Finite Group Databasebeta

Subgroups — #27295757

Order 13568.

210 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20273478	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	#20272781	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
5	#16810680	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	#16810376	—	2	1	no	—
8	#16810374	—	2	1	no	—
9	#16810374	C₂²	1	2	yes	—
10	#16810374	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
13	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₅₃×#346	C₂²	1	2	yes	—
15	C₂²×#88322	—	4	1	no	—
16	C₄⋊#88322	—	2	2	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	—	4	1	no	—
19	C₂²⋊#88322	—	4	1	no	—
20	#13133128	—	4	1	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	4	1	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
40	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×#88322	—	4	2	no	—
43	C₂×#88322	—	4	2	no	—
44	C₂×#88322	—	8	2	no	—
45	C₁₆×C₅₃	—	2	4	no	—
46	C₂³×C₅₃	#106	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	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	—	2	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	2	1	no	—
60	C₈×C₅₃	#131	1	2	yes	—
61	#88322	—	8	2	no	—
62	#88322	—	8	2	no	—
63	#12028	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	#365	1	2	yes	—
67	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	1	no	—
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	#424	1	1	yes	characteristic
74	C₄×C₅₃	#430	1	1	yes	characteristic
75	#2163	D₅₃	1	1	yes	characteristic
76	#1568	—	53	1	no	—
77	C₂×#334	—	53	1	no	—
78	#1368	—	53	4	no	—
79	C₂×C₅₃	—	2	2	no	—
80	C₂×C₅₃	#1570	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
82	C₂×C₅₃	#2155	1	1	yes	characteristic
83	#432	—	106	1	no	—
84	C₂×C₄⋊C₈	—	106	1	no	—
85	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₁₆	Dic₅₃	1	2	yes	—
87	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
88	#346	C₂×D₅₃	1	2	yes	—
89	#335	—	106	1	no	—
90	#334	—	53	2	no	—
91	#334	—	53	2	no	—
92	#334	—	106	1	no	—
93	C₅₃	#12028	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	212	1	no	—
95	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
96	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
97	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
98	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
99	C₄⋊C₈	—	106	2	no	—
100	C₄⋊C₈	—	106	2	no	—
101	C₄⋊C₈	—	212	1	no	—
102	C₂²⋊C₈	—	212	1	no	—
103	#97	—	212	1	no	—
104	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
105	C₄×C₈	D₂₁₂	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
108	C₂²×C₄	#11719913	1	1	yes	characteristic
109	C₂²×C₄	#11719913	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	424	2	no	—
114	C₂×C₈	—	212	2	no	—
115	C₂×C₈	—	212	2	no	—
116	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
117	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
119	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
120	C₄²	#11719913	1	1	yes	characteristic
121	C₄²	#11719913	1	1	yes	characteristic
122	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
123	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₂³	#13133149	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#13133149	1	1	yes	characteristic
131	C₂×C₄	#13133149	1	1	yes	characteristic
132	C₂×C₄	#13133149	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
134	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
135	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	424	2	no	—
140	C₈	—	424	2	no	—
141	C₈	#13133224	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#16810446	1	2	yes	—
145	C₂²	C₂×#13133149	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#16810650	1	1	yes	characteristic
152	C₄	#16810672	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	#20273486	1	1	yes	characteristic
155	C₂	C₂×#16810446	1	1	yes	characteristic
156	C₂	#20276488	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2163	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#12028
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #20272781	C₂	semidirect
93. C₅₃	#12028	semidirect

Maximal subgroups

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

subgroup	index
#20273478	2
C₂×#16810374	2
#20272781	2
C₅₃×#2163	2
#12028	53