Finite Group Databasebeta

Subgroups — #20274837

Order 6784.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#13133152	C₂	1	1	yes	maximal, characteristic
2	#16810659	C₂	1	1	yes	maximal, characteristic
3	#16810579	C₂	1	1	yes	maximal, characteristic
4	#16810395	C₂	1	1	yes	maximal, characteristic
5	#16810393	C₂	1	1	yes	maximal, characteristic
6	#16810379	C₂	1	1	yes	maximal, characteristic
7	C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133176	C₂²	1	1	yes	characteristic
12	#13133152	—	2	1	no	—
13	#13133152	—	2	1	no	—
14	#13133150	—	2	1	no	—
15	#13133148	—	2	1	no	—
16	#13133128	C₄	1	2	yes	complemented
17	C₄×#88322	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	#117×C₅₃	C₂²	1	1	yes	characteristic
20	#117×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	1	no	—
23	M₁₆×C₅₃	—	2	1	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₂²⋊C₄×C₅₃	—	2	1	no	—
28	C₂²⋊C₄×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂²⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄×Dic₅₃	—	2	1	no	—
34	#11719917	—	2	1	no	—
35	#11719917	—	2	1	no	—
36	#11719917	—	2	1	no	—
37	#11719917	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	C₂×C₄	1	2	yes	—
40	C₂×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂×#88322	C₂×C₄	1	1	yes	characteristic
42	C₂×#88322	D₄	1	1	yes	characteristic
43	C₂×#88322	D₄	1	1	yes	characteristic
44	C₂×#88322	Q₈	1	1	yes	characteristic
45	C₂×#88322	Q₈	1	1	yes	characteristic
46	C₂³×C₅₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
55	C₂×Dic₅₃	—	4	1	no	—
56	C₈×C₅₃	—	4	1	no	—
57	#88322	—	2	2	no	—
58	#88322	—	4	1	no	—
59	#88322	C₄⋊C₄	1	1	yes	characteristic
60	#88322	C₄⋊C₄	1	1	yes	characteristic
61	#88322	C₄⋊C₄	1	1	yes	characteristic
62	#88322	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	#117	1	1	yes	characteristic
65	C₂²×C₅₃	#124	1	1	yes	characteristic
66	C₂²×C₅₃	#125	1	1	yes	characteristic
67	C₄×C₅₃	—	4	2	no	—
68	C₄×C₅₃	—	8	1	no	—
69	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₅₃	#117	1	1	yes	characteristic
71	C₄×C₅₃	#124	1	1	yes	characteristic
72	C₄×C₅₃	#125	1	1	yes	characteristic
73	Dic₅₃	—	8	1	no	—
74	#1833	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	#382	1	1	yes	characteristic
79	C₂×#108	—	53	1	no	—
80	#427	—	53	1	no	—
81	#405	—	53	1	no	—
82	#344	—	53	1	no	—
83	#344	—	53	1	no	—
84	H(3,ℤ/4)	—	53	1	no	—
85	H(3,ℤ/4)	D₅₃	1	1	yes	characteristic
86	C₅₃	#1833	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	53	1	no	—
88	C₂×M₁₆	—	53	1	no	—
89	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
90	C₂²×C₈	—	53	1	no	—
91	#117	—	53	1	no	—
92	#117	C₂×D₅₃	1	1	yes	characteristic
93	#117	C₂×D₅₃	1	1	yes	characteristic
94	#108	—	106	1	no	—
95	#108	—	106	1	no	—
96	#107	—	106	1	no	—
97	#106	—	106	1	no	—
98	#97	—	53	2	no	—
99	C₄×C₈	—	53	2	no	—
100	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	106	1	no	—
104	M₁₆	—	106	1	no	—
105	M₁₆	—	106	1	no	—
106	M₁₆	—	106	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	—	106	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	106	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	106	1	no	—
124	C₄²	C₄×D₅₃	1	2	yes	—
125	C₂³	#11719947	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	212	1	no	—
129	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	#11719945	1	1	yes	characteristic
131	C₂×C₄	#11719945	1	1	yes	characteristic
132	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
133	C₂×C₄	#11719947	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	53	1	no	—
137	C₈	—	53	1	no	—
138	C₈	—	53	1	no	—
139	C₈	—	53	1	no	—
140	C₈	—	106	2	no	—
141	C₈	—	212	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#13133173	1	1	yes	characteristic
144	C₂²	#13133203	1	1	yes	characteristic
145	C₂²	#13133207	1	1	yes	characteristic
146	C₄	—	4	2	no	—
147	C₄	—	8	1	no	—
148	C₄	—	424	1	no	—
149	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
150	C₄	#13133174	1	1	yes	characteristic
151	C₄	#13133203	1	1	yes	characteristic
152	C₄	#13133207	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16810487	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133174
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×H(3,ℤ/4)	C₂
socle	C₂×C₅₃	#382
hypercenter	H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	#1833
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
16. #13133128	C₄	semidirect
17. C₄×#88322	C₄	semidirect
86. C₅₃	#1833	semidirect