Subgroups — #27300145

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×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810426	C₂	1	2	yes	maximal
4	#20272806	C₂	1	2	yes	maximal
5	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	C₄×Dic₂₁₂	—	2	2	no	—
10	C₄×Dic₂₁₂	C₄	1	4	yes	complemented
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133142	—	2	1	no	—
13	C₄⋊#11719908	—	2	2	no	—
14	#16810426	—	2	4	no	—
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#106×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#382	—	2	1	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×#11719919	—	4	1	no	—
30	C₂×C₄×Dic₅₃	—	4	1	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	—	2	2	no	—
33	C₂×#11719908	—	2	2	no	—
34	#13133151	—	2	1	no	—
35	#13133151	C₂×C₄	1	2	yes	—
36	#13133142	—	2	4	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	#107×C₅₃	—	2	2	no	—
42	#107×C₅₃	—	2	2	no	—
43	#106×C₅₃	—	4	2	no	—
44	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	#95×C₅₃	—	4	1	no	—
47	#95×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂²×Dic₅₃	—	4	1	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	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	SD₁₆	1	2	yes	—
73	C₄⋊Dic₅₃	—	2	2	no	—
74	C₄⋊Dic₅₃	—	4	1	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	#11719908	—	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	—	8	2	no	—
86	C₂×C₄×C₅₃	—	8	2	no	—
87	C₂×C₄×C₅₃	—	8	2	no	—
88	C₂×C₄×C₅₃	#102	1	2	yes	—
89	C₂×C₄×C₅₃	#102	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
96	C₂×Dic₅₃	—	4	2	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	2	no	—
99	C₂×Dic₅₃	—	4	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	—	16	2	no	—
119	C₄×C₅₃	#380	1	1	yes	characteristic
120	C₄×C₅₃	#417	1	1	yes	characteristic
121	C₄×C₅₃	#461	1	1	yes	characteristic
122	C₄×C₅₃	#476	1	1	yes	characteristic
123	Dic₅₃	—	4	4	no	—
124	Dic₅₃	—	8	2	no	—
125	C₂×C₄×Q₁₆	—	53	1	no	—
126	C₂×C₄⋊C₁₆	—	53	1	no	—
127	C₂×#358	—	53	2	no	—
128	#1846	D₅₃	1	1	yes	characteristic
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	—	53	2	no	—
138	C₄×Q₁₆	—	53	4	no	—
139	C₄×Q₁₆	—	106	2	no	—
140	C₂×#107	—	53	1	no	—
141	C₂×#107	C₂×D₅₃	1	2	yes	—
142	C₂×#106	—	2	1	no	—
143	C₂×#103	—	106	1	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	#382	—	2	1	no	—
146	C₄⋊C₁₆	—	106	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	C₂²⋊D₅₃	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	—	2	2	no	—
161	C₂×C₄⋊C₄	—	106	1	no	—
162	C₂×C₄⋊C₄	—	212	1	no	—
163	C₂×C₄²	—	212	1	no	—
164	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂×C₁₆	—	106	2	no	—
166	C₂×C₁₆	—	106	2	no	—
167	C₂×C₁₆	—	106	2	no	—
168	#107	—	2	2	no	—
169	#107	—	2	2	no	—
170	#107	—	53	2	no	—
171	#107	—	106	1	no	—
172	#106	—	4	2	no	—
173	#103	—	106	4	no	—
174	C₄×C₈	C₄×D₅₃	1	2	yes	—
175	C₄×C₈	D₂₁₂	1	2	yes	—
176	#95	—	4	1	no	—
177	#95	—	4	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	—	4	2	no	—
185	C₂²×C₄	—	212	1	no	—
186	C₂²×C₄	—	212	1	no	—
187	C₂²×C₄	#11719922	1	1	yes	characteristic
188	C₂²×C₄	#11719922	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅₃	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	C₂×D₂₁₂	1	1	yes	characteristic
199	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	#11719947	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
202	C₂×C₈	#11719949	1	1	yes	characteristic
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	212	4	no	—
207	C₄⋊C₄	—	212	1	no	—
208	C₄²	—	212	4	no	—
209	C₄²	#11719921	1	2	yes	—
210	C₄²	#11719924	1	2	yes	—
211	C₁₆	—	212	2	no	—
212	C₂³	C₂×#11719922	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	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	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	#13133137	1	2	yes	—
228	C₂×C₄	#13133137	1	2	yes	—
229	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
230	C₂×C₄	#13133188	1	1	yes	characteristic
231	C₂×C₄	#13133193	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
233	C₂×C₄	#13133243	1	1	yes	characteristic
234	C₂×C₄	#13133248	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	#13133192	1	1	yes	characteristic
239	C₈	#13133197	1	1	yes	characteristic
240	C₂²	C₂×#13133137	1	1	yes	characteristic
241	C₂²	#16810815	1	1	yes	characteristic
242	C₂²	#16810900	1	1	yes	characteristic
243	C₂²	#16811053	1	2	yes	—
244	C₂²	#16811063	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	16	2	no	—
248	C₄	—	424	2	no	—
249	C₄	—	212	4	no	—
250	C₄	#16810480	1	1	yes	characteristic
251	C₄	#16810622	1	1	yes	characteristic
252	C₄	#16810825	1	1	yes	characteristic
253	C₄	#16810910	1	1	yes	characteristic
254	C₂	#20274712	1	1	yes	characteristic
255	C₂	#20276420	1	1	yes	characteristic
256	C₂	#20276434	1	1	yes	characteristic
257	C₂	#20276774	1	2	yes	—
258	C₂	#20276876	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1846	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#12814
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
10. 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×C₄⋊#11719908	2
C₂×#16810426	2
#20272806	2
C₅₃×#1846	2
#12814	53