Subgroups — #28302037

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×#88324⋊C₈	C₂	1	2	yes	maximal
4	#20289537	C₂	1	2	yes	maximal
5	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	C₄	1	2	yes	—
7	C₂×C₄⋊#88324	C₄	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	C₂×C₄×#88324	C₂²	1	2	yes	—
10	#16811489	C₄	1	2	yes	—
11	#16811489	C₄	1	2	yes	—
12	#88324⋊C₈	—	2	4	no	—
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	C₂×C₄	1	4	yes	—
21	C₂²×#88324	C₂×C₄	1	4	yes	—
22	C₄⋊#88324	—	2	4	no	—
23	C₄⋊#88324	C₈	1	8	yes	complemented
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	2	no	—
26	C₄×#88324	—	2	2	no	—
27	C₄×#88324	—	2	2	no	—
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
36	C₄×#88322	—	2	2	no	—
37	C₈×Dic₅₃	—	2	2	no	—
38	C₈×Dic₅₃	—	2	1	no	—
39	C₈×Dic₅₃	—	2	1	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₈×Dic₅₃	D₄	1	2	yes	—
42	C₈×Dic₅₃	Q₈	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
52	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×#88324	—	2	4	no	—
55	C₂×#88324	—	2	4	no	—
56	C₂×#88324	—	2	4	no	—
57	C₂×#88324	—	2	4	no	—
58	C₂×#88324	—	2	4	no	—
59	C₂×#88324	C₂×C₈	1	8	yes	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
66	C₄²×C₅₃	M₁₆	1	2	yes	—
67	C₄²×C₅₃	M₁₆	1	2	yes	—
68	C₄×Dic₅₃	—	2	1	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	C₄×Dic₅₃	—	2	2	no	—
72	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
73	C₄×Dic₅₃	M₁₆	1	4	yes	—
74	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
76	C₄×Dic₅₃	#41	1	1	yes	characteristic
77	C₄×Dic₅₃	#41	1	1	yes	characteristic
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	C₄⋊C₄	1	2	yes	—
83	C₂×#88322	M₁₆	1	2	yes	—
84	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
88	C₂×C₄×C₅₃	#97	1	2	yes	—
89	C₂×C₄×C₅₃	#97	1	2	yes	—
90	C₂×C₄×C₅₃	#97	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×Dic₅₃	—	2	1	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	2	1	no	—
101	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
102	C₂×Dic₅₃	#97	1	2	yes	—
103	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
106	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
107	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
108	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
109	C₂×Dic₅₃	#131	1	2	yes	—
110	C₂×Dic₅₃	#131	1	2	yes	—
111	#88324	—	2	8	no	—
112	#88324	—	4	4	no	—
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	—	2	2	no	—
115	#88322	—	2	2	no	—
116	#88322	—	2	2	no	—
117	#8261	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
122	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
123	C₂²×C₅₃	#405	1	1	yes	characteristic
124	C₂²×C₅₃	#405	1	1	yes	characteristic
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	#322	1	4	yes	complemented
127	C₄×C₅₃	#423	1	1	yes	characteristic
128	C₄×C₅₃	#423	1	1	yes	characteristic
129	C₄×C₅₃	#432	1	1	yes	characteristic
130	C₄×C₅₃	#432	1	1	yes	characteristic
131	Dic₅₃	—	2	2	no	—
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	C₈×D₄	1	2	yes	—
134	Dic₅₃	#436	1	2	yes	—
135	Dic₅₃	C₈×Q₈	1	2	yes	—
136	Dic₅₃	#446	1	2	yes	—
137	C₄×C₄⋊C₈	—	53	1	no	—
138	C₄×C₄⋊C₈	—	53	1	no	—
139	C₄²×C₈	—	53	1	no	—
140	C₂×#322	—	53	2	no	—
141	#1273	—	53	2	no	—
142	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
143	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
144	C₂×C₅₃	#1451	1	1	yes	characteristic
145	C₂×C₅₃	#1452	1	1	yes	characteristic
146	C₂×C₅₃	#1767	1	1	yes	characteristic
147	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₅₃	#1770	1	1	yes	characteristic
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄×C₈	—	53	1	no	—
152	C₂×C₄×C₈	—	53	2	no	—
153	C₂×C₄×C₈	—	53	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	C₂×C₄×C₈	#6157	1	1	yes	characteristic
