Finite Group Databasebeta

Subgroups — #20276665

Order 6784.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16811423	C₂	1	1	yes	maximal, characteristic
2	#16811064	C₂	1	2	yes	maximal, complemented
3	#16810432	C₂	1	2	yes	maximal, complemented
4	#16810405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	—	2	1	no	—
7	#13133250	—	2	2	no	—
8	#13133245	—	2	2	no	—
9	#13133239	—	2	1	no	—
10	#13133239	C₂²	1	2	yes	—
11	#13133163	—	2	2	no	—
12	#13133160	—	2	2	no	—
13	#13133157	C₂²	1	2	yes	complemented
14	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₅₃	—	2	1	no	—
17	#106×C₅₃	C₂²	1	1	yes	characteristic, complemented
18	#103×C₅₃	—	2	1	no	—
19	C₂×Q₈×C₅₃	—	2	1	no	—
20	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₅₃	—	4	1	no	—
22	#11719945	—	2	2	no	—
23	#11719945	—	2	2	no	—
24	#11719945	—	4	1	no	—
25	#11719945	—	4	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×C₄×D₅₃	—	4	1	no	—
28	C₂×Dic₁₀₆	—	2	1	no	—
29	Q₁₆×C₅₃	—	2	2	no	—
30	M₁₆×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄⋊C₄×C₅₃	—	4	1	no	—
35	C₂²⋊C₄×C₅₃	—	4	1	no	—
36	Dic₂₁₂	—	2	1	no	—
37	Dic₂₁₂	D₄	1	2	yes	—
38	D₄₂₄	—	2	1	no	—
39	D₄₂₄	D₄	1	2	yes	—
40	#11719913	—	2	2	no	—
41	#11719913	—	2	2	no	—
42	#11719908	—	2	2	no	—
43	C₂³×C₅₃	D₈	1	1	yes	characteristic
44	C₂²×D₅₃	—	4	1	no	—
45	Q₈×C₅₃	—	4	1	no	—
46	Q₈×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	—	2	1	no	—
49	C₂×C₄×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₅₃	—	4	2	no	—
54	C₂²⋊D₅₃	—	4	2	no	—
55	C₂×Dic₅₃	—	4	1	no	—
56	D₂₁₂	—	2	2	no	—
57	D₂₁₂	—	4	1	no	—
58	C₄×D₅₃	—	4	2	no	—
59	C₄×D₅₃	—	4	2	no	—
60	Dic₁₀₆	—	2	2	no	—
61	Dic₁₀₆	—	4	1	no	—
62	C₈×C₅₃	—	2	1	no	—
63	C₈×C₅₃	C₂×D₄	1	2	yes	—
64	C₂²×C₅₃	—	2	1	no	—
65	C₂²×C₅₃	—	4	1	no	—
66	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×D₅₃	—	4	2	no	—
68	C₂×D₅₃	—	8	2	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	8	1	no	—
71	C₄×C₅₃	—	8	1	no	—
72	C₄×C₅₃	#120	1	1	yes	characteristic
73	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	Dic₅₃	—	4	2	no	—
75	#2191	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	4	1	no	—
78	C₂×C₅₃	#447	1	1	yes	characteristic
79	D₅₃	—	8	2	no	—
80	#575	—	53	1	no	—
81	#510	—	53	2	no	—
82	#470	D₅₃	1	1	yes	characteristic, complemented
83	#360	—	53	2	no	—
84	#349	—	53	1	no	—
85	C₅₃	#2191	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	106	1	no	—
87	#137	—	106	2	no	—
88	AGL(1,ℤ/8)	—	106	2	no	—
89	#135	—	53	2	no	—
90	#135	—	106	1	no	—
91	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
92	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
93	C₂²⋊Q₈	—	2	1	no	—
94	Q₃₂	—	106	2	no	—
95	SD₃₂	—	106	2	no	—
96	M₃₂	—	53	2	no	—
97	#106	C₂×D₅₃	1	1	yes	characteristic, complemented
98	#103	—	2	1	no	—
99	#41	—	106	2	no	—
100	#41	—	106	2	no	—
101	#41	—	212	1	no	—
102	#41	—	212	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	106	1	no	—
105	C₂×D₄	—	106	1	no	—
106	C₂×D₄	—	212	1	no	—
107	C₂²×C₄	—	212	1	no	—
108	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	53	2	no	—
111	Q₁₆	—	106	1	no	—
112	SD₁₆	—	106	2	no	—
113	SD₁₆	—	106	2	no	—
114	D₈	—	53	2	no	—
115	D₈	—	106	1	no	—
116	M₁₆	—	2	2	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	—	4	1	no	—
121	C₂²⋊C₄	—	4	1	no	—
122	C₁₆	—	106	2	no	—
123	C₂³	—	212	1	no	—
124	C₂³	#11719922	1	1	yes	characteristic
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	106	2	no	—
128	Q₈	—	212	1	no	—
129	D₄	—	106	2	no	—
130	D₄	—	212	1	no	—
131	D₄	—	212	2	no	—
132	D₄	—	212	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	212	2	no	—
138	C₂×C₄	—	212	1	no	—
139	C₂×C₄	—	212	2	no	—
140	C₂×C₄	#11719922	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	D₄×D₅₃	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	424	2	no	—
147	C₂²	—	212	2	no	—
148	C₂²	C₂×#11719922	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	8	1	no	—
151	C₄	—	8	1	no	—
152	C₄	—	212	2	no	—
153	C₄	#13133185	1	1	yes	characteristic
154	C₄	#13133242	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	4	1	no	—
157	C₂	—	424	2	no	—
158	C₂	#16810756	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810756
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#470	C₂
socle	C₂×C₅₃	#447
hypercenter	#470	D₅₃
hypocenter	C₅₃	#2191
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #470

Complements

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

N	H ≅ G/N	type
2. #16811064	C₂	semidirect
3. #16810432	C₂	semidirect
4. #16810405	C₂	semidirect
5. C₅₃×#470	C₂	semidirect
13. #13133157	C₂²	semidirect
14. C₂×Q₁₆×C₅₃	C₂²	semidirect
17. #106×C₅₃	C₂²	semidirect
82. #470	D₅₃	semidirect
85. C₅₃	#2191	semidirect
91. C₂×Q₁₆	C₂×D₅₃	semidirect
97. #106	C₂×D₅₃	semidirect