Finite Group Databasebeta

Subgroups — #20286508

Order 6784.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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	#16811429	C₂	1	4	yes	maximal, complemented
2	#16810883	C₂	1	4	yes	maximal, complemented
3	#16810805	C₂	1	2	yes	maximal, complemented
4	#16810790	C₂	1	2	yes	maximal, complemented
5	#16810741	C₂	1	2	yes	maximal, complemented
6	C₅₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
7	#13133265	C₂²	1	2	yes	—
8	#13133262	C₂²	1	2	yes	complemented
9	#13133248	C₂²	1	4	yes	complemented
10	#13133246	—	2	4	no	—
11	#13133244	—	2	4	no	—
12	#13133243	C₂²	1	4	yes	—
13	#13133239	C₂²	1	2	yes	—
14	#13133238	—	2	4	no	—
15	SD₁₆×D₅₃	—	2	4	no	—
16	#13133223	C₂²	1	4	yes	complemented
17	#13133153	C₂²	1	2	yes	complemented
18	#13133145	C₂²	1	4	yes	complemented
19	#13133136	C₂²	1	2	yes	complemented
20	#13133134	C₂²	1	2	yes	complemented
21	C₂×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
22	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
23	#127×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
26	C₂×D₄×C₅₃	—	4	1	no	—
27	C₂×D₄×C₅₃	C₂³	1	2	yes	—
28	C₂×C₂²⋊D₅₃	—	4	2	no	—
29	#11719949	—	2	4	no	—
30	Q₈×D₅₃	—	2	4	no	—
31	#11719947	—	2	4	no	—
32	D₄×D₅₃	—	2	4	no	—
33	#11719945	—	4	2	no	—
34	#11719945	C₂³	1	4	yes	—
35	SD₁₆×C₅₃	—	2	4	no	—
36	SD₁₆×C₅₃	—	2	4	no	—
37	C₂×C₈×C₅₃	C₂³	1	2	yes	—
38	C₄⋊C₄×C₅₃	—	4	1	no	—
39	C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#11719925	—	2	4	no	—
41	#11719924	—	2	4	no	—
42	#11719923	—	2	4	no	—
43	#11719922	—	2	4	no	—
44	#11719917	C₂³	1	4	yes	—
45	Dic₂₁₂	D₄	1	2	yes	complemented
46	D₄₂₄	D₄	1	2	yes	complemented
47	#11719913	D₄	1	4	yes	complemented
48	#11719912	—	2	4	no	—
49	C₈×D₅₃	—	2	4	no	—
50	C₂³×C₅₃	—	2	2	no	—
51	C₂²×D₅₃	—	4	2	no	—
52	Q₈×C₅₃	—	2	4	no	—
53	D₄×C₅₃	—	2	4	no	—
54	D₄×C₅₃	—	8	2	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²⋊D₅₃	—	2	4	no	—
59	C₂²⋊D₅₃	—	4	4	no	—
60	C₂×Dic₅₃	—	4	2	no	—
61	D₂₁₂	—	4	2	no	—
62	D₂₁₂	C₂×D₄	1	4	yes	—
63	C₄×D₅₃	—	2	4	no	—
64	C₄×D₅₃	—	4	4	no	—
65	Dic₁₀₆	—	4	2	no	—
66	Dic₁₀₆	C₂×D₄	1	4	yes	—
67	C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₈×C₅₃	C₂×D₄	1	2	yes	—
69	#88322	—	2	4	no	—
70	C₂²×C₅₃	—	4	2	no	—
71	C₂²×C₅₃	—	8	2	no	—
72	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×D₅₃	—	2	4	no	—
74	C₂×D₅₃	—	16	2	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	4	2	no	—
77	C₄×C₅₃	C₂²×D₄	1	2	yes	—
78	Dic₅₃	—	2	4	no	—
79	#3291	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	8	2	no	—
82	C₂×C₅₃	D₄²	1	1	yes	characteristic
83	D₅₃	—	4	4	no	—
84	#577	—	53	4	no	—
85	#492	D₅₃	1	1	yes	characteristic, complemented
86	#471	—	53	4	no	—
87	#456	—	53	2	no	—
88	#453	—	53	2	no	—
89	#443	—	53	2	no	—
