Subgroups — #27402928

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#20275471	C₂	1	2	yes	maximal
4	#20275470	C₂	1	4	yes	maximal
5	#20274873	C₂	1	2	yes	maximal
6	#107×Dic₅₃	C₂	1	2	yes	maximal
7	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1845	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×#13133151	C₂²	1	1	yes	characteristic
13	C₂×#13133151	C₂²	1	1	yes	characteristic
14	C₂×#13133150	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×#107×C₅₃	C₂²	1	2	yes	—
26	C₂×#107×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	#13133151	—	2	2	no	—
44	#13133151	Q₈	1	4	yes	complemented
45	#13133150	—	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	#107×C₅₃	—	2	4	no	—
58	#107×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	D₈	1	4	yes	—
70	C₂×Dic₁₀₆	Q₁₆	1	4	yes	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	1	no	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	4	4	no	—
80	C₄⋊C₄×C₅₃	—	8	2	no	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	C₄⋊C₄	1	4	yes	—
87	C₄⋊Dic₅₃	C₂×Q₈	1	4	yes	—
88	#11719919	—	4	8	no	—
89	#11719919	—	8	4	no	—
90	#11719919	C₄⋊C₄	1	8	yes	—
91	C₄×Dic₅₃	—	4	4	no	—
92	C₄×Dic₅₃	—	4	8	no	—
93	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
94	C₄×Dic₅₃	C₂×Q₈	1	4	yes	—
95	C₂×#88322	—	2	2	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	C₂²×C₄	1	4	yes	—
98	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	2	no	—
115	C₂×Dic₅₃	—	2	2	no	—
116	C₂×Dic₅₃	—	2	4	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₂×Dic₅₃	—	4	2	no	—
120	C₂×Dic₅₃	—	4	2	no	—
121	C₂×Dic₅₃	—	4	2	no	—
122	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×Dic₅₃	C₂×D₈	1	4	yes	—
125	C₂×Dic₅₃	C₂×Q₁₆	1	4	yes	—
126	Dic₁₀₆	#107	1	16	yes	complemented
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	C₄×Q₈	1	4	yes	complemented
129	#88322	—	2	4	no	—
130	Q₈×#107	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₅₃	#550	1	1	yes	characteristic
134	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
135	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
136	C₂²×C₅₃	#576	1	1	yes	characteristic
137	C₂²×C₅₃	#578	1	1	yes	characteristic
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	8	4	no	—
140	C₄×C₅₃	C₂×#107	1	4	yes	—
141	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₅₃	#520	1	1	yes	characteristic
143	C₄×C₅₃	#557	1	1	yes	characteristic
144	C₄×C₅₃	Q₈²	1	1	yes	characteristic
145	Dic₅₃	—	2	4	no	—
146	Dic₅₃	—	4	4	no	—
147	Dic₅₃	—	8	2	no	—
148	Dic₅₃	C₂×#107	1	8	yes	—
149	C₂×C₈×Q₈	—	53	1	no	—
150	Q₈×C₄⋊C₄	—	53	2	no	—
151	#1947	—	53	4	no	—
152	#1947	—	53	2	no	—
153	#1845	—	53	2	no	—
154	#1845	D₅₃	1	1	yes	characteristic
155	C₄×#107	—	53	2	no	—
156	C₄×#107	—	53	1	no	—
157	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
159	C₂×C₅₃	#2913	1	1	yes	characteristic
160	C₂×C₅₃	Q₈×D₈	1	1	yes	characteristic
161	C₂×C₅₃	Q₈×Q₁₆	1	1	yes	characteristic
162	C₂×C₅₃	#3391	1	1	yes	characteristic
163	C₂×C₅₃	#3395	1	1	yes	characteristic
164	C₂×C₄×Q₈	—	53	1	no	—
165	C₂×C₄×Q₈	—	106	2	no	—
166	C₈×Q₈	—	53	8	no	—
167	C₂×#107	—	53	1	no	—
168	C₂×#107	—	53	1	no	—
169	C₂×#107	—	53	4	no	—
170	C₂×#107	C₂×D₅₃	1	2	yes	—
171	C₂×#107	C₂×D₅₃	1	2	yes	—
172	C₂×C₄⋊C₈	—	53	2	no	—
173	C₂×C₄⋊C₈	—	53	1	no	—
174	C₂×C₄×C₈	—	53	2	no	—
175	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
176	#389	—	106	4	no	—
177	#389	—	106	2	no	—
178	#384	—	53	4	no	—
179	#384	C₂×D₅₃	1	2	yes	—
180	C₄×C₄⋊C₄	—	53	4	no	—
181	C₄×C₄⋊C₄	—	53	2	no	—
182	#340	—	53	2	no	—
183	#340	—	53	4	no	—
184	C₅₃	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	53	1	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	C₄×Q₈	—	53	4	no	—
