Subgroups — #27300393

Order 13568.

404 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274753	C₂	1	1	yes	maximal, characteristic
3	#20272810	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810587	—	2	1	no	—
8	C₂²⋊#11719908	—	2	1	no	—
9	C₂²⋊#11719908	—	2	1	no	—
10	#16810388	—	2	1	no	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₅₃	—	2	1	no	—
13	C₅₃×#427	—	2	1	no	—
14	C₅₃×#427	—	2	1	no	—
15	C₅₃×#427	C₄	1	2	yes	—
16	C₅₃×#427	C₂²	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#106×C₅₃	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	4	no	—
23	#13133153	—	2	4	no	—
24	C₂²⋊#88322	—	4	1	no	—
25	C₂²⋊#88322	—	4	1	no	—
26	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	#117×C₅₃	—	2	2	no	—
35	#117×C₅₃	—	2	2	no	—
36	#117×C₅₃	—	4	1	no	—
37	#117×C₅₃	—	4	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
41	C₂×C₄²×C₅₃	—	4	1	no	—
42	#107×C₅₃	—	2	1	no	—
43	#107×C₅₃	—	2	1	no	—
44	#107×C₅₃	C₂×C₄	1	2	yes	—
45	#107×C₅₃	C₂×C₄	1	2	yes	—
46	#106×C₅₃	—	2	2	no	—
47	#106×C₅₃	—	2	2	no	—
48	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	4	yes	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₄⋊C₄×C₅₃	—	4	1	no	—
73	C₂²⋊C₄×C₅₃	—	4	2	no	—
74	C₂²⋊C₄×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	#11719917	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	#11719908	—	4	2	no	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	4	1	no	—
84	C₂³×C₅₃	#102	1	1	yes	characteristic
85	C₂³×C₅₃	#102	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#123	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#137	1	2	yes	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	#117	1	2	yes	—
107	C₈×C₅₃	#123	1	2	yes	—
108	#88322	—	8	2	no	—
109	#12869	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	4	4	no	—
118	C₂²×C₅₃	#359	1	2	yes	—
119	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
120	C₂²×C₅₃	#483	1	2	yes	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₅₃	—	8	2	no	—
125	C₄×C₅₃	#381	1	1	yes	characteristic
126	C₄×C₅₃	#419	1	1	yes	characteristic
127	C₄×C₅₃	#480	1	2	yes	—
128	C₂×#427	D₅₃	1	1	yes	characteristic
129	C₂×C₂²⋊C₁₆	—	53	1	no	—
130	#1810	—	53	1	no	—
131	#1382	—	53	4	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	—	2	2	no	—
134	C₂×C₅₃	—	4	1	no	—
135	C₂×C₅₃	#1823	1	1	yes	characteristic
136	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
137	C₂×C₅₃	#2143	1	1	yes	characteristic
138	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
139	C₂×#117	—	2	1	no	—
140	C₂²×C₁₆	—	53	2	no	—
141	C₂×#108	—	53	2	no	—
142	#427	—	2	1	no	—
143	#427	—	2	1	no	—
144	#427	Dic₅₃	1	2	yes	—
145	#427	C₂×D₅₃	1	2	yes	—
146	C₂×#107	Dic₅₃	1	1	yes	characteristic
147	C₂×#107	Dic₅₃	1	1	yes	characteristic
148	C₂×#106	—	2	1	no	—
149	#407	—	106	1	no	—
150	C₂²⋊C₁₆	—	106	1	no	—
151	C₂²⋊C₁₆	—	106	1	no	—
152	#341	—	106	1	no	—
153	C₅₃	#12869	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂×M₁₆	—	106	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	4	1	no	—
166	#117	—	4	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	—	4	1	no	—
171	C₂×C₁₆	—	106	4	no	—
172	C₂×C₁₆	—	106	2	no	—
173	#108	—	106	4	no	—
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	#107	C₂×Dic₅₃	1	2	yes	—
177	#107	C₂×Dic₅₃	1	2	yes	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	C₂²⋊C₈	—	212	1	no	—
181	C₂²⋊C₈	—	212	1	no	—
182	C₂⁴	#11719923	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	#11719922	1	1	yes	characteristic
188	C₂²×C₄	#11719922	1	1	yes	characteristic
189	C₂²×C₄	#11719923	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
193	M₁₆	—	212	2	no	—
194	M₁₆	—	212	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₈	#11719945	1	4	yes	—
204	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	212	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	#13133138	1	1	yes	characteristic
219	C₂³	#13133138	1	1	yes	characteristic
220	C₂³	C₂×#11719923	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#13133138	1	1	yes	characteristic
233	C₂×C₄	#13133138	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	#13133200	1	2	yes	—
236	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
237	C₂×C₄	#13133248	1	2	yes	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	424	2	no	—
241	C₈	#13133176	1	2	yes	—
242	C₈	#13133200	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	#16810430	1	2	yes	—
252	C₂²	C₂×#13133138	1	1	yes	characteristic
253	C₂²	#16810953	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	#16810486	1	1	yes	characteristic
259	C₄	#16810632	1	1	yes	characteristic
260	C₄	#16810936	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	#20274801	1	1	yes	characteristic
265	C₂	C₂×#16810430	1	1	yes	characteristic
266	C₂	#20276443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	#12869
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #20272810	C₂	semidirect
153. C₅₃	#12869	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
#20274753	2
#20272810	2
C₂×C₅₃×#427	2
#12869	53