Subgroups — #27303161

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810427	C₂	1	2	yes	maximal
3	#20272806	C₂	1	2	yes	maximal
4	#20272793	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	—	2	1	no	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	#16811028	—	2	4	no	—
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	C₂²	1	2	yes	—
11	#16810427	—	2	4	no	—
12	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#391	—	2	1	no	—
17	C₂²×Dic₁₀₆	—	4	1	no	—
18	#13133217	—	2	4	no	—
19	#13133217	—	4	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	4	1	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	#13133151	—	2	4	no	—
26	#13133151	—	2	2	no	—
27	#13133151	D₄	1	4	yes	—
28	C₂²×Q₈×C₅₃	—	2	1	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	Q₈	1	2	yes	complemented
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
37	#107×C₅₃	—	2	2	no	—
38	#103×C₅₃	—	4	2	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	#95×C₅₃	—	4	1	no	—
41	#95×C₅₃	—	4	1	no	—
42	C₂×Q₈×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	4	1	no	—
44	C₂×Q₈×C₅₃	—	4	1	no	—
45	C₂×Q₈×C₅₃	—	4	1	no	—
46	C₂×Q₈×C₅₃	—	4	1	no	—
47	C₂×Q₈×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	—	4	1	no	—
49	C₂²×C₄×C₅₃	—	4	1	no	—
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×Dic₁₀₆	—	4	4	no	—
55	C₂×Dic₁₀₆	—	8	2	no	—
56	C₂×Dic₁₀₆	—	8	2	no	—
57	Q₁₆×C₅₃	—	2	4	no	—
58	Q₁₆×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	#41	1	2	yes	—
65	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	2	1	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄⋊Dic₅₃	—	2	4	no	—
71	C₄⋊Dic₅₃	—	4	4	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	#11719908	—	4	2	no	—
74	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	4	2	no	—
77	Q₈×C₅₃	—	8	1	no	—
78	Q₈×C₅₃	—	8	1	no	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	8	1	no	—
85	C₂×C₄×C₅₃	—	8	1	no	—
86	C₂×C₄×C₅₃	—	8	1	no	—
87	C₂×C₄×C₅₃	—	8	1	no	—
88	C₂×C₄×C₅₃	—	8	1	no	—
89	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
90	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
91	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
97	C₂×Dic₅₃	—	4	4	no	—
98	C₂×Dic₅₃	—	8	2	no	—
99	C₂×Dic₅₃	—	8	2	no	—
100	Dic₁₀₆	—	8	8	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₅₃	#124	1	2	yes	—
104	#13364	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	#447	1	1	yes	characteristic
106	C₂²×C₅₃	#474	1	1	yes	characteristic
107	C₂²×C₅₃	#486	1	1	yes	characteristic
108	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
110	C₂²×C₅₃	#509	1	1	yes	characteristic
111	C₂²×C₅₃	#510	1	1	yes	characteristic
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	2	no	—
114	C₄×C₅₃	—	8	2	no	—
115	C₄×C₅₃	—	16	1	no	—
116	C₄×C₅₃	#393	1	1	yes	characteristic
117	C₄×C₅₃	#448	1	1	yes	characteristic
118	C₄×C₅₃	#476	1	1	yes	characteristic
119	C₄×C₅₃	#489	1	1	yes	characteristic
120	Dic₅₃	—	8	4	no	—
121	C₂×#500	—	53	1	no	—
122	C₂×#358	—	53	2	no	—
123	#1969	D₅₃	1	1	yes	characteristic
124	#1380	—	53	2	no	—
125	#1374	—	53	1	no	—
126	C₂×C₅₃	#2023	1	1	yes	characteristic
127	C₂×C₅₃	#2187	1	1	yes	characteristic
128	C₂×C₅₃	#2188	1	1	yes	characteristic
129	C₂×C₅₃	#2225	1	1	yes	characteristic
130	C₂×C₅₃	#2227	1	1	yes	characteristic
131	C₂×C₅₃	#2241	1	1	yes	characteristic
132	C₂×C₅₃	#2243	1	1	yes	characteristic
133	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
