Subgroups — #27282665

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275434	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810376	—	2	2	no	—
12	#16810345	—	2	2	no	—
13	C₂×#106×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#340	C₄	1	1	yes	characteristic
17	C₅₃×#340	C₄	1	1	yes	characteristic
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	D₄	1	1	yes	characteristic
21	C₂²×#88322	D₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	4	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#106×C₅₃	—	2	4	no	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	#95×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₄⋊C₄×C₅₃	—	2	2	no	—
49	C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
52	C₄²×C₅₃	—	4	2	no	—
53	C₄²×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₅₃	#41	1	1	yes	characteristic
58	C₂×#88322	—	2	2	no	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₅₃	#131	1	2	yes	—
85	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
86	C₂×C₄×C₅₃	#137	1	2	yes	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	—	2	2	no	—
89	#88322	—	2	4	no	—
90	#88322	—	4	1	no	—
91	#88322	—	4	1	no	—
92	#88322	—	8	2	no	—
93	#9600	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#389	1	1	yes	characteristic
95	C₂²×C₅₃	#435	1	2	yes	—
96	C₂²×C₅₃	#440	1	2	yes	—
97	C₂²×C₅₃	#443	1	1	yes	characteristic
98	C₂²×C₅₃	#443	1	1	yes	characteristic
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	4	4	no	—
102	C₄×C₅₃	—	8	2	no	—
103	C₄×C₅₃	#423	1	1	yes	characteristic
104	C₄×C₅₃	#424	1	1	yes	characteristic
105	C₄×C₅₃	#446	1	2	yes	—
106	C₄×C₅₃	#469	1	1	yes	characteristic
107	C₄×C₅₃	#470	1	1	yes	characteristic
108	C₄×C₅₃	#501	1	2	yes	—
109	#1939	—	53	2	no	—
110	C₄×#106	D₅₃	1	1	yes	characteristic
111	C₂×#335	—	53	1	no	—
112	C₂×#322	—	53	1	no	—
113	#1279	—	53	2	no	—
114	C₂×C₅₃	#1585	1	2	yes	—
115	C₂×C₅₃	#1588	1	2	yes	—
116	C₂×C₅₃	#1940	1	1	yes	characteristic
117	C₂×C₅₃	#1944	1	1	yes	characteristic
118	C₂×C₅₃	#1945	1	1	yes	characteristic
119	C₂×#106	Dic₅₃	1	2	yes	—
120	C₂×C₄⋊C₈	—	53	2	no	—
121	C₂×C₄⋊C₈	—	106	2	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	C₂×C₄×C₈	—	53	1	no	—
124	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
126	#340	Dic₅₃	1	1	yes	characteristic
127	#340	Dic₅₃	1	1	yes	characteristic
128	#336	—	106	2	no	—
129	#335	—	106	2	no	—
130	#322	—	106	2	no	—
131	C₅₃	#9600	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	106	1	no	—
135	C₂²×C₈	—	106	2	no	—
136	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	#106	—	2	4	no	—
143	C₄⋊C₈	—	106	2	no	—
144	C₄⋊C₈	—	106	2	no	—
145	C₄⋊C₈	—	212	4	no	—
146	C₄×C₈	—	53	2	no	—
147	C₄×C₈	—	106	1	no	—
148	C₄×C₈	—	106	1	no	—
149	C₄×C₈	—	106	1	no	—
150	C₄×C₈	Dic₁₀₆	1	2	yes	—
151	C₄×C₈	D₂₁₂	1	2	yes	—
152	#95	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
156	C₂²×C₄	#11719947	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	106	2	no	—
161	C₂×C₈	—	106	2	no	—
162	C₂×C₈	—	106	2	no	—
163	C₂×C₈	—	106	1	no	—
164	C₂×C₈	—	106	1	no	—
165	C₂×C₈	—	106	2	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
171	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	#11719917	1	4	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
180	C₄²	Q₈×D₅₃	1	1	yes	characteristic
181	C₄²	#11719949	1	1	yes	characteristic
182	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
192	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
193	C₂×C₄	#13133192	1	1	yes	characteristic
194	C₂×C₄	#13133197	1	1	yes	characteristic
195	C₂×C₄	#13133208	1	1	yes	characteristic
196	C₂×C₄	#13133216	1	1	yes	characteristic
197	C₂×C₄	C₂×#11719917	1	2	yes	—
198	C₂×C₄	#13133224	1	2	yes	—
199	C₂×C₄	#13133244	1	2	yes	—
200	C₂×C₄	#13133246	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	106	4	no	—
204	C₈	—	424	2	no	—
205	C₈	—	212	1	no	—
206	C₈	—	212	1	no	—
207	C₂²	#16810522	1	1	yes	characteristic
208	C₂²	#16810695	1	2	yes	—
209	C₂²	#16810724	1	2	yes	—
210	C₂²	#16810740	1	1	yes	characteristic
211	C₂²	#16810740	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	#16810647	1	1	yes	characteristic
217	C₄	#16810650	1	1	yes	characteristic
218	C₄	#16810754	1	2	yes	—
219	C₄	#16810877	1	1	yes	characteristic
220	C₄	#16810882	1	1	yes	characteristic
221	C₄	#16811030	1	2	yes	—
222	C₂	#20273573	1	2	yes	—
223	C₂	#20273594	1	2	yes	—
224	C₂	#20275439	1	1	yes	characteristic
225	C₂	#20275459	1	1	yes	characteristic
226	C₂	#20275464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#106×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#106	D₅₃
hypocenter	C₅₃	#9600
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
131. C₅₃	#9600	semidirect

Maximal subgroups

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

subgroup	index
#20275434	2
C₂×#16810376	2
C₂×#16810345	2
#20272556	2
C₄×#106×C₅₃	2
#9600	53