Subgroups — #28310735

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811510	C₂	1	2	yes	maximal
3	C₄×C₈×#6157	C₂	1	1	yes	maximal, characteristic
4	(C₂×#6157)⋊C₁₆	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
7	C₄×D₅₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×#6157	C₄	1	1	yes	characteristic
9	C₂×C₈×#6157	C₄	1	1	yes	characteristic
10	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
11	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
12	C₄²×#6157	C₄	1	1	yes	characteristic
13	#16811544	C₄	1	2	yes	complemented
14	C₄⋊#11719910	C₄	1	2	yes	complemented
15	#16811510	—	2	4	no	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊#11719908	C₄	1	1	yes	characteristic, complemented
19	#16810441	C₄	1	2	yes	complemented
20	C₅₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
21	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×#6157	C₈	1	2	yes	—
24	C₂×C₄×#6157	C₈	1	2	yes	—
25	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
27	D₅₃⋊C₁₆	—	2	2	no	—
28	D₅₃⋊C₁₆	—	2	2	no	—
29	C₂×#11719910	C₂×C₄	1	4	yes	—
30	C₄×#88324	C₈	1	2	yes	—
31	C₄×#88324	C₈	1	2	yes	—
32	C₈×#6157	—	2	2	no	—
33	C₈×#6157	—	2	1	no	—
34	C₈×#6157	—	2	1	no	—
35	C₈×#6157	—	2	2	no	—
36	C₈×#6157	D₄	1	2	yes	—
37	C₈×#6157	Q₈	1	2	yes	—
38	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
40	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#11719908	C₂×C₄	1	2	yes	—
42	C₁₆×D₅₃	—	2	2	no	—
43	C₁₆×D₅₃	—	2	2	no	—
44	C₄×#88322	C₂×C₄	1	1	yes	characteristic
45	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×#6157	C₂×C₈	1	2	yes	—
50	C₂²×#6157	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×#88324	C₂×C₈	1	2	yes	—
55	C₂×#88324	C₂×C₈	1	2	yes	—
56	C₄×#6157	—	2	4	no	—
57	C₄×#6157	—	2	4	no	—
58	C₄×#6157	—	2	1	no	—
59	C₄×#6157	—	2	1	no	—
60	C₄×#6157	C₄⋊C₄	1	2	yes	—
61	C₄×#6157	M₁₆	1	2	yes	—
62	D₅₃⋊C₈	—	2	1	no	—
63	D₅₃⋊C₈	—	2	1	no	—
64	D₅₃⋊C₈	C₄⋊C₄	1	2	yes	—
65	D₅₃⋊C₈	M₁₆	1	2	yes	—
66	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₄²	1	1	yes	characteristic
69	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
70	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
71	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂×#88322	C₄²	1	1	yes	characteristic
73	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
74	C₈×D₅₃	—	2	1	no	—
75	C₈×D₅₃	—	2	1	no	—
76	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
77	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₈×D₅₃	#41	1	1	yes	characteristic
79	C₈×D₅₃	#41	1	1	yes	characteristic
80	#11719910	—	2	4	no	—
81	C₁₆×C₅₃	—	2	2	no	—
82	#11719908	—	2	2	no	—
83	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×#6157	—	2	2	no	—
85	C₂×#6157	—	2	2	no	—
86	C₂×#6157	—	2	2	no	—
87	C₂×#6157	—	2	2	no	—
88	C₂×#6157	C₄⋊C₈	1	4	yes	—
89	C₂×#6157	M₃₂	1	4	yes	—
90	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
94	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
95	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₄×D₅₃	—	2	2	no	—
97	C₄×D₅₃	M₃₂	1	4	yes	—
98	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₄×D₅₃	#117	1	1	yes	characteristic
100	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
102	#88324	—	2	2	no	—
103	#88324	C₄⋊C₈	1	4	yes	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
107	#88322	—	2	1	no	—
108	#88322	C₄×D₄	1	1	yes	characteristic
109	#88322	C₄×Q₈	1	1	yes	characteristic
110	#12653	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂×D₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂×D₅₃	#431	1	1	yes	characteristic
114	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
115	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
116	#6157	—	2	4	no	—
117	#6157	—	2	4	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	#346	1	2	yes	complemented
120	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
122	Dic₅₃	—	2	1	no	—
123	Dic₅₃	#346	1	2	yes	complemented
124	Dic₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	Dic₅₃	#431	1	1	yes	characteristic
