Finite Group Databasebeta

Subgroups — #27304982

Order 13568.

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

normal subgroups only

This group has 176 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	#20276989	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272917	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272916	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272910	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272893	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1415	C₂	1	1	yes	maximal, characteristic
8	#16811027	—	2	1	no	—
9	D₈×Dic₅₃	—	2	1	no	—
10	C₂×#13133123	C₂²	1	1	yes	characteristic
11	#16810449	C₂²	1	1	yes	characteristic
12	#16810449	C₂²	1	1	yes	characteristic
13	#16810448	—	2	1	no	—
14	#16810424	—	2	1	no	—
15	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#366	C₂²	1	1	yes	characteristic
19	#13133216	—	4	1	no	—
20	D₄×Dic₅₃	—	4	1	no	—
21	C₂×#11719908	D₄	1	1	yes	characteristic
22	#13133151	—	2	1	no	—
23	#13133151	—	2	1	no	—
24	#13133150	—	4	1	no	—
25	#13133138	—	4	1	no	—
26	C₈×Dic₅₃	D₄	1	1	yes	characteristic
27	#13133123	—	2	1	no	—
28	C₂×D₈×C₅₃	—	2	1	no	—
29	D₁₆×C₅₃	—	2	1	no	—
30	D₁₆×C₅₃	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
32	#107×C₅₃	—	2	1	no	—
33	C₃₂×C₅₃	—	2	1	no	—
34	C₂×D₄×C₅₃	—	4	1	no	—
35	C₂²×Dic₅₃	—	8	1	no	—
36	C₂×Dic₁₀₆	—	8	1	no	—
37	D₈×C₅₃	—	2	2	no	—
38	D₈×C₅₃	—	4	1	no	—
39	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₅₃	—	4	1	no	—
41	C₂²⋊Dic₅₃	—	8	1	no	—
42	C₄⋊Dic₅₃	—	4	1	no	—
43	C₄⋊Dic₅₃	—	4	1	no	—
44	#11719919	—	8	1	no	—
45	C₄×Dic₅₃	D₈	1	1	yes	characteristic
46	C₂×#88322	D₈	1	1	yes	characteristic
47	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₅₃	#41	1	1	yes	characteristic
49	#11719908	—	2	1	no	—
50	C₂³×C₅₃	—	8	1	no	—
51	D₄×C₅₃	—	4	2	no	—
52	D₄×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	—	16	2	no	—
58	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
59	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
60	Dic₁₀₆	—	16	2	no	—
61	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₈×C₅₃	#137	1	1	yes	characteristic
63	#88322	—	2	1	no	—
64	#13705	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	8	2	no	—
66	C₂²×C₅₃	—	16	1	no	—
67	C₂²×C₅₃	—	16	1	no	—
68	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₅₃	—	16	1	no	—
70	C₄×C₅₃	#474	1	1	yes	characteristic
71	C₄×C₅₃	#510	1	1	yes	characteristic
72	Dic₅₃	—	2	1	no	—
73	Dic₅₃	—	16	1	no	—
74	Dic₅₃	—	16	1	no	—
75	Dic₅₃	SD₆₄	1	2	yes	—
76	#2252	—	53	1	no	—
77	C₄×D₁₆	—	53	1	no	—
78	#1424	—	53	1	no	—
79	#1424	—	53	1	no	—
80	C₄⋊C₃₂	—	53	1	no	—
81	#1415	—	53	1	no	—
82	#1415	D₅₃	1	1	yes	characteristic
83	C₂×C₅₃	—	16	2	no	—
84	C₂×C₅₃	#2224	1	1	yes	characteristic
85	C₂×C₅₃	C₂×SD₆₄	1	1	yes	characteristic
86	C₂×C₅₃	#2263	1	1	yes	characteristic
87	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
88	#500	—	106	1	no	—
89	C₄×D₈	—	106	1	no	—
90	C₂×C₃₂	—	53	1	no	—
91	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
92	#366	—	53	1	no	—
93	#366	—	53	1	no	—
94	#366	—	106	1	no	—
95	#366	C₂×D₅₃	1	1	yes	characteristic
96	#357	—	106	1	no	—
97	C₄×C₁₆	—	53	1	no	—
98	C₅₃	#13705	1	1	yes	characteristic, complemented, Sylow
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	212	1	no	—
101	C₄×D₄	—	212	1	no	—
102	D₁₆	—	2	1	no	—
103	D₁₆	Dic₁₀₆	1	2	yes	complemented
104	C₂×C₁₆	—	53	1	no	—
105	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
106	#107	—	2	1	no	—
107	#107	—	106	1	no	—
108	#107	—	106	1	no	—
109	#107	—	212	1	no	—
110	#102	—	212	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₃₂	—	2	1	no	—
113	C₃₂	—	106	1	no	—
114	C₂×Q₈	—	424	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂²×C₄	—	424	1	no	—
117	D₈	—	2	2	no	—
118	D₈	—	4	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	424	1	no	—
123	C₄⋊C₄	—	212	1	no	—
124	C₄⋊C₄	—	212	1	no	—
125	C₂²⋊C₄	—	424	1	no	—
126	C₄²	—	53	1	no	—
127	C₁₆	—	106	1	no	—
128	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
129	C₁₆	#11719947	1	1	yes	characteristic
130	C₂³	—	8	1	no	—
131	Q₈	—	848	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	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	—	106	1	no	—
142	C₈	#13133194	1	1	yes	characteristic
143	C₈	#13133246	1	1	yes	characteristic
144	C₂²	—	8	2	no	—
145	C₂²	—	16	1	no	—
146	C₂²	—	16	1	no	—
147	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	53	2	no	—
150	C₄	—	848	1	no	—
151	C₄	—	848	1	no	—
152	C₄	—	106	1	no	—
153	C₄	#16810901	1	1	yes	characteristic
154	C₄	#16811062	1	1	yes	characteristic
155	C₂	—	16	2	no	—
156	C₂	#20276850	1	1	yes	characteristic
157	C₂	SD₆₄×D₅₃	1	1	yes	characteristic
158	C₂	#20277026	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator, cosocle	C₂×C₁₆×C₅₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1415	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂×D₁₆×C₅₃	C₂²
hypercenter	#1415	D₅₃
hypocenter	C₅₃	#13705
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1415

Complements

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

N	H ≅ G/N	type
1. #20276989	C₂	semidirect
3. #20272917	C₂	semidirect
4. #20272916	C₂	semidirect
5. #20272910	C₂	semidirect
30. D₁₆×C₅₃	Q₈	semidirect
98. C₅₃	#13705	semidirect
103. D₁₆	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20276989	2
D₁₆×Dic₅₃	2
#20272917	2
#20272916	2
#20272910	2
#20272893	2
C₅₃×#1415	2
#13705	53