Finite Group Databasebeta

Subgroups — #28311407

Order 13568.

274 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 100 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	#20290656	C₂	1	1	yes	maximal, characteristic
2	#20290655	C₂	1	1	yes	maximal, characteristic
3	#20289775	C₂	1	4	yes	maximal
4	#346×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133288	—	2	1	no	—
6	C₂×#13133288	C₂²	1	2	yes	—
7	C₂×#13133288	C₂²	1	2	yes	—
8	C₂×#13133287	—	2	1	no	—
9	#16811628	—	2	1	no	—
10	#16811627	—	2	1	no	—
11	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
12	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16810399	C₄	1	1	yes	characteristic
14	#16810398	C₄	1	2	yes	—
15	C₅₃×#346	C₄	1	1	yes	characteristic
16	C₂×#11719936	—	2	2	no	—
17	C₂×#11719936	—	2	2	no	—
18	#13133288	—	2	4	no	—
19	#13133288	—	2	4	no	—
20	#13133288	—	4	2	no	—
21	#13133287	—	4	2	no	—
22	#13133284	—	4	2	no	—
23	C₄⋊#88324	—	4	2	no	—
24	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
26	C₄²×D₅₃	D₄	1	1	yes	characteristic
27	C₂×#11719908	C₂×C₄	1	2	yes	—
28	C₁₆×D₅₃	—	2	2	no	—
29	C₁₆×D₅₃	—	2	2	no	—
30	C₄×#88322	Q₈	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
36	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×#88324	—	4	2	no	—
38	C₂×#88324	—	4	2	no	—
39	#11719936	—	4	2	no	—
40	#11719936	—	4	2	no	—
41	#11719936	—	4	2	no	—
42	#11719936	—	4	2	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₅₃	D₈	1	2	yes	—
48	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
49	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
50	C₈×D₅₃	SD₁₆	1	2	yes	—
51	C₈×D₅₃	SD₁₆	1	2	yes	—
52	C₈×D₅₃	#41	1	2	yes	—
53	C₈×D₅₃	#41	1	2	yes	—
54	C₁₆×C₅₃	—	2	2	no	—
55	#11719908	—	2	2	no	—
56	C₂²×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#107	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×Dic₅₃	#107	1	2	yes	—
60	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
61	C₄×D₅₃	—	4	2	no	—
62	C₄×D₅₃	#124	1	1	yes	characteristic
63	C₄×D₅₃	#124	1	1	yes	characteristic
64	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	#88324	—	8	2	no	—
67	#88324	—	8	2	no	—
68	C₈×C₅₃	#106	1	2	yes	—
69	C₈×C₅₃	#117	1	2	yes	—
70	#88322	#106	1	2	yes	—
71	#88322	#125	1	2	yes	—
72	#12892	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
74	C₂×D₅₃	—	2	2	no	—
75	C₂×D₅₃	#487	1	1	yes	characteristic
76	C₂×D₅₃	#487	1	1	yes	characteristic
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	#382	1	1	yes	characteristic
79	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
80	Dic₅₃	—	2	2	no	—
81	Dic₅₃	#486	1	1	yes	characteristic
82	Dic₅₃	#486	1	1	yes	characteristic
83	C₂×#346	—	53	1	no	—
84	#1848	—	53	1	no	—
85	#1848	—	53	1	no	—
86	#1381	—	53	4	no	—
87	C₂×C₅₃	#1834	1	1	yes	characteristic
88	C₂×C₅₃	#2161	1	2	yes	—
89	D₅₃	—	2	1	no	—
90	D₅₃	—	2	1	no	—
91	C₂²×C₁₆	—	53	2	no	—
92	C₂×#108	—	53	2	no	—
93	C₂×#108	—	53	2	no	—
94	C₂×#108	—	106	1	no	—
95	C₂×#108	—	106	1	no	—
96	#423	—	106	1	no	—
97	#423	—	106	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	#346	—	53	2	no	—
100	#346	—	53	1	no	—
101	#346	#6157	1	1	yes	characteristic
102	C₅₃	#12892	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	106	2	no	—
104	C₂×M₁₆	—	106	2	no	—
105	C₂²×C₈	—	53	1	no	—
106	C₂²×C₈	—	53	1	no	—
107	C₂×C₄²	—	53	1	no	—
108	C₂×C₁₆	—	53	2	no	—
109	C₂×C₁₆	—	106	2	no	—
110	C₂×C₁₆	—	106	2	no	—
111	C₂×C₁₆	C₂×#6157	1	2	yes	—
112	#108	—	106	4	no	—
113	#108	—	106	4	no	—
114	#108	—	212	2	no	—
115	#108	—	212	2	no	—
116	C₄⋊C₈	—	212	2	no	—
117	C₄⋊C₈	—	212	2	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	C₂×#6157	1	1	yes	characteristic
122	C₂²×C₄	—	53	1	no	—
123	C₂²×C₄	—	53	2	no	—
124	M₁₆	—	212	2	no	—
125	M₁₆	—	212	2	no	—
126	M₁₆	—	212	2	no	—
127	M₁₆	—	212	2	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	—	53	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	212	2	no	—
135	C₂×C₈	—	212	2	no	—
136	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
137	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
138	C₄²	—	53	2	no	—
139	C₄²	—	53	1	no	—
140	C₄²	C₄⋊#6157	1	1	yes	characteristic
141	C₁₆	—	2	2	no	—
142	C₁₆	—	106	2	no	—
143	C₂³	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	2	no	—
150	C₂×C₄	—	212	2	no	—
151	C₂×C₄	#13133286	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
153	C₈	—	53	2	no	—
154	C₈	—	53	2	no	—
155	C₈	—	424	2	no	—
156	C₈	—	424	2	no	—
157	C₈	#13133285	1	2	yes	—
158	C₈	#13133296	1	2	yes	—
159	C₂²	—	53	1	no	—
160	C₂²	—	53	1	no	—
161	C₂²	—	106	2	no	—
162	C₂²	C₂×#13133286	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	53	1	no	—
165	C₄	—	53	1	no	—
166	C₄	—	106	2	no	—
167	C₄	#16811567	1	1	yes	characteristic
168	C₄	C₂×#13133285	1	1	yes	characteristic
169	C₂	—	106	1	no	—
170	C₂	—	106	1	no	—
171	C₂	#20290636	1	1	yes	characteristic
172	C₂	#20291071	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133286
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#346	C₄
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂²×D₅₃	C₂×Q₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#12892
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
102. C₅₃	#12892	semidirect

Maximal subgroups

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

subgroup	index
#20290656	2
#20290655	2
#20289775	2
#346×D₅₃	2
#12892	53