159	C₄³	—	53	1	no	—
160	#336	—	53	2	no	—
161	#336	—	53	2	no	—
162	#322	—	106	4	no	—
163	C₅₃	#8261	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₈	—	53	4	no	—
165	C₂²×C₈	—	53	1	no	—
166	C₂²×C₈	—	53	1	no	—
167	C₂²×C₈	—	53	4	no	—
168	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
169	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
170	C₂×C₄²	—	53	1	no	—
171	C₂×C₄²	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	—	53	1	no	—
174	C₂×C₄²	—	53	1	no	—
175	C₂×C₄²	—	53	1	no	—
176	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
177	C₄⋊C₈	—	53	8	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	53	2	no	—
181	C₄×C₈	—	53	2	no	—
182	C₄×C₈	—	106	1	no	—
183	C₄×C₈	—	106	1	no	—
184	C₄×C₈	—	106	2	no	—
185	C₄×C₈	—	106	2	no	—
186	C₄×C₈	—	106	2	no	—
187	C₄×C₈	—	106	2	no	—
188	C₄×C₈	—	106	2	no	—
189	C₄×C₈	—	106	2	no	—
190	C₄×C₈	—	106	2	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
196	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
197	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	53	8	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	106	4	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	—	106	1	no	—
208	C₂×C₈	—	106	1	no	—
209	C₂×C₈	—	106	1	no	—
210	C₂×C₈	—	106	1	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	106	4	no	—
214	C₂×C₈	#11719936	1	2	yes	—
215	C₂×C₈	C₄⋊#6157	1	2	yes	—
216	C₄²	—	53	4	no	—
217	C₄²	—	53	4	no	—
218	C₄²	—	53	1	no	—
219	C₄²	—	53	1	no	—
220	C₄²	—	53	1	no	—
221	C₄²	—	53	1	no	—
222	C₄²	—	106	2	no	—
223	C₄²	—	106	2	no	—
224	C₄²	—	106	1	no	—
225	C₄²	—	106	1	no	—
226	C₄²	#11719936	1	2	yes	—
227	C₄²	#11719940	1	2	yes	—
228	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	53	1	no	—
233	C₂×C₄	—	53	1	no	—
234	C₂×C₄	—	53	1	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	53	2	no	—
237	C₂×C₄	—	53	2	no	—
238	C₂×C₄	—	53	2	no	—
239	C₂×C₄	—	53	2	no	—
240	C₂×C₄	—	53	2	no	—
241	C₂×C₄	—	106	1	no	—
242	C₂×C₄	—	106	1	no	—
243	C₂×C₄	—	106	1	no	—
244	C₂×C₄	—	106	1	no	—
245	C₂×C₄	C₈×#6157	1	2	yes	—
246	C₂×C₄	#13133271	1	2	yes	—
247	C₂×C₄	#13133272	1	2	yes	—
248	C₂×C₄	#13133272	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
250	C₂×C₄	#13133296	1	1	yes	characteristic
251	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
252	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
253	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	106	8	no	—
258	C₈	—	106	2	no	—
259	C₈	—	106	2	no	—
260	C₈	—	212	4	no	—
261	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
262	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
263	C₂²	C₂×#13133271	1	1	yes	characteristic
264	C₂²	C₂×#13133272	1	1	yes	characteristic
265	C₂²	C₄×#11719936	1	1	yes	characteristic
266	C₂²	#16811586	1	1	yes	characteristic
267	C₂²	#16811586	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	53	2	no	—
270	C₄	—	53	2	no	—
271	C₄	—	53	2	no	—
272	C₄	—	53	2	no	—
273	C₄	—	106	2	no	—
274	C₄	—	106	2	no	—
275	C₄	#88324⋊C₈	1	4	yes	complemented
276	C₄	#16811626	1	1	yes	characteristic
277	C₄	#16811627	1	1	yes	characteristic
278	C₄	#16811644	1	1	yes	characteristic
279	C₄	#16811645	1	1	yes	characteristic
280	C₂	C₂×#88324⋊C₈	1	1	yes	characteristic
281	C₂	C₈×#11719936	1	1	yes	characteristic
282	C₂	#20289925	1	1	yes	characteristic
283	C₂	#20289928	1	1	yes	characteristic
284	C₂	#20290501	1	1	yes	characteristic
285	C₂	C₈×C₄⋊#6157	1	1	yes	characteristic
286	C₂	#20290509	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8261
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
23. C₄⋊#88324	C₈	semidirect
126. C₄×C₅₃	#322	semidirect
163. C₅₃	#8261	semidirect
275. C₄	#88324⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88324	2
C₂×#88324⋊C₈	2
#20289537	2
C₄×C₈×Dic₅₃	2
#8261	53