Finite Group Databasebeta

Subgroups — #27304711

Order 13568.

182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20276601	C₂	1	1	yes	maximal, characteristic
2	#20276586	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272921	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272908	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272902	C₂	1	1	yes	maximal, characteristic
6	#20272850	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2264	C₂	1	1	yes	maximal, characteristic
8	#16810740	—	2	1	no	—
9	Q₁₆×Dic₅₃	—	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	#16810430	—	2	1	no	—
14	#16810427	—	2	1	no	—
15	#16810398	C₂²	1	1	yes	characteristic
16	C₅₃×#508	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×Q₆₄	C₄	1	2	yes	complemented
19	C₅₃×SD₆₄	C₄	1	2	yes	complemented
20	C₅₃×M₆₄	C₂²	1	1	yes	characteristic
21	#13133224	—	4	1	no	—
22	Q₈×Dic₅₃	—	4	1	no	—
23	C₂×#11719908	D₄	1	1	yes	characteristic
24	#13133153	—	2	1	no	—
25	#13133151	—	2	1	no	—
26	#13133144	—	4	1	no	—
27	#13133141	—	4	1	no	—
28	C₈×Dic₅₃	D₄	1	1	yes	characteristic
29	#13133123	—	2	1	no	—
30	#135×C₅₃	—	2	1	no	—
31	C₂×Q₁₆×C₅₃	—	2	1	no	—
32	Q₃₂×C₅₃	—	2	1	no	—
33	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
34	Q₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
35	SD₃₂×C₅₃	—	2	1	no	—
36	D₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
38	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
39	#41×C₅₃	—	4	1	no	—
40	C₂×Q₈×C₅₃	—	4	1	no	—
41	Q₁₆×C₅₃	—	2	1	no	—
42	Q₁₆×C₅₃	—	2	2	no	—
43	Q₁₆×C₅₃	—	4	1	no	—
44	SD₁₆×C₅₃	—	4	1	no	—
45	D₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Dic₅₃	—	4	1	no	—
48	C₄⋊Dic₅₃	—	8	1	no	—
49	C₄×Dic₅₃	—	8	1	no	—
50	C₄×Dic₅₃	D₈	1	1	yes	characteristic
51	#11719917	—	4	1	no	—
52	#11719917	—	8	1	no	—
53	C₂×#88322	—	8	1	no	—
54	C₂×#88322	D₈	1	1	yes	characteristic
55	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₅₃	#41	1	1	yes	characteristic
57	#11719908	—	2	1	no	—
58	Q₈×C₅₃	—	4	2	no	—
59	Q₈×C₅₃	—	4	1	no	—
60	Q₈×C₅₃	—	8	1	no	—
61	D₄×C₅₃	—	4	1	no	—
62	D₄×C₅₃	—	8	1	no	—
63	C₂×C₄×C₅₃	—	8	1	no	—
64	C₂×C₄×C₅₃	—	8	1	no	—
65	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	1	no	—
67	C₂×Dic₅₃	—	8	1	no	—
68	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₈×C₅₃	#135	1	1	yes	characteristic
70	#88322	—	2	1	no	—
71	#88322	—	8	1	no	—
72	#13657	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	—	8	1	no	—
74	C₂²×C₅₃	#509	1	1	yes	characteristic
75	C₄×C₅₃	—	8	1	no	—
76	C₄×C₅₃	—	8	2	no	—
77	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
78	C₄×C₅₃	#509	1	1	yes	characteristic
79	Dic₅₃	—	4	1	no	—
80	Dic₅₃	—	16	1	no	—
81	#2264	D₅₃	1	1	yes	characteristic
82	#2179	—	53	1	no	—
83	#2176	—	53	1	no	—
84	#1426	—	53	1	no	—
85	#1420	—	53	1	no	—
86	#1418	—	53	1	no	—
87	#1398	—	53	1	no	—
88	C₂×C₅₃	—	2	1	no	—
89	C₂×C₅₃	—	16	1	no	—
90	C₂×C₅₃	#2177	1	1	yes	characteristic
