Finite Group Databasebeta

Subgroups — #27304714

Order 13568.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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	#20277027	C₂	1	1	yes	maximal, characteristic
2	#20276601	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276590	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272920	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272906	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272850	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
9	#16810740	—	2	1	no	—
10	#16810705	—	2	1	no	—
11	#16810460	C₄	1	2	yes	complemented
12	#16810458	C₄	1	2	yes	complemented
13	#16810457	C₂²	1	1	yes	characteristic
14	#16810453	C₂²	1	1	yes	characteristic
15	#16810430	—	2	1	no	—
16	#16810423	—	2	1	no	—
17	#16810398	C₂²	1	1	yes	characteristic
18	C₅₃×#508	C₂²	1	1	yes	characteristic
19	C₅₃×M₆₄	C₂²	1	1	yes	characteristic
20	C₅₃×#367	C₂²	1	1	yes	characteristic
21	C₂×D₄₂₄	—	2	1	no	—
22	#13133224	—	4	1	no	—
23	#13133179	—	4	1	no	—
24	D₈₄₈	—	2	1	no	—
25	D₈₄₈	C₂×C₄	1	2	yes	—
26	C₂×#11719908	D₄	1	1	yes	characteristic
27	#13133153	—	2	1	no	—
28	#13133144	—	4	1	no	—
29	#13133137	—	4	1	no	—
30	C₈×Dic₅₃	D₄	1	1	yes	characteristic
31	#13133123	C₂×C₄	1	2	yes	complemented
32	#135×C₅₃	—	2	1	no	—
33	Q₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
34	SD₃₂×C₅₃	—	2	1	no	—
35	D₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
37	#107×C₅₃	—	2	1	no	—
38	C₃₂×C₅₃	—	2	1	no	—
39	#41×C₅₃	—	4	1	no	—
40	C₂×D₂₁₂	—	4	1	no	—
41	C₂×C₄×D₅₃	—	8	1	no	—
42	Q₁₆×C₅₃	—	2	1	no	—
43	SD₁₆×C₅₃	—	4	1	no	—
44	D₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	4	1	no	—
47	#11719921	—	8	1	no	—
48	C₄×Dic₅₃	D₈	1	1	yes	characteristic
49	#11719917	—	4	1	no	—
50	#11719917	—	8	1	no	—
51	C₂×#88322	—	8	1	no	—
52	C₂×#88322	D₈	1	1	yes	characteristic
53	D₄₂₄	—	2	2	no	—
54	D₄₂₄	—	4	1	no	—
55	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₁₆×C₅₃	#41	1	1	yes	characteristic
57	#11719908	—	2	1	no	—
58	C₂²×D₅₃	—	8	1	no	—
59	Q₈×C₅₃	—	4	1	no	—
60	D₄×C₅₃	—	4	1	no	—
61	D₄×C₅₃	—	8	1	no	—
62	C₂×C₄×C₅₃	—	8	1	no	—
63	C₂×C₄×C₅₃	—	8	1	no	—
64	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×Dic₅₃	—	2	1	no	—
66	D₂₁₂	—	4	2	no	—
67	D₂₁₂	—	8	1	no	—
68	C₄×D₅₃	—	16	2	no	—
69	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₈×C₅₃	#135	1	1	yes	characteristic
71	#88322	—	2	1	no	—
72	#88322	—	8	1	no	—
73	#13658	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	8	1	no	—
75	C₂²×C₅₃	#509	1	1	yes	characteristic
76	C₂×D₅₃	—	8	2	no	—
77	C₂×D₅₃	—	16	1	no	—
78	C₂×D₅₃	—	16	1	no	—
79	C₄×C₅₃	—	8	1	no	—
80	C₄×C₅₃	—	16	1	no	—
81	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
82	C₄×C₅₃	#509	1	1	yes	characteristic
83	Dic₅₃	—	4	1	no	—
84	#2263	—	53	1	no	—
85	#2179	—	53	1	no	—
86	#2177	—	53	1	no	—
87	#1426	—	53	1	no	—
88	#1419	—	53	1	no	—
89	#1418	D₅₃	1	1	yes	characteristic, complemented
90	#1398	—	53	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	—	16	1	no	—