190	C₄×Q₈	—	53	4	no	—
191	C₄×Q₈	—	212	8	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	53	2	no	—
195	C₂×C₄⋊C₄	—	53	1	no	—
196	C₂×C₄⋊C₄	—	53	2	no	—
197	C₂×C₄⋊C₄	—	53	2	no	—
198	C₂×C₄⋊C₄	—	106	4	no	—
199	C₂×C₄⋊C₄	—	106	4	no	—
200	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
201	C₂×C₄²	—	53	2	no	—
202	C₂×C₄²	—	106	4	no	—
203	C₂×C₄²	—	106	2	no	—
204	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
205	#107	—	2	4	no	—
206	#107	—	4	2	no	—
207	#107	—	53	4	no	—
208	#107	—	106	2	no	—
209	#107	—	212	4	no	—
210	C₄⋊C₈	—	53	8	no	—
211	C₄⋊C₈	—	53	4	no	—
212	C₄×C₈	—	53	8	no	—
213	C₄×C₈	C₄×D₅₃	1	4	yes	—
214	#95	—	106	4	no	—
215	#95	—	106	2	no	—
216	C₂×Q₈	—	53	4	no	—
217	C₂×Q₈	—	53	4	no	—
218	C₂×Q₈	—	53	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	53	2	no	—
221	C₂²×C₄	—	53	2	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	—	106	2	no	—
224	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	53	4	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
234	C₂×C₈	#11719947	1	2	yes	—
235	C₂×C₈	Q₈×D₅₃	1	2	yes	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	8	2	no	—
240	C₄⋊C₄	—	53	8	no	—
241	C₄⋊C₄	—	53	4	no	—
242	C₄⋊C₄	—	53	4	no	—
243	C₄⋊C₄	—	106	2	no	—
244	C₄⋊C₄	—	106	2	no	—
245	C₄⋊C₄	—	106	2	no	—
246	C₄⋊C₄	—	424	4	no	—
247	C₄⋊C₄	—	212	8	no	—
248	C₄²	—	53	4	no	—
249	C₄²	—	53	4	no	—
250	C₄²	—	212	8	no	—
251	C₄²	—	212	4	no	—
252	C₄²	C₂×C₄×D₅₃	1	2	yes	—
253	C₄²	Q₈×D₅₃	1	2	yes	—
254	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
255	Q₈	—	53	16	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	53	4	no	—
262	C₂×C₄	—	53	4	no	—
263	C₂×C₄	—	53	4	no	—
264	C₂×C₄	—	53	4	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	4	no	—
268	C₂×C₄	—	212	2	no	—
269	C₂×C₄	—	212	2	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	—	212	2	no	—
272	C₂×C₄	—	212	2	no	—
273	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
274	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
275	C₂×C₄	D₈×D₅₃	1	2	yes	—
276	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
277	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
281	C₂×C₄	#13133262	1	1	yes	characteristic
282	C₂×C₄	#13133265	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	106	4	no	—
285	C₈	#13133182	1	4	yes	complemented
286	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
287	C₂²	#16811288	1	1	yes	characteristic
288	C₂²	#16811293	1	1	yes	characteristic
289	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
290	C₂²	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
291	C₂²	#16811424	1	1	yes	characteristic
292	C₂²	#16811436	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	8	4	no	—
295	C₄	—	53	8	no	—
296	C₄	—	106	4	no	—
297	C₄	—	424	2	no	—
298	C₄	—	212	4	no	—
299	C₄	#107×D₅₃	1	4	yes	—
300	C₄	C₂×#13133182	1	1	yes	characteristic
301	C₄	#16811104	1	1	yes	characteristic
302	C₄	#16811348	1	1	yes	characteristic
303	C₄	#16811353	1	1	yes	characteristic
304	C₂	#20277627	1	1	yes	characteristic
305	C₂	C₂×#107×D₅₃	1	1	yes	characteristic
306	C₂	#20283311	1	1	yes	characteristic
307	C₂	#20287567	1	1	yes	characteristic
308	C₂	#20287598	1	1	yes	characteristic
309	C₂	#20287688	1	1	yes	characteristic
310	C₂	#20287719	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1845	D₅₃
hypocenter	C₅₃	Q₈×#107
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₁₀₆	C₄	semidirect
44. #13133151	Q₈	semidirect
126. Dic₁₀₆	#107	semidirect
128. C₈×C₅₃	C₄×Q₈	semidirect
184. C₅₃	Q₈×#107	semidirect
285. 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
#20275471	2
#20275470	2
#20274873	2
#107×Dic₅₃	2
C₄×#13133151	2
C₅₃×#1845	2
Q₈×#107	53