Finite Group Databasebeta

Subgroups — #20276172

Order 6784.

204 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16810672	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810628	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133134	C₂	1	1	yes	maximal, characteristic
4	#16810394	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810393	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810379	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#418	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	#13133176	C₂²	1	1	yes	characteristic
12	#13133153	—	2	2	no	—
13	#13133138	—	2	2	no	—
14	#13133134	—	2	2	no	—
15	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	#117×C₅₃	C₂²	1	1	yes	characteristic
18	#102×C₅₃	—	2	2	no	—
19	C₂⁴×C₅₃	—	2	1	no	—
20	C₂×D₄×C₅₃	—	2	1	no	—
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₄×C₅₃	D₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	M₁₆×C₅₃	—	2	1	no	—
28	M₁₆×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	D₄	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂²⋊C₄×C₅₃	—	2	1	no	—
32	C₄²×C₅₃	—	2	1	no	—
33	C₂²⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄×Dic₅₃	—	2	1	no	—
36	#11719917	—	2	1	no	—
37	#11719917	—	2	1	no	—
38	#11719917	—	2	1	no	—
39	#11719917	—	2	1	no	—
40	#11719917	—	2	1	no	—
41	#11719917	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂³×C₅₃	—	2	2	no	—
46	C₂³×C₅₃	—	4	1	no	—
47	C₂³×C₅₃	—	4	1	no	—
48	C₂³×C₅₃	—	8	2	no	—
49	C₂³×C₅₃	#41	1	1	yes	characteristic
50	D₄×C₅₃	—	2	2	no	—
51	D₄×C₅₃	—	2	2	no	—
52	D₄×C₅₃	—	4	1	no	—
53	D₄×C₅₃	—	4	1	no	—
54	C₂×C₄×C₅₃	—	4	1	no	—
55	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂×Dic₅₃	—	4	1	no	—
62	C₈×C₅₃	—	2	2	no	—
63	#88322	—	4	1	no	—
64	#88322	—	4	1	no	—
65	#88322	—	4	1	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	—	4	2	no	—
68	C₂²×C₅₃	—	8	2	no	—
69	C₂²×C₅₃	—	8	2	no	—
70	C₂²×C₅₃	—	8	2	no	—
71	C₂²×C₅₃	#123	1	1	yes	characteristic
72	C₂²×C₅₃	#124	1	1	yes	characteristic
73	C₂²×C₅₃	#126	1	1	yes	characteristic
74	C₄×C₅₃	—	8	1	no	—
75	C₄×C₅₃	#121	1	1	yes	characteristic
76	C₄×C₅₃	#123	1	1	yes	characteristic
77	C₄×C₅₃	#123	1	1	yes	characteristic
78	C₄×C₅₃	#126	1	1	yes	characteristic
79	Dic₅₃	—	8	1	no	—
80	#2081	—	53	1	no	maximal, Sylow
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	—	2	1	no	—
83	C₂×C₅₃	—	2	1	no	—
84	C₂×C₅₃	—	8	2	no	—
85	C₂×C₅₃	#397	1	1	yes	characteristic
86	#430	—	53	1	no	—
87	#418	—	53	1	no	—
88	#418	D₅₃	1	1	yes	characteristic
89	C₂×#100	—	53	1	no	—
90	#344	—	53	1	no	—
91	#344	—	53	1	no	—
92	H(3,ℤ/4)	—	53	1	no	—
93	C₅₃	#2081	1	1	yes	characteristic, complemented, Sylow
94	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
95	C₂×M₁₆	—	53	1	no	—
96	C₂×M₁₆	—	53	1	no	—
97	C₂×M₁₆	—	53	1	no	—
98	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
99	#117	—	53	1	no	—
100	#117	C₂×D₅₃	1	1	yes	characteristic
101	#108	—	106	2	no	—
102	#102	—	2	2	no	—
103	#102	—	106	2	no	—
104	#100	—	106	2	no	—
105	C₂⁴	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	D₂₁₂	1	2	yes	—
112	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	106	1	no	—
116	M₁₆	—	106	1	no	—
117	M₁₆	—	106	1	no	—
118	M₁₆	—	106	1	no	—
119	M₁₆	—	106	1	no	—
120	M₁₆	—	106	1	no	—
121	C₂×C₈	—	106	1	no	—
122	C₂×C₈	—	106	1	no	—
123	C₂×C₈	—	106	1	no	—
124	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	106	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	106	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	106	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	8	2	no	—
135	C₂³	#11719947	1	1	yes	characteristic
136	D₄	—	2	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	212	1	no	—
142	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
143	C₂×C₄	#11719945	1	1	yes	characteristic
144	C₂×C₄	#11719947	1	1	yes	characteristic
145	C₂×C₄	#11719947	1	1	yes	characteristic
146	C₂×C₄	#11719947	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	212	1	no	—
150	C₈	—	212	1	no	—
151	C₈	—	212	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	#13133201	1	1	yes	characteristic
158	C₂²	#13133204	1	1	yes	characteristic
159	C₂²	#13133210	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	424	1	no	—
162	C₄	#13133191	1	1	yes	characteristic
163	C₄	#13133198	1	1	yes	characteristic
164	C₄	#13133201	1	1	yes	characteristic
165	C₄	#13133210	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	8	2	no	—
170	C₂	#16810548	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810548
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#418	C₂
socle	C₂×C₅₃	#397
hypercenter	#418	D₅₃
hypocenter	C₅₃	#2081
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
1. #16810672	C₂	semidirect
4. #16810394	C₂	semidirect
5. #16810393	C₂	semidirect
6. #16810379	C₂	semidirect
93. C₅₃	#2081	semidirect