90	C₅₃	#3291	1	1	yes	characteristic, complemented, Sylow
91	2₋¹⁺⁴	—	53	2	no	—
92	H(5,𝔽₂)	—	53	2	no	—
93	#137	—	53	4	no	—
94	#137	—	106	4	no	—
95	AGL(1,ℤ/8)	—	53	4	no	—
96	AGL(1,ℤ/8)	—	106	4	no	—
97	#135	—	53	2	no	—
98	#135	—	106	4	no	—
99	C₂×SD₁₆	—	106	4	no	—
100	C₂×SD₁₆	C₂×D₅₃	1	4	yes	complemented
101	#131	—	53	4	no	—
102	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
103	#127	C₂×D₅₃	1	1	yes	characteristic
104	#108	—	53	2	no	—
105	#104	—	53	4	no	—
106	#101	—	53	2	no	—
107	#100	—	53	2	no	—
108	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
109	#41	—	53	4	no	—
110	#41	—	106	4	no	—
111	#41	—	106	4	no	—
112	#41	—	212	2	no	—
113	C₂×Q₈	—	106	4	no	—
114	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	106	4	no	—
117	C₂×D₄	—	212	2	no	—
118	C₂×D₄	C₂²×D₅₃	1	2	yes	—
119	Q₁₆	—	53	2	no	—
120	Q₁₆	—	106	4	no	—
121	SD₁₆	—	2	4	no	—
122	SD₁₆	—	2	4	no	—
123	SD₁₆	—	53	4	no	—
124	SD₁₆	—	106	4	no	—
125	SD₁₆	—	106	4	no	—
126	D₈	—	53	2	no	—
127	D₈	—	106	4	no	—
128	M₁₆	—	53	4	no	—
129	M₁₆	—	106	4	no	—
130	C₂×C₈	—	106	4	no	—
131	C₂×C₈	C₂²×D₅₃	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	—	212	2	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	53	4	no	—
138	Q₈	—	212	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	8	2	no	—
141	D₄	—	53	4	no	—
142	D₄	—	106	4	no	—
143	D₄	—	212	2	no	—
144	D₄	—	212	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	106	4	no	—
148	C₂×C₄	—	212	4	no	—
149	C₂×C₄	—	212	2	no	—
150	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
151	C₈	—	106	4	no	—
152	C₈	D₄×D₅₃	1	2	yes	—
153	C₈	D₄×D₅₃	1	2	yes	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	848	2	no	—
157	C₂²	—	106	4	no	—
158	C₂²	#13133262	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	106	4	no	—
162	C₄	C₂×D₄×D₅₃	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	212	4	no	—
166	C₂	#16811258	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811258
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂×C₅₃	D₄²
hypercenter	#492	D₅₃
hypocenter	C₅₃	#3291
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #16811429	C₂	semidirect
2. #16810883	C₂	semidirect
3. #16810805	C₂	semidirect
4. #16810790	C₂	semidirect
5. #16810741	C₂	semidirect
6. C₅₃×#492	C₂	semidirect
8. #13133262	C₂²	semidirect
9. #13133248	C₂²	semidirect
16. #13133223	C₂²	semidirect
17. #13133153	C₂²	semidirect
18. #13133145	C₂²	semidirect
19. #13133136	C₂²	semidirect
20. #13133134	C₂²	semidirect
21. C₂×SD₁₆×C₅₃	C₂²	semidirect
22. C₄⋊Q₈×C₅₃	C₂²	semidirect
24. C₄×C₈×C₅₃	C₂²	semidirect
45. Dic₂₁₂	D₄	semidirect
46. D₄₂₄	D₄	semidirect
47. #11719913	D₄	semidirect
85. #492	D₅₃	semidirect
90. C₅₃	#3291	semidirect
100. C₂×SD₁₆	C₂×D₅₃	semidirect
102. C₄⋊Q₈	C₂×D₅₃	semidirect
108. C₄×C₈	C₂×D₅₃	semidirect