Subgroups — #27270984

Order 13568.

238 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	—	2	1	no	—
5	C₄⋊#11719908	C₄	1	2	yes	—
6	C₂²⋊#11719908	—	2	1	no	—
7	C₄×#11719908	—	2	1	no	—
8	C₄×#11719908	—	2	1	no	—
9	C₄×#11719908	C₄	1	2	yes	—
10	C₅₃×#431	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#322	—	2	1	no	—
14	C₈²×C₅₃	—	2	1	no	—
15	C₂×#11719908	—	2	4	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×M₁₆×C₅₃	C₈	1	2	yes	—
19	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
23	C₂²⋊C₈×C₅₃	—	2	1	no	—
24	#97×C₅₃	—	2	1	no	—
25	C₄×C₈×C₅₃	—	2	1	no	—
26	C₄×C₈×C₅₃	—	2	1	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
34	M₁₆×C₅₃	—	2	2	no	—
35	M₁₆×C₅₃	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₄²	1	1	yes	characteristic
45	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
48	#11719908	—	2	4	no	—
49	#11719908	—	4	2	no	—
50	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	—	2	1	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	—	2	4	no	—
65	C₈×C₅₃	—	4	2	no	—
66	C₈×C₅₃	#104	1	2	yes	—
67	C₈×C₅₃	#108	1	2	yes	—
68	#7163	—	53	1	no	maximal, Sylow
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	—	2	1	no	—
71	C₂²×C₅₃	#336	1	1	yes	characteristic
72	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#339	1	1	yes	characteristic
76	C₄×C₅₃	#341	1	1	yes	characteristic
77	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
78	C₄×C₅₃	#346	1	1	yes	characteristic
79	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
80	C₄×C₅₃	#350	1	2	yes	—
81	#2162	—	53	1	no	—
82	C₂×C₄×C₁₆	—	53	1	no	—
83	C₈×M₁₆	D₅₃	1	1	yes	characteristic
84	C₂×C₅₃	—	2	2	no	—
85	C₂×C₅₃	#1278	1	1	yes	characteristic
86	C₂×C₅₃	#1374	1	1	yes	characteristic
87	C₂×C₅₃	#1379	1	1	yes	characteristic
88	C₂²×C₁₆	—	106	1	no	—
89	#431	Dic₅₃	1	1	yes	characteristic
90	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	C₄⋊C₁₆	—	53	2	no	—
93	C₂²⋊C₁₆	—	106	1	no	—
94	C₄×C₁₆	—	53	2	no	—
95	C₄×C₁₆	—	106	1	no	—
96	C₄×C₁₆	—	106	1	no	—
97	#322	—	2	1	no	—
98	C₈²	—	2	1	no	—
99	C₅₃	#7163	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	#88322	1	2	yes	—
101	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
102	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂×C₁₆	—	106	4	no	—
105	C₂×C₁₆	—	106	2	no	—
106	C₂×C₁₆	—	106	2	no	—
107	C₄⋊C₈	#88322	1	2	yes	—
108	C₂²⋊C₈	—	2	1	no	—
109	#97	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	—	2	1	no	—
112	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
113	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
117	C₂²×C₄	#11719917	1	1	yes	characteristic
118	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
119	M₁₆	—	2	2	no	—
120	M₁₆	#11719908	1	4	yes	complemented
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	#11719919	1	1	yes	characteristic
126	C₂×C₈	#11719919	1	1	yes	characteristic
127	C₂×C₈	#11719921	1	1	yes	characteristic
128	C₂×C₈	#11719921	1	1	yes	characteristic
129	C₄²	C₂×#88322	1	1	yes	characteristic
130	C₄²	#11719917	1	1	yes	characteristic
131	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
132	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
133	C₁₆	—	106	4	no	—
134	C₁₆	—	212	2	no	—
135	C₂³	C₄⋊#88322	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	#13133124	1	1	yes	characteristic
140	C₂×C₄	C₄×#88322	1	1	yes	characteristic
141	C₂×C₄	#13133128	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
144	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
145	C₂×C₄	C₂×#11719908	1	2	yes	—
146	C₂×C₄	#13133157	1	2	yes	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	4	no	—
150	C₈	—	4	2	no	—
151	C₈	#13133143	1	2	yes	—
152	C₈	#13133152	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#16810378	1	1	yes	characteristic
156	C₂²	C₄⋊#11719908	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#16810384	1	1	yes	characteristic
160	C₄	#16810387	1	1	yes	characteristic
161	C₄	C₄×#11719908	1	1	yes	characteristic
162	C₄	#16810399	1	1	yes	characteristic
163	C₄	C₂²⋊#11719908	1	2	yes	—
164	C₄	#16810407	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#20272553	1	1	yes	characteristic
167	C₂	#20272793	1	1	yes	characteristic
168	C₂	#20272804	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#7163
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
35. M₁₆×C₅₃	C₁₆	semidirect
99. C₅₃	#7163	semidirect
120. M₁₆	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
C₂×C₄×#11719908	2
C₈×M₁₆×C₅₃	2
#7163	53