Finite Group Databasebeta

Subgroups — #27300484

Order 13568.

274 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272808	C₂	1	4	yes	maximal
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	#16810399	C₄	1	4	yes	—
10	C₂×#108×C₅₃	—	2	1	no	—
11	C₂×#108×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#11719917	—	2	2	no	—
15	C₂×#11719908	—	2	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	C₂×C₄	1	4	yes	—
18	#13133153	—	2	4	no	—
19	#13133153	—	4	2	no	—
20	C₄⋊#88322	—	4	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	#108×C₅₃	—	2	4	no	—
26	#108×C₅₃	—	4	2	no	—
27	C₄⋊C₈×C₅₃	—	4	2	no	—
28	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₅₃	—	4	2	no	—
33	M₁₆×C₅₃	—	4	2	no	—
34	C₂×C₈×C₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₂×C₈×C₅₃	#41	1	2	yes	—
42	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₄²×C₅₃	D₈	1	2	yes	—
44	#11719917	—	4	2	no	—
45	#11719917	—	4	2	no	—
46	C₂×#88322	—	4	2	no	—
47	#11719908	—	2	4	no	—
48	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	#107	1	4	yes	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₅₃	—	8	2	no	—
58	C₈×C₅₃	#106	1	4	yes	—
59	C₈×C₅₃	#117	1	2	yes	—
60	C₈×C₅₃	#125	1	2	yes	—
61	#88322	—	8	2	no	—
62	#12892	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
65	C₂²×C₅₃	#487	1	1	yes	characteristic
66	C₂²×C₅₃	#487	1	1	yes	characteristic
67	C₄×C₅₃	—	2	4	no	—
68	C₄×C₅₃	#382	1	1	yes	characteristic
69	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
70	C₄×C₅₃	#486	1	1	yes	characteristic
71	C₄×C₅₃	#486	1	1	yes	characteristic
72	C₂×#346	—	53	1	no	—
73	#1848	—	53	1	no	—
74	#1848	D₅₃	1	1	yes	characteristic
75	#1381	—	53	4	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	#1834	1	1	yes	characteristic
79	C₂×C₅₃	#2161	1	2	yes	—
80	C₂²×C₁₆	—	53	2	no	—
81	C₂×#108	—	2	1	no	—
82	C₂×#108	—	53	2	no	—
83	C₂×#108	—	106	1	no	—
84	C₂×#108	C₂×D₅₃	1	2	yes	—
85	#423	—	2	1	no	—
86	#423	—	106	1	no	—
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	#346	—	53	4	no	—
89	C₅₃	#12892	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	106	2	no	—
92	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₂×C₁₆	—	53	4	no	—
96	C₂×C₁₆	—	106	2	no	—
97	C₂×C₁₆	—	106	2	no	—
98	#108	—	2	4	no	—
99	#108	—	4	2	no	—
100	#108	—	106	4	no	—
101	#108	—	212	2	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	212	2	no	—
104	C₄×C₈	Dic₁₀₆	1	2	yes	—
105	C₄×C₈	C₄×D₅₃	1	2	yes	—
106	C₂²×C₄	#11719925	1	2	yes	—
107	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	212	2	no	—
111	M₁₆	—	212	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	212	2	no	—
114	C₂×C₈	#11719919	1	2	yes	—
115	C₂×C₈	#11719923	1	2	yes	—
116	C₂×C₈	#11719924	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
119	C₂×C₈	#11719947	1	2	yes	—
120	C₂×C₈	#11719949	1	2	yes	—
121	C₄²	#11719919	1	2	yes	—
122	C₄²	#11719922	1	2	yes	—
123	C₁₆	—	106	4	no	—
124	C₂³	C₂×#11719925	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#13133150	1	4	yes	—
127	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
128	C₂×C₄	#13133204	1	1	yes	characteristic
129	C₂×C₄	#13133206	1	1	yes	characteristic
130	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
131	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
132	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
133	C₈	—	8	2	no	—
134	C₈	—	424	2	no	—
135	C₈	#13133148	1	4	yes	—
136	C₈	#13133175	1	2	yes	—
137	C₈	#13133208	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#13133150	1	1	yes	characteristic
140	C₂²	#16810973	1	1	yes	characteristic
141	C₂²	#16810975	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	#16810488	1	1	yes	characteristic
144	C₄	C₂×#13133148	1	1	yes	characteristic
145	C₄	#16810969	1	1	yes	characteristic
146	C₄	#16810971	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#20274842	1	1	yes	characteristic
150	C₂	#20276509	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#12892
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
89. C₅₃	#12892	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
#20274887	2
#20272808	2
C₅₃×#1848	2
#12892	53