Subgroups — #27537603

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#20276622	C₂	1	4	yes	maximal, complemented
3	#20276620	C₂	1	2	yes	maximal, complemented
4	#20276617	C₂	1	2	yes	maximal, complemented
5	#20276470	C₂	1	2	yes	maximal
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₅₃	C₂	1	2	yes	maximal
8	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀₆	C₄	1	2	yes	—
13	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
15	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	C₈×Dic₁₀₆	C₂²	1	2	yes	—
18	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
19	C₂×#13133146	C₄	1	2	yes	—
20	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
21	#16810441	C₂²	1	4	yes	—
22	#16810441	C₂²	1	4	yes	—
23	#16810399	C₂²	1	2	yes	—
24	#16810398	C₂²	1	4	yes	—
25	C₄×#11719908	C₂²	1	2	yes	—
26	C₁₆×Dic₅₃	C₂²	1	4	yes	—
27	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
34	C₄×Dic₁₀₆	C₈	1	2	yes	—
35	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
36	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
40	C₂×#11719919	C₈	1	2	yes	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
42	#13133157	—	2	2	no	—
43	#13133157	Q₈	1	4	yes	complemented
44	C₂×#11719908	C₂³	1	2	yes	—
45	C₂×#11719908	C₂³	1	2	yes	—
46	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
47	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
48	C₄⋊#88322	C₂×C₄	1	2	yes	—
49	#13133146	C₂×C₄	1	2	yes	—
50	#13133146	C₂×C₄	1	2	yes	—
51	#13133146	C₂×C₄	1	2	yes	—
52	#13133146	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₃	C₂³	1	2	yes	—
56	C₈×Dic₅₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	M₃₂×C₅₃	—	2	4	no	—
61	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
70	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₀₆	—	2	4	no	—
72	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
73	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
74	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	2	yes	—
81	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
87	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
88	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
89	#11719919	C₂×C₈	1	2	yes	—
90	#11719919	C₂×C₈	1	2	yes	—
91	#11719919	C₂×C₈	1	2	yes	—
92	#11719919	C₂×C₈	1	2	yes	—
93	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
95	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	C₂²×C₄	1	2	yes	—
99	C₂×#88322	C₂²×C₄	1	2	yes	—
100	C₁₆×C₅₃	—	2	4	no	—
101	#11719908	—	2	2	no	—
102	#11719908	C₂×Q₈	1	4	yes	complemented
103	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#131	1	2	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
117	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
118	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
120	Dic₁₀₆	—	2	2	no	—
121	Dic₁₀₆	—	2	4	no	—
122	Dic₁₀₆	M₃₂	1	4	yes	complemented
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	C₄×Q₈	1	2	yes	—
126	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
127	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
128	#88322	—	2	2	no	—
129	#88322	—	2	2	no	—
130	Q₈×M₃₂	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
134	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	C₈×Q₈	1	2	yes	—
139	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
140	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
142	Dic₅₃	—	2	2	no	—
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	—	2	2	no	—
145	Dic₅₃	C₂×M₃₂	1	4	yes	—
146	C₂×C₈×Q₈	—	53	1	no	—
147	#2183	—	53	2	no	—
148	#2183	—	53	4	no	—
149	Q₈×C₁₆	—	53	2	no	—
150	#2150	—	53	2	no	—
151	#2150	D₅₃	1	1	yes	characteristic
152	C₄×M₃₂	—	53	2	no	—
153	C₄×M₃₂	—	53	1	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
156	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
157	C₂×C₅₃	#3407	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	53	1	no	—
159	C₂×M₃₂	—	53	1	no	—
