Subgroups — #28328444

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291129	C₂	1	2	yes	maximal, complemented
2	#20291093	C₂	1	4	yes	maximal, complemented
3	D₄×#11719910	C₂	1	2	yes	maximal, complemented
4	#20291075	C₂	1	1	yes	maximal, characteristic, complemented
5	#20291073	C₂	1	2	yes	maximal, complemented
6	C₂×#16811509	C₂	1	2	yes	maximal, complemented
7	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#11719947	C₂	1	1	yes	maximal, characteristic
9	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
10	C₂²×#11719910	C₂²	1	4	yes	—
11	#16811544	C₂²	1	4	yes	—
12	C₄⋊#11719910	C₂²	1	2	yes	—
13	#16811518	C₂²	1	2	yes	—
14	C₂²⋊#11719910	C₂²	1	4	yes	—
15	#16811509	C₂²	1	1	yes	characteristic
16	#16811509	C₂²	1	2	yes	—
17	#16811509	C₂²	1	4	yes	—
18	#16811509	C₂²	1	1	yes	characteristic
19	C₄×#11719910	C₂²	1	2	yes	—
20	C₄×#11719947	C₄	1	1	yes	characteristic
21	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
22	#16810747	C₂²	1	1	yes	characteristic
23	D₄×#88322	C₂²	1	1	yes	characteristic
24	C₈×C₂²⋊D₅₃	C₄	1	2	yes	—
25	#16810686	C₂²	1	2	yes	—
26	#16810675	C₄	1	1	yes	characteristic
27	#16810673	C₄	1	2	yes	—
28	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
29	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
30	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic
31	D₅₃⋊C₁₆	—	2	4	no	—
32	C₂×#11719910	—	2	8	no	—
33	C₂×#11719910	C₂³	1	4	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	2	yes	—
36	C₂×#11719947	C₈	1	1	yes	characteristic
37	C₂²×#88322	C₂³	1	2	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
40	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
41	#13133182	C₈	1	1	yes	characteristic
42	D₄×Dic₅₃	C₈	1	1	yes	characteristic
43	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
44	#13133177	C₈	1	2	yes	—
45	#13133175	C₈	1	1	yes	characteristic
46	#13133173	C₈	1	2	yes	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
48	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
50	#13133146	C₂×C₄	1	2	yes	—
51	#13133146	C₂×C₄	1	2	yes	—
52	C₂²⋊#88322	C₂×C₄	1	2	yes	—
53	#13133129	C₂×C₄	1	2	yes	—
54	C₄×#88322	C₂³	1	1	yes	characteristic
55	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
58	C₈×Dic₅₃	C₂³	1	2	yes	—
59	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
68	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
69	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
70	#11719947	C₁₆	1	8	yes	complemented
71	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
83	#11719921	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
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	1	yes	characteristic
89	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
91	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	C₂²×C₄	1	2	yes	—
95	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
96	C₂×#88322	C₂⁴	1	1	yes	characteristic
97	C₈×D₅₃	—	2	2	no	—
98	C₈×D₅₃	—	2	2	no	—
99	#11719910	—	2	4	no	—
100	#11719910	—	2	2	no	—
101	#11719910	#41	1	4	yes	complemented
102	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
103	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
104	D₄×C₅₃	C₂×C₁₆	1	4	yes	complemented
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₅₃	C₂×C₁₆	1	8	yes	complemented
112	C₂×Dic₅₃	—	2	4	no	—
113	C₂×Dic₅₃	C₂×C₁₆	1	8	yes	complemented
114	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
116	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
117	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₄×D₅₃	—	2	1	no	—
120	C₄×D₅₃	—	2	1	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	C₂×C₁₆	1	4	yes	complemented
123	Dic₁₀₆	C₂×C₁₆	1	4	yes	complemented
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	#131	1	2	yes	—
127	#88322	—	2	2	no	—
128	#88322	—	2	1	no	—
129	#88322	C₂×#41	1	2	yes	—
130	C₁₆×#41	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	C₂²×C₁₆	1	4	yes	—
134	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₅₃	—	2	1	no	—
136	C₂×D₅₃	—	2	1	no	—
137	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
141	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	—	2	1	no	—
145	Dic₅₃	C₂²×C₁₆	1	4	yes	—
146	Dic₅₃	C₂²×C₁₆	1	2	yes	—
147	Dic₅₃	#504	1	2	yes	—
148	C₈×#41	—	53	1	no	—
149	Q₈×C₁₆	—	53	2	no	—
150	D₄×C₁₆	—	53	2	no	—
151	D₄×C₁₆	—	53	4	no	—
152	#2162	—	53	2	no	—
153	#2162	—	53	1	no	—
154	C₂×C₄×C₁₆	—	53	1	no	—
155	C₂×C₄×C₁₆	—	53	2	no	—
156	C₂×C₅₃	—	2	4	no	—
157	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
158	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
