Subgroups — #28302039

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 274 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	C₄²×#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	C₄	1	2	yes	—
9	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
10	C₂×C₄×#88324	C₂²	1	2	yes	—
11	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
12	#16811489	C₄	1	2	yes	—
13	#88324⋊C₈	—	2	4	no	—
14	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810377	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	C₂×C₄	1	2	yes	—
20	C₂²×#88324	C₂×C₄	1	4	yes	—
21	C₂²×#88324	C₂×C₄	1	2	yes	—
22	C₄⋊#88324	C₈	1	8	yes	complemented
23	C₄×#88324	—	2	4	no	—
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	—	2	2	no	—
28	C₄×#88324	—	2	4	no	—
29	C₄×#88324	D₄	1	2	yes	—
30	C₄×#88324	Q₈	1	2	yes	—
31	C₂²×#88322	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
38	C₄⋊#88322	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×#88324	—	2	2	no	—
52	C₂×#88324	—	2	4	no	—
53	C₂×#88324	—	2	2	no	—
54	C₂×#88324	—	2	2	no	—
55	C₂×#88324	—	2	2	no	—
56	C₂×#88324	—	2	4	no	—
57	C₂×#88324	C₄⋊C₄	1	4	yes	—
58	C₂×#88324	C₂×C₈	1	8	yes	—
59	C₂×#88324	M₁₆	1	4	yes	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	M₁₆	1	2	yes	—
64	C₄²×C₅₃	M₁₆	1	2	yes	—
65	C₄×Dic₅₃	—	2	1	no	—
66	C₄×Dic₅₃	—	2	1	no	—
67	C₄×Dic₅₃	—	2	2	no	—
68	C₄×Dic₅₃	—	2	2	no	—
69	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
70	C₄×Dic₅₃	M₁₆	1	4	yes	—
71	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₄×Dic₅₃	#41	1	1	yes	characteristic
74	C₄×Dic₅₃	#41	1	1	yes	characteristic
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₅₃	#97	1	2	yes	—
83	C₂×C₄×C₅₃	#97	1	2	yes	—
84	C₂×C₄×C₅₃	#97	1	2	yes	—
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
91	C₂×Dic₅₃	—	2	1	no	—
92	C₂×Dic₅₃	—	2	1	no	—
93	C₂×Dic₅₃	—	2	1	no	—
94	C₂×Dic₅₃	—	2	1	no	—
95	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
96	C₂×Dic₅₃	#97	1	2	yes	—
97	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×Dic₅₃	#117	1	1	yes	characteristic
99	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
102	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
103	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
104	C₂×Dic₅₃	#131	1	2	yes	—
105	#88324	—	2	4	no	—
106	#88324	—	2	4	no	—
107	#88324	—	2	8	no	—
108	C₈×C₅₃	—	4	2	no	—
109	#88322	—	4	2	no	—
110	#8261	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
115	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
116	C₂²×C₅₃	#405	1	1	yes	characteristic
117	C₂²×C₅₃	#405	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	#322	1	4	yes	complemented
120	C₄×C₅₃	#436	1	1	yes	characteristic
121	C₄×C₅₃	#436	1	1	yes	characteristic
122	C₄×C₅₃	#446	1	1	yes	characteristic
123	C₄×C₅₃	#446	1	1	yes	characteristic
124	Dic₅₃	—	2	2	no	—
125	Dic₅₃	—	2	2	no	—
126	Dic₅₃	#423	1	2	yes	—
127	Dic₅₃	#432	1	2	yes	—
128	Dic₅₃	C₈×D₄	1	2	yes	—
129	Dic₅₃	C₈×Q₈	1	2	yes	—
130	C₄×C₄⋊C₈	—	53	1	no	—
131	C₄×C₄⋊C₈	—	53	1	no	—
132	C₄²×C₈	—	53	1	no	—
133	C₂×#322	—	53	2	no	—
134	#1273	—	53	2	no	—
135	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
136	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
137	C₂×C₅₃	#1451	1	1	yes	characteristic
138	C₂×C₅₃	#1452	1	1	yes	characteristic
139	C₂×C₅₃	#1767	1	1	yes	characteristic
140	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₅₃	#1770	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×C₄⋊C₈	—	53	1	no	—
144	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	2	no	—
147	C₂×C₄×C₈	—	53	2	no	—
148	C₂×C₄×C₈	—	53	2	no	—
