Finite Group Databasebeta

Subgroups — #20273479

Order 6784.

166 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 71 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	#16810679	C₂	1	1	yes	maximal, characteristic
2	#16810645	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719912	C₂	1	1	yes	maximal, characteristic
4	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719912	—	2	1	no	—
9	C₂×#11719912	C₄	1	1	yes	characteristic
10	C₂×#11719912	C₄	1	1	yes	characteristic
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	#13133146	—	2	1	no	—
15	#13133129	—	2	1	no	—
16	#13133127	—	2	1	no	—
17	#13133127	C₄	1	1	yes	characteristic
18	#13133127	C₄	1	1	yes	characteristic
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₅₃	D₄	1	1	yes	characteristic
33	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
34	C₂×#88322	—	2	1	no	—
35	C₂×#88322	—	2	1	no	—
36	C₂×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	#11719912	—	2	2	no	—
39	#11719912	—	2	2	no	—
40	#11719912	—	4	2	no	—
41	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄×D₅₃	—	2	1	no	—
49	C₄×D₅₃	—	2	1	no	—
50	C₄×D₅₃	—	2	1	no	—
51	C₄×D₅₃	—	2	1	no	—
52	C₄×D₅₃	—	4	1	no	—
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	—	4	1	no	—
56	C₈×C₅₃	M₁₆	1	2	yes	complemented
57	#88322	—	2	1	no	—
58	#88322	—	4	1	no	—
59	#88322	—	4	1	no	—
60	#88322	M₁₆	1	2	yes	complemented
61	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	1	no	—
64	C₂×D₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₅₃	#137	1	1	yes	characteristic
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	—	2	1	no	—
72	Dic₅₃	—	2	1	no	—
73	#1569	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#423	1	1	yes	characteristic
75	C₂×C₅₃	#428	1	1	yes	characteristic
76	C₂×C₅₃	#430	1	1	yes	characteristic
77	D₅₃	—	4	1	no	—
78	#432	—	53	1	no	—
79	#423	—	53	1	no	—
80	C₄×M₁₆	—	53	1	no	—
81	#335	—	53	1	no	—
82	#335	—	53	1	no	—
83	#334	—	53	1	no	—
84	#334	D₅₃	1	1	yes	characteristic, complemented
85	C₅₃	#1569	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	53	1	no	—
87	C₂×M₁₆	—	53	1	no	—
88	C₂×M₁₆	—	106	1	no	—
89	C₂×C₄²	—	53	1	no	—
90	C₄⋊C₈	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	106	1	no	—
93	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
95	C₂²⋊C₈	—	106	1	no	—
96	#97	—	53	1	no	—
97	#97	—	53	1	no	—
98	#97	—	106	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	1	no	—
104	M₁₆	—	106	2	no	—
105	M₁₆	—	106	2	no	—
106	M₁₆	—	212	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	106	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
114	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₄²	—	53	1	no	—
116	C₄²	—	53	1	no	—
117	C₄²	—	53	1	no	—
118	C₄²	C₂²×D₅₃	1	1	yes	characteristic
119	C₂³	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	106	1	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	53	2	no	—
135	C₈	—	106	1	no	—
136	C₈	—	212	1	no	—
137	C₈	—	212	1	no	—
138	C₈	#11719912	1	2	yes	complemented
139	C₂²	—	106	1	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	1	no	—
142	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	106	1	no	—
145	C₄	—	106	1	no	—
146	C₄	—	106	1	no	—
147	C₄	M₁₆×D₅₃	1	1	yes	characteristic
148	C₄	C₂×#11719912	1	1	yes	characteristic
149	C₄	#13133244	1	1	yes	characteristic
150	C₄	#13133246	1	1	yes	characteristic
151	C₂	—	212	1	no	—
152	C₂	#16810645	1	1	yes	characteristic
153	C₂	#16810662	1	1	yes	characteristic
154	C₂	#16810670	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#334	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#334	D₅₃
hypocenter	C₅₃	#1569
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
4. #16810376	C₂	semidirect
5. #16810375	C₂	semidirect
6. #16810373	C₂	semidirect
7. C₅₃×#334	C₂	semidirect
56. C₈×C₅₃	M₁₆	semidirect
60. #88322	M₁₆	semidirect
84. #334	D₅₃	semidirect
85. C₅₃	#1569	semidirect
138. C₈	#11719912	semidirect