	no	—
206	#95	—	106	2	no	—
207	C₂×Q₈	—	53	4	no	—
208	C₂×Q₈	—	53	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	53	2	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	—	106	2	no	—
213	C₂²×C₄	—	106	2	no	—
214	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	53	4	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
224	C₂×C₈	#11719947	1	2	yes	—
225	C₂×C₈	Q₈×D₅₃	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	8	2	no	—
230	C₄⋊C₄	—	53	8	no	—
231	C₄⋊C₄	—	53	4	no	—
232	C₄⋊C₄	—	53	4	no	—
233	C₄⋊C₄	—	106	2	no	—
234	C₄⋊C₄	—	106	2	no	—
235	C₄⋊C₄	—	106	2	no	—
236	C₄⋊C₄	—	424	4	no	—
237	C₄⋊C₄	—	212	8	no	—
238	C₄²	—	53	4	no	—
239	C₄²	—	53	4	no	—
240	C₄²	—	212	8	no	—
241	C₄²	—	212	4	no	—
242	C₄²	C₂×C₄×D₅₃	1	2	yes	—
243	C₄²	Q₈×D₅₃	1	2	yes	—
244	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
245	Q₈	—	53	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	53	4	no	—
251	C₂×C₄	—	53	8	no	—
252	C₂×C₄	—	53	4	no	—
253	C₂×C₄	—	106	4	no	—
254	C₂×C₄	—	106	4	no	—
255	C₂×C₄	—	212	2	no	—
256	C₂×C₄	—	212	2	no	—
257	C₂×C₄	—	212	2	no	—
258	C₂×C₄	—	212	2	no	—
259	C₂×C₄	—	212	2	no	—
260	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
262	C₂×C₄	SD₁₆×D₅₃	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
267	C₂×C₄	#13133262	1	1	yes	characteristic
268	C₂×C₄	#13133265	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	106	4	no	—
271	C₈	#13133182	1	4	yes	complemented
272	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
273	C₂²	#16811288	1	1	yes	characteristic
274	C₂²	#16811293	1	1	yes	characteristic
275	C₂²	C₂×SD₁₆×D₅₃	1	2	yes	—
276	C₂²	#16811429	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	8	4	no	—
279	C₄	—	53	8	no	—
280	C₄	—	106	4	no	—
281	C₄	—	424	2	no	—
282	C₄	—	212	4	no	—
283	C₄	#106×D₅₃	1	4	yes	—
284	C₄	C₂×#13133182	1	1	yes	characteristic
285	C₄	#16811104	1	1	yes	characteristic
286	C₄	#16811348	1	1	yes	characteristic
287	C₄	#16811353	1	1	yes	characteristic
288	C₂	#20277627	1	1	yes	characteristic
289	C₂	C₂×#106×D₅₃	1	1	yes	characteristic
290	C₂	#20283306	1	1	yes	characteristic
291	C₂	#20287572	1	2	yes	—
292	C₂	#20287693	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1844	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1844	D₅₃
hypocenter	C₅₃	Q₈×#106
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₀₆	C₄	semidirect
44. #13133149	Q₈	semidirect
121. Dic₁₀₆	#106	semidirect
123. C₈×C₅₃	C₄×Q₈	semidirect
175. C₅₃	Q₈×#106	semidirect
271. C₈	#13133182	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
#20277627	2
#20275476	2
#20275475	2
#20274870	2
#106×Dic₅₃	2
C₄×#13133149	2
C₅₃×#1844	2
Q₈×#106	53