Finite Group Databasebeta

Subgroups — #27304978

Order 13568.

164 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20276989	C₂	1	1	yes	maximal, characteristic
2	#20276573	C₂	1	1	yes	maximal, characteristic
3	#20272916	C₂	1	1	yes	maximal, characteristic
4	#20272910	C₂	1	1	yes	maximal, characteristic
5	#20272897	C₂	1	1	yes	maximal, characteristic
6	#20272895	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1424	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
9	#16811027	—	2	1	no	—
10	#16810717	—	2	1	no	—
11	C₂×#13133123	C₂²	1	1	yes	characteristic
12	#16810449	C₂²	1	1	yes	characteristic
13	#16810448	—	2	1	no	—
14	#16810426	—	2	1	no	—
15	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#366	C₂²	1	1	yes	characteristic
18	C₅₃×#366	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₂	—	2	1	no	—
20	#13133216	—	4	1	no	—
21	#13133182	—	4	1	no	—
22	Dic₄₂₄	—	2	1	no	—
23	Dic₄₂₄	Q₈	1	2	yes	complemented
24	C₂×#11719908	D₄	1	1	yes	characteristic
25	#13133151	—	2	1	no	—
26	#13133150	—	4	1	no	—
27	#13133140	—	4	1	no	—
28	C₈×Dic₅₃	D₄	1	1	yes	characteristic
29	#13133123	—	2	1	no	—
30	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
31	#107×C₅₃	—	2	1	no	—
32	#107×C₅₃	—	2	1	no	—
33	C₃₂×C₅₃	—	2	1	no	—
34	C₂×Dic₁₀₆	—	4	1	no	—
35	C₂×Dic₁₀₆	—	8	1	no	—
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	—	4	1	no	—
38	C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₄⋊Dic₅₃	—	4	1	no	—
40	#11719919	—	8	1	no	—
41	#11719919	—	8	1	no	—
42	C₄×Dic₅₃	—	8	1	no	—
43	C₄×Dic₅₃	D₈	1	1	yes	characteristic
44	C₂×#88322	D₈	1	1	yes	characteristic
45	Dic₂₁₂	—	2	2	no	—
46	Dic₂₁₂	—	4	1	no	—
47	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₅₃	#41	1	1	yes	characteristic
49	#11719908	—	2	1	no	—
50	C₂×C₄×C₅₃	—	8	1	no	—
51	C₂×C₄×C₅₃	—	8	1	no	—
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₅₃	—	8	1	no	—
54	C₂×Dic₅₃	—	8	1	no	—
55	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
56	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
57	Dic₁₀₆	—	4	2	no	—
58	Dic₁₀₆	—	8	1	no	—
59	Dic₁₀₆	—	16	2	no	—
60	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₅₃	#137	1	1	yes	characteristic
62	#88322	—	2	1	no	—
63	#13704	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
65	C₄×C₅₃	—	16	1	no	—
66	C₄×C₅₃	—	16	1	no	—
67	C₄×C₅₃	#474	1	1	yes	characteristic
68	C₄×C₅₃	#510	1	1	yes	characteristic
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	8	2	no	—
71	Dic₅₃	—	16	1	no	—
72	Dic₅₃	SD₆₄	1	2	yes	—
73	#2252	—	53	1	no	—
74	C₄×Q₃₂	—	53	1	no	—
75	#1424	—	53	1	no	—
76	#1424	D₅₃	1	1	yes	characteristic
77	C₄⋊C₃₂	—	53	1	no	—
78	#1416	—	53	1	no	—
79	#1416	—	53	1	no	—
80	C₂×C₅₃	#2224	1	1	yes	characteristic
81	C₂×C₅₃	C₂×SD₆₄	1	1	yes	characteristic
82	C₂×C₅₃	#2264	1	1	yes	characteristic
83	C₂×Q₃₂	—	53	1	no	—
84	#500	—	106	1	no	—
85	C₄×Q₁₆	—	106	1	no	—
86	C₂×C₃₂	—	53	1	no	—
87	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
88	#366	—	53	1	no	—
89	#366	—	106	1	no	—
90	#366	C₂×D₅₃	1	1	yes	characteristic
91	#366	C₂×D₅₃	1	1	yes	characteristic
92	#358	—	106	1	no	—
93	C₄×C₁₆	—	53	1	no	—
94	C₅₃	#13704	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₁₆	—	106	1	no	—
96	C₄⋊Q₈	—	212	1	no	—
97	C₄×Q₈	—	212	1	no	—
98	Q₃₂	—	53	2	no	—
99	Q₃₂	—	106	1	no	—
100	C₂×C₁₆	—	53	1	no	—
101	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	2	1	no	—
104	#107	—	106	1	no	—
105	#107	—	212	1	no	—
106	#103	—	212	1	no	—
107	C₄×C₈	—	53	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	106	1	no	—
110	C₂×Q₈	—	424	1	no	—
111	C₂×Q₈	—	212	1	no	—
112	Q₁₆	—	106	2	no	—
113	Q₁₆	—	212	1	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	424	1	no	—
119	C₄⋊C₄	—	424	1	no	—
120	C₄⋊C₄	—	212	1	no	—
121	C₄²	—	53	1	no	—
122	C₄²	—	424	1	no	—
123	C₁₆	—	106	1	no	—
124	C₁₆	#11719945	1	1	yes	characteristic
125	C₁₆	Q₈×D₅₃	1	1	yes	characteristic
126	Q₈	—	848	2	no	—
127	Q₈	—	424	1	no	—
128	Q₈	—	212	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	53	1	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	424	1	no	—
134	C₂×C₄	—	424	1	no	—
135	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
136	C₈	—	106	1	no	—
137	C₈	#13133195	1	1	yes	characteristic
138	C₈	#13133249	1	1	yes	characteristic
139	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
140	C₄	—	16	1	no	—
141	C₄	—	16	1	no	—
142	C₄	—	53	2	no	—
143	C₄	—	848	1	no	—
144	C₄	—	106	1	no	—
145	C₄	—	424	2	no	—
146	C₄	#16810902	1	1	yes	characteristic
147	C₄	#16811065	1	1	yes	characteristic
148	C₂	#20276851	1	1	yes	characteristic
149	C₂	SD₆₄×D₅₃	1	1	yes	characteristic
150	C₂	#20277031	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1424	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
hypercenter	#1424	D₅₃
hypocenter	C₅₃	#13704
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
23. Dic₄₂₄	Q₈	semidirect
94. C₅₃	#13704	semidirect

Maximal subgroups

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

subgroup	index
#20276989	2
#20276573	2
#20272916	2
#20272910	2
#20272897	2
#20272895	2
C₅₃×#1424	2
#13704	53