Finite Group Databasebeta

Subgroups — #27298484

Order 13568.

172 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 55 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	#20276991	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276892	C₂	1	1	yes	maximal, characteristic
3	#20276888	C₂	1	1	yes	maximal, characteristic
4	#20276551	C₂	1	1	yes	maximal, characteristic
5	#20272790	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272777	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1366	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811041	—	2	1	no	—
9	#16811025	C₂²	1	1	yes	characteristic
10	#16810741	C₂²	1	1	yes	characteristic
11	#16810450	—	2	1	no	—
12	#16810448	—	2	1	no	—
13	#16810446	—	2	1	no	—
14	#16810445	—	2	1	no	—
15	#16810444	C₂²	1	1	yes	characteristic
16	#16810434	—	2	1	no	—
17	#16810432	—	2	1	no	—
18	#16810408	—	2	1	no	—
19	C₄×#11719908	C₂²	1	1	yes	characteristic
20	C₅₃×#499	C₂²	1	1	yes	characteristic
21	C₅₃×#364	C₂²	1	1	yes	characteristic
22	C₅₃×#364	C₂²	1	1	yes	characteristic
23	#13133239	D₄	1	1	yes	characteristic
24	#13133223	—	2	1	no	—
25	#13133209	—	2	1	no	—
26	#13133157	—	2	1	no	—
27	#13133156	—	4	1	no	—
28	C₂×#11719908	D₄	1	1	yes	characteristic
29	C₂×#11719908	D₄	1	1	yes	characteristic
30	C₁₆×D₅₃	—	4	1	no	—
31	#13133153	D₄	1	1	yes	characteristic
32	#13133151	—	2	1	no	—
33	#13133149	—	2	1	no	—
34	#13133145	—	2	1	no	—
35	#125×C₅₃	—	2	1	no	—
36	M₃₂×C₅₃	—	2	1	no	—
37	M₃₂×C₅₃	—	2	1	no	—
38	#107×C₅₃	—	2	1	no	—
39	#106×C₅₃	—	2	1	no	—
40	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#11719945	—	2	1	no	—
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	4	1	no	—
45	C₄⋊C₄×C₅₃	—	8	1	no	—
46	C₄²×C₅₃	#41	1	1	yes	characteristic
47	C₄⋊Dic₅₃	—	4	1	no	—
48	#11719919	—	8	1	no	—
49	#11719917	—	2	1	no	—
50	Dic₂₁₂	Q₁₆	1	1	yes	characteristic
51	D₄₂₄	Q₁₆	1	1	yes	characteristic
52	#11719913	—	2	1	no	—
53	#11719912	—	4	1	no	—
54	C₈×D₅₃	—	4	1	no	—
55	C₁₆×C₅₃	—	4	1	no	—
56	C₁₆×C₅₃	—	4	1	no	—
57	#11719908	—	2	1	no	—
58	#11719908	—	4	1	no	—
59	#11719908	SD₁₆	1	1	yes	characteristic
60	#11719908	SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	8	1	no	—
63	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
64	C₂²⋊D₅₃	—	4	1	no	—
65	C₂×Dic₅₃	—	8	1	no	—
66	D₂₁₂	—	2	1	no	—
67	C₄×D₅₃	—	4	1	no	—
68	Dic₁₀₆	—	2	1	no	—
69	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₈×C₅₃	#135	1	1	yes	characteristic
72	C₈×C₅₃	#137	1	1	yes	characteristic
73	#88322	—	4	1	no	—
74	#12463	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	#472	1	1	yes	characteristic
76	C₂×D₅₃	—	4	1	no	—
77	C₄×C₅₃	—	4	1	no	—
78	C₄×C₅₃	—	16	1	no	—
79	C₄×C₅₃	#461	1	1	yes	characteristic
80	C₄×C₅₃	#467	1	1	yes	characteristic
81	Dic₅₃	—	4	1	no	—
82	Dic₅₃	—	16	1	no	—
83	#2253	—	53	1	no	—
84	#2231	—	53	1	no	—
85	#2231	—	53	1	no	—
86	#2170	—	53	1	no	—
87	#1373	—	53	1	no	—
