Finite Group Databasebeta

Subgroups — #28299176

Order 13568.

178 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×#16811555	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133267	C₂	1	1	yes	maximal, characteristic
3	#20276314	C₂	1	1	yes	maximal, characteristic
4	#16811555	—	2	1	no	—
5	#16811555	C₄	1	2	yes	complemented
6	C₂×#13133267	—	2	1	no	—
7	C₂×#13133267	—	2	1	no	—
8	C₂×#13133267	—	2	1	no	—
9	C₂×#13133267	C₄	1	2	yes	complemented
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	#16810578	C₄	1	1	yes	characteristic
12	#16810398	—	2	1	no	—
13	C₁₆×Dic₅₃	—	2	1	no	—
14	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
15	#13133267	—	2	2	no	—
16	#13133267	—	2	2	no	—
17	C₂×#11719917	C₈	1	1	yes	characteristic
18	#13133176	C₈	1	1	yes	characteristic
19	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
20	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719908	D₄	1	1	yes	characteristic
22	C₂×#11719908	D₄	1	1	yes	characteristic
23	C₂×#11719908	D₄	1	1	yes	characteristic
24	C₂×#11719908	Q₈	1	1	yes	characteristic
25	#13133127	—	2	1	no	—
26	C₈×Dic₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₅₃	—	2	1	no	—
29	M₃₂×C₅₃	C₈	1	2	yes	—
30	C₂×C₁₆×C₅₃	—	2	1	no	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
38	C₂²⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	C₁₆	1	2	yes	—
40	C₄×Dic₅₃	—	2	1	no	—
41	#11719917	—	2	1	no	—
42	#11719917	C₁₆	1	2	yes	—
43	C₂×#88322	—	2	1	no	—
44	C₁₆×C₅₃	—	2	2	no	—
45	#11719908	C₂²⋊C₄	1	1	yes	characteristic
46	#11719908	C₂²⋊C₄	1	1	yes	characteristic
47	#11719908	C₄⋊C₄	1	1	yes	characteristic
48	#11719908	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
56	C₂×Dic₅₃	—	2	2	no	—
57	C₈×C₅₃	#95	1	1	yes	characteristic
58	C₈×C₅₃	#97	1	1	yes	characteristic
59	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
60	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	#88322	—	2	2	no	—
62	#7470	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#346	1	1	yes	characteristic
65	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
67	C₄×C₅₃	#336	1	1	yes	characteristic
68	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
69	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
70	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
71	Dic₅₃	—	4	2	no	—
72	C₂×M₆₄	—	53	1	no	—
73	C₂²×C₃₂	—	53	1	no	—
74	#2108	—	53	1	no	—
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	#1374	1	1	yes	characteristic
78	C₂×C₅₃	#1401	1	2	yes	—
79	C₂×M₃₂	#6157	1	1	yes	characteristic
80	C₂²×C₁₆	—	53	1	no	—
81	#405	—	53	1	no	—
82	M₆₄	—	53	2	no	—
83	M₆₄	—	106	1	no	—
84	C₂×C₃₂	—	53	2	no	—
85	C₂×C₃₂	—	106	1	no	—
86	C₂×C₃₂	—	106	1	no	—
87	C₂×C₃₂	—	106	1	no	—
88	#346	—	106	1	no	—
89	C₄×C₁₆	—	106	1	no	—
90	C₅₃	#7470	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	53	1	no	—
92	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
93	#117	—	53	1	no	—
94	M₃₂	—	2	1	no	—
95	M₃₂	#88324	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	53	1	no	—
98	C₂×C₁₆	—	53	1	no	—
99	C₂×C₁₆	—	53	1	no	—
100	C₂×C₁₆	—	53	1	no	—
101	C₂×C₁₆	—	53	1	no	—
102	C₂×C₁₆	—	53	1	no	—
103	#97	—	106	1	no	—
104	C₄×C₈	—	106	1	no	—
105	C₃₂	—	106	2	no	—
106	C₃₂	—	106	2	no	—
107	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
108	M₁₆	—	53	2	no	—
109	M₁₆	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	#11719936	1	1	yes	characteristic
112	C₂×C₈	C₄×#6157	1	1	yes	characteristic
113	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
114	C₂×C₈	C₂×#88324	1	1	yes	characteristic
115	C₂×C₈	#11719940	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
117	C₄⋊C₄	—	53	2	no	—
118	C₂²⋊C₄	—	106	1	no	—
119	C₄²	—	106	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	53	1	no	—
122	C₁₆	—	53	1	no	—
123	C₁₆	—	53	1	no	—
124	C₁₆	—	53	1	no	—
125	C₂³	#13133292	1	1	yes	characteristic
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	C₄×#88324	1	1	yes	characteristic
128	C₂×C₄	#13133274	1	1	yes	characteristic
129	C₂×C₄	#13133284	1	1	yes	characteristic
130	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
131	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
132	C₂×C₄	#13133292	1	1	yes	characteristic
133	C₈	—	106	2	no	—
134	C₈	(C₂×#6157)⋊C₄	1	1	yes	characteristic
135	C₈	#13133272	1	1	yes	characteristic
136	C₈	C₂²⋊#88324	1	1	yes	characteristic
137	C₈	C₄⋊#88324	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	#16811512	1	1	yes	characteristic
140	C₂²	#16811518	1	1	yes	characteristic
141	C₂²	#16811544	1	1	yes	characteristic
142	C₄	—	212	2	no	—
143	C₄	#16811489	1	1	yes	characteristic
144	C₄	#16811509	1	1	yes	characteristic
145	C₄	#16811518	1	1	yes	characteristic
146	C₄	#16811544	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#20289754	1	1	yes	characteristic
150	C₂	#20289833	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6157
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂²⋊C₁₆
prime radical	C₂³×C₅₃	M₃₂
cosocle	C₂²×#11719908	C₂²
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	#7470
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
5. #16811555	C₄	semidirect
9. C₂×#13133267	C₄	semidirect
90. C₅₃	#7470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811555	2
C₂²×#13133267	2
#20276314	2
#7470	53