Subgroups — #27300149

Order 13568.

430 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	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810428	C₂	1	2	yes	maximal
3	#20274879	C₂	1	1	yes	maximal, characteristic
4	#20272807	C₂	1	2	yes	maximal
5	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₄×Dic₂₁₂	—	2	2	no	—
9	C₄×Dic₂₁₂	C₄	1	4	yes	complemented
10	C₂×#13133151	C₂²	1	2	yes	—
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133149	—	2	1	no	—
13	C₂×#13133142	—	2	1	no	—
14	#16810489	—	2	1	no	—
15	#16810428	—	2	4	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×Dic₁₀₆	—	2	1	no	—
20	C₂×Dic₂₁₂	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	1	no	—
22	C₂×Dic₂₁₂	—	2	1	no	—
23	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
24	C₂×Dic₂₁₂	D₄	1	2	yes	—
25	C₂×Dic₂₁₂	Q₈	1	2	yes	—
26	C₄×Dic₁₀₆	—	2	4	no	—
27	C₄×Dic₁₀₆	—	4	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#11719919	—	4	1	no	—
31	C₂×C₄×Dic₅₃	—	4	1	no	—
32	#13133151	—	2	1	no	—
33	#13133151	—	2	2	no	—
34	#13133151	—	2	2	no	—
35	#13133151	C₂×C₄	1	2	yes	—
36	#13133149	—	4	2	no	—
37	#13133142	—	2	4	no	—
38	#13133124	—	4	1	no	—
39	#13133124	—	4	1	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	1	no	—
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	2	2	no	—
56	C₂×Dic₁₀₆	—	2	2	no	—
57	C₂×Dic₁₀₆	—	4	1	no	—
58	C₂×Dic₁₀₆	—	4	1	no	—
59	C₂×Dic₁₀₆	—	4	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	SD₁₆	1	2	yes	—
71	C₄⋊Dic₅₃	—	2	2	no	—
72	C₄⋊Dic₅₃	—	4	1	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	#11719919	—	4	4	no	—
76	C₄×Dic₅₃	—	4	4	no	—
77	Dic₂₁₂	—	2	2	no	—
78	Dic₂₁₂	—	2	2	no	—
79	Dic₂₁₂	—	2	2	no	—
80	Dic₂₁₂	C₄⋊C₄	1	4	yes	—
81	C₁₆×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	#102	1	2	yes	—
86	C₂×C₄×C₅₃	#102	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
93	C₂×Dic₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	—	4	2	no	—
96	C₂×Dic₅₃	—	4	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	C₂×Dic₅₃	—	8	2	no	—
99	C₂×Dic₅₃	—	8	2	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	Dic₁₀₆	—	2	4	no	—
102	Dic₁₀₆	—	4	2	no	—
103	Dic₁₀₆	—	4	2	no	—
104	Dic₁₀₆	—	4	2	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₅₃	#117	1	1	yes	characteristic
108	C₈×C₅₃	#121	1	1	yes	characteristic
109	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
110	#12814	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
112	C₂²×C₅₃	#459	1	1	yes	characteristic
113	C₂²×C₅₃	#474	1	1	yes	characteristic
114	C₂²×C₅₃	#508	1	2	yes	—
115	C₂²×C₅₃	#510	1	2	yes	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	#380	1	1	yes	characteristic
119	C₄×C₅₃	#417	1	1	yes	characteristic
120	C₄×C₅₃	#461	1	1	yes	characteristic
121	C₄×C₅₃	#476	1	1	yes	characteristic
122	Dic₅₃	—	4	4	no	—
123	Dic₅₃	—	8	2	no	—
124	Dic₅₃	—	16	2	no	—
125	C₂×C₄×Q₁₆	—	53	1	no	—
126	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
127	C₂×#358	—	53	2	no	—
128	#1846	—	53	1	no	—
129	#1380	—	53	2	no	—
130	C₂×C₅₃	#1802	1	1	yes	characteristic
131	C₂×C₅₃	#2138	1	1	yes	characteristic
132	C₂×C₅₃	#2141	1	1	yes	characteristic
133	C₂×C₅₃	#2211	1	2	yes	—
134	C₂×C₅₃	#2229	1	2	yes	—
135	C₂²×Q₁₆	—	53	1	no	—
136	C₂×C₄×Q₈	—	106	1	no	—
137	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
138	C₄×Q₁₆	—	53	4	no	—
139	C₄×Q₁₆	—	106	2	no	—
140	C₂×#107	—	53	1	no	—
141	C₂×#107	—	53	2	no	—
142	C₂×#106	—	106	1	no	—
143	C₂×#103	—	106	1	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	#382	—	106	1	no	—
