Finite Group Databasebeta

Subgroups — #20274380

Order 6784.

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

normal subgroups only

This group has 242 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	#16810999	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810999	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810995	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810970	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810969	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719923	—	2	2	no	—
9	C₂×#11719922	—	2	2	no	—
10	C₂×#11719922	—	2	2	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133214	C₂²	1	1	yes	characteristic
13	#13133139	—	2	2	no	—
14	#13133138	—	2	2	no	—
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₅₃	—	2	2	no	—
18	#127×C₅₃	C₂²	1	1	yes	characteristic
19	#127×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×D₂₁₂	—	2	1	no	—
30	C₂×Dic₁₀₆	—	2	1	no	—
31	D₈×C₅₃	—	4	4	no	—
32	C₂×C₈×C₅₃	D₄	1	2	yes	—
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	#11719923	—	4	4	no	—
35	#11719922	—	4	4	no	—
36	#11719922	—	4	4	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×#88322	D₄	1	2	yes	—
40	C₂×#88322	D₄	1	2	yes	—
41	C₂³×C₅₃	—	4	1	no	—
42	C₂³×C₅₃	—	4	1	no	—
43	C₂²×D₅₃	—	4	1	no	—
44	D₄×C₅₃	—	4	2	no	—
45	D₄×C₅₃	—	4	2	no	—
46	D₄×C₅₃	—	4	2	no	—
47	D₄×C₅₃	—	4	2	no	—
48	D₄×C₅₃	—	4	2	no	—
49	D₄×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₅₃	—	4	1	no	—
54	D₂₁₂	—	4	2	no	—
55	D₂₁₂	—	4	2	no	—
56	D₂₁₂	—	4	2	no	—
57	Dic₁₀₆	—	4	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	#88322	—	2	2	no	—
60	#88322	D₈	1	4	yes	complemented
61	C₂²×C₅₃	—	8	1	no	—
62	C₂²×C₅₃	—	8	1	no	—
63	C₂²×C₅₃	—	8	1	no	—
64	C₂²×C₅₃	—	8	1	no	—
65	C₂²×C₅₃	—	8	1	no	—
66	C₂²×C₅₃	—	8	1	no	—
67	C₂²×C₅₃	#127	1	1	yes	characteristic
68	C₂×D₅₃	—	8	1	no	—
69	C₂×D₅₃	—	8	1	no	—
70	C₂×D₅₃	—	8	1	no	—
71	C₄×C₅₃	C₂×D₈	1	2	yes	—
72	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₅₃	—	8	1	no	—
75	#1721	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	16	1	no	—
77	C₂×C₅₃	—	16	1	no	—
78	C₂×C₅₃	#493	1	1	yes	characteristic
79	C₂×C₅₃	#496	1	1	yes	characteristic
80	C₂×C₅₃	#496	1	1	yes	characteristic
81	D₅₃	—	16	1	no	—
82	#493	—	53	1	no	—
83	#493	—	53	1	no	—
84	#493	D₅₃	1	1	yes	characteristic, complemented
85	#492	—	53	1	no	—
86	#486	—	53	1	no	—
87	#486	—	53	1	no	—
88	#322	—	53	1	no	—
89	C₅₃	#1721	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	106	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	106	2	no	—
93	C₂×D₈	—	106	2	no	—
94	C₄⋊Q₈	—	53	1	no	—
95	#127	—	53	1	no	—
96	#127	C₂×D₅₃	1	1	yes	characteristic
97	#127	C₂×D₅₃	1	1	yes	characteristic
98	#102	—	106	2	no	—
99	#102	—	106	2	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×Q₈	—	106	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	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	—	106	1	no	—
111	C₂×D₄	—	106	1	no	—
112	C₂×D₄	—	106	1	no	—
113	SD₁₆	—	212	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	212	4	no	—
116	D₈	—	212	4	no	—
117	C₂×C₈	—	53	2	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
120	C₄⋊C₄	—	106	1	no	—
121	C₄⋊C₄	—	106	1	no	—
122	C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	212	1	no	—
126	Q₈	—	212	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	212	2	no	—
134	D₄	—	212	2	no	—
135	D₄	—	212	2	no	—
136	C₂×C₄	—	212	1	no	—
137	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	53	4	no	—
142	C₈	—	106	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	424	1	no	—
150	C₂²	—	424	1	no	—
151	C₂²	—	424	1	no	—
152	C₂²	#13133213	1	1	yes	characteristic
153	C₄	—	424	1	no	—
154	C₄	D₈×D₅₃	1	2	yes	—
155	C₄	#13133241	1	2	yes	—
156	C₄	#13133243	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	848	1	no	—
160	C₂	#16810998	1	1	yes	characteristic
161	C₂	#16811010	1	1	yes	characteristic
162	C₂	#16811011	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	#127
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1721
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #16810999	C₂	semidirect
2. #16810999	C₂	semidirect
3. #16810995	C₂	semidirect
4. #16810970	C₂	semidirect
5. #16810969	C₂	semidirect
6. #16810345	C₂	semidirect
7. C₅₃×#493	C₂	semidirect
60. #88322	D₈	semidirect
84. #493	D₅₃	semidirect
89. C₅₃	#1721	semidirect