Subgroups — #28310233

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#20289757	C₂	1	2	yes	maximal, complemented
3	#20289756	C₂	1	2	yes	maximal, complemented
4	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133292	C₄	1	4	yes	complemented
6	D₅₃⋊M₃₂	—	2	4	no	—
7	D₅₃⋊M₃₂	—	2	4	no	—
8	C₂×#13133291	C₂²	1	4	yes	—
9	C₂×D₅₃⋊C₁₆	C₄	1	4	yes	complemented
10	C₂²×#11719912	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133174	C₄	1	1	yes	characteristic
13	#16810577	—	2	4	no	—
14	C₂×#13133128	C₂²	1	1	yes	characteristic
15	C₂×#13133127	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
18	#13133292	—	2	8	no	—
19	#13133291	—	2	4	no	—
20	#13133291	—	2	4	no	—
21	D₅₃⋊C₁₆	—	2	8	no	—
22	C₂×#11719910	C₂×C₄	1	8	yes	—
23	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#11719912	—	2	2	no	—
25	C₂×#11719912	—	2	2	no	—
26	C₂×#11719912	C₈	1	4	yes	—
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	D₄	1	4	yes	—
32	C₂×C₈×D₅₃	D₄	1	2	yes	—
33	C₂×C₈×D₅₃	Q₈	1	2	yes	—
34	#13133174	—	2	2	no	—
35	#13133174	C₈	1	4	yes	—
36	C₂×#11719919	—	2	1	no	—
37	C₂×#11719921	—	2	1	no	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	#13133128	—	2	2	no	—
40	#13133127	—	2	2	no	—
41	C₈×Dic₅₃	—	2	2	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	—	2	2	no	—
46	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	1	no	—
50	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₄²	1	2	yes	—
52	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	C₂×C₈	1	2	yes	—
66	#11719921	—	2	4	no	—
67	#11719919	—	2	4	no	—
68	C₄×Dic₅₃	—	2	1	no	—
69	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
70	C₂×#88322	—	2	1	no	—
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	C₂×C₈	1	2	yes	—
73	C₂×#88322	C₂²×C₄	1	2	yes	—
74	C₂×#88322	C₂×D₄	1	1	yes	characteristic
75	C₂×#88322	C₂×D₄	1	2	yes	—
76	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
77	#11719912	—	2	8	no	—
78	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
79	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
80	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
81	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
82	#11719910	—	2	8	no	—
83	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×D₅₃	—	4	2	no	—
85	C₂²×D₅₃	#97	1	2	yes	—
86	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
87	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×Dic₅₃	—	2	2	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	#97	1	2	yes	—
99	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
100	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
101	C₄×D₅₃	#95	1	4	yes	—
102	C₄×D₅₃	C₄×C₈	1	4	yes	—
103	C₄×D₅₃	C₂²⋊C₈	1	4	yes	—
104	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	#95	1	2	yes	—
107	C₈×C₅₃	#95	1	2	yes	—
108	#88322	—	2	2	no	—
109	#88322	C₂×C₂²⋊C₄	1	2	yes	—
110	#88322	C₂×C₄⋊C₄	1	2	yes	—
111	#12547	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂×D₅₃	—	2	4	no	—
117	C₂×D₅₃	—	4	2	no	—
118	C₂×D₅₃	—	4	2	no	—
119	C₂×D₅₃	#336	1	4	yes	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
123	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	Dic₅₃	—	4	2	no	—
126	Dic₅₃	#336	1	4	yes	—
127	C₂²×M₃₂	—	53	2	no	—
128	C₂×#405	—	53	1	no	—
129	#1375	—	53	2	no	—
130	#1375	—	53	2	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
134	C₂×C₅₃	#2110	1	2	yes	—
135	D₅₃	—	2	4	no	—
136	C₂²×M₁₆	—	53	1	no	—
137	C₂³×C₈	—	53	1	no	—
138	C₂×#117	—	53	1	no	—
139	C₂×M₃₂	—	53	4	no	—
140	C₂×M₃₂	—	53	4	no	—
141	C₂×M₃₂	—	106	4	no	—
142	C₂×M₃₂	—	106	4	no	—
143	C₂²×C₁₆	—	53	4	no	—
