Finite Group Databasebeta

Subgroups — #28309956

Order 13568.

258 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 88 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	#20291094	C₂	1	2	yes	maximal, complemented
2	#20289744	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289666	C₂	1	2	yes	maximal, complemented
4	C₈×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273509	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	—	2	2	no	—
7	C₄⋊#11719910	C₂²	1	2	yes	—
8	#16811518	—	2	2	no	—
9	C₄×#11719910	C₂²	1	1	yes	characteristic
10	C₄×#11719910	C₂²	1	1	yes	characteristic
11	C₄×D₄₂₄	C₄	1	1	yes	characteristic
12	#16810688	C₂²	1	2	yes	—
13	#16810350	C₄	1	2	yes	—
14	C₈×#88322	C₂²	1	1	yes	characteristic
15	C₅₃×#335	C₄	1	1	yes	characteristic
16	D₅₃⋊C₁₆	—	4	4	no	—
17	C₂×#11719910	—	2	1	no	—
18	C₂×#11719910	—	2	2	no	—
19	C₂×#11719910	D₄	1	1	yes	characteristic
20	C₂×#11719910	D₄	1	1	yes	characteristic
21	C₂×D₄₂₄	C₈	1	1	yes	characteristic
22	C₂×C₈×D₅₃	—	2	2	no	—
23	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
24	#13133151	C₈	1	1	yes	characteristic
25	#13133139	C₈	1	2	yes	—
26	#13133129	—	2	2	no	—
27	C₄×#88322	C₂×C₄	1	1	yes	characteristic
28	C₄×#88322	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
32	C₂×C₄×D₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	#11719921	—	2	2	no	—
38	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	C₂×D₄	1	1	yes	characteristic
41	C₂×#88322	#41	1	1	yes	characteristic
42	D₄₂₄	C₁₆	1	4	yes	complemented
43	C₈×D₅₃	—	4	2	no	—
44	C₈×D₅₃	—	4	2	no	—
45	#11719910	—	2	2	no	—
46	#11719910	—	2	1	no	—
47	#11719910	—	4	2	no	—
48	#11719910	D₈	1	2	yes	complemented
49	C₂²×D₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	2	no	—
54	D₂₁₂	—	2	2	no	—
55	D₂₁₂	C₂×C₁₆	1	4	yes	complemented
56	C₄×D₅₃	—	4	2	no	—
57	C₄×D₅₃	—	4	2	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	4	2	no	—
60	C₈×C₅₃	C₂×C₁₆	1	2	yes	complemented
61	#88322	—	2	2	no	—
62	#88322	—	2	1	no	—
63	#88322	C₂×D₈	1	1	yes	characteristic
64	#88322	#135	1	1	yes	characteristic
65	C₁₆×D₈	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
67	C₂×D₅₃	—	2	4	no	—
68	C₂×D₅₃	—	4	2	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
72	C₄×C₅₃	#443	1	1	yes	characteristic
73	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₄×C₅₃	#504	1	1	yes	characteristic
75	Dic₅₃	—	4	2	no	—
76	D₄×C₁₆	—	53	2	no	—
77	C₈×D₈	—	53	1	no	—
78	#1370	—	53	1	no	—
79	D₄⋊C₁₆	—	53	2	no	—
80	C₈×C₁₆	—	53	1	no	—
81	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
82	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
83	C₂×C₅₃	#2170	1	1	yes	characteristic
84	D₅₃	—	4	4	no	—
85	C₂²×C₁₆	—	106	2	no	—
86	C₄×D₈	—	53	1	no	—
87	C₈×D₄	—	53	2	no	—
88	C₄⋊C₁₆	—	53	2	no	—
89	C₂²⋊C₁₆	—	106	2	no	—
90	C₄×C₁₆	—	53	1	no	—
91	C₄×C₁₆	—	53	1	no	—
92	#335	#6157	1	1	yes	characteristic
93	D₄⋊C₈	—	53	2	no	—
94	C₈²	—	53	1	no	—
95	C₅₃	C₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
96	C₂×D₈	—	53	1	no	—
97	C₂²×C₈	—	106	2	no	—
98	C₄×D₄	—	53	2	no	—
99	C₂×C₁₆	—	53	1	no	—
100	C₂×C₁₆	—	53	1	no	—
101	C₂×C₁₆	—	106	1	no	—
102	C₂×C₁₆	—	106	2	no	—
103	C₂×C₁₆	—	212	4	no	—
104	#107	—	53	1	no	—
105	C₄⋊C₈	C₂×#6157	1	2	yes	—
106	#102	—	53	2	no	—
107	C₂²⋊C₈	—	106	2	no	—
108	C₄×C₈	—	53	1	no	—
109	C₄×C₈	—	53	1	no	—
110	C₄×C₈	C₂×#6157	1	1	yes	characteristic
111	C₂×D₄	—	53	2	no	—
112	C₂²×C₄	—	106	2	no	—
113	D₈	—	53	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	53	1	no	—
116	C₂×C₈	—	53	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₂×C₈	—	212	2	no	—
119	C₂×C₈	—	212	2	no	—
120	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
121	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
122	C₄⋊C₄	—	53	2	no	—
123	C₂²⋊C₄	—	106	2	no	—
124	C₄²	C₂²×#6157	1	1	yes	characteristic
125	C₁₆	—	53	2	no	—
126	C₁₆	—	106	1	no	—
127	C₁₆	—	106	2	no	—
128	C₁₆	—	212	2	no	—
129	C₂³	—	106	2	no	—
130	D₄	—	53	4	no	—
131	D₄	—	106	2	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	—	212	2	no	—
134	C₂×C₄	—	212	2	no	—
135	C₂×C₄	D₄×#6157	1	1	yes	characteristic
136	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
137	C₂×C₄	#13133305	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	2	no	—
140	C₈	—	53	1	no	—
141	C₈	—	53	1	no	—
142	C₈	—	106	2	no	—
143	C₈	—	106	1	no	—
144	C₈	D₅₃⋊C₁₆	1	2	yes	complemented
145	C₂²	—	106	4	no	—
146	C₂²	—	212	2	no	—
147	C₂²	—	212	2	no	—
148	C₂²	#16811651	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	212	2	no	—
151	C₄	D₈×#6157	1	1	yes	characteristic
152	C₄	#16811667	1	1	yes	characteristic
153	C₄	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
154	C₄	#16811682	1	1	yes	characteristic
155	C₂	—	212	4	no	—
156	C₂	#20290296	1	1	yes	characteristic
157	C₂	#20291094	1	1	yes	characteristic
158	C₂	#20291104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#335	C₄
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₄₂₄	C₈
cosocle	C₄×#88322	C₂³
hypercenter	#335	#6157
hypocenter	C₅₃	C₁₆×D₈
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
1. #20291094	C₂	semidirect
2. #20289744	C₂	semidirect
3. #20289666	C₂	semidirect
4. C₈×#11719910	C₂	semidirect
42. D₄₂₄	C₁₆	semidirect
48. #11719910	D₈	semidirect
55. D₂₁₂	C₂×C₁₆	semidirect
60. C₈×C₅₃	C₂×C₁₆	semidirect
95. C₅₃	C₁₆×D₈	semidirect
144. C₈	D₅₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20291094	2
#20289744	2
#20289666	2
C₈×#11719910	2
#20273509	2
C₁₆×D₈	53