Finite Group Databasebeta

Subgroups — #28310208

Order 13568.

350 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 168 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	#20290938	C₂	1	1	yes	maximal, characteristic
2	#20290936	C₂	1	2	yes	maximal
3	#20290926	C₂	1	1	yes	maximal, characteristic
4	#20290924	C₂	1	2	yes	maximal
5	#20274469	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂×#13133291	C₂²	1	1	yes	characteristic
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	#16811516	C₄	1	4	yes	complemented
10	#16811515	C₄	1	8	yes	complemented
11	#16811512	C₄	1	2	yes	complemented
12	#16811511	C₄	1	4	yes	complemented
13	#16811509	C₄	1	2	yes	complemented
14	C₄×#11719910	C₄	1	4	yes	complemented
15	#16810581	C₂²	1	1	yes	characteristic
16	#16810579	C₂²	1	2	yes	—
17	#16810381	C₄	1	1	yes	characteristic
18	#16810379	C₄	1	2	yes	—
19	C₅₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
20	#13133291	C₂×C₄	1	4	yes	—
21	#13133291	C₂×C₄	1	4	yes	—
22	#13133291	C₂×C₄	1	4	yes	—
23	C₂×#11719910	C₂×C₄	1	2	yes	—
24	C₂×#11719910	C₂×C₄	1	1	yes	characteristic
25	C₂×#11719910	C₂×C₄	1	2	yes	—
26	C₂×#11719910	C₂×C₄	1	1	yes	characteristic
27	C₂×#11719910	C₂×C₄	1	4	yes	—
28	C₂×#11719910	C₂×C₄	1	2	yes	—
29	C₂×#11719917	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂³	1	1	yes	characteristic
31	#13133176	C₂×C₄	1	1	yes	characteristic
32	#13133128	C₂×C₄	1	4	yes	—
33	#13133127	C₂×C₄	1	2	yes	—
34	C₄×#88322	C₂×C₄	1	4	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
37	#117×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₄²	1	2	yes	—
41	C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂²⋊C₄×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	C₄²	1	4	yes	—
44	C₂²⋊Dic₅₃	—	2	1	no	—
45	C₄⋊Dic₅₃	—	2	1	no	—
46	C₄×Dic₅₃	C₄²	1	2	yes	—
47	#11719917	—	2	2	no	—
48	#11719917	—	2	2	no	—
49	C₂×#88322	C₄²	1	4	yes	—
50	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
51	C₂×#88322	C₂²×C₄	1	2	yes	—
52	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
53	C₂×#88322	C₂²×C₄	1	2	yes	—
54	#11719910	C₄²	1	1	yes	characteristic
55	#11719910	C₄²	1	2	yes	—
56	#11719910	C₄²	1	1	yes	characteristic
57	#11719910	C₄²	1	8	yes	—
58	#11719910	C₄²	1	4	yes	—
59	C₂³×C₅₃	#131	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#131	1	2	yes	—
64	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
65	C₂×Dic₅₃	—	2	2	no	—
66	C₈×C₅₃	—	2	2	no	—
67	#88322	—	2	4	no	—
68	#88322	C₂×C₄²	1	1	yes	characteristic
69	#88322	C₂×C₄²	1	1	yes	characteristic
70	#88322	C₂×C₄²	1	2	yes	—
71	#12542	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	#405	1	2	yes	—
74	C₂²×C₅₃	#405	1	1	yes	characteristic
75	C₄×C₅₃	—	4	4	no	—
76	C₄×C₅₃	C₄³	1	1	yes	characteristic
77	C₄×C₅₃	#405	1	1	yes	characteristic
78	C₄×C₅₃	#405	1	2	yes	—
79	Dic₅₃	—	4	2	no	—
80	#2110	—	53	2	no	—
81	#2110	—	53	1	no	—
82	#2108	—	53	2	no	—
83	#2108	—	53	1	no	—
84	#1747	—	53	1	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	#1726	1	1	yes	characteristic
88	C₂×M₃₂	—	53	2	no	—
89	C₂×M₃₂	—	53	1	no	—
90	C₂²×C₁₆	—	53	1	no	—
91	#405	—	53	1	no	—
92	#405	—	53	2	no	—
93	#347	—	53	8	no	—
94	#347	—	53	4	no	—
95	#346	—	53	4	no	—
96	#346	—	53	2	no	—
97	C₄×C₁₆	—	53	4	no	—
98	C₄×C₁₆	—	53	2	no	—
99	#338	—	53	1	no	—
100	H(3,ℤ/4)	—	53	2	no	—
101	H(3,ℤ/4)	#6157	1	1	yes	characteristic
102	C₅₃	#12542	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	53	2	no	—
104	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
105	C₂²×C₈	—	53	1	no	—
106	#117	—	53	1	no	—
107	#117	C₂×#6157	1	2	yes	—
108	M₃₂	—	53	4	no	—
109	M₃₂	—	53	4	no	—
110	M₃₂	—	53	4	no	—
111	C₂×C₁₆	—	53	2	no	—
112	C₂×C₁₆	—	53	1	no	—
113	C₂×C₁₆	—	53	4	no	—
114	C₂×C₁₆	—	53	2	no	—
115	C₂×C₁₆	—	53	2	no	—
116	C₂×C₁₆	—	53	1	no	—
117	#97	—	53	2	no	—
118	#97	—	53	4	no	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	4	no	—
121	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	106	2	no	—
124	M₁₆	—	106	2	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	2	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	2	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	C₄×#6157	1	2	yes	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	106	1	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	106	1	no	—
135	C₄²	—	53	2	no	—
136	C₄²	C₄×#6157	1	4	yes	—
137	C₁₆	—	53	1	no	—
138	C₁₆	—	53	2	no	—
139	C₁₆	—	53	1	no	—
140	C₁₆	—	53	8	no	—
141	C₁₆	—	53	4	no	—
142	C₂³	#13133304	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	106	2	no	—
145	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
147	C₂×C₄	#13133304	1	2	yes	—
148	C₂×C₄	#13133305	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	53	1	no	—
151	C₈	—	53	1	no	—
152	C₈	—	53	2	no	—
153	C₈	—	106	4	no	—
154	C₂²	—	4	1	no	—
155	C₂²	#16811587	1	2	yes	—
156	C₂²	#16811588	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	—	212	2	no	—
159	C₄	C₄²×#6157	1	1	yes	characteristic
160	C₄	#16811587	1	1	yes	characteristic
161	C₄	#16811588	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	2	1	no	—
164	C₂	#20290372	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#6157
commutator	C₄×C₅₃	C₄³
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×H(3,ℤ/4)	C₄
socle	C₂×C₅₃	#1726
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×#88322	C₂³
hypercenter	H(3,ℤ/4)	#6157
hypocenter	C₅₃	#12542
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
9. #16811516	C₄	semidirect
10. #16811515	C₄	semidirect
11. #16811512	C₄	semidirect
12. #16811511	C₄	semidirect
13. #16811509	C₄	semidirect
14. C₄×#11719910	C₄	semidirect
102. C₅₃	#12542	semidirect

Maximal subgroups

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

subgroup	index
#20290938	2
#20290936	2
#20290926	2
#20290924	2
#20274469	2
#12542	53