Subgroups — #28328441

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	#20291132	C₂	1	4	yes	maximal, complemented
2	#20291130	C₂	1	2	yes	maximal, complemented
3	#20291129	C₂	1	2	yes	maximal, complemented
4	#20291048	C₂	1	1	yes	maximal, characteristic
5	#20291048	C₂	1	2	yes	maximal
6	#20290922	C₂	1	2	yes	maximal
7	C₄×#13133292	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133292	C₂²	1	1	yes	characteristic
10	C₂×#13133292	C₂²	1	2	yes	—
11	C₂×#13133292	C₂²	1	1	yes	characteristic
12	#16811544	C₂²	1	4	yes	—
13	#16811544	C₂²	1	2	yes	—
14	C₄⋊#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	4	yes	—
16	#16811512	C₂²	1	2	yes	—
17	#16811511	C₂²	1	4	yes	—
18	#16811509	C₂²	1	4	yes	—
19	C₄×#11719910	C₂²	1	2	yes	—
20	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
21	Q₈×#88322	C₂²	1	1	yes	characteristic
22	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
23	C₈×Dic₁₀₆	C₄	1	2	yes	—
24	#16810747	C₂²	1	1	yes	characteristic
25	#16810747	C₂²	1	2	yes	—
26	C₄⋊C₈×D₅₃	C₄	1	1	yes	characteristic
27	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
28	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₅₃	C₂²	1	2	yes	—
30	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
31	#13133292	—	2	2	no	—
32	#13133292	—	2	4	no	—
33	#13133292	Q₈	1	4	yes	complemented
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	4	yes	—
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
38	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
41	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
42	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
44	#13133182	C₈	1	2	yes	—
45	#13133182	C₈	1	1	yes	characteristic
46	C₄⋊C₄×D₅₃	C₈	1	1	yes	characteristic
47	C₄⋊C₄×D₅₃	C₈	1	2	yes	—
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₅₃	C₂×C₄	1	2	yes	—
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
52	#13133146	C₂×C₄	1	2	yes	—
53	#13133146	C₂×C₄	1	2	yes	—
54	#13133146	C₂×C₄	1	2	yes	—
55	C₄×#88322	C₂³	1	1	yes	characteristic
56	C₄×#88322	C₂³	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₅₃	C₂³	1	2	yes	—
60	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
61	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
67	Q₈×D₅₃	—	2	4	no	—
68	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₅₃	C₂²×C₄	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	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
83	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
84	#11719919	C₂×C₈	1	2	yes	—
85	#11719919	C₂×C₈	1	2	yes	—
86	#11719919	C₂×C₈	1	2	yes	—
87	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
88	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
89	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
91	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
93	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
94	C₂×#88322	C₂²×C₄	1	2	yes	—
95	C₂×#88322	C₂⁴	1	1	yes	characteristic
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	2	4	no	—
98	C₈×D₅₃	C₂×Q₈	1	2	yes	—
99	C₈×D₅₃	#41	1	2	yes	—
100	#11719910	—	2	2	no	—
101	#11719910	—	2	4	no	—
102	#11719910	C₂×Q₈	1	4	yes	complemented
103	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
112	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
115	C₄×D₅₃	—	2	2	no	—
116	C₄×D₅₃	—	2	4	no	—
117	C₄×D₅₃	—	4	1	no	—
118	C₄×D₅₃	—	4	2	no	—
119	C₄×D₅₃	C₄×Q₈	1	2	yes	—
120	C₄×D₅₃	#131	1	2	yes	—
121	Dic₁₀₆	—	2	4	no	—
122	Dic₁₀₆	M₃₂	1	4	yes	complemented
123	C₈×C₅₃	—	2	2	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	C₄×Q₈	1	2	yes	—
126	#88322	—	2	1	no	—
127	#88322	—	2	2	no	—
128	#88322	C₂²×Q₈	1	1	yes	characteristic
129	#88322	C₂×#41	1	1	yes	characteristic
130	Q₈×M₃₂	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
132	C₂×D₅₃	—	2	1	no	—
133	C₂×D₅₃	—	2	1	no	—
134	C₂×D₅₃	C₈×Q₈	1	2	yes	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
139	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
141	Dic₅₃	—	2	1	no	—
142	Dic₅₃	—	2	2	no	—
143	Dic₅₃	—	2	1	no	—
144	Dic₅₃	C₈×Q₈	1	2	yes	—
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	1	no	—
151	#2150	—	53	2	no	—
