Finite Group Databasebeta

Subgroups — #28298700

Order 13568.

208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₅₃⋊#364	C₂	1	1	yes	maximal, characteristic
2	#20290655	C₂	1	1	yes	maximal, characteristic
3	#364×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊M₃₂	—	2	1	no	—
5	C₂×#13133288	—	2	1	no	—
6	C₂×#13133287	—	2	1	no	—
7	#16811628	C₄	1	2	yes	—
8	#16811546	C₄	1	2	yes	—
9	#16811545	—	2	1	no	—
10	M₃₂×D₅₃	—	2	1	no	—
11	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	#16810444	C₄	1	1	yes	characteristic
13	#16810443	—	2	1	no	—
14	C₅₃×#364	C₄	1	1	yes	characteristic
15	C₂×#11719936	—	2	2	no	—
16	#13133292	—	4	1	no	—
17	#13133291	—	2	2	no	—
18	D₅₃⋊C₁₆	—	4	1	no	—
19	#13133288	—	4	1	no	—
20	#13133288	—	4	1	no	—
21	#13133287	—	4	1	no	—
22	#13133287	—	4	1	no	—
23	#13133284	—	4	2	no	—
24	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
26	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
27	#13133157	—	2	1	no	—
28	#13133156	—	4	1	no	—
29	C₁₆×D₅₃	—	4	1	no	—
30	C₄×#88322	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	D₄	1	1	yes	characteristic
32	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
33	M₃₂×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₅₃	—	2	1	no	—
36	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×#88324	—	4	2	no	—
38	#11719936	—	4	2	no	—
39	#11719936	—	4	2	no	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₅₃	C₄²	1	1	yes	characteristic
43	C₄×Dic₅₃	—	2	1	no	—
44	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
47	C₈×D₅₃	D₈	1	1	yes	characteristic
48	C₈×D₅₃	D₈	1	1	yes	characteristic
49	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
50	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
51	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
52	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
53	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
54	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
55	#11719910	—	4	2	no	—
56	C₁₆×C₅₃	—	4	1	no	—
57	#11719908	—	4	1	no	—
58	C₂²×D₅₃	#101	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	1	no	—
62	C₂×Dic₅₃	#100	1	1	yes	characteristic
63	C₄×D₅₃	—	4	2	no	—
64	C₄×D₅₃	#102	1	1	yes	characteristic
65	C₄×D₅₃	#102	1	1	yes	characteristic
66	C₄×D₅₃	#103	1	1	yes	characteristic
67	C₄×D₅₃	#103	1	1	yes	characteristic
68	#88324	—	8	2	no	—
69	C₈×C₅₃	#102	1	1	yes	characteristic
70	C₈×C₅₃	#103	1	1	yes	characteristic
71	C₈×C₅₃	#106	1	1	yes	characteristic
72	C₈×C₅₃	#107	1	1	yes	characteristic
73	#88322	#102	1	1	yes	characteristic
74	#88322	#103	1	1	yes	characteristic
75	#88322	#106	1	1	yes	characteristic
76	#88322	#107	1	1	yes	characteristic
77	#7273	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	#343	1	1	yes	characteristic
79	C₂×D₅₃	—	4	1	no	—
80	C₂×D₅₃	#332	1	1	yes	characteristic
81	C₂×D₅₃	#332	1	1	yes	characteristic
82	C₄×C₅₃	—	4	1	no	—
83	C₄×C₅₃	#340	1	1	yes	characteristic
84	C₄×C₅₃	#340	1	1	yes	characteristic
85	Dic₅₃	—	2	2	no	—
86	Dic₅₃	#331	1	1	yes	characteristic
87	Dic₅₃	#333	1	1	yes	characteristic
88	C₂×#364	—	53	1	no	—
89	C₂×#364	—	53	1	no	—
90	#1848	—	53	1	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	#1296	1	1	yes	characteristic
93	D₅₃	—	2	1	no	—
94	D₅₃	—	2	1	no	—
95	C₂×M₃₂	—	106	1	no	—
96	C₂×M₃₂	—	106	1	no	—
97	C₂×#108	—	106	1	no	—
98	C₂×#108	—	106	1	no	—
99	#423	—	53	2	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	#364	—	53	2	no	—
102	#364	—	53	1	no	—
103	#364	—	106	1	no	—
104	#364	—	106	1	no	—
105	#364	#6157	1	1	yes	characteristic
106	C₅₃	#7273	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	106	2	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	—	53	1	no	—
110	C₂×C₄²	—	53	1	no	—
111	M₃₂	—	2	1	no	—
112	M₃₂	—	106	2	no	—
113	M₃₂	—	106	1	no	—
114	M₃₂	—	212	1	no	—
115	M₃₂	—	212	1	no	—
116	C₂×C₁₆	—	212	1	no	—
117	C₂×C₁₆	—	212	1	no	—
118	#108	—	212	1	no	—
119	#108	—	212	1	no	—
120	#108	—	212	1	no	—
121	#108	—	212	1	no	—
122	C₄⋊C₈	—	212	2	no	—
123	C₄×C₈	—	53	1	no	—
124	C₄×C₈	—	53	1	no	—
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	C₂×#6157	1	1	yes	characteristic
127	C₂²×C₄	—	53	1	no	—
128	C₂²×C₄	—	106	1	no	—
129	M₁₆	—	212	2	no	—
130	M₁₆	—	212	2	no	—
131	C₂×C₈	—	53	1	no	—
132	C₂×C₈	—	53	1	no	—
133	C₂×C₈	—	53	1	no	—
134	C₂×C₈	—	53	1	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	1	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	—	53	1	no	—
140	C₂×C₈	—	53	1	no	—
141	C₂×C₈	—	212	2	no	—
142	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
143	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
144	C₄²	—	53	1	no	—
145	C₄²	—	106	1	no	—
146	C₄²	C₄×#6157	1	1	yes	characteristic
147	C₁₆	—	4	1	no	—
148	C₁₆	—	212	2	no	—
149	C₁₆	—	212	1	no	—
150	C₂³	—	53	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	1	no	—
154	C₂×C₄	—	53	1	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	53	1	no	—
157	C₂×C₄	—	106	1	no	—
158	C₂×C₄	—	212	2	no	—
159	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
160	C₈	—	53	1	no	—
161	C₈	—	53	1	no	—
162	C₈	—	53	1	no	—
163	C₈	—	53	1	no	—
164	C₈	—	424	2	no	—
165	C₈	#13133279	1	1	yes	characteristic
166	C₈	#13133280	1	1	yes	characteristic
167	C₈	#13133285	1	1	yes	characteristic
168	C₈	#13133286	1	1	yes	characteristic
169	C₂²	—	53	1	no	—
170	C₂²	—	53	1	no	—
171	C₂²	—	212	1	no	—
172	C₂²	#16811502	1	1	yes	characteristic
173	C₄	—	4	1	no	—
174	C₄	—	53	1	no	—
175	C₄	—	53	1	no	—
176	C₄	—	106	2	no	—
177	C₄	#16811496	1	1	yes	characteristic
178	C₄	#16811497	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	106	1	no	—
181	C₂	—	106	1	no	—
182	C₂	#20289594	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289594
commutator	C₄²×C₅₃	C₄²
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#364	C₄
socle	C₂×C₅₃	#1296
prime radical	C₂²×D₅₃	#101
cosocle	C₄×C₈×D₅₃	C₂²
hypercenter	#364	#6157
hypocenter	C₅₃	#7273
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
106. C₅₃	#7273	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#364	2
#20290655	2
#364×D₅₃	2
#7273	53