88	#1366	—	53	1	no	—
89	#1366	D₅₃	1	1	yes	characteristic, complemented
90	C₂×C₅₃	—	2	1	no	—
91	C₂×C₅₃	#1675	1	1	yes	characteristic
92	D₅₃	—	8	1	no	—
93	#504	—	106	1	no	—
94	#499	—	53	1	no	—
95	#499	C₂×D₅₃	1	1	yes	characteristic
96	#443	—	53	1	no	—
97	#367	—	106	1	no	—
98	#366	—	106	1	no	—
99	#365	—	106	1	no	—
100	#365	—	106	1	no	—
101	#364	—	53	1	no	—
102	#364	C₂×D₅₃	1	1	yes	characteristic
103	#364	C₂×D₅₃	1	1	yes	characteristic
104	#360	—	106	1	no	—
105	#360	—	106	1	no	—
106	#350	—	106	1	no	—
107	C₄×C₁₆	—	53	1	no	—
108	C₅₃	#12463	1	1	yes	characteristic, complemented, Sylow
109	#135	—	53	1	no	—
110	#131	—	106	1	no	—
111	#125	—	2	1	no	—
112	#125	—	106	1	no	—
113	M₃₂	—	2	1	no	—
114	M₃₂	—	2	1	no	—
115	M₃₂	—	106	1	no	—
116	M₃₂	—	212	1	no	—
117	C₂×C₁₆	—	53	1	no	—
118	C₂×C₁₆	—	53	1	no	—
119	C₂×C₁₆	—	212	1	no	—
120	#108	—	53	1	no	—
121	#107	—	2	1	no	—
122	#107	—	106	1	no	—
123	#106	—	2	1	no	—
124	#106	—	106	1	no	—
125	#104	—	106	1	no	—
126	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	#41	—	106	1	no	—
128	Q₁₆	—	53	1	no	—
129	SD₁₆	—	106	1	no	—
130	D₈	—	53	1	no	—
131	M₁₆	—	106	1	no	—
132	M₁₆	—	212	1	no	—
133	C₂×C₈	—	212	1	no	—
134	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
135	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	8	1	no	—
138	C₄⋊C₄	—	424	1	no	—
139	C₄⋊C₄	—	212	1	no	—
140	C₄²	#11719945	1	1	yes	characteristic
141	C₁₆	—	4	1	no	—
142	C₁₆	—	4	1	no	—
143	C₁₆	—	53	1	no	—
144	C₁₆	—	53	1	no	—
145	C₁₆	—	106	1	no	—
146	C₁₆	—	212	1	no	—
147	Q₈	—	106	1	no	—
148	D₄	—	106	1	no	—
149	D₄	—	212	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	424	1	no	—
153	C₂×C₄	—	212	1	no	—
154	C₂×C₄	#13133190	1	1	yes	characteristic
155	C₈	—	212	1	no	—
156	C₈	SD₁₆×D₅₃	1	1	yes	characteristic
157	C₈	Q₁₆×D₅₃	1	1	yes	characteristic
158	C₈	#13133238	1	1	yes	characteristic
159	C₈	#13133249	1	1	yes	characteristic
160	C₂²	—	212	1	no	—
161	C₂²	#16810892	1	1	yes	characteristic
162	C₄	—	4	1	no	—
163	C₄	—	16	1	no	—
164	C₄	—	848	1	no	—
165	C₄	—	212	1	no	—
166	C₄	#16810827	1	1	yes	characteristic
167	C₄	#16810863	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	424	1	no	—
170	C₂	#20274113	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20274113
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1366	C₂
socle	C₂×C₅₃	#1675
prime radical	#13133239	D₄
hypercenter	#1366	D₅₃
hypocenter	C₅₃	#12463
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1366

Complements

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

N	H ≅ G/N	type
1. #20276991	C₂	semidirect
5. #20272790	C₂	semidirect
6. #20272777	C₂	semidirect
7. C₅₃×#1366	C₂	semidirect
89. #1366	D₅₃	semidirect
108. C₅₃	#12463	semidirect

Maximal subgroups

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

subgroup	index
#20276991	2
#20276892	2
#20276888	2
#20276551	2
#20272790	2
#20272777	2
C₅₃×#1366	2
#12463	53