Subgroups — #27277738

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	2	yes	maximal
2	#20272560	C₂	1	2	yes	maximal
3	#20272544	C₂	1	2	yes	maximal
4	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	4	yes	—
6	#16810376	—	2	2	no	—
7	#16810374	—	2	2	no	—
8	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
9	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#376	C₂²	1	1	yes	characteristic
15	C₄⋊#88322	—	2	4	no	—
16	C₂²⋊#88322	—	2	4	no	—
17	C₄×#88322	—	2	4	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	#97×C₅₃	—	2	2	no	—
26	#97×C₅₃	—	2	2	no	—
27	#97×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
34	#95×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	#41	1	2	yes	—
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	—	2	1	no	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	C₂×C₈	1	2	yes	—
56	C₄²×C₅₃	M₁₆	1	2	yes	—
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	2	yes	—
60	C₂×#88322	—	4	4	no	—
61	C₂³×C₅₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	2	no	—
65	C₂×C₄×C₅₃	—	4	2	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#124	1	2	yes	—
70	C₂×C₄×C₅₃	#125	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₅₃	—	2	1	no	—
76	C₈×C₅₃	—	2	1	no	—
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	—	2	2	no	—
79	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
80	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
81	#88322	—	8	4	no	—
82	#8780	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	#382	1	1	yes	characteristic
85	C₂²×C₅₃	#431	1	1	yes	characteristic
86	C₂²×C₅₃	#432	1	1	yes	characteristic
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	4	2	no	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₅₃	#423	1	1	yes	characteristic
93	C₄×C₅₃	#431	1	1	yes	characteristic
94	C₄×C₅₃	#432	1	1	yes	characteristic
95	#1726	D₅₃	1	1	yes	characteristic
96	#1564	—	53	2	no	—
97	#1281	—	53	2	no	—
98	#1274	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1831	1	1	yes	characteristic
102	C₂×C₅₃	#1833	1	2	yes	—
103	#431	—	53	4	no	—
104	C₂×#97	Dic₅₃	1	1	yes	characteristic
105	C₂×#97	Dic₅₃	1	1	yes	characteristic
106	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
107	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
108	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#376	C₂×D₅₃	1	1	yes	characteristic
111	#335	—	106	2	no	—
112	#334	—	106	2	no	—
113	C₅₃	#8780	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
119	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
121	C₄⋊C₈	—	106	4	no	—
122	C₂²⋊C₈	—	106	4	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	106	4	no	—
128	C₄×C₈	#88322	1	4	yes	complemented
129	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
130	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
131	C₄×C₈	D₂₁₂	1	1	yes	characteristic
132	C₄×C₈	D₂₁₂	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	#11719945	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	212	4	no	—
145	C₂×C₈	C₂×#88322	1	2	yes	—
146	C₂×C₈	C₂×#88322	1	2	yes	—
147	C₂×C₈	#11719917	1	2	yes	—
148	C₂×C₈	#11719917	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
152	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×#88322	1	2	yes	—
156	C₄²	#11719917	1	2	yes	—
157	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
158	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
159	C₄²	#11719945	1	2	yes	—
160	C₂³	#13133176	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	#13133176	1	1	yes	characteristic
168	C₂×C₄	#13133205	1	2	yes	—
169	C₂×C₄	#13133209	1	2	yes	—
170	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	424	4	no	—
179	C₈	C₄⋊#88322	1	2	yes	—
180	C₈	C₄⋊#88322	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#16810489	1	1	yes	characteristic
183	C₂²	#16810676	1	1	yes	characteristic
184	C₂²	#16810680	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
190	C₄	#16810647	1	1	yes	characteristic
191	C₄	#16810676	1	1	yes	characteristic
192	C₄	#16810680	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#20274832	1	1	yes	characteristic
196	C₂	#20274839	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#8780
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
29. C₄×C₈×C₅₃	C₈	semidirect
113. C₅₃	#8780	semidirect
128. C₄×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
#20272560	2
#20272544	2
C₅₃×#1726	2
#8780	53