160	C₂×M₃₂	—	53	1	no	—
161	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
162	C₈×Q₈	—	53	2	no	—
163	C₈×Q₈	—	53	1	no	—
164	C₈×Q₈	—	53	1	no	—
165	C₈×Q₈	—	53	1	no	—
166	C₈×Q₈	—	53	2	no	—
167	C₈×Q₈	—	53	1	no	—
168	C₂×C₄⋊C₈	—	53	2	no	—
169	C₂×C₄⋊C₈	—	53	1	no	—
170	C₂×C₄×C₈	—	53	2	no	—
171	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₄⋊C₁₆	—	53	4	no	—
173	C₄⋊C₁₆	—	53	4	no	—
174	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
175	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
176	#346	—	53	4	no	—
177	#346	—	53	2	no	—
178	C₄×C₁₆	—	53	4	no	—
179	C₄×C₁₆	—	53	2	no	—
180	C₅₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	53	1	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄×Q₈	—	53	1	no	—
186	C₄×Q₈	—	53	2	no	—
187	C₄×Q₈	—	53	1	no	—
188	C₄×Q₈	—	53	1	no	—
189	C₄×Q₈	—	53	2	no	—
190	C₄×Q₈	—	53	1	no	—
191	C₂×C₄⋊C₄	—	53	2	no	—
192	C₂×C₄⋊C₄	—	53	1	no	—
193	C₂×C₄²	—	53	2	no	—
194	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
195	M₃₂	—	2	4	no	—
196	M₃₂	—	53	4	no	—
197	M₃₂	—	106	2	no	—
198	C₂×C₁₆	—	53	2	no	—
199	C₂×C₁₆	—	53	2	no	—
200	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
201	C₄⋊C₈	—	53	2	no	—
202	C₄⋊C₈	—	53	2	no	—
203	C₄⋊C₈	—	53	2	no	—
204	C₄⋊C₈	—	53	2	no	—
205	C₄⋊C₈	—	53	1	no	—
206	C₄⋊C₈	—	53	2	no	—
207	C₄⋊C₈	—	53	1	no	—
208	C₄×C₈	—	53	2	no	—
209	C₄×C₈	—	53	2	no	—
210	C₄×C₈	—	53	2	no	—
211	C₄×C₈	—	53	2	no	—
212	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
213	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₂×Q₈	—	53	1	no	—
217	C₂×Q₈	—	53	2	no	—
218	C₂×Q₈	—	53	1	no	—
219	C₂×Q₈	—	106	4	no	—
220	C₂²×C₄	—	53	2	no	—
221	C₂²×C₄	—	53	2	no	—
222	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₈	#11719947	1	2	yes	—
233	C₂×C₈	Q₈×D₅₃	1	2	yes	—
234	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
235	C₄⋊C₄	—	53	2	no	—
236	C₄⋊C₄	—	53	2	no	—
237	C₄⋊C₄	—	53	2	no	—
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	53	2	no	—
240	C₄⋊C₄	—	53	1	no	—
241	C₄⋊C₄	—	53	1	no	—
242	C₄²	—	53	2	no	—
243	C₄²	—	53	2	no	—
244	C₄²	—	53	2	no	—
245	C₄²	—	53	2	no	—
246	C₄²	C₈×D₅₃	1	1	yes	characteristic
247	C₄²	C₈×D₅₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₁₆	—	2	4	no	—
251	C₁₆	—	53	4	no	—
252	C₁₆	—	106	2	no	—
253	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
254	Q₈	—	53	4	no	—
255	Q₈	—	106	4	no	—
256	Q₈	—	106	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	53	2	no	—
261	C₂×C₄	—	53	2	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	106	4	no	—
264	C₂×C₄	—	212	2	no	—
265	C₂×C₄	#13133182	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
271	C₂×C₄	#13133225	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	—	106	2	no	—
276	C₈	—	106	2	no	—
277	C₈	#13133182	1	2	yes	—
278	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
279	C₈	C₂×#11719947	1	1	yes	characteristic
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	#16810747	1	2	yes	—
283	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	53	4	no	—
288	C₄	—	106	2	no	—
289	C₄	—	106	2	no	—
290	C₄	—	106	2	no	—
291	C₄	#16810747	1	2	yes	—
292	C₄	M₃₂×D₅₃	1	2	yes	—
293	C₄	C₂×#13133182	1	1	yes	characteristic
294	C₄	C₂×#13133225	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	C₂×#16810747	1	1	yes	characteristic
297	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
298	C₂	#20287817	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2150	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2150	D₅₃
hypocenter	C₅₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #20276622	C₂	semidirect
3. #20276620	C₂	semidirect
4. #20276617	C₂	semidirect
43. #13133157	Q₈	semidirect
102. #11719908	C₂×Q₈	semidirect
122. Dic₁₀₆	M₃₂	semidirect
180. C₅₃	Q₈×M₃₂	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
#20276622	2
#20276620	2
#20276617	2
#20276470	2
C₄×#13133157	2
M₃₂×Dic₅₃	2
C₅₃×#2150	2
Q₈×M₃₂	53