152	C₄×M₃₂	—	53	1	no	—
153	C₄×M₃₂	—	53	2	no	—
154	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
155	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
156	C₂×C₅₃	#3407	1	1	yes	characteristic
157	D₅₃	—	2	2	no	—
158	C₂×C₄×Q₈	—	53	1	no	—
159	C₂×M₃₂	—	53	1	no	—
160	C₂×M₃₂	—	53	1	no	—
161	C₂×M₃₂	—	53	2	no	—
162	C₈×Q₈	—	53	1	no	—
163	C₈×Q₈	—	53	1	no	—
164	C₈×Q₈	—	53	2	no	—
165	C₈×Q₈	—	53	2	no	—
166	C₈×Q₈	—	53	1	no	—
167	C₈×Q₈	#6157	1	1	yes	characteristic
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	—	53	1	no	—
172	C₄⋊C₁₆	—	53	2	no	—
173	C₄⋊C₁₆	—	53	4	no	—
174	C₄⋊C₁₆	—	53	4	no	—
175	C₄⋊C₁₆	—	53	2	no	—
176	#346	—	53	4	no	—
177	#346	—	53	2	no	—
178	C₄×C₁₆	—	53	2	no	—
179	C₄×C₁₆	—	53	4	no	—
180	C₅₃	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	1	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₄×Q₈	—	53	1	no	—
186	C₄×Q₈	—	53	1	no	—
187	C₄×Q₈	—	53	2	no	—
188	C₄×Q₈	—	53	1	no	—
189	C₄×Q₈	—	53	2	no	—
190	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
191	C₂×C₄⋊C₄	—	53	2	no	—
192	C₂×C₄⋊C₄	—	53	1	no	—
193	C₂×C₄²	—	53	2	no	—
194	C₂×C₄²	—	53	1	no	—
195	M₃₂	—	53	4	no	—
196	M₃₂	—	106	2	no	—
197	M₃₂	—	106	4	no	—
198	C₂×C₁₆	—	53	2	no	—
199	C₂×C₁₆	—	53	2	no	—
200	C₂×C₁₆	—	53	4	no	—
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	C₂×#6157	1	1	yes	characteristic
207	C₄⋊C₈	C₂×#6157	1	2	yes	—
208	C₄×C₈	—	53	2	no	—
209	C₄×C₈	—	53	1	no	—
210	C₄×C₈	—	53	2	no	—
211	C₄×C₈	—	53	1	no	—
212	C₄×C₈	—	53	1	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	C₂×#6157	1	2	yes	—
215	C₄×C₈	C₂×#6157	1	1	yes	characteristic
216	C₂×Q₈	—	53	1	no	—
217	C₂×Q₈	—	53	2	no	—
218	C₂×Q₈	—	106	4	no	—
219	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	2	no	—
222	C₂²×C₄	—	53	1	no	—
223	C₂²×C₄	—	53	2	no	—
224	C₂²×C₄	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	2	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	2	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	C₂²×#6157	1	2	yes	—
233	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
234	C₂×C₈	C₂²×#6157	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	1	no	—
240	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
241	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
242	C₄²	—	53	2	no	—
243	C₄²	—	53	2	no	—
244	C₄²	—	53	1	no	—
245	C₄²	—	53	1	no	—
246	C₄²	—	53	2	no	—
247	C₄²	—	53	1	no	—
248	C₄²	C₂²×#6157	1	2	yes	—
249	C₄²	C₂²×#6157	1	1	yes	characteristic
250	C₁₆	—	53	4	no	—
251	C₁₆	—	106	2	no	—
252	C₁₆	—	106	4	no	—
253	C₂³	—	53	1	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	53	4	no	—
256	Q₈	—	106	4	no	—
257	C₂×C₄	—	53	1	no	—
258	C₂×C₄	—	53	2	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	53	2	no	—
261	C₂×C₄	—	53	2	no	—
262	C₂×C₄	—	53	1	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	106	4	no	—
265	C₂×C₄	—	106	2	no	—
266	C₂×C₄	—	212	2	no	—
267	C₂×C₄	—	212	1	no	—
268	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
269	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
271	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
272	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	1	no	—
275	C₈	—	53	1	no	—
276	C₈	—	53	1	no	—
277	C₈	—	106	1	no	—
278	C₈	—	106	2	no	—
279	C₈	Q₈×#6157	1	2	yes	—
280	C₂²	—	53	2	no	—
281	C₂²	—	106	1	no	—
282	C₂²	—	106	1	no	—
283	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	53	4	no	—
288	C₄	—	53	2	no	—
289	C₄	—	106	1	no	—
290	C₄	—	106	2	no	—
291	C₄	—	106	1	no	—
292	C₄	D₅₃⋊M₃₂	1	2	yes	—
293	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
294	C₄	D₅₃⋊#131	1	1	yes	characteristic
295	C₂	—	106	2	no	—
296	C₂	Q₈×D₅₃⋊C₈	1	1	yes	characteristic
297	C₂	C₂×D₅₃⋊M₃₂	1	1	yes	characteristic
298	C₂	#20291683	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₈×Q₈×C₅₃	C₄
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	Q₈×M₃₂
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291130	C₂	semidirect
3. #20291129	C₂	semidirect
33. #13133292	Q₈	semidirect
102. #11719910	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
#20291132	2
#20291130	2
#20291129	2
#20291048	2
#20290922	2
C₄×#13133292	2
C₈×Q₈×D₅₃	2
Q₈×M₃₂	53