Finite Group Databasebeta

Subgroups — #20276789

Order 6784.

170 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16811058	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810998	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810740	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810443	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810437	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810436	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#509	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
9	C₂×#11719922	—	2	1	no	—
10	#13133224	—	2	1	no	—
11	#13133213	—	2	1	no	—
12	#13133160	—	2	1	no	—
13	#13133158	—	2	1	no	—
14	#13133157	C₂²	1	1	yes	characteristic
15	#13133153	C₂²	1	1	yes	characteristic
16	#13133144	—	2	1	no	—
17	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#135×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
20	SD₃₂×C₅₃	—	2	1	no	—
21	D₁₆×C₅₃	—	2	1	no	—
22	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
23	#41×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×C₂²⋊D₅₃	—	4	2	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	Q₁₆×C₅₃	D₄	1	1	yes	characteristic, complemented
28	SD₁₆×C₅₃	—	2	1	no	—
29	D₈×C₅₃	—	2	1	no	—
30	D₈×C₅₃	—	2	1	no	—
31	D₈×C₅₃	D₄	1	1	yes	characteristic, complemented
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	#11719922	—	4	1	no	—
34	#11719922	—	4	1	no	—
35	C₄×Dic₅₃	D₄	1	1	yes	characteristic
36	#11719917	—	2	1	no	—
37	#11719917	—	4	1	no	—
38	C₂×#88322	—	4	1	no	—
39	C₂×#88322	D₄	1	1	yes	characteristic
40	D₄₂₄	—	2	1	no	—
41	D₄₂₄	—	2	1	no	—
42	C₁₆×C₅₃	—	2	1	no	—
43	#11719908	—	2	1	no	—
44	C₂³×C₅₃	—	4	1	no	—
45	C₂²×D₅₃	—	4	1	no	—
46	Q₈×C₅₃	—	2	1	no	—
47	D₄×C₅₃	—	2	1	no	—
48	D₄×C₅₃	—	4	1	no	—
49	D₄×C₅₃	—	4	1	no	—
50	D₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₅₃	—	8	4	no	—
54	C₂×Dic₅₃	—	2	1	no	—
55	D₂₁₂	—	4	1	no	—
56	D₂₁₂	—	4	1	no	—
57	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₈×C₅₃	#41	1	1	yes	characteristic
59	#88322	—	4	1	no	—
60	#88322	D₈	1	1	yes	characteristic
61	#88322	D₈	1	1	yes	characteristic
62	C₂²×C₅₃	—	4	1	no	—
63	C₂²×C₅₃	—	8	2	no	—
64	C₂²×C₅₃	—	8	1	no	—
65	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×D₅₃	—	8	2	no	—
67	C₂×D₅₃	—	8	1	no	—
68	C₄×C₅₃	—	4	1	no	—
69	C₄×C₅₃	#121	1	1	yes	characteristic
70	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
71	Dic₅₃	—	2	2	no	—
72	#2213	—	53	1	no	maximal, Sylow
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	—	8	1	no	—
75	C₂×C₅₃	—	16	1	no	—
76	C₂×C₅₃	#459	1	1	yes	characteristic
77	D₅₃	—	16	1	no	—
78	#509	—	53	1	no	—
79	#509	D₅₃	1	1	yes	characteristic, complemented
80	#493	—	53	1	no	—
81	#443	—	53	1	no	—
82	#364	—	53	1	no	—
83	#361	—	53	1	no	—
84	#361	—	53	1	no	—
85	C₅₃	#2213	1	1	yes	characteristic, complemented, Sylow
86	#135	C₂×D₅₃	1	1	yes	characteristic
87	C₂×D₈	—	53	1	no	—
88	C₂×D₈	—	106	1	no	—
89	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
90	#131	—	106	1	no	—
91	#127	—	106	1	no	—
92	SD₃₂	—	2	1	no	—
93	SD₃₂	—	106	1	no	—
94	D₁₆	—	2	1	no	—
95	D₁₆	—	106	1	no	—
96	M₃₂	—	53	1	no	—
97	M₃₂	C₂×D₅₃	1	1	yes	characteristic
98	#108	—	53	1	no	—
99	#104	—	106	1	no	—
100	C₄×C₈	—	53	1	no	—
101	#41	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	106	1	no	—
104	C₂×D₄	—	212	2	no	—
105	Q₁₆	C₂²⋊D₅₃	1	1	yes	characteristic, complemented
106	SD₁₆	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	106	1	no	—
110	D₈	—	106	1	no	—
111	D₈	—	212	1	no	—
112	D₈	—	212	1	no	—
113	D₈	C₂²⋊D₅₃	1	1	yes	characteristic, complemented
114	M₁₆	—	106	1	no	—
115	M₁₆	—	212	1	no	—
116	C₂×C₈	—	53	1	no	—
117	C₂×C₈	—	212	1	no	—
118	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
119	C₄²	—	53	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	106	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	212	1	no	—
124	Q₈	—	2	1	no	—
125	D₄	—	2	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	424	4	no	—
130	D₄	—	212	1	no	—
131	D₄	—	212	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	106	1	no	—
134	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
135	C₈	—	53	1	no	—
136	C₈	—	53	1	no	—
137	C₈	—	212	1	no	—
138	C₈	#11719947	1	1	yes	characteristic
139	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	424	2	no	—
144	C₂²	—	424	1	no	—
145	C₂²	#13133241	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	106	2	no	—
148	C₄	#13133189	1	1	yes	characteristic
149	C₄	D₈×D₅₃	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	8	1	no	—
152	C₂	—	16	1	no	—
153	C₂	—	848	1	no	—
154	C₂	#16810816	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810816
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#509	C₂
socle	C₂×C₅₃	#459
hypercenter	#509	D₅₃
hypocenter	C₅₃	#2213
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

Complements

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

N	H ≅ G/N	type
1. #16811058	C₂	semidirect
2. #16810998	C₂	semidirect
3. #16810740	C₂	semidirect
4. #16810443	C₂	semidirect
5. #16810437	C₂	semidirect
6. #16810436	C₂	semidirect
7. C₅₃×#509	C₂	semidirect
27. Q₁₆×C₅₃	D₄	semidirect
31. D₈×C₅₃	D₄	semidirect
79. #509	D₅₃	semidirect
85. C₅₃	#2213	semidirect
105. Q₁₆	C₂²⋊D₅₃	semidirect
113. D₈	C₂²⋊D₅₃	semidirect