Finite Group Databasebeta

Subgroups — #28299194

Order 13568.

178 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 64 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	D₅₃⋊M₆₄	C₂	1	1	yes	maximal, characteristic
2	#20291066	C₂	1	1	yes	maximal, characteristic
3	#367×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811633	—	2	1	no	—
5	#16811556	C₄	1	2	yes	complemented
6	#16811555	—	2	1	no	—
7	D₅₃⋊C₃₂	—	2	1	no	—
8	#16811550	—	2	1	no	—
9	#16811549	C₄	1	2	yes	complemented
10	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
11	#107×D₅₃	—	2	1	no	—
12	#16810451	C₄	1	1	yes	characteristic, complemented
13	#16810450	—	2	1	no	—
14	C₅₃×#367	C₄	1	1	yes	characteristic, complemented
15	#13133296	—	4	1	no	—
16	#13133286	—	2	2	no	—
17	#13133285	—	4	1	no	—
18	#13133267	—	2	2	no	—
19	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
20	C₄⋊C₄×D₅₃	—	4	1	no	—
21	C₂×#11719908	D₄	1	1	yes	characteristic
22	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₁₆×D₅₃	D₄	1	1	yes	characteristic
25	C₁₆×D₅₃	Q₈	1	1	yes	characteristic
26	#13133151	—	2	1	no	—
27	#13133150	—	4	1	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#107×C₅₃	—	2	1	no	—
30	C₂×C₄×D₅₃	—	8	1	no	—
31	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
32	C₂²⋊#6157	—	8	1	no	—
33	C₄⋊#6157	—	4	2	no	—
34	C₄×#6157	—	8	1	no	—
35	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	4	1	no	—
37	C₄⋊Dic₅₃	—	4	1	no	—
38	#11719919	—	8	1	no	—
39	C₂×#88322	SD₁₆	1	1	yes	characteristic
40	C₈×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
43	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
44	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
45	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	#11719908	C₂²⋊C₄	1	1	yes	characteristic
47	#11719908	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×D₅₃	D₁₆	1	1	yes	characteristic
49	C₂×#6157	—	8	2	no	—
50	C₂×C₄×C₅₃	—	8	1	no	—
51	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
52	C₂×Dic₅₃	—	8	1	no	—
53	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
54	C₄×D₅₃	—	16	2	no	—
55	C₄×D₅₃	#102	1	1	yes	characteristic
56	C₄×D₅₃	#107	1	1	yes	characteristic
57	C₄×D₅₃	SD₃₂	1	1	yes	characteristic
58	C₄×D₅₃	Q₃₂	1	1	yes	characteristic
59	C₈×C₅₃	#95	1	1	yes	characteristic
60	C₈×C₅₃	#103	1	1	yes	characteristic
61	#88322	#103	1	1	yes	characteristic
62	#88322	#106	1	1	yes	characteristic
63	#7477	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	#357	1	1	yes	characteristic
65	C₂×D₅₃	—	2	2	no	—
66	C₂×D₅₃	#357	1	1	yes	characteristic
67	C₂×D₅₃	#366	1	1	yes	characteristic
68	#6157	—	16	2	no	—
69	C₄×C₅₃	—	16	1	no	—
70	C₄×C₅₃	#340	1	1	yes	characteristic
71	C₄×C₅₃	#358	1	1	yes	characteristic
72	Dic₅₃	—	16	1	no	—
73	Dic₅₃	#358	1	1	yes	characteristic
74	Dic₅₃	#367	1	1	yes	characteristic
75	C₂×M₆₄	—	53	1	no	—
76	#2158	—	53	1	no	—
77	C₂×#367	—	53	1	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	#1380	1	1	yes	characteristic
80	D₅₃	—	2	1	no	—
81	D₅₃	#1418	1	2	yes	—
82	C₂²×C₁₆	—	53	1	no	—
83	#427	—	106	1	no	—
84	C₂×#107	—	106	1	no	—
85	M₆₄	—	53	2	no	—
86	M₆₄	—	106	1	no	—
87	C₂×C₃₂	—	106	1	no	—
88	#367	—	53	1	no	—
89	#367	—	106	1	no	—
90	#367	—	106	1	no	—
91	#367	#6157	1	1	yes	characteristic, complemented
92	#366	—	53	2	no	—
93	C₅₃	#7477	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	53	1	no	—
95	#117	—	212	1	no	—
96	C₂×C₄⋊C₄	—	212	1	no	—
97	C₂×C₁₆	—	53	1	no	—
98	C₂×C₁₆	—	53	1	no	—
99	C₂×C₁₆	—	53	1	no	—
100	C₂×C₁₆	—	53	1	no	—
101	C₂×C₁₆	—	53	1	no	—
102	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#107	—	106	2	no	—
105	#107	—	106	1	no	—
106	#107	—	212	1	no	—
107	#106	—	212	1	no	—
108	C₃₂	—	106	2	no	—
109	C₂²×C₄	—	53	1	no	—
110	C₂²×C₄	—	424	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	—	53	1	no	—
116	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	424	1	no	—
119	C₄⋊C₄	—	212	2	no	—
120	C₄⋊C₄	—	212	1	no	—
121	C₂²⋊C₄	—	424	1	no	—
122	C₄²	—	424	1	no	—
123	C₁₆	—	53	1	no	—
124	C₁₆	—	53	1	no	—
125	C₁₆	C₄×#6157	1	1	yes	characteristic
126	C₁₆	C₄⋊#6157	1	1	yes	characteristic
127	C₂³	—	53	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	53	1	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	848	2	no	—
135	C₂×C₄	—	424	2	no	—
136	C₂×C₄	—	424	1	no	—
137	C₂×C₄	#13133279	1	1	yes	characteristic
138	C₈	—	53	1	no	—
139	C₈	—	53	1	no	—
140	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
141	C₈	#13133280	1	1	yes	characteristic
142	C₂²	—	53	1	no	—
143	C₂²	—	53	1	no	—
144	C₂²	—	106	2	no	—
145	C₂²	#16811535	1	1	yes	characteristic
146	C₄	—	16	1	no	—
147	C₄	—	53	1	no	—
148	C₄	—	53	1	no	—
149	C₄	—	848	1	no	—
150	C₄	—	848	2	no	—
151	C₄	#16811496	1	1	yes	characteristic
152	C₄	#16811536	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	53	2	no	—
155	C₂	—	106	1	no	—
156	C₂	#20289772	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289772
commutator	C₁₆×C₅₃	C₄²
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₅₃×#367	C₄
socle	C₂×C₅₃	#1380
prime radical	C₂²×D₅₃	D₁₆
cosocle	C₂×C₁₆×D₅₃	C₂²
hypercenter	#367	#6157
hypocenter	C₅₃	#7477
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
5. #16811556	C₄	semidirect
9. #16811549	C₄	semidirect
12. #16810451	C₄	semidirect
14. C₅₃×#367	C₄	semidirect
91. #367	#6157	semidirect
93. C₅₃	#7477	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊M₆₄	2
#20291066	2
#367×D₅₃	2
#7477	53