Subgroups — #27270927

Order 13568.

246 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
2	#20272998	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	C₄	1	2	yes	—
5	#16810676	C₄	1	2	yes	—
6	#16810399	—	2	1	no	—
7	C₄×#11719908	—	2	1	no	—
8	#16810344	—	2	1	no	—
9	C₈×#88322	—	2	1	no	—
10	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#346	—	2	1	no	—
13	C₅₃×#346	—	2	1	no	—
14	#13133157	—	2	2	no	—
15	#13133157	C₈	1	4	yes	complemented
16	C₂×#11719908	—	2	2	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₂²⋊#88322	C₈	1	4	yes	—
19	C₄×#88322	—	2	2	no	—
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×C₁₆×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
39	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
40	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	M₁₆	1	2	yes	—
43	C₂×#88322	—	2	4	no	—
44	C₁₆×C₅₃	—	2	4	no	—
45	#11719908	—	2	4	no	—
46	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
56	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
57	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
58	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
59	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
60	#88322	—	4	4	no	—
61	#7140	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	C₈²	1	1	yes	characteristic
64	C₂²×C₅₃	#322	1	1	yes	characteristic
65	C₂²×C₅₃	#336	1	1	yes	characteristic
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	#322	1	1	yes	characteristic
69	C₄×C₅₃	#322	1	1	yes	characteristic
70	C₄×C₅₃	#336	1	1	yes	characteristic
71	C₄×C₅₃	#336	1	1	yes	characteristic
72	C₄×M₃₂	—	53	1	no	—
73	C₂×#346	D₅₃	1	1	yes	characteristic
74	#1450	—	53	1	no	—
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#1273	1	1	yes	characteristic
78	C₂×C₅₃	#1378	1	2	yes	—
79	C₂×M₃₂	—	53	2	no	—
80	C₂²×C₁₆	Dic₅₃	1	2	yes	—
81	#431	—	53	2	no	—
82	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	#346	—	2	1	no	—
84	#346	—	2	1	no	—
85	#346	—	106	1	no	—
86	C₄×C₁₆	—	106	1	no	—
87	#322	—	106	1	no	—
88	C₈²	—	106	1	no	—
89	C₅₃	#7140	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
93	M₃₂	—	53	4	no	—
94	M₃₂	—	106	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	106	2	no	—
98	C₂×C₁₆	#88322	1	4	yes	complemented
99	C₄⋊C₈	—	106	2	no	—
100	C₂²⋊C₈	—	53	4	no	—
101	C₄×C₈	—	106	2	no	—
102	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
103	C₄×C₈	D₂₁₂	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
106	C₂²×C₄	C₈×D₅₃	1	2	yes	—
107	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
108	C₂×C₈	—	106	4	no	—
109	C₂×C₈	C₈×D₅₃	1	2	yes	—
110	C₂×C₈	#11719912	1	2	yes	—
111	C₂×C₈	C₂×#88322	1	2	yes	—
112	C₂×C₈	#11719917	1	2	yes	—
113	C₂×C₈	#11719919	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₂×C₈	#11719921	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
117	C₄²	#11719912	1	2	yes	—
118	C₄²	#11719919	1	1	yes	characteristic
119	C₄²	#11719921	1	1	yes	characteristic
120	C₁₆	—	2	4	no	—
121	C₁₆	—	106	4	no	—
122	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	#13133124	1	1	yes	characteristic
125	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄	C₄×#88322	1	1	yes	characteristic
127	C₂×C₄	#13133127	1	1	yes	characteristic
128	C₂×C₄	#13133127	1	1	yes	characteristic
129	C₂×C₄	#13133128	1	1	yes	characteristic
130	C₂×C₄	#13133129	1	2	yes	—
131	C₂×C₄	#13133146	1	2	yes	—
132	C₈	—	212	4	no	—
133	C₈	#13133129	1	2	yes	—
134	C₈	C₂²⋊#88322	1	2	yes	—
135	C₈	#13133146	1	2	yes	—
136	C₈	C₄⋊#88322	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₈×#88322	1	1	yes	characteristic
139	C₂²	#16810345	1	1	yes	characteristic
140	C₂²	#16810377	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	#16810344	1	1	yes	characteristic
144	C₄	#16810345	1	1	yes	characteristic
145	C₄	#16810377	1	1	yes	characteristic
146	C₄	#16810378	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#20272542	1	1	yes	characteristic
150	C₂	#20272800	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₈²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	#336
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#7140
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
15. #13133157	C₈	semidirect
25. C₂×C₁₆×C₅₃	C₈	semidirect
89. C₅₃	#7140	semidirect
98. C₂×C₁₆	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133157	2
#20272998	2
C₂×C₅₃×#346	2
#7140	53