Finite Group Databasebeta

Subgroups — #28309930

Order 13568.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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	#20290275	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811484	C₂	1	1	yes	maximal, characteristic
3	#20289741	C₂	1	2	yes	maximal, complemented
4	#20289739	C₂	1	2	yes	maximal, complemented
5	#20276515	C₂	1	1	yes	maximal, characteristic
6	#16811644	—	2	1	no	—
7	C₂×C₄⋊#88324	—	2	1	no	—
8	#16811487	—	2	1	no	—
9	#16811484	—	2	1	no	—
10	#16811484	C₂²	1	1	yes	characteristic
11	#16811484	C₂²	1	2	yes	—
12	#16811484	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810441	C₄	1	2	yes	—
15	C₂²⋊#11719908	C₄	1	1	yes	characteristic
16	#16810398	C₂²	1	2	yes	—
17	C₅₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
18	C₂²×#88324	—	4	1	no	—
19	C₄⋊#88324	—	2	2	no	—
20	C₄⋊#88324	—	2	1	no	—
21	C₄⋊#88324	—	2	1	no	—
22	C₄⋊#88324	—	4	1	no	—
23	C₂²⋊#88324	—	4	1	no	—
24	#13133272	—	4	1	no	—
25	C₂²×#88322	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×#11719908	C₂×C₄	1	2	yes	—
28	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₅₃	D₄	1	1	yes	characteristic
30	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
37	C₂×#88324	—	4	1	no	—
38	C₂×#88324	—	4	1	no	—
39	C₂×#88324	—	4	2	no	—
40	C₂×#88324	—	8	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
46	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
47	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×#88322	—	4	1	no	—
49	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
51	C₁₆×C₅₃	—	2	2	no	—
52	#11719908	—	2	2	no	—
53	C₂³×C₅₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×Dic₅₃	—	2	2	no	—
59	C₂×Dic₅₃	—	4	1	no	—
60	C₂×Dic₅₃	#106	1	1	yes	characteristic
61	C₂×Dic₅₃	#106	1	1	yes	characteristic
62	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
64	#88324	—	8	1	no	—
65	#88324	—	8	1	no	—
66	#88324	—	8	2	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	#131	1	2	yes	—
69	#88322	—	2	1	no	—
70	#88322	—	2	1	no	—
71	#12028	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	#365	1	2	yes	—
75	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	#424	1	1	yes	characteristic
78	C₄×C₅₃	#430	1	1	yes	characteristic
79	Dic₅₃	—	2	1	no	—
80	Dic₅₃	—	2	1	no	—
81	Dic₅₃	—	2	1	no	—
82	Dic₅₃	—	2	1	no	—
83	#2163	—	53	1	no	—
84	#1568	—	53	1	no	—
85	C₂×#334	—	53	1	no	—
86	#1368	—	53	2	no	—
87	#1368	—	53	2	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	#1570	1	1	yes	characteristic
90	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
91	C₂×C₅₃	#2155	1	1	yes	characteristic
92	#432	—	106	1	no	—
93	C₂×C₄⋊C₈	—	106	1	no	—
94	C₂×C₄×C₈	—	53	1	no	—
95	C₄⋊C₁₆	—	53	2	no	—
96	C₂²⋊C₁₆	—	53	1	no	—
97	C₂²⋊C₁₆	#6157	1	1	yes	characteristic
98	#346	—	53	2	no	—
99	#335	—	106	1	no	—
100	#334	—	53	2	no	—
101	#334	—	53	1	no	—
102	#334	—	53	1	no	—
103	#334	—	106	1	no	—
104	C₅₃	#12028	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₈	—	53	1	no	—
106	C₂²×C₈	—	212	1	no	—
107	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
108	C₂×C₄²	—	53	1	no	—
109	C₂×C₁₆	—	53	2	no	—
110	C₂×C₁₆	C₂×#6157	1	2	yes	—
111	C₄⋊C₈	—	106	2	no	—
112	C₄⋊C₈	—	106	1	no	—
113	C₄⋊C₈	—	106	1	no	—
114	C₄⋊C₈	—	212	1	no	—
115	C₂²⋊C₈	—	212	1	no	—
116	#97	—	212	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	1	no	—
120	C₄×C₈	—	53	1	no	—
121	C₂²×C₄	—	53	1	no	—
122	C₂²×C₄	—	53	1	no	—
123	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	424	2	no	—
128	C₂×C₈	—	212	1	no	—
129	C₂×C₈	—	212	1	no	—
130	C₂×C₈	—	212	1	no	—
131	C₂×C₈	—	212	2	no	—
132	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
133	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
134	C₄²	—	53	1	no	—
135	C₄²	—	53	1	no	—
136	C₄²	—	53	1	no	—
137	C₄²	—	53	1	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	106	2	no	—
140	C₂³	#13133285	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	106	2	no	—
148	C₂×C₄	—	212	1	no	—
149	C₂×C₄	#13133285	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	106	1	no	—
153	C₈	—	106	1	no	—
154	C₈	—	424	2	no	—
155	C₈	—	424	1	no	—
156	C₈	—	424	1	no	—
157	C₈	#13133304	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#16811547	1	2	yes	—
161	C₂²	C₂×#13133285	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	106	1	no	—
164	C₄	—	106	1	no	—
165	C₄	—	106	1	no	—
166	C₄	—	106	1	no	—
167	C₄	#16811629	1	1	yes	characteristic
168	C₄	#16811638	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#20290282	1	1	yes	characteristic
171	C₂	C₂×#16811547	1	1	yes	characteristic
172	C₂	#20291063	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133285
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×C₂²⋊C₁₆	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂²⋊C₁₆	#6157
hypocenter	C₅₃	#12028
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #20289741	C₂	semidirect
4. #20289739	C₂	semidirect
104. C₅₃	#12028	semidirect

Maximal subgroups

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

subgroup	index
#20290275	2
C₂×#16811484	2
#20289741	2
#20289739	2
#20276515	2
#12028	53