Subgroups — #27293799

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×#13133157	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	complemented
9	#16810399	—	2	1	no	—
10	C₄×#11719908	C₂²	1	2	yes	complemented
11	C₂³×C₈×C₅₃	—	2	1	no	—
12	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
13	C₈×D₄×C₅₃	—	2	1	no	—
14	C₈×D₄×C₅₃	—	2	1	no	—
15	C₈×D₄×C₅₃	C₄	1	2	yes	—
16	C₈×D₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	2	4	no	—
21	#13133157	—	4	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	—	4	2	no	—
30	C₂²×C₈×C₅₃	—	4	2	no	—
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₄×D₄×C₅₃	—	2	1	no	—
34	C₄×D₄×C₅₃	—	2	1	no	—
35	C₄×D₄×C₅₃	C₈	1	2	yes	—
36	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	1	no	—
41	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₂²⋊C₈×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	—	2	1	no	—
49	C₂×D₄×C₅₃	—	2	4	no	—
50	C₂×D₄×C₅₃	—	2	1	no	—
51	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	—	4	2	no	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	4	4	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	D₈	1	2	yes	—
69	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
74	C₂²⋊C₄×C₅₃	—	2	2	no	—
75	C₂²⋊C₄×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	#11719908	—	2	4	no	—
81	#11719908	—	4	2	no	—
82	C₂³×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
87	D₄×C₅₃	—	2	4	no	—
88	D₄×C₅₃	—	2	2	no	—
89	D₄×C₅₃	M₃₂	1	4	yes	complemented
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#102	1	2	yes	—
101	C₂×C₄×C₅₃	#104	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	4	2	no	—
108	C₈×C₅₃	#102	1	2	yes	—
109	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
110	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
111	#11583	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	4	8	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
119	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
125	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
127	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
128	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
129	#2150	—	53	1	no	—
130	C₄×M₃₂	—	53	1	no	—
131	D₄⋊C₁₆	—	53	4	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	4	4	no	—
134	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
135	C₂×C₅₃	#2112	1	1	yes	characteristic
136	C₂×C₅₃	#2117	1	1	yes	characteristic
137	C₂³×C₈	—	2	1	no	—
138	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
139	C₂×M₃₂	—	106	1	no	—
140	C₂×M₃₂	—	106	1	no	—
141	C₈×D₄	—	2	1	no	—
142	C₈×D₄	—	2	1	no	—
143	C₈×D₄	Dic₅₃	1	2	yes	—
144	C₈×D₄	C₂×D₅₃	1	2	yes	—
145	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	C₄⋊C₁₆	—	53	2	no	—
149	C₄⋊C₁₆	—	106	1	no	—
150	#346	—	106	1	no	—
151	C₄×C₁₆	—	53	2	no	—
152	C₅₃	#11583	1	1	yes	characteristic, complemented, Sylow
153	C₂²×D₄	#88322	1	1	yes	characteristic
154	C₂³×C₄	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	4	2	no	—
159	C₂²×C₈	—	4	2	no	—
160	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	#88322	1	2	yes	—
165	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
166	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
169	M₃₂	—	106	4	no	—
170	M₃₂	—	212	2	no	—
171	C₂×C₁₆	—	106	2	no	—
172	C₂×C₁₆	—	106	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₂⁴	—	2	1	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂×#88322	1	2	yes	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
191	C₂²×C₄	#11719917	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	#11719922	1	2	yes	—
200	C₂×C₈	#11719923	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	C₂×#88322	1	2	yes	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	C₂×#88322	1	1	yes	characteristic
210	C₄²	#11719917	1	1	yes	characteristic
211	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
213	C₁₆	—	106	4	no	—
214	C₁₆	—	212	2	no	—
215	C₂³	—	2	4	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₂²⋊#88322	1	1	yes	characteristic
220	D₄	—	2	4	no	—
221	D₄	—	2	2	no	—
222	D₄	#13133157	1	4	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
231	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
232	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
233	C₂×C₄	#13133138	1	2	yes	—
234	C₂×C₄	#13133144	1	2	yes	—
235	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	#13133138	1	2	yes	—
242	C₈	C₂×#11719922	1	1	yes	characteristic
243	C₈	C₂×#11719923	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	D₄⋊#88322	1	2	yes	—
251	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	D₄⋊#88322	1	2	yes	—
257	C₄	C₂×#13133138	1	1	yes	characteristic
258	C₄	C₂×#13133144	1	1	yes	characteristic
259	C₄	C₂×#13133157	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×D₄⋊#88322	1	1	yes	characteristic
263	C₂	#20276326	1	1	yes	characteristic
264	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#11583
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. C₄×#13133157	C₂	semidirect
3. D₄⋊#11719908	C₂	semidirect
8. C₄⋊#11719908	C₂²	semidirect
10. C₄×#11719908	C₂²	semidirect
89. D₄×C₅₃	M₃₂	semidirect
152. C₅₃	#11583	semidirect
222. D₄	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₄×#13133157	2
D₄⋊#11719908	2
C₂×C₈×D₄×C₅₃	2
#11583	53