134	C₂×C₄⋊Q₈	—	106	1	no	—
135	C₂²×C₁₆	—	53	2	no	—
136	#500	—	106	4	no	—
137	C₂×#107	—	53	2	no	—
138	C₂×#107	—	106	1	no	—
139	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
140	C₂×#103	—	2	1	no	—
141	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
142	#391	—	2	1	no	—
143	#358	—	106	4	no	—
144	C₅₃	#13364	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	212	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	Dic₁₀₆	1	2	yes	complemented
151	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊Q₈	—	106	4	no	—
154	C₄⋊Q₈	—	212	2	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	106	2	no	—
157	C₂×C₄⋊C₄	—	212	1	no	—
158	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₁₆	—	106	2	no	—
160	C₂×C₁₆	—	106	2	no	—
161	C₂×C₁₆	—	106	2	no	—
162	#107	—	2	2	no	—
163	#107	—	53	4	no	—
164	#107	—	106	4	no	—
165	#107	—	106	2	no	—
166	#103	—	4	2	no	—
167	C₄×C₈	—	2	2	no	—
168	#95	—	4	1	no	—
169	#95	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	424	2	no	—
177	C₂×Q₈	—	424	2	no	—
178	C₂×Q₈	—	212	4	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	212	2	no	—
182	C₂²×C₄	#11719922	1	1	yes	characteristic
183	C₂²×C₄	#11719922	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	#11719923	1	2	yes	—
190	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
191	C₂×C₈	#11719945	1	1	yes	characteristic
192	C₂×C₈	D₄×D₅₃	1	2	yes	—
193	C₂×C₈	#11719947	1	2	yes	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	106	4	no	—
197	C₄⋊C₄	—	212	4	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₁₆	—	212	2	no	—
202	C₂³	C₂×#11719922	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	1	no	—
206	Q₈	—	8	1	no	—
207	Q₈	—	424	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	424	2	no	—
219	C₂×C₄	—	424	2	no	—
220	C₂×C₄	—	212	4	no	—
221	C₂×C₄	#13133160	1	2	yes	—
222	C₂×C₄	#13133163	1	2	yes	—
223	C₂×C₄	#13133185	1	1	yes	characteristic
224	C₂×C₄	#13133196	1	1	yes	characteristic
225	C₂×C₄	#13133204	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
227	C₂×C₄	#13133242	1	1	yes	characteristic
228	C₂×C₄	#13133247	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	#13133194	1	2	yes	—
231	C₈	#13133203	1	2	yes	—
232	C₂²	#16810756	1	1	yes	characteristic
233	C₂²	#16810903	1	1	yes	characteristic
234	C₂²	#16810969	1	1	yes	characteristic
235	C₂²	C₂×#13133160	1	1	yes	characteristic
236	C₂²	C₂×#13133163	1	1	yes	characteristic
237	C₂²	#16811058	1	1	yes	characteristic
238	C₂²	#16811063	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	2	no	—
242	C₄	—	16	1	no	—
243	C₄	—	424	4	no	—
244	C₄	#16810535	1	1	yes	characteristic
245	C₄	#16810761	1	1	yes	characteristic
246	C₄	#16810913	1	1	yes	characteristic
247	C₄	#16810982	1	1	yes	characteristic
248	C₂	#20275851	1	1	yes	characteristic
249	C₂	#20276643	1	1	yes	characteristic
250	C₂	#20276648	1	1	yes	characteristic
251	C₂	#20276857	1	1	yes	characteristic
252	C₂	#20276867	1	1	yes	characteristic
253	C₂	#20276942	1	1	yes	characteristic
254	C₂	#20276951	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1969	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#1969	D₅₃
hypocenter	C₅₃	#13364
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₅₃	Q₈	semidirect
144. C₅₃	#13364	semidirect
150. C₂×Q₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
C₂×#16810427	2
#20272806	2
#20272793	2
C₅₃×#1969	2
#13364	53