126	C₂×C₄⋊C₁₆	—	53	1	no	—
127	C₂×C₄⋊C₁₆	—	53	1	no	—
128	C₂×#346	—	53	2	no	—
129	C₄²×C₈	—	53	1	no	—
130	#1374	—	53	2	no	—
131	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
133	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
134	D₅₃	#2150	1	2	yes	—
135	D₅₃	#2163	1	2	yes	—
136	C₂²×C₁₆	—	53	2	no	—
137	C₂²×C₁₆	—	53	2	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₄³	—	53	1	no	—
145	C₄⋊C₁₆	—	53	2	no	—
146	C₄⋊C₁₆	—	53	2	no	—
147	C₄⋊C₁₆	—	53	2	no	—
148	C₄⋊C₁₆	—	53	1	no	—
149	C₄⋊C₁₆	#6157	1	1	yes	characteristic, complemented
150	#346	—	106	4	no	—
151	C₅₃	#12653	1	1	yes	characteristic, complemented, Sylow
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	1	no	—
156	C₂×C₄²	—	53	1	no	—
157	C₂×C₄²	—	53	1	no	—
158	C₂×C₄²	—	53	1	no	—
159	C₂×C₄²	—	53	2	no	—
160	C₂×C₄²	—	53	2	no	—
161	C₂×C₁₆	—	53	4	no	—
162	C₂×C₁₆	—	53	2	no	—
163	C₂×C₁₆	—	106	2	no	—
164	C₂×C₁₆	—	106	2	no	—
165	C₂×C₁₆	—	106	2	no	—
166	C₂×C₁₆	—	106	2	no	—
167	C₂×C₁₆	C₂×#6157	1	2	yes	—
168	C₄×C₈	—	53	2	no	—
169	C₄×C₈	—	53	2	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	53	2	no	—
172	C₄×C₈	—	53	1	no	—
173	C₄×C₈	—	53	1	no	—
174	C₄×C₈	—	53	1	no	—
175	C₄×C₈	—	106	2	no	—
176	C₄×C₈	—	106	2	no	—
177	C₄×C₈	—	106	1	no	—
178	C₄×C₈	—	106	1	no	—
179	C₄×C₈	C₂×#6157	1	1	yes	characteristic
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	1	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	—	53	1	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	53	2	no	—
187	C₂×C₈	—	53	2	no	—
188	C₂×C₈	—	53	1	no	—
189	C₂×C₈	—	53	1	no	—
190	C₂×C₈	—	53	1	no	—
191	C₂×C₈	—	53	1	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	1	no	—
195	C₂×C₈	—	106	1	no	—
196	C₂×C₈	—	106	1	no	—
197	C₂×C₈	—	106	1	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	C₄×#6157	1	1	yes	characteristic
200	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
201	C₄²	—	53	2	no	—
202	C₄²	—	53	2	no	—
203	C₄²	—	53	1	no	—
204	C₄²	—	53	1	no	—
205	C₄²	—	53	1	no	—
206	C₄²	—	106	1	no	—
207	C₄²	—	106	1	no	—
208	C₄²	—	106	4	no	—
209	C₄²	—	106	4	no	—
210	C₄²	C₄×#6157	1	1	yes	characteristic
211	C₁₆	—	2	2	no	—
212	C₁₆	—	106	4	no	—
213	C₁₆	—	106	2	no	—
214	C₂³	—	53	1	no	—
215	C₂×C₄	—	53	1	no	—
216	C₂×C₄	—	53	1	no	—
217	C₂×C₄	—	53	1	no	—
218	C₂×C₄	—	53	1	no	—
219	C₂×C₄	—	53	1	no	—
220	C₂×C₄	—	53	1	no	—
221	C₂×C₄	—	53	4	no	—
222	C₂×C₄	—	53	4	no	—
223	C₂×C₄	—	53	4	no	—
224	C₂×C₄	—	53	1	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	106	2	no	—
227	C₂×C₄	—	106	2	no	—
228	C₂×C₄	—	106	2	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	C₈×#6157	1	1	yes	characteristic
231	C₂×C₄	C₈×#6157	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	53	4	no	—
235	C₈	—	53	1	no	—
236	C₈	—	53	1	no	—
237	C₈	—	106	2	no	—
238	C₈	—	106	1	no	—
239	C₈	D₄×#6157	1	1	yes	characteristic
240	C₈	Q₈×#6157	1	1	yes	characteristic
241	C₂²	—	53	2	no	—
242	C₂²	—	53	2	no	—
243	C₂²	—	53	1	no	—
244	C₂²	—	53	1	no	—
245	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	53	2	no	—
248	C₄	—	53	1	no	—
249	C₄	—	53	1	no	—
250	C₄	—	106	4	no	—
251	C₄	—	106	4	no	—
252	C₄	—	106	1	no	—
253	C₄	#16811510	1	2	yes	complemented
254	C₄	C₄⋊C₄×#6157	1	1	yes	characteristic
255	C₄	M₁₆×#6157	1	1	yes	characteristic
256	C₂	—	53	2	no	—
257	C₂	—	53	2	no	—
258	C₂	C₄⋊C₈×#6157	1	1	yes	characteristic
259	C₂	C₂×#16811510	1	1	yes	characteristic
260	C₂	M₃₂×#6157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×C₄⋊C₁₆	C₄
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₄⋊C₁₆	#6157
hypocenter	C₅₃	#12653
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. #16811544	C₄	semidirect
14. C₄⋊#11719910	C₄	semidirect
18. C₄⋊#11719908	C₄	semidirect
19. #16810441	C₄	semidirect
20. C₅₃×C₄⋊C₁₆	C₄	semidirect
119. C₄×C₅₃	#346	semidirect
123. Dic₅₃	#346	semidirect
149. C₄⋊C₁₆	#6157	semidirect
151. C₅₃	#12653	semidirect
253. C₄	#16811510	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊(C₄⋊C₁₆)	2
C₂×#16811510	2
C₄×C₈×#6157	2
(C₂×#6157)⋊C₁₆	2
C₄⋊C₁₆×D₅₃	2
#12653	53