Finite Group Databasebeta

Subgroups — #20290267

Order 6784.

180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811642	C₂	1	1	yes	maximal, characteristic
2	D₅₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#16811487	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811486	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811485	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811484	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810575	C₂	1	1	yes	maximal, characteristic
8	C₂×D₅₃⋊C₈	—	2	1	no	—
9	#13133284	—	2	1	no	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	1	yes	characteristic
12	C₄⋊#88324	C₂²	1	1	yes	characteristic
13	C₄⋊#88324	C₂²	1	1	yes	characteristic
14	#13133274	—	2	1	no	—
15	C₄×#88324	—	2	1	no	—
16	C₂×#11719912	C₄	1	1	yes	characteristic
17	C₂×#11719912	C₄	1	1	yes	characteristic
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	#13133128	C₄	1	1	yes	characteristic
20	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	#97×C₅₃	C₄	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×#88324	—	2	1	no	—
27	C₂×#88324	—	2	1	no	—
28	C₂×#88324	—	2	1	no	—
29	C₂×#88324	—	2	1	no	—
30	D₅₃⋊C₈	—	4	2	no	—
31	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	C₄×Dic₅₃	Q₈	1	1	yes	characteristic
36	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
37	C₂×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂×#88322	C₂×C₄	1	1	yes	characteristic
39	#11719912	—	2	2	no	—
40	#11719912	C₈	1	4	yes	complemented
41	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₅₃	—	2	1	no	—
49	C₄×D₅₃	—	2	1	no	—
50	C₄×D₅₃	—	4	1	no	—
51	C₄×D₅₃	C₂×C₈	1	2	yes	—
52	C₄×D₅₃	M₁₆	1	2	yes	—
53	#88324	—	4	1	no	—
54	#88324	—	4	1	no	—
55	#88324	—	4	1	no	—
56	#88324	—	4	1	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	C₂×C₈	1	2	yes	complemented
59	#88322	—	2	1	no	—
60	#88322	C₂×C₈	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	C₄⋊C₈	1	2	yes	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	C₄⋊C₈	1	2	yes	—
71	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₅₃	#137	1	1	yes	characteristic
73	#1565	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₅₃	#428	1	1	yes	characteristic
76	C₂×C₅₃	#430	1	1	yes	characteristic
77	D₅₃	—	2	2	no	—
78	#431	—	53	1	no	—
79	C₂×C₄⋊C₈	—	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	—	53	1	no	—
85	C₅₃	#1565	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₁₆	—	53	1	no	—
87	C₂×M₁₆	—	53	1	no	—
88	C₂²×C₈	—	106	1	no	—
89	C₂×C₄²	—	53	1	no	—
90	C₄⋊C₈	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	53	1	no	—
94	C₄⋊C₈	—	106	1	no	—
95	C₂²⋊C₈	—	106	1	no	—
96	#97	—	53	1	no	—
97	#97	#6157	1	1	yes	characteristic
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	—	106	1	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	53	1	no	—
104	M₁₆	—	53	4	no	—
105	M₁₆	—	106	2	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	106	1	no	—
112	C₂×C₈	—	212	2	no	—
113	C₂×C₈	C₂×#6157	1	1	yes	characteristic
114	C₂×C₈	C₂×#6157	1	1	yes	characteristic
115	C₄²	—	53	1	no	—
116	C₄²	—	53	1	no	—
117	C₄²	—	53	1	no	—
118	C₄²	C₂×#6157	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	2	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	106	1	no	—
126	C₂×C₄	—	106	1	no	—
127	C₂×C₄	—	212	1	no	—
128	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
129	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
130	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	53	2	no	—
133	C₈	—	106	1	no	—
134	C₈	—	212	1	no	—
135	C₈	—	212	1	no	—
136	C₈	—	212	1	no	—
137	C₈	—	212	1	no	—
138	C₈	D₅₃⋊C₈	1	2	yes	complemented
139	C₂²	—	53	2	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	1	no	—
142	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	53	2	no	—
146	C₄	—	53	1	no	—
147	C₄	—	53	1	no	—
148	C₄	—	106	1	no	—
149	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
151	C₂	—	106	2	no	—
152	C₂	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
153	C₂	#16811634	1	1	yes	characteristic
154	C₂	#16811638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#97	#6157
hypocenter	C₅₃	#1565
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
3. #16811487	C₂	semidirect
4. #16811486	C₂	semidirect
5. #16811485	C₂	semidirect
6. #16811484	C₂	semidirect
40. #11719912	C₈	semidirect
58. C₈×C₅₃	C₂×C₈	semidirect
60. #88322	C₂×C₈	semidirect
85. C₅₃	#1565	semidirect
138. C₈	D₅₃⋊C₈	semidirect