Finite Group Databasebeta

Subgroups — #20276683

Order 6784.

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

normal subgroups only

This group has 216 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	#16811425	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133160	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810895	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810437	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×#510	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133263	—	2	1	no	—
9	#13133248	—	2	1	no	—
10	#13133244	—	2	2	no	—
11	#13133242	—	2	1	no	—
12	#13133240	—	2	1	no	—
13	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
14	D₈×D₅₃	—	2	1	no	—
15	#13133223	C₂²	1	1	yes	characteristic
16	#13133160	—	2	2	no	—
17	C₂×#11719908	C₂²	1	1	yes	characteristic, complemented
18	#13133153	C₂²	1	1	yes	characteristic, complemented
19	#13133136	—	2	1	no	—
20	#135×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
22	Q₃₂×C₅₃	—	2	1	no	—
23	SD₃₂×C₅₃	—	2	1	no	—
24	M₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
25	#41×C₅₃	—	2	1	no	—
26	C₂×Q₈×C₅₃	—	2	1	no	—
27	#11719949	—	2	2	no	—
28	D₄×D₅₃	—	2	2	no	—
29	D₄×D₅₃	—	4	2	no	—
30	#11719945	—	4	1	no	—
31	#11719945	D₄	1	1	yes	characteristic
32	C₂×D₂₁₂	—	2	1	no	—
33	C₂×D₂₁₂	—	4	1	no	—
34	Q₁₆×C₅₃	—	2	1	no	—
35	Q₁₆×C₅₃	—	2	1	no	—
36	Q₁₆×C₅₃	D₄	1	1	yes	characteristic
37	SD₁₆×C₅₃	—	2	1	no	—
38	D₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#11719925	—	4	1	no	—
41	#11719924	—	2	2	no	—
42	#11719924	—	4	1	no	—
43	#11719922	—	2	2	no	—
44	#11719917	—	2	1	no	—
45	#11719917	D₄	1	1	yes	characteristic
46	D₄₂₄	—	2	1	no	—
47	D₄₂₄	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	C₂²×D₅₃	—	4	2	no	—
53	C₂²×D₅₃	—	4	1	no	—
54	Q₈×C₅₃	—	2	1	no	—
55	Q₈×C₅₃	—	4	1	no	—
56	Q₈×C₅₃	—	4	1	no	—
57	D₄×C₅₃	—	2	1	no	—
58	D₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	—	4	1	no	—
60	C₂×C₄×C₅₃	—	4	1	no	—
61	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₅₃	—	2	1	no	—
63	C₂²⋊D₅₃	—	4	2	no	—
64	D₂₁₂	—	2	2	no	—
65	D₂₁₂	—	4	2	no	—
66	D₂₁₂	—	4	1	no	—
67	D₂₁₂	D₈	1	1	yes	characteristic
68	C₄×D₅₃	—	2	1	no	—
69	C₄×D₅₃	—	4	2	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	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₅₃	—	2	1	no	—
78	C₂×D₅₃	—	4	2	no	—
79	C₂×D₅₃	—	16	2	no	—
80	C₂×D₅₃	—	16	1	no	—
81	C₄×C₅₃	—	4	1	no	—
82	C₄×C₅₃	—	8	1	no	—
83	C₄×C₅₃	#120	1	1	yes	characteristic
84	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
85	Dic₅₃	—	2	1	no	—
86	#2194	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	—	2	1	no	—
88	C₂×C₅₃	—	8	1	no	—
89	C₂×C₅₃	#447	1	1	yes	characteristic
90	D₅₃	—	4	1	no	—
91	D₅₃	—	8	2	no	—
92	#576	—	53	1	no	—
93	#510	D₅₃	1	1	yes	characteristic, complemented
94	C₂×SD₃₂	—	53	1	no	—
95	#473	—	53	1	no	—
96	#361	—	53	1	no	—
97	#359	—	53	1	no	—
98	#350	—	53	1	no	—
