Subgroups — #28305586

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290812	C₂	1	2	yes	maximal
2	C₂×#16811484	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811464	C₂	1	1	yes	maximal, characteristic
4	#20289551	C₂	1	2	yes	maximal
5	#107×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	—	2	2	no	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
10	#16811489	—	2	2	no	—
11	#16811484	—	2	2	no	—
12	#16811464	—	2	2	no	—
13	C₂×#13133151	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
16	#16810385	C₄	1	2	yes	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	—	2	1	no	—
19	C₂²×#88324	—	2	2	no	—
20	C₂²×#88324	D₄	1	2	yes	—
21	C₄⋊#88324	—	2	2	no	—
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	—	4	4	no	—
24	C₄×#88324	—	2	1	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	—	2	1	no	—
27	C₄×#88324	Q₈	1	2	yes	—
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	—	2	2	no	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133151	—	2	2	no	—
33	C₈×Dic₅₃	D₄	1	2	yes	—
34	C₈×Dic₅₃	Q₈	1	2	yes	—
35	#13133124	—	2	2	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	#107×C₅₃	C₈	1	4	yes	complemented
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	2	no	—
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	—	2	2	no	—
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	4	1	no	—
49	C₂×#88324	—	4	1	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×#88324	—	4	2	no	—
52	C₂×#88324	—	4	2	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
55	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
56	C₄⋊C₄×C₅₃	—	2	2	no	—
57	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₅₃	#41	1	1	yes	characteristic
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	C₄⋊C₄	1	2	yes	—
69	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	#121	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₅₃	#125	1	1	yes	characteristic
87	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×Dic₅₃	#137	1	2	yes	—
90	#88324	—	4	2	no	—
91	#88324	—	4	2	no	—
92	#88324	—	8	2	no	—
93	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
94	#88322	—	2	2	no	—
95	#9595	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#389	1	1	yes	characteristic
97	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
98	C₂²×C₅₃	#436	1	1	yes	characteristic
99	C₂²×C₅₃	#441	1	1	yes	characteristic
100	C₂²×C₅₃	#442	1	1	yes	characteristic
101	C₂²×C₅₃	#444	1	1	yes	characteristic
102	C₂²×C₅₃	#444	1	1	yes	characteristic
103	C₄×C₅₃	—	4	4	no	—
104	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₅₃	#424	1	1	yes	characteristic
106	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
107	C₄×C₅₃	#446	1	1	yes	characteristic
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	2	1	no	—
110	Dic₅₃	—	8	2	no	—
111	Dic₅₃	#468	1	2	yes	—
112	Dic₅₃	#501	1	2	yes	—
113	#1939	—	53	2	no	—
114	C₄×#107	—	53	1	no	—
115	C₂×#334	—	53	1	no	—
116	C₂×#322	—	53	1	no	—
117	#1279	—	53	2	no	—
118	C₂×C₅₃	#1579	1	1	yes	characteristic
119	C₂×C₅₃	#1580	1	1	yes	characteristic
120	C₂×C₅₃	#1593	1	1	yes	characteristic
121	C₂×C₅₃	#1594	1	1	yes	characteristic
122	C₂×C₅₃	#1939	1	1	yes	characteristic
123	C₂×C₅₃	#1943	1	1	yes	characteristic
124	C₂×C₅₃	#1946	1	1	yes	characteristic
125	C₂×#107	—	53	1	no	—
126	C₂×#107	#6157	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	53	2	no	—
128	C₂×C₄⋊C₈	—	106	2	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₄×C₄⋊C₄	—	53	2	no	—
133	#340	—	53	2	no	—
134	#336	—	106	2	no	—
135	#334	—	106	2	no	—
136	#322	—	106	2	no	—
137	C₅₃	#9595	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	53	2	no	—
140	C₂²×C₈	—	106	1	no	—
141	C₂²×C₈	—	106	2	no	—
142	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	53	2	no	—
144	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	106	2	no	—
147	#107	—	106	2	no	—
148	#107	#88324	1	4	yes	complemented
149	C₄⋊C₈	—	106	2	no	—
150	C₄⋊C₈	—	106	2	no	—
151	C₄⋊C₈	—	212	4	no	—
152	C₄×C₈	—	53	2	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	106	1	no	—
156	C₄×C₈	—	106	1	no	—
157	C₄×C₈	—	106	1	no	—
158	#95	—	106	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	53	1	no	—
162	C₂²×C₄	—	106	2	no	—
163	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	—	212	1	no	—
175	C₂×C₈	—	212	1	no	—
176	C₂×C₈	#11719936	1	2	yes	—
177	C₂×C₈	C₄⋊#6157	1	2	yes	—
178	C₂×C₈	C₂×#88324	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	106	2	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₄⋊C₄	C₂×#88324	1	4	yes	—
183	C₄²	—	53	1	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	1	no	—
186	C₄²	—	53	1	no	—
187	C₄²	—	212	2	no	—
188	C₄²	—	212	2	no	—
189	C₂³	D₄×#6157	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	53	1	no	—
199	C₂×C₄	—	106	1	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	—	212	2	no	—
203	C₂×C₄	—	212	2	no	—
204	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
205	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
206	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
208	C₂×C₄	#13133304	1	1	yes	characteristic
209	C₂×C₄	#13133305	1	1	yes	characteristic
210	C₈	—	106	2	no	—
211	C₈	—	424	2	no	—
212	C₈	—	212	2	no	—
213	C₈	—	212	2	no	—
214	C₈	C₄⋊#88324	1	4	yes	complemented
215	C₂²	#16811574	1	1	yes	characteristic
216	C₂²	D₄×#88324	1	1	yes	characteristic
217	C₂²	#16811657	1	1	yes	characteristic
218	C₂²	#16811663	1	1	yes	characteristic
219	C₂²	#16811664	1	1	yes	characteristic
220	C₂²	#16811669	1	1	yes	characteristic
221	C₂²	#16811670	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	—	53	2	no	—
224	C₄	—	53	2	no	—
225	C₄	—	106	1	no	—
226	C₄	—	106	1	no	—
227	C₄	—	424	2	no	—
228	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
229	C₄	#16811630	1	1	yes	characteristic
230	C₄	Q₈×#88324	1	1	yes	characteristic
231	C₄	#16811673	1	1	yes	characteristic
232	C₂	#20290308	1	1	yes	characteristic
233	C₂	#20290311	1	1	yes	characteristic
234	C₂	#20290361	1	1	yes	characteristic
235	C₂	#20290364	1	1	yes	characteristic
236	C₂	#20290812	1	1	yes	characteristic
237	C₂	#20290820	1	1	yes	characteristic
238	C₂	#20290824	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#107×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#107	#6157
hypocenter	C₅₃	#9595
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
38. #107×C₅₃	C₈	semidirect
93. C₈×C₅₃	C₄⋊C₈	semidirect
137. C₅₃	#9595	semidirect
148. #107	#88324	semidirect
214. C₈	C₄⋊#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290812	2
C₂×#16811484	2
C₂×#16811464	2
#20289551	2
#107×Dic₅₃	2
#9595	53