160	D₅₃	—	2	2	no	—
161	C₄×#41	—	53	1	no	—
162	C₂²×C₁₆	—	53	4	no	—
163	C₂²×C₁₆	—	53	2	no	—
164	C₈×Q₈	—	53	1	no	—
165	C₈×Q₈	—	53	1	no	—
166	C₈×D₄	—	53	2	no	—
167	C₈×D₄	—	53	1	no	—
168	C₈×D₄	—	53	2	no	—
169	C₈×D₄	#6157	1	1	yes	characteristic
170	#431	—	53	1	no	—
171	#431	—	53	2	no	—
172	C₂×C₄×C₈	—	53	2	no	—
173	C₂×C₄×C₈	—	53	1	no	—
174	C₄⋊C₁₆	—	53	2	no	—
175	C₄⋊C₁₆	—	53	4	no	—
176	C₂²⋊C₁₆	—	53	4	no	—
177	C₂²⋊C₁₆	—	53	2	no	—
178	C₄×C₁₆	—	53	1	no	—
179	C₄×C₁₆	—	53	1	no	—
180	C₄×C₁₆	—	53	2	no	—
181	C₄×C₁₆	—	53	2	no	—
182	C₄×C₁₆	—	53	4	no	—
183	C₅₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	53	1	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	1	no	—
187	C₂²×C₈	—	53	1	no	—
188	C₂²×C₈	C₂×#6157	1	2	yes	—
189	C₄×Q₈	—	53	1	no	—
190	C₄×Q₈	—	53	1	no	—
191	C₄×D₄	—	53	1	no	—
192	C₄×D₄	—	53	2	no	—
193	C₄×D₄	—	53	2	no	—
194	C₄×D₄	C₂×#6157	1	1	yes	characteristic
195	#117	—	53	1	no	—
196	#117	—	53	2	no	—
197	C₂×C₄²	—	53	2	no	—
198	C₂×C₄²	—	53	1	no	—
199	C₂×C₁₆	—	53	2	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	—	53	4	no	—
202	C₂×C₁₆	—	106	8	no	—
203	C₂×C₁₆	—	106	4	no	—
204	C₄⋊C₈	—	53	2	no	—
205	C₄⋊C₈	—	53	2	no	—
206	C₄⋊C₈	—	53	1	no	—
207	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
208	C₂²⋊C₈	—	53	2	no	—
209	C₂²⋊C₈	—	53	2	no	—
210	C₂²⋊C₈	C₂×#6157	1	2	yes	—
211	C₄×C₈	—	53	2	no	—
212	C₄×C₈	—	53	1	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	53	1	no	—
216	C₄×C₈	—	53	1	no	—
217	C₄×C₈	C₂×#6157	1	1	yes	characteristic
218	#41	—	53	8	no	—
219	C₂×Q₈	—	53	1	no	—
220	C₂×D₄	—	53	2	no	—
221	C₂×D₄	D₅₃⋊C₈	1	1	yes	characteristic
222	C₂²×C₄	—	53	2	no	—
223	C₂²×C₄	—	53	1	no	—
224	C₂²×C₄	—	53	2	no	—
225	C₂²×C₄	—	53	1	no	—
226	C₂²×C₄	—	53	1	no	—
227	C₂²×C₄	C₂²×#6157	1	2	yes	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	106	4	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	C₂²×#6157	1	2	yes	—
236	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
237	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	53	2	no	—
240	C₄⋊C₄	—	53	1	no	—
241	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
242	C₂²⋊C₄	—	53	2	no	—
243	C₂²⋊C₄	—	53	2	no	—
244	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
245	C₄²	—	53	2	no	—
246	C₄²	—	53	1	no	—
247	C₄²	—	53	2	no	—
248	C₄²	—	53	1	no	—
249	C₄²	—	53	1	no	—
250	C₄²	—	53	2	no	—
251	C₄²	C₂²×#6157	1	1	yes	characteristic
252	C₁₆	—	53	4	no	—
253	C₁₆	—	106	4	no	—
254	C₁₆	—	106	2	no	—
255	C₂³	—	53	1	no	—
256	C₂³	C₂×D₅₃⋊C₈	1	2	yes	—
257	Q₈	—	53	4	no	—
258	D₄	—	53	8	no	—
259	D₄	D₅₃⋊C₁₆	1	4	yes	complemented
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	53	8	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	53	4	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	53	1	no	—
269	C₂×C₄	—	106	2	no	—
270	C₂×C₄	—	106	4	no	—
271	C₂×C₄	—	106	1	no	—
272	C₂×C₄	—	106	1	no	—
273	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
274	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
275	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₈	—	53	2	no	—
280	C₈	—	106	2	no	—
281	C₈	—	106	1	no	—
282	C₈	#13133304	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	53	2	no	—
286	C₂²	—	106	1	no	—
287	C₂²	—	106	1	no	—
288	C₂²	C₂×D₅₃⋊C₁₆	1	4	yes	—
289	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	53	2	no	—
293	C₄	—	53	4	no	—
294	C₄	—	53	2	no	—
295	C₄	—	106	2	no	—
296	C₄	—	106	1	no	—
297	C₄	C₂×D₅₃⋊C₁₆	1	2	yes	—
298	C₄	#41×#6157	1	1	yes	characteristic
299	C₄	C₂×#13133304	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	106	2	no	—
302	C₂	#20291520	1	1	yes	characteristic
303	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic
304	C₂	D₅₃⋊#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×D₄×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#11719947	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×D₄	#6157
hypocenter	C₅₃	C₁₆×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20291129	C₂	semidirect
2. #20291093	C₂	semidirect
3. D₄×#11719910	C₂	semidirect
4. #20291075	C₂	semidirect
5. #20291073	C₂	semidirect
6. C₂×#16811509	C₂	semidirect
7. C₄×D₅₃⋊C₁₆	C₂	semidirect
70. #11719947	C₁₆	semidirect
101. #11719910	#41	semidirect
104. D₄×C₅₃	C₂×C₁₆	semidirect
111. C₂²⋊D₅₃	C₂×C₁₆	semidirect
113. C₂×Dic₅₃	C₂×C₁₆	semidirect
122. C₄×D₅₃	C₂×C₁₆	semidirect
123. Dic₁₀₆	C₂×C₁₆	semidirect
183. C₅₃	C₁₆×#41	semidirect
259. D₄	D₅₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20291129	2
#20291093	2
D₄×#11719910	2
#20291075	2
#20291073	2
C₂×#16811509	2
C₄×D₅₃⋊C₁₆	2
C₈×#11719947	2
C₁₆×#41	53