144	#405	—	106	4	no	—
145	C₂×#97	—	53	1	no	—
146	C₂×#97	—	53	1	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₂×C₄×C₈	#6157	1	1	yes	characteristic
149	C₅₃	#12547	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	—	53	1	no	—
151	C₂×M₁₆	—	53	4	no	—
152	C₂×M₁₆	—	106	2	no	—
153	C₂×M₁₆	—	106	2	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	2	no	—
157	C₂²×C₈	—	53	2	no	—
158	C₂²×C₈	—	53	2	no	—
159	C₂²×C₈	—	53	4	no	—
160	C₂²×C₈	—	53	1	no	—
161	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
162	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
163	#117	—	53	4	no	—
164	#117	—	106	2	no	—
165	C₂×C₄⋊C₄	—	106	1	no	—
166	C₂×C₂²⋊C₄	—	106	1	no	—
167	C₂×C₄²	—	53	1	no	—
168	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
169	M₃₂	—	106	4	no	—
170	M₃₂	—	106	4	no	—
171	M₃₂	—	106	8	no	—
172	C₂×C₁₆	—	53	8	no	—
173	C₂×C₁₆	—	106	8	no	—
174	#97	—	106	2	no	—
175	#97	—	106	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	106	2	no	—
178	C₂⁴	—	53	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	53	1	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	—	53	2	no	—
185	C₂²×C₄	—	53	2	no	—
186	C₂²×C₄	—	53	2	no	—
187	C₂²×C₄	—	106	1	no	—
188	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
189	M₁₆	—	106	8	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	1	no	—
195	C₂×C₈	—	53	1	no	—
196	C₂×C₈	—	53	4	no	—
197	C₂×C₈	—	53	4	no	—
198	C₂×C₈	—	53	4	no	—
199	C₂×C₈	—	53	4	no	—
200	C₂×C₈	—	53	2	no	—
201	C₂×C₈	—	106	1	no	—
202	C₂×C₈	—	106	1	no	—
203	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
204	C₂×C₈	C₄×#6157	1	1	yes	characteristic
205	C₂×C₈	C₄×#6157	1	1	yes	characteristic
206	C₂×C₈	C₄⋊#6157	1	2	yes	—
207	C₂×C₈	C₂²⋊#6157	1	2	yes	—
208	C₄⋊C₄	—	106	4	no	—
209	C₂²⋊C₄	—	106	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	53	2	no	—
212	C₄²	—	106	1	no	—
213	C₄²	D₅₃⋊C₈	1	2	yes	—
214	C₁₆	—	106	8	no	—
215	C₂³	—	53	2	no	—
216	C₂³	—	53	2	no	—
217	C₂³	—	53	2	no	—
218	C₂³	—	212	2	no	—
219	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	53	2	no	—
223	C₂×C₄	—	53	2	no	—
224	C₂×C₄	—	53	2	no	—
225	C₂×C₄	—	53	4	no	—
226	C₂×C₄	—	53	4	no	—
227	C₂×C₄	—	53	4	no	—
228	C₂×C₄	—	53	4	no	—
229	C₂×C₄	—	106	2	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
232	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
234	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
236	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	53	2	no	—
239	C₈	—	53	2	no	—
240	C₈	—	106	2	no	—
241	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
242	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	53	4	no	—
245	C₂²	—	106	4	no	—
246	C₂²	—	212	2	no	—
247	C₂²	—	212	2	no	—
248	C₂²	#16811582	1	1	yes	characteristic
249	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
250	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
251	C₄	—	4	2	no	—
252	C₄	—	53	4	no	—
253	C₄	—	212	2	no	—
254	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
255	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
256	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
257	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	106	4	no	—
261	C₂	#20290376	1	1	yes	characteristic
262	C₂	#20290937	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12547
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289757	C₂	semidirect
3. #20289756	C₂	semidirect
5. C₂×#13133292	C₄	semidirect
9. C₂×D₅₃⋊C₁₆	C₄	semidirect
149. C₅₃	#12547	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊M₃₂	2
#20289757	2
#20289756	2
C₂×#16810577	2
#12547	53