93	C₂×C₅₃	#2177	1	1	yes	characteristic
94	D₅₃	—	16	2	no	—
95	#508	C₂×D₅₃	1	1	yes	characteristic
96	C₂×D₁₆	—	53	1	no	—
97	#443	—	106	1	no	—
98	C₄×D₈	—	106	1	no	—
99	SD₆₄	—	53	2	no	—
100	D₃₂	—	53	2	no	—
101	M₆₄	—	53	1	no	—
102	M₆₄	C₂×D₅₃	1	1	yes	characteristic
103	#368	—	53	1	no	—
104	#367	C₂×D₅₃	1	1	yes	characteristic
105	#359	—	106	1	no	—
106	#357	—	106	1	no	—
107	#346	—	53	1	no	—
108	C₅₃	#13658	1	1	yes	characteristic, complemented, Sylow
109	#135	—	2	1	no	—
110	C₂×D₈	—	106	1	no	—
111	#131	—	212	1	no	—
112	C₄×D₄	—	212	1	no	—
113	Q₃₂	C₄×D₅₃	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	53	2	no	—
116	D₁₆	—	106	1	no	—
117	D₁₆	C₄×D₅₃	1	1	yes	characteristic, complemented
118	C₂×C₁₆	—	53	1	no	—
119	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
120	#108	—	106	1	no	—
121	#107	—	2	1	no	—
122	#104	—	212	1	no	—
123	#102	—	212	1	no	—
124	C₄×C₈	—	53	1	no	—
125	C₃₂	—	2	1	no	—
126	C₃₂	—	53	2	no	—
127	#41	—	4	1	no	—
128	C₂×D₄	—	212	1	no	—
129	C₂²×C₄	—	424	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	4	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	106	2	no	—
134	D₈	—	212	1	no	—
135	M₁₆	—	424	1	no	—
136	M₁₆	—	212	1	no	—
137	C₂×C₈	—	53	1	no	—
138	C₂×C₈	—	424	1	no	—
139	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₂²⋊C₄	—	424	1	no	—
142	C₄²	—	53	1	no	—
143	C₁₆	—	106	1	no	—
144	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₁₆	#11719947	1	1	yes	characteristic
146	C₂³	—	424	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	424	1	no	—
151	D₄	—	212	2	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	848	2	no	—
155	C₂×C₄	—	106	1	no	—
156	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
157	C₈	—	106	1	no	—
158	C₈	—	424	1	no	—
159	C₈	#13133177	1	1	yes	characteristic
160	C₈	#13133238	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	848	1	no	—
163	C₂²	—	848	1	no	—
164	C₂²	—	424	2	no	—
165	C₂²	#16811057	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	16	1	no	—
168	C₄	—	212	1	no	—
169	C₄	#16810703	1	1	yes	characteristic
170	C₄	#16811060	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	848	2	no	—
174	C₂	#20276588	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20276588
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1418	C₂
socle	C₂×C₅₃	#2177
prime radical	#20277027	C₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1418	D₅₃
hypocenter	C₅₃	#13658
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
2. #20276601	C₂	semidirect
3. #20276590	C₂	semidirect
4. #20272920	C₂	semidirect
5. #20272906	C₂	semidirect
6. #20272850	C₂	semidirect
7. C₅₃×#1418	C₂	semidirect
11. #16810460	C₄	semidirect
12. #16810458	C₄	semidirect
31. #13133123	C₂×C₄	semidirect
33. Q₃₂×C₅₃	C₂×C₄	semidirect
35. D₁₆×C₅₃	C₂×C₄	semidirect
89. #1418	D₅₃	semidirect
108. C₅₃	#13658	semidirect
113. Q₃₂	C₄×D₅₃	semidirect
117. D₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20277027	2
#20276601	2
#20276590	2
#20272920	2
#20272906	2
#20272850	2
C₅₃×#1418	2
#13658	53