91	#508	C₂×D₅₃	1	1	yes	characteristic
92	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
93	#443	—	106	1	no	—
94	C₄×Q₁₆	—	106	1	no	—
95	Q₆₄	Dic₅₃	1	2	yes	complemented
96	SD₆₄	Dic₅₃	1	2	yes	complemented
97	M₆₄	—	53	1	no	—
98	M₆₄	C₂×D₅₃	1	1	yes	characteristic
99	#368	—	53	1	no	—
100	#367	—	53	1	no	—
101	#359	—	106	1	no	—
102	#358	—	106	1	no	—
103	#346	—	53	1	no	—
104	C₅₃	#13657	1	1	yes	characteristic, complemented, Sylow
105	#135	—	2	1	no	—
106	C₂×Q₁₆	—	2	1	no	—
107	#131	—	212	1	no	—
108	C₄×Q₈	—	212	1	no	—
109	Q₃₂	—	2	1	no	—
110	Q₃₂	C₂×Dic₅₃	1	2	yes	—
111	Q₃₂	C₂×Dic₅₃	1	1	yes	characteristic
112	SD₃₂	—	2	1	no	—
113	D₁₆	C₂×Dic₅₃	1	1	yes	characteristic
114	C₂×C₁₆	—	53	1	no	—
115	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
116	#108	—	106	1	no	—
117	#107	—	106	1	no	—
118	#104	—	212	1	no	—
119	#103	—	212	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₃₂	—	106	1	no	—
122	C₃₂	C₂×Dic₅₃	1	2	yes	—
123	#41	—	4	1	no	—
124	C₂×Q₈	—	4	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	2	2	no	—
127	Q₁₆	—	4	1	no	—
128	SD₁₆	—	4	1	no	—
129	D₈	—	2	1	no	—
130	M₁₆	—	424	1	no	—
131	M₁₆	—	212	1	no	—
132	C₂×C₈	—	53	1	no	—
133	C₂×C₈	—	424	1	no	—
134	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
135	C₄⋊C₄	—	424	1	no	—
136	C₄⋊C₄	—	212	1	no	—
137	C₄²	—	53	1	no	—
138	C₄²	—	424	1	no	—
139	C₁₆	—	106	1	no	—
140	C₁₆	#11719947	1	1	yes	characteristic
141	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
142	Q₈	—	4	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	8	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	106	1	no	—
150	C₂×C₄	—	424	1	no	—
151	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
152	C₈	—	106	1	no	—
153	C₈	—	424	1	no	—
154	C₈	D₄×Dic₅₃	1	1	yes	characteristic
155	C₈	#13133236	1	1	yes	characteristic
156	C₂²	—	8	1	no	—
157	C₂²	#16811057	1	1	yes	characteristic
158	C₄	—	8	1	no	—
159	C₄	—	8	2	no	—
160	C₄	—	848	1	no	—
161	C₄	—	212	1	no	—
162	C₄	D₈×Dic₅₃	1	1	yes	characteristic
163	C₄	#16811057	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	16	1	no	—
166	C₂	#20276591	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20276591
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#2264	C₂
socle	C₂×C₅₃	#2177
prime radical	C₅₃×#508	C₂²
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#2264	D₅₃
hypocenter	C₅₃	#13657
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2264

Complements

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

N	H ≅ G/N	type
2. #20276586	C₂	semidirect
3. #20272921	C₂	semidirect
4. #20272908	C₂	semidirect
6. #20272850	C₂	semidirect
18. C₅₃×Q₆₄	C₄	semidirect
19. C₅₃×SD₆₄	C₄	semidirect
95. Q₆₄	Dic₅₃	semidirect
96. SD₆₄	Dic₅₃	semidirect
104. C₅₃	#13657	semidirect

Maximal subgroups

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

subgroup	index
#20276601	2
#20276586	2
#20272921	2
#20272908	2
#20272902	2
#20272850	2
C₅₃×#2264	2
#13657	53