149	C₂×C₄×C₈	—	53	1	no	—
150	C₂×C₄×C₈	—	53	2	no	—
151	C₄³	—	53	1	no	—
152	#336	—	53	2	no	—
153	#336	—	53	2	no	—
154	#322	—	106	4	no	—
155	C₅₃	#8261	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₈	—	53	4	no	—
157	C₂²×C₈	—	53	2	no	—
158	C₂²×C₈	—	53	2	no	—
159	C₂²×C₈	—	53	2	no	—
160	C₂²×C₈	C₂×#6157	1	2	yes	—
161	C₂×C₄²	—	53	1	no	—
162	C₂×C₄²	—	53	1	no	—
163	C₂×C₄²	—	53	1	no	—
164	C₂×C₄²	—	53	1	no	—
165	C₂×C₄²	—	53	1	no	—
166	C₂×C₄²	—	53	1	no	—
167	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	53	8	no	—
170	C₄⋊C₈	—	106	2	no	—
171	C₄×C₈	—	53	2	no	—
172	C₄×C₈	—	53	2	no	—
173	C₄×C₈	—	106	2	no	—
174	C₄×C₈	—	106	2	no	—
175	C₄×C₈	—	106	2	no	—
176	C₄×C₈	—	106	4	no	—
177	C₄×C₈	—	106	1	no	—
178	C₄×C₈	—	106	2	no	—
179	C₄×C₈	—	106	1	no	—
180	C₄×C₈	—	106	4	no	—
181	C₂²×C₄	—	53	1	no	—
182	C₂²×C₄	—	53	1	no	—
183	C₂²×C₄	—	53	1	no	—
184	C₂²×C₄	—	53	1	no	—
185	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
186	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
187	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	53	4	no	—
192	C₂×C₈	—	53	8	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	—	106	4	no	—
196	C₂×C₈	—	106	2	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	106	4	no	—
200	C₂×C₈	—	106	2	no	—
201	C₂×C₈	—	106	2	no	—
202	C₂×C₈	—	106	2	no	—
203	C₄²	—	53	4	no	—
204	C₄²	—	53	4	no	—
205	C₄²	—	53	1	no	—
206	C₄²	—	53	1	no	—
207	C₄²	—	53	1	no	—
208	C₄²	—	53	1	no	—
209	C₄²	—	106	2	no	—
210	C₄²	—	106	2	no	—
211	C₄²	—	106	1	no	—
212	C₄²	—	106	1	no	—
213	C₄²	#11719936	1	2	yes	—
214	C₄²	#11719940	1	2	yes	—
215	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	53	1	no	—
219	C₂×C₄	—	53	1	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	1	no	—
222	C₂×C₄	—	53	2	no	—
223	C₂×C₄	—	53	2	no	—
224	C₂×C₄	—	53	2	no	—
225	C₂×C₄	—	53	2	no	—
226	C₂×C₄	—	53	2	no	—
227	C₂×C₄	—	53	2	no	—
228	C₂×C₄	—	106	1	no	—
229	C₂×C₄	—	106	1	no	—
230	C₂×C₄	—	106	1	no	—
231	C₂×C₄	—	106	1	no	—
232	C₂×C₄	C₈×#6157	1	2	yes	—
233	C₂×C₄	#13133271	1	2	yes	—
234	C₂×C₄	#13133272	1	2	yes	—
235	C₂×C₄	#13133272	1	2	yes	—
236	C₂×C₄	D₄×#6157	1	1	yes	characteristic
237	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
240	C₂×C₄	#13133304	1	1	yes	characteristic
241	C₂×C₄	#13133305	1	1	yes	characteristic
242	C₈	—	4	2	no	—
243	C₈	—	106	4	no	—
244	C₈	—	106	4	no	—
245	C₈	—	106	8	no	—
246	C₈	—	212	2	no	—
247	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
248	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
249	C₂²	C₂×#13133271	1	1	yes	characteristic
250	C₂²	C₂×#13133272	1	1	yes	characteristic
251	C₂²	M₁₆×#6157	1	1	yes	characteristic
252	C₂²	#16811588	1	1	yes	characteristic
253	C₂²	#16811589	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	53	2	no	—
256	C₄	—	53	2	no	—
257	C₄	—	53	2	no	—
258	C₄	—	53	2	no	—
259	C₄	—	106	2	no	—
260	C₄	—	106	2	no	—
261	C₄	#88324⋊C₈	1	4	yes	complemented
262	C₄	#16811657	1	1	yes	characteristic
263	C₄	#16811658	1	1	yes	characteristic
264	C₄	#16811673	1	1	yes	characteristic
265	C₄	#16811674	1	1	yes	characteristic
266	C₂	C₂×#88324⋊C₈	1	1	yes	characteristic
267	C₂	#20289917	1	1	yes	characteristic
268	C₂	#20289922	1	1	yes	characteristic
269	C₂	#20289931	1	1	yes	characteristic
270	C₂	#20290502	1	1	yes	characteristic
271	C₂	#20290508	1	1	yes	characteristic
272	C₂	#20290511	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
22. C₄⋊#88324	C₈	semidirect
119. C₄×C₅₃	#322	semidirect
155. C₅₃	#8261	semidirect
261. 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
C₄²×#88324	2
C₂×#88324⋊C₈	2
#20289537	2
C₄⋊C₈×Dic₅₃	2
#8261	53