Subgroups — #27300610

Order 13568.

404 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810453	C₂	1	2	yes	maximal, complemented
3	#20274753	C₂	1	1	yes	maximal, characteristic
4	#20272809	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	1	no	—
7	C₂×#13133153	—	2	4	no	—
8	C₂×#13133153	—	2	1	no	—
9	C₂×#13133153	C₂²	1	1	yes	characteristic
10	C₂×#13133153	C₂²	1	1	yes	characteristic
11	C₂×#13133153	C₂²	1	2	yes	—
12	#16810587	—	2	1	no	—
13	#16810453	—	2	4	no	—
14	#16810388	—	2	1	no	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
18	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
19	C₂×#11719917	—	2	4	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	1	no	—
22	C₂×#11719917	—	2	1	no	—
23	C₂×#11719917	—	4	2	no	—
24	C₂×#11719917	—	4	1	no	—
25	C₂²×#88322	—	4	1	no	—
26	#13133153	—	2	4	no	—
27	#13133153	—	2	2	no	—
28	#13133153	—	2	2	no	—
29	#13133153	—	2	2	no	—
30	#13133153	D₄	1	4	yes	—
31	C₂²⋊#88322	—	4	1	no	—
32	C₂²⋊#88322	—	4	1	no	—
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	#11719917	—	2	4	no	—
69	#11719917	—	4	4	no	—
70	#11719917	—	4	2	no	—
71	#11719917	—	4	1	no	—
72	#11719917	—	4	1	no	—
73	#11719917	—	4	1	no	—
74	#11719917	—	4	1	no	—
75	C₂×#88322	—	4	4	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	1	no	—
78	C₂×#88322	—	4	1	no	—
79	C₁₆×C₅₃	—	2	4	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	4	1	no	—
84	C₂³×C₅₃	#107	1	1	yes	characteristic
85	C₂³×C₅₃	#107	1	1	yes	characteristic
86	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	1	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	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	C₄×D₄	1	2	yes	—
108	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	#88322	—	4	4	no	—
110	#88322	—	8	1	no	—
111	#88322	—	8	1	no	—
112	#12922	—	53	1	no	maximal, Sylow
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	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	4	2	no	—
121	C₂²×C₅₃	—	4	2	no	—
122	C₂²×C₅₃	#368	1	2	yes	complemented
123	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
124	C₂²×C₅₃	#451	1	1	yes	characteristic
125	C₂²×C₅₃	#484	1	1	yes	characteristic
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	#385	1	1	yes	characteristic
129	C₄×C₅₃	#428	1	1	yes	characteristic
130	C₄×C₅₃	#447	1	1	yes	characteristic
131	C₄×C₅₃	#480	1	1	yes	characteristic
132	C₂²×#108	—	53	1	no	—
133	C₂×#368	—	53	2	no	—
134	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
135	#1810	—	53	1	no	—
136	#1381	—	53	2	no	—
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	4	1	no	—
141	C₂×C₅₃	#1852	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#368	1	1	yes	characteristic
143	C₂×C₅₃	#2161	1	1	yes	characteristic
144	C₂²×M₁₆	—	106	1	no	—
145	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
147	C₂×#108	—	53	2	no	—
148	C₂×#108	—	53	1	no	—
149	C₂×#108	—	53	1	no	—
150	C₂×#108	—	106	4	no	—
151	C₂×#108	—	106	1	no	—
152	#407	—	106	1	no	—
153	#368	—	106	4	no	—
154	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
155	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
156	#341	—	106	1	no	—
157	C₅₃	#12922	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
159	C₂×M₁₆	—	106	4	no	—
160	C₂×M₁₆	—	106	2	no	—
161	C₂×M₁₆	—	106	1	no	—
162	C₂×M₁₆	—	106	1	no	—
163	C₂×M₁₆	—	212	2	no	—
164	C₂×M₁₆	—	212	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	212	1	no	—
168	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
169	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
170	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
177	#108	—	53	4	no	—
178	#108	—	106	4	no	—
179	#108	—	106	2	no	—
180	#108	—	106	2	no	—
181	#108	—	106	2	no	—
182	C₂²⋊C₈	—	212	1	no	—
183	C₂²⋊C₈	—	212	1	no	—
184	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
190	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
191	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
195	M₁₆	—	106	4	no	—
196	M₁₆	—	212	1	no	—
197	M₁₆	—	212	1	no	—
198	M₁₆	—	212	1	no	—
199	M₁₆	—	212	1	no	—
200	M₁₆	—	212	4	no	—
201	M₁₆	—	212	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	212	1	no	—
209	C₂×C₈	—	212	1	no	—
210	C₂×C₈	—	212	4	no	—
211	C₂×C₈	—	212	2	no	—
212	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
215	C₂×C₈	D₄×D₅₃	1	2	yes	—
216	C₂×C₈	#11719947	1	2	yes	—
217	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
218	C₁₆	—	2	4	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	1	no	—
223	C₂³	#13133151	1	1	yes	characteristic
224	C₂³	#13133151	1	1	yes	characteristic
225	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	#13133151	1	1	yes	characteristic
237	C₂×C₄	#13133151	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	#13133187	1	1	yes	characteristic
240	C₂×C₄	#13133201	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
242	C₂×C₄	#13133245	1	1	yes	characteristic
243	C₂×C₄	#13133250	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	424	1	no	—
247	C₈	—	424	1	no	—
248	C₈	—	212	4	no	—
249	C₈	D₄×Dic₅₃	1	2	yes	—
250	C₈	#13133194	1	2	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#16810453	1	2	yes	complemented
261	C₂²	C₂×#13133151	1	1	yes	characteristic
262	C₂²	#16810781	1	1	yes	characteristic
263	C₂²	#16810959	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	#16810498	1	1	yes	characteristic
267	C₄	#16810665	1	1	yes	characteristic
268	C₄	#16810759	1	1	yes	characteristic
269	C₄	#16810937	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	4	1	no	—
274	C₂	#20274905	1	1	yes	characteristic
275	C₂	C₂×#16810453	1	1	yes	characteristic
276	C₂	#20276510	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#12922
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810453	C₂	semidirect
4. #20272809	C₂	semidirect
122. C₂²×C₅₃	#368	semidirect
157. C₅₃	#12922	semidirect
260. C₂²	#16810453	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810453	2
#20274753	2
#20272809	2
C₂×C₅₃×C₂²⋊C₁₆	2
#12922	53