Finite Group Databasebeta

Subgroups — #20274245

Order 6784.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16810910	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810815	C₂	1	1	yes	maximal, characteristic
3	#16810702	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719922	—	2	1	no	—
9	C₂×#11719912	—	2	1	no	—
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133148	—	2	1	no	—
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	#13133138	—	2	1	no	—
16	#13133137	—	2	1	no	—
17	#13133129	—	2	1	no	—
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	#127×C₅₃	C₂²	1	1	yes	characteristic
20	#102×C₅₃	—	2	1	no	—
21	#102×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₂₁₂	D₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#11719922	—	4	2	no	—
35	#11719921	—	2	1	no	—
36	C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	#11719912	—	4	2	no	—
41	C₂³×C₅₃	—	4	1	no	—
42	C₂²×D₅₃	—	2	1	no	—
43	Q₈×C₅₃	—	8	1	no	—
44	D₄×C₅₃	—	4	1	no	—
45	D₄×C₅₃	—	4	1	no	—
46	D₄×C₅₃	—	4	1	no	—
47	D₄×C₅₃	—	4	1	no	—
48	D₄×C₅₃	—	8	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×Dic₅₃	—	2	1	no	—
54	D₂₁₂	—	2	1	no	—
55	D₂₁₂	—	2	1	no	—
56	C₄×D₅₃	—	4	1	no	—
57	C₄×D₅₃	—	4	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	2	1	no	—
60	#88322	—	4	1	no	—
61	#88322	—	4	1	no	—
62	#88322	—	4	1	no	—
63	C₂²×C₅₃	—	8	1	no	—
64	C₂²×C₅₃	—	8	1	no	—
65	C₂²×C₅₃	—	8	1	no	—
66	C₂²×C₅₃	#121	1	1	yes	characteristic
67	C₂×D₅₃	—	4	1	no	—
68	C₂×D₅₃	—	4	1	no	—
69	C₂×D₅₃	—	4	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	8	1	no	—
72	C₄×C₅₃	#135	1	1	yes	characteristic
73	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₅₃	#137	1	1	yes	characteristic
76	Dic₅₃	—	4	1	no	—
77	#1697	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	16	1	no	—
79	C₂×C₅₃	#463	1	1	yes	characteristic
80	C₂×C₅₃	#468	1	1	yes	characteristic
81	C₂×C₅₃	#472	1	1	yes	characteristic
82	D₅₃	—	8	1	no	—
83	#486	D₅₃	1	1	yes	characteristic, complemented
84	#476	—	53	1	no	—
85	#459	—	53	1	no	—
86	#436	—	53	1	no	—
87	#335	—	53	1	no	—
88	#333	—	53	1	no	—
89	#331	—	53	1	no	—
90	C₅₃	#1697	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	106	1	no	—
92	C₂×M₁₆	—	106	1	no	—
93	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
94	#127	C₂×D₅₃	1	1	yes	characteristic
95	C₄×D₄	—	53	1	no	—
96	#106	—	106	1	no	—
97	C₄⋊C₈	—	53	1	no	—
98	C₄⋊C₈	—	53	1	no	—
99	C₄⋊C₈	—	53	1	no	—
100	#102	—	2	1	no	—
101	#102	—	2	1	no	—
102	#102	—	106	1	no	—
103	#102	—	106	1	no	—
104	C₂²⋊C₈	—	106	1	no	—
105	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	53	1	no	—
111	C₂²×C₄	—	106	1	no	—
112	D₈	—	212	2	no	—
113	M₁₆	—	212	2	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	53	1	no	—
122	C₂²⋊C₄	—	106	1	no	—
123	C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₂³	—	4	1	no	—
125	C₂³	—	106	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	106	1	no	—
133	D₄	—	106	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	106	1	no	—
136	C₂×C₄	—	212	1	no	—
137	C₂×C₄	—	212	1	no	—
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	#11719947	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	212	1	no	—
144	C₈	—	212	1	no	—
145	C₈	—	212	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	212	1	no	—
150	C₂²	—	212	1	no	—
151	C₂²	—	212	1	no	—
152	C₂²	#13133188	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	212	1	no	—
156	C₄	#13133237	1	1	yes	characteristic
157	C₄	#13133242	1	1	yes	characteristic
158	C₄	#13133244	1	1	yes	characteristic
159	C₄	#13133246	1	1	yes	characteristic
160	C₂	—	16	1	no	—
161	C₂	—	424	1	no	—
162	C₂	#16810835	1	1	yes	characteristic
163	C₂	#16810866	1	1	yes	characteristic
164	C₂	#16810890	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133188
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#486	C₂
socle	C₂²×C₅₃	#121
hypercenter	#486	D₅₃
hypocenter	C₅₃	#1697
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

Complements

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

N	H ≅ G/N	type
1. #16810910	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810376	C₂	semidirect
5. #16810372	C₂	semidirect
6. #16810367	C₂	semidirect
7. C₅₃×#486	C₂	semidirect
83. #486	D₅₃	semidirect
90. C₅₃	#1697	semidirect