Subgroups — #27270957

Order 13568.

230 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	2	yes	maximal
2	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	C₄⋊#11719908	—	2	2	no	—
5	C₄⋊#11719908	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#322	—	2	2	no	—
10	#13133157	—	4	2	no	—
11	#13133157	—	4	2	no	—
12	C₂×#11719908	—	2	4	no	—
13	C₂²×C₈×C₅₃	—	2	1	no	—
14	C₂²×C₈×C₅₃	C₈	1	2	yes	—
15	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₅₃	—	2	1	no	—
19	C₄×C₈×C₅₃	—	2	1	no	—
20	C₄×C₈×C₅₃	—	2	1	no	—
21	C₄×C₈×C₅₃	C₈	1	2	yes	—
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	4	2	no	—
32	C₂×C₈×C₅₃	—	4	2	no	—
33	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
38	C₄²×C₅₃	C₄²	1	1	yes	characteristic
39	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
42	#11719908	—	4	4	no	—
43	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	2	no	—
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
56	C₈×C₅₃	—	2	2	no	—
57	C₈×C₅₃	—	4	2	no	—
58	C₈×C₅₃	—	4	2	no	—
59	C₈×C₅₃	#102	1	2	yes	—
60	C₈×C₅₃	#103	1	2	yes	—
61	#7152	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	2	1	no	—
64	C₂²×C₅₃	#334	1	1	yes	characteristic
65	C₂²×C₅₃	#335	1	1	yes	characteristic
66	C₂²×C₅₃	#336	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	#334	1	1	yes	characteristic
70	C₄×C₅₃	#335	1	1	yes	characteristic
71	C₄×C₅₃	#340	1	1	yes	characteristic
72	C₄×C₅₃	#341	1	1	yes	characteristic
73	C₄×C₅₃	#347	1	2	yes	—
74	C₄×C₅₃	#349	1	2	yes	—
75	#2150	—	53	2	no	—
76	C₂×#322	D₅₃	1	1	yes	characteristic
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	#1277	1	1	yes	characteristic
80	C₂×C₅₃	#1375	1	1	yes	characteristic
81	C₂×C₅₃	#1377	1	1	yes	characteristic
82	C₂×M₃₂	—	106	2	no	—
83	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
84	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₁₆	—	53	4	no	—
87	C₄⋊C₁₆	—	106	2	no	—
88	#322	—	2	2	no	—
89	C₅₃	#7152	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	#88322	1	2	yes	—
92	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
93	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
95	M₃₂	—	212	2	no	—
96	M₃₂	—	212	2	no	—
97	C₂×C₁₆	—	106	4	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	#88322	1	2	yes	—
102	C₄×C₈	C₄×D₅₃	1	2	yes	—
103	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
104	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
105	C₂²×C₄	#11719917	1	1	yes	characteristic
106	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	#11719913	1	2	yes	—
114	C₂×C₈	D₄₂₄	1	1	yes	characteristic
115	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
116	C₂×C₈	#11719919	1	2	yes	—
117	C₂×C₈	#11719921	1	2	yes	—
118	C₄²	C₂×#88322	1	1	yes	characteristic
119	C₄²	#11719917	1	1	yes	characteristic
120	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
121	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
122	C₁₆	—	212	4	no	—
123	C₂³	C₄⋊#88322	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	#13133124	1	1	yes	characteristic
127	C₂×C₄	C₄×#88322	1	1	yes	characteristic
128	C₂×C₄	#13133128	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
131	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
132	C₂×C₄	#13133149	1	1	yes	characteristic
133	C₂×C₄	#13133151	1	1	yes	characteristic
134	C₂×C₄	#13133153	1	1	yes	characteristic
135	C₂×C₄	#13133153	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	#13133139	1	2	yes	—
140	C₈	#13133142	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#16810374	1	1	yes	characteristic
144	C₂²	#16810376	1	1	yes	characteristic
145	C₂²	#16810378	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	#16810374	1	1	yes	characteristic
149	C₄	#16810376	1	1	yes	characteristic
150	C₄	#16810386	1	1	yes	characteristic
151	C₄	#16810388	1	1	yes	characteristic
152	C₄	#16810401	1	2	yes	—
153	C₄	#16810405	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#20272550	1	1	yes	characteristic
157	C₂	#20272795	1	1	yes	characteristic
158	C₂	#20272799	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#7152
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
89. C₅₃	#7152	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×C₅₃×#322	2
#7152	53