99	C₅₃	#2194	1	1	yes	characteristic, complemented, Sylow
100	H(5,𝔽₂)	—	106	1	no	—
101	#137	—	106	1	no	—
102	AGL(1,ℤ/8)	—	106	1	no	—
103	AGL(1,ℤ/8)	—	106	2	no	—
104	#135	—	106	1	no	—
105	#135	C₂×D₅₃	1	1	yes	characteristic
106	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
107	C₂×D₈	—	53	1	no	—
108	C₂×D₈	—	106	1	no	—
109	#131	—	53	1	no	—
110	Q₃₂	—	2	1	no	—
111	SD₃₂	—	2	1	no	—
112	SD₃₂	—	106	2	no	—
113	M₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
114	C₂×C₁₆	—	53	1	no	—
115	#108	—	53	1	no	—
116	#101	—	106	1	no	—
117	#41	—	2	1	no	—
118	#41	—	53	1	no	—
119	#41	—	106	2	no	—
120	#41	—	212	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×D₄	—	106	2	no	—
123	C₂×D₄	—	106	1	no	—
124	C₂×D₄	—	212	1	no	—
125	C₂×D₄	—	212	2	no	—
126	Q₁₆	—	2	1	no	—
127	Q₁₆	—	2	1	no	—
128	Q₁₆	—	212	1	no	—
129	Q₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
130	SD₁₆	—	2	1	no	—
131	SD₁₆	—	106	2	no	—
132	SD₁₆	—	212	1	no	—
133	D₈	—	53	2	no	—
134	D₈	—	106	2	no	—
135	D₈	—	106	1	no	—
136	D₈	C₂²⋊D₅₃	1	1	yes	characteristic
137	M₁₆	—	53	1	no	—
138	M₁₆	—	106	1	no	—
139	M₁₆	—	106	1	no	—
140	C₂×C₈	—	106	1	no	—
141	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	—	106	1	no	—
144	C₂³	—	212	2	no	—
145	C₂³	—	212	1	no	—
146	Q₈	—	2	1	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	53	1	no	—
150	D₄	—	2	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	53	1	no	—
153	D₄	—	106	1	no	—
154	D₄	—	106	2	no	—
155	D₄	—	212	2	no	—
156	D₄	—	212	2	no	—
157	D₄	—	212	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	106	1	no	—
161	C₂×C₄	—	212	2	no	—
162	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
163	C₈	—	106	1	no	—
164	C₈	—	212	1	no	—
165	C₈	D₄×D₅₃	1	1	yes	characteristic
166	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂²	—	4	1	no	—
168	C₂²	—	848	1	no	—
169	C₂²	—	848	2	no	—
170	C₂²	—	106	1	no	—
171	C₂²	—	212	2	no	—
172	C₂²	#13133241	1	1	yes	characteristic
173	C₄	—	4	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	106	1	no	—
176	C₄	#13133185	1	1	yes	characteristic
177	C₄	D₈×D₅₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	8	1	no	—
180	C₂	—	424	2	no	—
181	C₂	—	212	1	no	—
182	C₂	#16810755	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810755
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#510	C₂
socle	C₂×C₅₃	#447
hypercenter	#510	D₅₃
hypocenter	C₅₃	#2194
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

Complements

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

N	H ≅ G/N	type
2. C₂×#13133160	C₂	semidirect
3. #16810895	C₂	semidirect
4. #16810437	C₂	semidirect
5. #16810430	C₂	semidirect
6. #16810408	C₂	semidirect
7. C₅₃×#510	C₂	semidirect
17. C₂×#11719908	C₂²	semidirect
18. #13133153	C₂²	semidirect
21. C₂×Q₁₆×C₅₃	C₂²	semidirect
24. M₃₂×C₅₃	C₂²	semidirect
93. #510	D₅₃	semidirect
99. C₅₃	#2194	semidirect
106. C₂×Q₁₆	C₂×D₅₃	semidirect
113. M₃₂	C₂×D₅₃	semidirect