146	C₄⋊C₁₆	—	2	2	no	—
147	#358	—	106	4	no	—
148	C₅₃	#12814	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	106	1	no	—
150	C₂×Q₁₆	—	53	2	no	—
151	C₂×Q₁₆	—	53	2	no	—
152	C₂×Q₁₆	—	53	2	no	—
153	C₂×Q₁₆	—	106	1	no	—
154	C₂×Q₁₆	—	106	1	no	—
155	C₂×Q₁₆	—	106	1	no	—
156	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
158	C₄×Q₈	—	106	4	no	—
159	C₄×Q₈	—	212	2	no	—
160	C₂×C₄⋊C₄	—	106	1	no	—
161	C₂×C₄⋊C₄	—	106	2	no	—
162	C₂×C₄⋊C₄	—	212	1	no	—
163	C₂×C₄²	—	212	1	no	—
164	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	53	2	no	—
169	#107	—	106	1	no	—
170	#107	—	106	2	no	—
171	#107	—	106	2	no	—
172	#106	—	212	2	no	—
173	#103	—	106	4	no	—
174	C₄×C₈	C₄×D₅₃	1	2	yes	—
175	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
176	#95	—	212	1	no	—
177	#95	—	212	1	no	—
178	C₂×Q₈	—	106	2	no	—
179	C₂×Q₈	—	106	2	no	—
180	C₂×Q₈	—	106	2	no	—
181	C₂×Q₈	—	212	1	no	—
182	C₂×Q₈	—	212	1	no	—
183	C₂×Q₈	—	212	1	no	—
184	C₂²×C₄	—	212	2	no	—
185	C₂²×C₄	—	212	1	no	—
186	C₂²×C₄	—	212	1	no	—
187	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
188	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	Q₁₆	—	53	4	no	—
191	Q₁₆	—	106	2	no	—
192	Q₁₆	—	106	2	no	—
193	Q₁₆	—	106	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	#11719921	1	2	yes	—
197	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	#11719947	1	1	yes	characteristic
200	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
201	C₂×C₈	#11719949	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	106	2	no	—
204	C₄⋊C₄	—	212	2	no	—
205	C₄⋊C₄	—	212	2	no	—
206	C₄⋊C₄	—	212	4	no	—
207	C₄⋊C₄	—	212	1	no	—
208	C₄²	—	212	4	no	—
209	C₄²	#11719913	1	2	yes	—
210	C₄²	#11719921	1	2	yes	—
211	C₁₆	—	4	2	no	—
212	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
213	Q₈	—	106	4	no	—
214	Q₈	—	212	2	no	—
215	Q₈	—	212	2	no	—
216	Q₈	—	212	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	424	2	no	—
220	C₂×C₄	—	424	2	no	—
221	C₂×C₄	—	424	2	no	—
222	C₂×C₄	—	424	2	no	—
223	C₂×C₄	—	212	2	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	—	212	2	no	—
226	C₂×C₄	—	212	2	no	—
227	C₂×C₄	#13133139	1	2	yes	—
228	C₂×C₄	#13133139	1	2	yes	—
229	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
230	C₂×C₄	#13133192	1	1	yes	characteristic
231	C₂×C₄	#13133197	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
233	C₂×C₄	#13133245	1	1	yes	characteristic
234	C₂×C₄	#13133250	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
237	C₈	#13133175	1	1	yes	characteristic
238	C₈	#13133188	1	1	yes	characteristic
239	C₈	#13133193	1	1	yes	characteristic
240	C₂²	C₂×#13133139	1	1	yes	characteristic
241	C₂²	#16810819	1	1	yes	characteristic
242	C₂²	#16810904	1	1	yes	characteristic
243	C₂²	#16811055	1	2	yes	—
244	C₂²	#16811066	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	848	2	no	—
248	C₄	—	424	2	no	—
249	C₄	—	212	4	no	—
250	C₄	#16810480	1	1	yes	characteristic
251	C₄	#16810625	1	1	yes	characteristic
252	C₄	#16810829	1	1	yes	characteristic
253	C₄	#16810914	1	1	yes	characteristic
254	C₂	#20274716	1	1	yes	characteristic
255	C₂	#20276423	1	1	yes	characteristic
256	C₂	#20276436	1	1	yes	characteristic
257	C₂	#20276778	1	2	yes	—
258	C₂	#20276880	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12814
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₂₁₂	C₄	semidirect
148. C₅₃	#12814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
C₂×#16810428	2
#20274879	2
#20272807	2
C₂×C₅₃×C₄⋊C₁₆	2
#12814	53