Finite Group Databasebeta

Subgroups — #27304596

Order 13568.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#16810456	C₂	1	1	yes	maximal, characteristic
2	#20276496	C₂	1	1	yes	maximal, characteristic
3	#366×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272917	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272916	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272914	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
9	#16810660	—	2	1	no	—
10	#107×D₅₃	—	2	1	no	—
11	#16810456	C₄	1	4	yes	complemented
12	C₂×#13133123	C₂²	1	1	yes	characteristic
13	#16810450	—	2	1	no	—
14	#16810449	C₂²	1	1	yes	characteristic
15	#16810449	C₂²	1	1	yes	characteristic
16	#16810448	—	2	1	no	—
17	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#366	C₂²	1	1	yes	characteristic
19	C₅₃×#366	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
21	#13133175	—	4	1	no	—
22	C₄⋊C₄×D₅₃	—	4	1	no	—
23	C₂×#11719908	D₄	1	1	yes	characteristic
24	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
25	C₁₆×D₅₃	Q₈	1	2	yes	—
26	#13133151	—	2	1	no	—
27	#13133151	—	2	1	no	—
28	#13133150	—	4	1	no	—
29	#13133148	—	4	1	no	—
30	#13133123	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
32	#107×C₅₃	—	2	1	no	—
33	#107×C₅₃	—	2	1	no	—
34	C₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₅₃	—	8	1	no	—
36	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₄⋊C₄×C₅₃	—	4	1	no	—
40	#11719921	—	8	1	no	—
41	C₄⋊Dic₅₃	—	4	1	no	—
42	C₄⋊Dic₅₃	—	4	1	no	—
43	#11719919	—	8	1	no	—
44	C₄×Dic₅₃	—	8	1	no	—
45	C₂×#88322	D₈	1	1	yes	characteristic
46	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
47	C₈×D₅₃	Q₁₆	1	2	yes	—
48	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	#11719908	C₄⋊C₄	1	2	yes	—
51	C₂²×D₅₃	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	8	1	no	—
53	C₂×C₄×C₅₃	—	8	1	no	—
54	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₅₃	—	8	1	no	—
56	C₂×Dic₅₃	—	8	1	no	—
57	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
58	C₄×D₅₃	—	16	2	no	—
59	C₄×D₅₃	#107	1	2	yes	—
60	C₄×D₅₃	Q₃₂	1	2	yes	—
61	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
63	#88322	#107	1	2	yes	—
64	#13634	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
66	C₂×D₅₃	—	2	1	no	—
67	C₂×D₅₃	—	2	1	no	—
68	C₂×D₅₃	#366	1	2	yes	—
69	C₄×C₅₃	—	16	1	no	—
70	C₄×C₅₃	—	16	1	no	—
71	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
72	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
73	Dic₅₃	—	16	1	no	—
74	Dic₅₃	—	16	1	no	—
75	Dic₅₃	#366	1	2	yes	—
76	C₂×M₆₄	—	53	1	no	—
77	#2158	—	53	1	no	—
78	C₂×#366	—	53	1	no	—
79	#1424	—	53	1	no	—
80	#1424	—	53	1	no	—
81	#1423	—	53	1	no	—
82	#1423	D₅₃	1	1	yes	characteristic, complemented
83	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
84	C₂×C₅₃	#2263	1	1	yes	characteristic
85	C₂×C₅₃	#2264	1	1	yes	characteristic
86	D₅₃	—	2	2	no	—
87	C₂²×C₁₆	—	53	1	no	—
88	#427	—	106	1	no	—
89	C₂×#107	—	106	1	no	—
90	M₆₄	—	53	4	no	—
91	C₂×C₃₂	—	53	1	no	—
92	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
93	#367	—	106	1	no	—
94	#366	—	53	1	no	—
95	#366	—	53	1	no	—
96	#366	—	106	1	no	—
97	#366	C₂×D₅₃	1	1	yes	characteristic
98	#366	C₂×D₅₃	1	1	yes	characteristic
99	C₅₃	#13634	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	53	1	no	—
101	#117	—	212	1	no	—
102	C₂×C₄⋊C₄	—	212	1	no	—
103	C₂×C₁₆	—	53	1	no	—
104	C₂×C₁₆	—	53	2	no	—
105	C₂×C₁₆	—	53	2	no	—
106	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	106	1	no	—
110	#107	—	106	1	no	—
111	#107	—	212	1	no	—
112	#106	—	212	1	no	—
113	C₃₂	—	53	2	no	—
114	C₃₂	C₄×D₅₃	1	2	yes	complemented
115	C₂²×C₄	—	53	1	no	—
116	C₂²×C₄	—	424	1	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	424	1	no	—
124	C₄⋊C₄	—	212	1	no	—
125	C₄⋊C₄	—	212	1	no	—
126	C₂²⋊C₄	—	424	1	no	—
127	C₄²	—	424	1	no	—
128	C₁₆	—	53	2	no	—
129	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₁₆	Q₈×D₅₃	1	1	yes	characteristic
131	C₂³	—	53	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	2	no	—
136	C₂×C₄	—	53	2	no	—
137	C₂×C₄	—	848	2	no	—
138	C₂×C₄	—	424	1	no	—
139	C₂×C₄	—	424	1	no	—
140	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
141	C₈	—	53	2	no	—
142	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
143	C₈	Q₁₆×D₅₃	1	1	yes	characteristic
144	C₂²	—	53	2	no	—
145	C₂²	—	106	1	no	—
146	C₂²	—	106	1	no	—
147	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	53	2	no	—
151	C₄	—	848	1	no	—
152	C₄	—	848	1	no	—
153	C₄	#107×D₅₃	1	1	yes	characteristic
154	C₄	Q₃₂×D₅₃	1	1	yes	characteristic
155	C₂	—	106	2	no	—
156	C₂	#366×D₅₃	1	1	yes	characteristic
157	C₂	#20277026	1	1	yes	characteristic
158	C₂	#20277034	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1423	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂²×D₅₃	D₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1423	D₅₃
hypocenter	C₅₃	#13634
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

Complements

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

N	H ≅ G/N	type
4. #20272917	C₂	semidirect
5. #20272916	C₂	semidirect
6. #20272914	C₂	semidirect
7. C₅₃×#1423	C₂	semidirect
11. #16810456	C₄	semidirect
30. #13133123	C₂×C₄	semidirect
34. C₃₂×C₅₃	C₂×C₄	semidirect
82. #1423	D₅₃	semidirect
99. C₅₃	#13634	semidirect
114. C₃₂	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810456	2
#20276496	2
#366×D₅₃	2
#20272917	2
#20272916	2
#20272914	2
C₅₃×#1423	2
#13634	53