Subgroups — #27293841

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272676	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	#16810399	—	2	1	no	—
8	C₄×#11719908	C₂²	1	2	yes	complemented
9	#16810376	C₂²	1	2	yes	complemented
10	#16810374	—	2	1	no	—
11	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	—	2	1	no	—
13	C₄×D₈×C₅₃	—	2	1	no	—
14	C₄×D₈×C₅₃	—	2	1	no	—
15	C₄×D₈×C₅₃	C₄	1	2	yes	—
16	C₄×D₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#102×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	2	4	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₂²⋊#88322	—	4	1	no	—
24	C₄×#88322	—	4	1	no	—
25	C₂²×D₄×C₅₃	—	2	1	no	—
26	C₂³×C₄×C₅₃	—	4	1	no	—
27	C₂×D₈×C₅₃	—	2	1	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	C₈	1	4	yes	complemented
30	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×D₄×C₅₃	—	2	2	no	—
34	C₄×D₄×C₅₃	—	2	2	no	—
35	C₄×D₄×C₅₃	—	4	1	no	—
36	C₄×D₄×C₅₃	—	4	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	1	no	—
41	#107×C₅₃	C₂×C₄	1	2	yes	—
42	#102×C₅₃	—	2	2	no	—
43	#102×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₅₃	—	4	1	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	—	4	1	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	8	4	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
60	D₈×C₅₃	—	2	4	no	—
61	D₈×C₅₃	—	2	2	no	—
62	D₈×C₅₃	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₂²⋊C₄×C₅₃	—	4	2	no	—
73	C₂²⋊C₄×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅₃	D₈	1	1	yes	characteristic
76	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×#88322	—	4	2	no	—
79	#11719908	—	2	4	no	—
80	C₂³×C₅₃	—	4	4	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	—	8	2	no	—
83	C₂³×C₅₃	—	8	2	no	—
84	C₂³×C₅₃	#102	1	1	yes	characteristic
85	D₄×C₅₃	—	2	4	no	—
86	D₄×C₅₃	—	4	4	no	—
87	D₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	8	4	no	—
92	C₂×C₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#104	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₈×C₅₃	—	2	1	no	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
105	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
107	#88322	—	8	2	no	—
108	#11593	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	4	4	no	—
112	C₂²×C₅₃	—	8	8	no	—
113	C₂²×C₅₃	—	8	2	no	—
114	C₂²×C₅₃	—	8	2	no	—
115	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
116	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	8	2	no	—
121	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
122	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
124	C₄×C₅₃	#509	1	2	yes	—
125	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
126	C₄×M₃₂	—	53	1	no	—
127	#1564	—	53	1	no	—
128	#1331	—	53	4	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	—	8	4	no	—
131	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
132	C₂×C₅₃	#2140	1	1	yes	characteristic
133	C₂×C₅₃	#2143	1	1	yes	characteristic
134	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
135	C₂×C₄×D₄	—	2	1	no	—
136	C₂×M₃₂	—	106	1	no	—
137	C₄×D₈	—	2	1	no	—
138	C₄×D₈	—	2	1	no	—
139	C₄×D₈	Dic₅₃	1	2	yes	—
140	C₄×D₈	C₂×D₅₃	1	2	yes	—
141	#431	—	106	1	no	—
142	C₂×#107	Dic₅₃	1	1	yes	characteristic
143	C₂×#102	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	#346	—	106	1	no	—
146	C₄×C₁₆	—	53	2	no	—
147	#335	—	53	2	no	—
148	#334	—	106	1	no	—
149	C₅₃	#11593	1	1	yes	characteristic, complemented, Sylow
150	C₂²×D₄	—	2	1	no	—
151	C₂³×C₄	—	4	1	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	—	2	1	no	—
154	C₂×D₈	#88322	1	4	yes	complemented
155	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
156	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	4	1	no	—
161	C₄×D₄	—	4	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	—	4	1	no	—
164	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
165	M₃₂	—	106	4	no	—
166	C₂×C₁₆	—	106	2	no	—
167	#107	—	2	1	no	—
168	#107	C₂×Dic₅₃	1	2	yes	—
169	C₄⋊C₈	—	106	2	no	—
170	#102	—	2	2	no	—
171	#102	—	2	2	no	—
172	C₂²⋊C₈	—	212	1	no	—
173	C₄×C₈	—	212	1	no	—
174	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₂⁴	—	4	1	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	4	no	—
187	C₂²×C₄	#11719922	1	1	yes	characteristic
188	C₂²×C₄	#11719923	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	D₈	—	2	4	no	—
191	D₈	—	2	2	no	—
192	D₈	C₂×#88322	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	C₂×#88322	1	2	yes	—
196	C₂×C₈	#11719917	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
198	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₂²⋊C₄	—	4	2	no	—
205	C₄²	#11719922	1	1	yes	characteristic
206	C₄²	#11719923	1	1	yes	characteristic
207	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₁₆	—	106	4	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	8	2	no	—
213	C₂³	—	8	2	no	—
214	C₂³	#13133138	1	1	yes	characteristic
215	D₄	—	2	4	no	—
216	D₄	—	4	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	C₂²⋊#88322	1	2	yes	—
225	C₂×C₄	#13133138	1	1	yes	characteristic
226	C₂×C₄	#13133138	1	1	yes	characteristic
227	C₂×C₄	#13133138	1	1	yes	characteristic
228	C₂×C₄	#13133144	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	424	2	no	—
235	C₈	C₂²⋊#88322	1	2	yes	—
236	C₈	C₂²×#88322	1	1	yes	characteristic
237	C₈	C₂×#11719917	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	8	8	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	2	no	—
244	C₂²	D₄⋊#88322	1	2	yes	—
245	C₂²	C₂×#13133138	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	2	no	—
250	C₄	D₄⋊#88322	1	2	yes	—
251	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
252	C₄	C₂×#13133144	1	1	yes	characteristic
253	C₄	#16811059	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	8	4	no	—
256	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic
257	C₂	#20276431	1	1	yes	characteristic
258	C₂	#20276443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	#11593
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₄×#13133157	C₂	semidirect
2. #20273463	C₂	semidirect
3. #20272676	C₂	semidirect
8. C₄×#11719908	C₂²	semidirect
9. #16810376	C₂²	semidirect
29. C₂×D₈×C₅₃	C₈	semidirect
149. C₅₃	#11593	semidirect
154. C₂×D₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
#20273463	2
#20272676	2
C₂×C₄×D₈×C₅₃	2
#11593	53