Subgroups — #27392436

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275439	C₂	1	2	yes	maximal
2	#20275281	C₂	1	4	yes	maximal
3	#20274835	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#88322	C₂	1	4	yes	maximal
5	C₄⋊C₄×#88322	C₂	1	2	yes	maximal
6	#20274402	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#2402	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#88322	C₂²	1	4	yes	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×#13133128	C₂²	1	2	yes	—
11	C₂×#13133128	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	4	yes	—
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	#16810378	C₂²	1	4	yes	—
16	#16810378	C₂²	1	4	yes	—
17	#16810378	C₂²	1	4	yes	—
18	C₅₃×#400	C₄	1	4	yes	—
19	C₅₃×#398	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₅₃×#376	C₂²	1	1	yes	characteristic
23	C₂²×#88322	C₂³	1	4	yes	—
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	C₂²×#88322	C₂³	1	2	yes	—
26	C₂²×#88322	C₂³	1	1	yes	characteristic
27	C₄⋊#88322	—	2	4	no	—
28	C₄⋊#88322	—	2	8	no	—
29	#13133128	—	2	2	no	—
30	#13133128	—	2	4	no	—
31	C₄×#88322	—	2	4	no	—
32	C₄×#88322	—	2	8	no	—
33	C₄×#88322	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#95×C₅₃	C₂×C₄	1	4	yes	—
41	#95×C₅₃	C₂×C₄	1	2	yes	—
42	#95×C₅₃	C₂×C₄	1	4	yes	—
43	#95×C₅₃	C₂×C₄	1	4	yes	—
44	#95×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	—	2	8	no	—
51	C₄⋊C₄×C₅₃	—	2	8	no	—
52	C₄⋊C₄×C₅₃	—	2	8	no	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
58	C₄²×C₅₃	#41	1	2	yes	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	—	2	2	no	—
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	2	4	no	—
65	C₂×#88322	C₂×Q₈	1	2	yes	—
66	C₂×#88322	C₂×Q₈	1	2	yes	—
67	C₂×#88322	#41	1	4	yes	—
68	C₂×#88322	#41	1	4	yes	—
69	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	4	no	—
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	—	2	4	no	—
74	C₂×C₄×C₅₃	—	2	4	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
80	C₂×C₄×C₅₃	#131	1	8	yes	—
81	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
85	#88322	—	2	4	no	—
86	#88322	—	2	2	no	—
87	#88322	—	4	4	no	—
88	#88322	#125	1	4	yes	—
89	#21676	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂²×C₅₃	C₄×#41	1	2	yes	—
92	C₂²×C₅₃	C₂×#131	1	4	yes	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	2	8	no	—
95	C₄×C₅₃	—	4	4	no	—
96	C₄×C₅₃	—	4	4	no	—
97	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
98	C₄×C₅₃	#529	1	2	yes	—
99	C₄×C₅₃	#533	1	1	yes	characteristic
100	#2402	D₅₃	1	1	yes	characteristic
101	#1940	—	53	2	no	—
102	#1917	—	53	4	no	—
103	#1832	—	53	1	no	—
104	C₈×C₄⋊C₄	—	53	4	no	—
105	C₈×C₄⋊C₄	—	53	2	no	—
106	#1726	—	53	1	no	—
107	C₂×C₅₃	C₄×#125	1	1	yes	characteristic
108	C₂×C₅₃	#2960	1	2	yes	—
109	C₂×C₅₃	#2966	1	4	yes	—
110	C₂×C₄⋊C₈	—	53	2	no	—
111	C₂×C₄⋊C₈	—	53	4	no	—
112	C₂×#97	—	53	2	no	—
113	C₂×#97	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	—	53	2	no	—
116	C₂×C₄×C₈	—	53	4	no	—
117	#400	Dic₅₃	1	4	yes	—
118	#398	Dic₅₃	1	4	yes	—
119	C₄×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
120	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
121	#376	C₂×D₅₃	1	1	yes	characteristic
122	#336	—	53	4	no	—
123	#336	—	53	4	no	—
124	#336	—	53	4	no	—
125	C₅₃	#21676	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	53	2	no	—
129	C₂²×C₈	—	53	4	no	—
130	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
132	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
133	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
135	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
136	C₄⋊C₈	—	106	8	no	—
137	C₄⋊C₈	—	106	4	no	—
138	#97	—	106	4	no	—
139	#97	—	106	2	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	106	4	no	—
142	C₄×C₈	—	106	8	no	—
143	#95	C₂×Dic₅₃	1	4	yes	—
144	#95	C₂×Dic₅₃	1	2	yes	—
145	#95	C₂×Dic₅₃	1	4	yes	—
146	#95	C₂×Dic₅₃	1	4	yes	—
147	#95	C₂×Dic₅₃	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
151	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
152	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	53	4	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	106	2	no	—
160	C₂×C₈	—	106	4	no	—
161	C₂×C₈	—	106	4	no	—
162	C₂×C₈	—	106	4	no	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	8	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	Q₈×D₅₃	1	2	yes	—
171	C₄²	#11719949	1	2	yes	—
172	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	Q₈×Dic₅₃	1	4	yes	—
183	C₂×C₄	#13133224	1	8	yes	—
184	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
185	C₂×C₄	#41×D₅₃	1	2	yes	—
186	C₂×C₄	#41×D₅₃	1	2	yes	—
187	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
188	C₈	—	53	4	no	—
189	C₈	—	106	4	no	—
190	C₈	—	106	2	no	—
191	C₈	—	212	4	no	—
192	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
193	C₂²	#41×Dic₅₃	1	2	yes	—
194	C₂²	C₂×#13133224	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	#125×D₅₃	1	1	yes	characteristic
200	C₄	#16811158	1	2	yes	—
201	C₄	#16811179	1	1	yes	characteristic
202	C₂	#125×Dic₅₃	1	1	yes	characteristic
203	C₂	#20283650	1	2	yes	—
204	C₂	#20283691	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2402	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2402	D₅₃
hypocenter	C₅₃	#21676
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2402

Complements

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

N	H ≅ G/N	type
125. C₅₃	#21676	semidirect

Maximal subgroups

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

subgroup	index
#20275439	2
#20275281	2
#20274835	2
C₄⋊C₄×#88322	2
#20274402	2
C₅₃×#2402	2
#21676	53