Finite Group Databasebeta

Subgroups — #27305011

Order 13568.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276996	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276596	C₂	1	1	yes	maximal, characteristic
3	#20272921	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272920	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272912	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272902	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1418	C₂	1	1	yes	maximal, characteristic
8	#16811027	—	2	1	no	—
9	#16810740	—	2	1	no	—
10	#16810457	C₂²	1	1	yes	characteristic
11	#16810453	C₂²	1	1	yes	characteristic
12	#16810451	C₂²	1	1	yes	characteristic
13	#16810450	—	2	1	no	—
14	#16810430	—	2	1	no	—
15	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#508	C₂²	1	1	yes	characteristic
17	C₅₃×M₆₄	C₂²	1	1	yes	characteristic
18	C₅₃×#367	C₂²	1	1	yes	characteristic
19	#13133224	—	4	1	no	—
20	#13133216	—	4	1	no	—
21	C₂×#11719908	D₄	1	1	yes	characteristic
22	#13133153	—	2	1	no	—
23	#13133151	—	2	1	no	—
24	#13133150	—	4	1	no	—
25	#13133144	—	4	1	no	—
26	C₈×Dic₅₃	D₄	1	1	yes	characteristic
27	#13133123	—	2	1	no	—
28	#135×C₅₃	—	2	1	no	—
29	Q₃₂×C₅₃	Q₈	1	1	yes	characteristic, complemented
30	SD₃₂×C₅₃	—	2	1	no	—
31	D₁₆×C₅₃	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
33	#107×C₅₃	—	2	1	no	—
34	C₃₂×C₅₃	—	2	1	no	—
35	#41×C₅₃	—	4	1	no	—
36	C₂×Dic₁₀₆	—	8	1	no	—
37	Q₁₆×C₅₃	—	2	1	no	—
38	SD₁₆×C₅₃	—	4	1	no	—
39	D₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	—	4	1	no	—
42	C₄⋊Dic₅₃	—	4	1	no	—
43	#11719919	—	8	1	no	—
44	C₄×Dic₅₃	D₈	1	1	yes	characteristic
45	#11719917	—	4	1	no	—
46	#11719917	—	8	1	no	—
47	C₂×#88322	—	8	1	no	—
48	C₂×#88322	D₈	1	1	yes	characteristic
49	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₁₆×C₅₃	#41	1	1	yes	characteristic
51	#11719908	—	2	1	no	—
52	Q₈×C₅₃	—	4	1	no	—
53	D₄×C₅₃	—	4	1	no	—
54	D₄×C₅₃	—	8	1	no	—
55	C₂×C₄×C₅₃	—	8	1	no	—
56	C₂×C₄×C₅₃	—	8	1	no	—
57	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	1	no	—
59	C₂×Dic₅₃	—	8	1	no	—
60	Dic₁₀₆	—	16	2	no	—
61	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₅₃	#137	1	1	yes	characteristic
63	#88322	—	8	1	no	—
64	#88322	SD₃₂	1	1	yes	characteristic
65	#88322	SD₃₂	1	1	yes	characteristic
66	#13710	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	8	1	no	—
68	C₂²×C₅₃	#509	1	1	yes	characteristic
69	C₄×C₅₃	—	8	1	no	—
70	C₄×C₅₃	—	16	1	no	—
71	C₄×C₅₃	#474	1	1	yes	characteristic
72	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	Dic₅₃	—	16	1	no	—
75	#2254	—	53	1	no	—
76	#2178	—	53	1	no	—
77	#1426	—	53	1	no	—
78	#1426	—	53	1	no	—
79	#1422	—	53	1	no	—
80	#1418	—	53	1	no	—
81	#1418	D₅₃	1	1	yes	characteristic
82	C₂×C₅₃	—	2	1	no	—
83	C₂×C₅₃	—	16	1	no	—
84	C₂×C₅₃	#2226	1	1	yes	characteristic
85	#508	C₂×D₅₃	1	1	yes	characteristic
86	#500	—	106	1	no	—
87	#443	—	106	1	no	—
88	M₆₄	—	53	1	no	—
89	M₆₄	C₂×D₅₃	1	1	yes	characteristic
90	#368	—	53	1	no	—
91	#367	—	53	1	no	—
92	#367	—	106	1	no	—
93	#367	C₂×D₅₃	1	1	yes	characteristic
94	#359	—	106	1	no	—
95	C₄×C₁₆	—	53	1	no	—
96	C₅₃	#13710	1	1	yes	characteristic, complemented, Sylow
97	#135	—	2	1	no	—
98	#131	—	212	1	no	—
99	C₄⋊Q₈	—	212	1	no	—
100	Q₃₂	Dic₁₀₆	1	1	yes	characteristic, complemented
101	SD₃₂	—	2	1	no	—
102	D₁₆	Dic₁₀₆	1	1	yes	characteristic, complemented
103	C₂×C₁₆	—	53	1	no	—
104	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
105	#108	—	106	1	no	—
106	#107	—	2	1	no	—
107	#107	—	106	1	no	—
108	#107	—	212	1	no	—
109	#104	—	212	1	no	—
110	C₄×C₈	—	53	1	no	—
111	C₃₂	—	2	1	no	—
112	C₃₂	—	106	1	no	—
113	#41	—	4	1	no	—
114	C₂×Q₈	—	424	1	no	—
115	Q₁₆	—	2	1	no	—
116	SD₁₆	—	4	1	no	—
117	D₈	—	2	1	no	—
118	M₁₆	—	424	1	no	—
119	M₁₆	—	212	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	424	1	no	—
122	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	424	1	no	—
125	C₄⋊C₄	—	212	1	no	—
126	C₄²	—	53	1	no	—
127	C₁₆	—	106	1	no	—
128	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
129	C₁₆	#11719947	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	848	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	8	1	no	—
136	C₂×C₄	—	106	1	no	—
137	C₂×C₄	—	424	1	no	—
138	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
139	C₈	—	53	1	no	—
140	C₈	—	53	1	no	—
141	C₈	—	424	1	no	—
142	C₈	#13133194	1	1	yes	characteristic
143	C₈	#13133246	1	1	yes	characteristic
144	C₂²	—	8	1	no	—
145	C₂²	#16811057	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	16	1	no	—
148	C₄	—	848	1	no	—
149	C₄	—	106	2	no	—
150	C₄	#16810901	1	1	yes	characteristic
151	C₄	SD₃₂×D₅₃	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	16	1	no	—
154	C₂	#20276860	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20276860
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1418	C₂
socle	C₂×C₅₃	#2226
prime radical	C₅₃×#508	C₂²
hypercenter	#1418	D₅₃
hypocenter	C₅₃	#13710
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
1. #20276996	C₂	semidirect
3. #20272921	C₂	semidirect
4. #20272920	C₂	semidirect
5. #20272912	C₂	semidirect
29. Q₃₂×C₅₃	Q₈	semidirect
31. D₁₆×C₅₃	Q₈	semidirect
96. C₅₃	#13710	semidirect
100. Q₃₂	Dic₁₀₆	semidirect
102. D₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20276996	2
#20276596	2
#20272921	2
#20272920	2
#20272912	2
#20272902	2
C₅₃×#1418	2
#13710	53