Subgroups — #27270965

Order 13568.

234 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₄	1	2	yes	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	#16810399	—	2	1	no	—
7	#16810399	—	2	1	no	—
8	#16810374	—	2	1	no	—
9	#16810374	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#334	—	2	1	no	—
13	C₅₃×#334	C₄	1	2	yes	—
14	C₂²×#88322	—	4	1	no	—
15	C₂×#11719908	—	2	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	C₈	1	4	yes	complemented
18	C₄⋊#88322	—	2	2	no	—
19	C₄⋊#88322	—	4	1	no	—
20	C₂²×C₈×C₅₃	—	4	1	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	—	2	2	no	—
25	C₄⋊C₈×C₅₃	—	4	1	no	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	8	2	no	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
42	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	D₈	1	1	yes	characteristic
45	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
46	C₂×#88322	—	4	2	no	—
47	C₂×#88322	—	8	2	no	—
48	#11719908	—	2	4	no	—
49	C₂³×C₅₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
58	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#108	1	2	yes	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	—	2	1	no	—
62	C₈×C₅₃	—	8	2	no	—
63	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
64	C₈×C₅₃	#97	1	1	yes	characteristic
65	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
66	#88322	—	8	2	no	—
67	#7156	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#334	1	2	yes	—
69	C₂²×C₅₃	#340	1	1	yes	characteristic
70	C₂²×C₅₃	#365	1	2	yes	—
71	C₂²×C₅₃	#365	1	2	yes	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	#335	1	2	yes	—
77	C₄×C₅₃	#336	1	1	yes	characteristic
78	C₄×C₅₃	#341	1	1	yes	characteristic
79	C₂×#346	—	53	1	no	—
80	C₂×#334	—	53	1	no	—
81	C₂×#334	D₅₃	1	1	yes	characteristic
82	C₂×C₅₃	#1277	1	1	yes	characteristic
83	C₂×C₅₃	#1368	1	4	yes	—
84	C₂×C₅₃	#1383	1	1	yes	characteristic
85	C₂×C₅₃	#1383	1	1	yes	characteristic
86	C₂²×C₁₆	—	53	2	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄⋊C₈	—	106	1	no	—
89	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
90	#346	—	106	1	no	—
91	#346	—	106	1	no	—
92	#334	—	2	1	no	—
93	#334	—	53	2	no	—
94	#334	—	106	1	no	—
95	#334	Dic₅₃	1	2	yes	—
96	C₅₃	#7156	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	—	212	1	no	—
99	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
100	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₁₆	—	53	4	no	—
103	C₂×C₁₆	—	106	2	no	—
104	C₂×C₁₆	—	106	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	106	2	no	—
108	C₄⋊C₈	—	212	1	no	—
109	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
110	C₄×C₈	D₂₁₂	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²×C₄	#11719919	1	1	yes	characteristic
114	C₂²×C₄	#11719923	1	1	yes	characteristic
115	C₂²×C₄	#11719924	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	8	2	no	—
119	C₂×C₈	—	424	2	no	—
120	C₂×C₈	—	212	2	no	—
121	C₂×C₈	C₈×D₅₃	1	2	yes	—
122	C₂×C₈	#11719912	1	2	yes	—
123	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
124	C₂×C₈	#11719919	1	1	yes	characteristic
125	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
126	C₂×C₈	#11719921	1	1	yes	characteristic
127	C₄²	#11719921	1	1	yes	characteristic
128	C₄²	#11719922	1	1	yes	characteristic
129	C₄²	#11719925	1	1	yes	characteristic
130	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₁₆	—	106	4	no	—
132	C₂³	#13133148	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	#13133124	1	1	yes	characteristic
136	C₂×C₄	#13133137	1	1	yes	characteristic
137	C₂×C₄	#13133138	1	1	yes	characteristic
138	C₂×C₄	#13133140	1	1	yes	characteristic
139	C₂×C₄	#13133141	1	1	yes	characteristic
140	C₂×C₄	#13133146	1	2	yes	—
141	C₂×C₄	#13133150	1	1	yes	characteristic
142	C₂×C₄	#13133152	1	2	yes	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	2	no	—
146	C₈	—	424	2	no	—
147	C₈	C₈×Dic₅₃	1	1	yes	characteristic
148	C₈	#13133127	1	1	yes	characteristic
149	C₈	#13133129	1	2	yes	—
150	C₂²	#16810373	1	2	yes	—
151	C₂²	#16810385	1	1	yes	characteristic
152	C₂²	#16810445	1	2	yes	—
153	C₂²	#16810447	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#16810375	1	2	yes	—
159	C₄	#16810377	1	1	yes	characteristic
160	C₄	#16810387	1	1	yes	characteristic
161	C₂	#20272549	1	1	yes	characteristic
162	C₂	#20272780	1	4	yes	—
163	C₂	#20272813	1	1	yes	characteristic
164	C₂	#20272815	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133148
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#106
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#7156
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
17. C₂×#11719908	C₈	semidirect
96. C₅₃	#7156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
C₂×#16810374	2
C₂×C₅₃×#334	2
#7156	53