Finite Group Databasebeta

Subgroups — #20276676

Order 6784.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16811437	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810890	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810440	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810430	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810408	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#509	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133264	—	2	1	no	—
9	#13133247	—	2	1	no	—
10	#13133246	—	2	2	no	—
11	#13133243	—	2	1	no	—
12	#13133236	—	2	1	no	—
13	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
14	Q₁₆×D₅₃	—	2	1	no	—
15	#13133223	C₂²	1	1	yes	characteristic
16	#13133161	—	2	2	no	—
17	C₂×#11719908	C₂²	1	1	yes	characteristic
18	#13133153	C₂²	1	1	yes	characteristic
19	#13133134	—	2	1	no	—
20	#135×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
22	SD₃₂×C₅₃	—	2	1	no	—
23	D₁₆×C₅₃	—	2	1	no	—
24	M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	#41×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	Q₈×D₅₃	—	2	2	no	—
28	#11719947	—	2	2	no	—
29	#11719947	—	4	2	no	—
30	#11719945	—	4	1	no	—
31	#11719945	D₄	1	1	yes	characteristic
32	C₂×Dic₁₀₆	—	2	1	no	—
33	C₂×Dic₁₀₆	—	4	1	no	—
34	Q₁₆×C₅₃	D₄	1	1	yes	characteristic
35	SD₁₆×C₅₃	—	2	1	no	—
36	D₈×C₅₃	—	2	1	no	—
37	D₈×C₅₃	—	2	1	no	—
38	D₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#11719925	—	2	2	no	—
41	#11719923	—	2	2	no	—
42	#11719923	—	4	1	no	—
43	#11719922	—	4	1	no	—
44	#11719917	—	2	1	no	—
45	#11719917	D₄	1	1	yes	characteristic
46	Dic₂₁₂	—	2	1	no	—
47	Dic₂₁₂	D₄	1	2	yes	—
48	#11719912	—	2	1	no	—
49	C₈×D₅₃	—	2	1	no	—
50	C₁₆×C₅₃	—	2	1	no	—
51	#11719908	—	2	1	no	—
52	C₂³×C₅₃	—	4	1	no	—
53	Q₈×C₅₃	—	2	1	no	—
54	D₄×C₅₃	—	2	1	no	—
55	D₄×C₅₃	—	4	1	no	—
56	D₄×C₅₃	—	4	1	no	—
57	D₄×C₅₃	—	4	1	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₅₃	—	2	1	no	—
61	C₂²⋊D₅₃	—	4	2	no	—
62	C₂×Dic₅₃	—	4	2	no	—
63	C₂×Dic₅₃	—	4	1	no	—
64	D₂₁₂	D₈	1	1	yes	characteristic
65	C₄×D₅₃	—	2	1	no	—
66	C₄×D₅₃	—	4	2	no	—
67	Dic₁₀₆	—	2	2	no	—
68	Dic₁₀₆	—	4	2	no	—
69	Dic₁₀₆	—	4	1	no	—
70	Dic₁₀₆	D₈	1	1	yes	characteristic
71	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	#88322	—	2	1	no	—
74	#88322	—	4	1	no	—
75	C₂²×C₅₃	—	4	1	no	—
76	C₂²×C₅₃	—	8	1	no	—
77	C₂²×C₅₃	—	16	1	no	—
78	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₅₃	—	2	1	no	—
80	C₄×C₅₃	—	4	1	no	—
81	C₄×C₅₃	#120	1	1	yes	characteristic
82	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	Dic₅₃	—	2	1	no	—
84	Dic₅₃	—	4	2	no	—
85	#2193	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	8	1	no	—
88	C₂×C₅₃	—	16	1	no	—
89	C₂×C₅₃	#447	1	1	yes	characteristic
90	D₅₃	—	4	1	no	—
91	#578	—	53	1	no	—
92	#509	D₅₃	1	1	yes	characteristic, complemented
93	C₂×SD₃₂	—	53	1	no	—
94	#472	—	53	1	no	—
95	#362	—	53	1	no	—
96	#359	—	53	1	no	—
97	#350	—	53	1	no	—
98	C₅₃	#2193	1	1	yes	characteristic, complemented, Sylow
99	2₋¹⁺⁴	—	106	1	no	—
100	#137	—	106	1	no	—
101	#137	—	106	2	no	—
102	AGL(1,ℤ/8)	—	106	1	no	—
103	#135	—	106	1	no	—
104	#135	C₂×D₅₃	1	1	yes	characteristic
105	C₂×Q₁₆	—	53	1	no	—
106	C₂×Q₁₆	—	106	1	no	—
107	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
108	#131	—	53	1	no	—
109	SD₃₂	—	2	1	no	—
110	SD₃₂	—	106	2	no	—
111	D₁₆	—	2	1	no	—
112	M₃₂	C₂×D₅₃	1	1	yes	characteristic
113	C₂×C₁₆	—	53	1	no	—
114	#108	—	53	1	no	—
115	#100	—	106	1	no	—
116	#41	—	2	1	no	—
117	#41	—	53	1	no	—
118	#41	—	106	2	no	—
119	#41	—	212	1	no	—
120	#41	—	212	2	no	—
121	C₂×Q₈	—	106	2	no	—
122	C₂×Q₈	—	106	1	no	—
123	C₂×Q₈	—	212	1	no	—
124	C₂×D₄	—	2	1	no	—
125	Q₁₆	—	53	2	no	—
126	Q₁₆	—	106	2	no	—
127	Q₁₆	—	106	1	no	—
128	Q₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
129	SD₁₆	—	2	1	no	—
130	SD₁₆	—	106	2	no	—
131	SD₁₆	—	212	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	212	1	no	—
135	D₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	M₁₆	—	53	1	no	—
137	M₁₆	—	106	1	no	—
138	M₁₆	—	106	1	no	—
139	C₂×C₈	—	106	1	no	—
140	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	106	1	no	—
143	C₂³	—	4	1	no	—
144	Q₈	—	2	1	no	—
145	Q₈	—	53	1	no	—
146	Q₈	—	106	2	no	—
147	Q₈	—	212	2	no	—
148	Q₈	—	212	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	53	1	no	—
154	D₄	—	106	1	no	—
155	D₄	—	212	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	106	1	no	—
158	C₂×C₄	—	212	2	no	—
159	C₂×C₄	—	212	2	no	—
160	C₂×C₄	—	212	1	no	—
161	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
162	C₈	—	106	1	no	—
163	C₈	—	212	1	no	—
164	C₈	D₄×D₅₃	1	1	yes	characteristic
165	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂²	—	4	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	106	1	no	—
170	C₂²	#13133241	1	1	yes	characteristic
171	C₄	—	4	1	no	—
172	C₄	—	106	1	no	—
173	C₄	—	212	2	no	—
174	C₄	#13133185	1	1	yes	characteristic
175	C₄	D₈×D₅₃	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	8	1	no	—
178	C₂	—	16	1	no	—
179	C₂	—	212	1	no	—
180	C₂	#16810755	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810755
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#509	C₂
socle	C₂×C₅₃	#447
hypercenter	#509	D₅₃
hypocenter	C₅₃	#2193
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. #16811437	C₂	semidirect
2. C₂×#13133161	C₂	semidirect
3. #16810890	C₂	semidirect
4. #16810440	C₂	semidirect
5. #16810430	C₂	semidirect
6. #16810408	C₂	semidirect
7. C₅₃×#509	C₂	semidirect
92. #509	D₅₃	semidirect
98. C₅₃	#2193	semidirect