Finite Group Databasebeta

Subgroups — #28309833

Order 13568.

234 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 120 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	#20290262	C₂	1	2	yes	maximal
2	#20289740	C₂	1	2	yes	maximal, complemented
3	#20289739	C₂	1	2	yes	maximal, complemented
4	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
5	#16811639	—	2	2	no	—
6	#16811486	—	2	2	no	—
7	#16811484	C₂²	1	4	yes	—
8	C₂²×#11719908	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810398	C₄	1	1	yes	characteristic
11	#16810398	C₄	1	1	yes	characteristic
12	#16810398	C₂²	1	1	yes	characteristic
13	#16810398	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₄⋊#88324	—	2	4	no	—
16	C₂²⋊#88324	—	4	2	no	—
17	C₄×#88324	—	4	2	no	—
18	C₂²×#88322	Q₈	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
22	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₅₃	D₄	1	1	yes	characteristic
25	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
26	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₈	1	4	yes	complemented
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	SD₁₆	1	2	yes	—
33	C₂×#88324	—	4	4	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₅₃	SD₁₆	1	2	yes	—
40	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
44	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
45	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
46	#11719908	—	2	1	no	—
47	#11719908	—	2	1	no	—
48	C₂³×C₅₃	#106	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
50	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₅₃	#108	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×Dic₅₃	—	4	2	no	—
54	C₂×Dic₅₃	#106	1	2	yes	—
55	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
56	#88324	—	8	4	no	—
57	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
58	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	#88322	—	2	1	no	—
61	#88322	—	2	1	no	—
62	#11999	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#334	1	2	yes	—
65	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
66	C₄×C₅₃	#334	1	2	yes	—
67	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
69	Dic₅₃	—	2	2	no	—
70	Dic₅₃	—	2	2	no	—
71	C₂×#346	—	53	1	no	—
72	#1564	—	53	2	no	—
73	#1368	—	53	2	no	—
74	#1368	—	53	2	no	—
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
77	C₂×C₅₃	#2155	1	2	yes	—
78	C₂²×C₁₆	—	53	1	no	—
79	C₂²×C₁₆	#6157	1	1	yes	characteristic
80	#431	—	106	2	no	—
81	C₂×C₄×C₈	—	53	1	no	—
82	#346	—	53	1	no	—
83	#346	—	53	1	no	—
84	#346	—	53	1	no	—
85	#346	—	53	1	no	—
86	#335	—	106	2	no	—
87	#334	—	53	4	no	—
88	C₅₃	#11999	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	53	1	no	—
90	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
91	C₂×C₄²	—	53	1	no	—
92	C₂×C₁₆	—	53	1	no	—
93	C₂×C₁₆	—	53	1	no	—
94	C₂×C₁₆	—	106	2	no	—
95	C₂×C₁₆	#88324	1	4	yes	complemented
96	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
97	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
98	C₄⋊C₈	—	106	4	no	—
99	C₂²⋊C₈	—	212	2	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	53	1	no	—
103	C₄×C₈	—	53	1	no	—
104	C₄×C₈	—	212	2	no	—
105	C₂²×C₄	—	53	2	no	—
106	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	212	4	no	—
111	C₂×C₈	#11719936	1	2	yes	—
112	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
113	C₂×C₈	C₂×#88324	1	2	yes	—
114	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
115	C₄²	—	53	2	no	—
116	C₄²	—	53	1	no	—
117	C₄²	—	53	1	no	—
118	C₁₆	—	106	1	no	—
119	C₁₆	—	106	1	no	—
120	C₁₆	C₂×#88324	1	2	yes	—
121	C₁₆	C₂×#88324	1	2	yes	—
122	C₂³	#13133285	1	1	yes	characteristic
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	212	2	no	—
126	C₂×C₄	C₄⋊#88324	1	2	yes	—
127	C₂×C₄	#13133285	1	1	yes	characteristic
128	C₂×C₄	#13133287	1	2	yes	—
129	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
130	C₈	—	106	1	no	—
131	C₈	—	106	1	no	—
132	C₈	—	424	4	no	—
133	C₈	C₄⋊#88324	1	2	yes	—
134	C₈	C₂²×#88324	1	1	yes	characteristic
135	C₈	C₂×#11719936	1	1	yes	characteristic
136	C₂²	—	2	2	no	—
137	C₂²	#16811484	1	2	yes	—
138	C₂²	C₂×#13133285	1	1	yes	characteristic
139	C₄	—	106	2	no	—
140	C₄	—	106	2	no	—
141	C₄	#16811484	1	2	yes	—
142	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
143	C₄	C₂×#13133287	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₂×#16811484	1	1	yes	characteristic
146	C₂	#20291062	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133285
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#11999
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20289740	C₂	semidirect
3. #20289739	C₂	semidirect
28. C₂×C₁₆×C₅₃	C₈	semidirect
88. C₅₃	#11999	semidirect
95. C₂×C₁₆	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290262	2
#20289740	2
#20289739	2
C₂×#16810398	2
#11999	53