Finite Group Databasebeta

Subgroups — #20273620

Order 6784.

162 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11719924	C₂	1	1	yes	maximal, characteristic
3	#16810700	C₂	1	1	yes	maximal, characteristic
4	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810352	C₂	1	1	yes	maximal, characteristic
6	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	C₄	1	1	yes	characteristic
9	C₂×#11719912	—	2	1	no	—
10	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
11	#13133148	C₄	1	1	yes	characteristic
12	C₄⋊#88322	—	2	1	no	—
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	#13133141	C₄	1	1	yes	characteristic
15	#13133137	C₄	1	1	yes	characteristic
16	#13133129	—	2	1	no	—
17	#13133128	—	2	1	no	—
18	C₄×#88322	C₂²	1	1	yes	characteristic
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	—	2	1	no	—
31	C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	#11719924	—	2	2	no	—
33	#11719921	—	2	1	no	—
34	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂×#88322	D₄	1	1	yes	characteristic
39	C₂×#88322	D₄	1	1	yes	characteristic
40	#11719912	—	4	2	no	—
41	C₂²×D₅₃	—	2	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	1	no	—
50	D₂₁₂	—	2	1	no	—
51	D₂₁₂	—	2	1	no	—
52	C₄×D₅₃	—	4	1	no	—
53	C₄×D₅₃	—	4	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	—	4	1	no	—
56	#88322	—	2	1	no	—
57	#88322	—	2	1	no	—
58	#88322	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	M₁₆	1	2	yes	complemented
61	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	4	1	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	—	4	1	no	—
68	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₅₃	#131	1	1	yes	characteristic
70	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₅₃	#137	1	1	yes	characteristic
72	Dic₅₃	—	4	1	no	—
73	#1592	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#436	1	1	yes	characteristic
75	C₂×C₅₃	#440	1	1	yes	characteristic
76	C₂×C₅₃	#444	1	1	yes	characteristic
77	D₅₃	—	8	1	no	—
78	#446	—	53	1	no	—
79	C₄×SD₁₆	—	53	1	no	—
80	#436	—	53	1	no	—
81	#335	—	53	1	no	—
82	Q₈⋊C₈	D₅₃	1	1	yes	characteristic, complemented
83	D₄⋊C₈	—	53	1	no	—
84	#322	—	53	1	no	—
85	C₅₃	#1592	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	53	1	no	—
87	C₂×M₁₆	—	106	1	no	—
88	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
89	C₄×D₄	—	53	1	no	—
90	#106	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	106	1	no	—
93	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
94	#103	—	53	1	no	—
95	#102	—	53	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	#97	—	106	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
102	C₂×D₄	—	53	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	SD₁₆	—	106	2	no	—
105	M₁₆	—	212	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	53	1	no	—
115	C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	106	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₂³	—	106	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	D₄	—	106	1	no	—
123	D₄	—	106	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
131	C₂×C₄	#11719947	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	53	2	no	—
135	C₈	—	106	1	no	—
136	C₈	—	106	1	no	—
137	C₈	—	106	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	212	1	no	—
140	C₂²	—	212	1	no	—
141	C₂²	—	212	1	no	—
142	C₂²	#13133177	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	212	1	no	—
147	C₄	M₁₆×D₅₃	1	1	yes	characteristic
148	C₄	#13133223	1	1	yes	characteristic
149	C₄	#13133244	1	1	yes	characteristic
150	C₄	#13133249	1	1	yes	characteristic
151	C₂	—	424	1	no	—
152	C₂	#16810698	1	1	yes	characteristic
153	C₂	#16810726	1	1	yes	characteristic
154	C₂	#16810742	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×Q₈⋊C₈	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	Q₈⋊C₈	D₅₃
hypocenter	C₅₃	#1592
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
4. #16810375	C₂	semidirect
6. #16810345	C₂	semidirect
7. C₅₃×Q₈⋊C₈	C₂	semidirect
60. #88322	M₁₆	semidirect
82. Q₈⋊C₈	D₅₃	semidirect
85. C₅₃	#1592	semidirect