Finite Group Databasebeta

Subgroups — #20276663

Order 6784.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16811423	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811061	C₂	1	2	yes	maximal, complemented
3	#16810876	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810434	C₂	1	2	yes	maximal, complemented
5	C₅₃×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	—	2	1	no	—
7	#13133250	—	2	2	no	—
8	#13133245	—	2	2	no	—
9	#13133239	—	2	1	no	—
10	#13133239	C₂²	1	2	yes	—
11	C₂×D₄₂₄	C₂²	1	1	yes	characteristic, complemented
12	#13133191	—	2	1	no	—
13	#13133162	—	2	2	no	—
14	D₈₄₈	—	2	2	no	—
15	#13133149	C₂²	1	1	yes	characteristic, complemented
16	#13133139	—	2	1	no	—
17	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	M₃₂×C₅₃	C₂²	1	2	yes	complemented
19	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₅₃	—	4	1	no	—
21	C₂×C₂²⋊D₅₃	—	4	1	no	—
22	#11719945	—	2	2	no	—
23	#11719945	—	2	2	no	—
24	#11719945	—	4	1	no	—
25	#11719945	—	4	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×C₄×D₅₃	—	4	1	no	—
29	C₂×Dic₁₀₆	—	2	1	no	—
30	M₁₆×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	#11719921	—	4	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	Dic₂₁₂	—	2	1	no	—
36	Dic₂₁₂	D₄	1	2	yes	complemented
37	D₄₂₄	—	2	1	no	—
38	D₄₂₄	—	2	2	no	—
39	D₄₂₄	D₄	1	2	yes	complemented
40	#11719913	—	2	2	no	—
41	#11719913	—	2	2	no	—
42	C₁₆×C₅₃	—	2	2	no	—
43	C₂³×C₅₃	D₈	1	1	yes	characteristic
44	C₂²×D₅₃	—	4	1	no	—
45	C₂²×D₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₅₃	—	4	2	no	—
51	C₂²⋊D₅₃	—	4	2	no	—
52	C₂²⋊D₅₃	—	8	2	no	—
53	C₂×Dic₅₃	—	4	1	no	—
54	C₂×Dic₅₃	—	4	1	no	—
55	D₂₁₂	—	2	2	no	—
56	D₂₁₂	—	4	1	no	—
57	D₂₁₂	—	4	1	no	—
58	D₂₁₂	—	4	1	no	—
59	C₄×D₅₃	—	4	2	no	—
60	C₄×D₅₃	—	4	2	no	—
61	Dic₁₀₆	—	2	2	no	—
62	Dic₁₀₆	—	4	1	no	—
63	C₈×C₅₃	—	2	1	no	—
64	C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×D₅₃	—	4	2	no	—
69	C₂×D₅₃	—	8	2	no	—
70	C₂×D₅₃	—	8	1	no	—
71	C₂×D₅₃	—	8	1	no	—
72	C₂×D₅₃	—	8	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#120	1	1	yes	characteristic
75	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₅₃	—	4	2	no	—
77	Dic₅₃	—	8	1	no	—
78	#2190	—	53	1	no	maximal, Sylow
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	4	1	no	—
81	C₂×C₅₃	#447	1	1	yes	characteristic
82	D₅₃	—	8	2	no	—
83	D₅₃	—	16	1	no	—
84	#575	—	53	1	no	—
85	#509	—	53	2	no	—
86	#469	—	53	1	no	—
87	#360	—	53	2	no	—
88	#349	D₅₃	1	1	yes	characteristic, complemented
89	C₅₃	#2190	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	106	1	no	—
91	#137	—	106	2	no	—
92	AGL(1,ℤ/8)	—	106	2	no	—
93	#135	—	53	2	no	—
94	#135	—	106	1	no	—
95	C₂×D₈	—	53	1	no	—
96	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
97	#121	—	106	1	no	—
98	SD₃₂	—	106	2	no	—
99	D₁₆	—	106	2	no	—
100	M₃₂	C₂×D₅₃	1	2	yes	complemented
101	#106	—	53	1	no	—
102	#102	—	106	1	no	—
103	#41	—	106	2	no	—
104	#41	—	106	2	no	—
105	#41	—	212	1	no	—
106	#41	—	212	1	no	—
107	C₂×Q₈	—	106	1	no	—
108	C₂×D₄	—	106	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂×D₄	—	212	1	no	—
111	C₂×D₄	—	212	1	no	—
112	C₂²×C₄	—	212	1	no	—
113	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
114	Q₁₆	—	53	2	no	—
115	Q₁₆	—	106	1	no	—
116	SD₁₆	—	106	2	no	—
117	SD₁₆	—	106	2	no	—
118	D₈	—	53	2	no	—
119	D₈	—	106	1	no	—
120	D₈	—	106	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	D₂₁₂	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	106	1	no	—
125	C₂²⋊C₄	—	212	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	212	1	no	—
128	C₂³	—	212	1	no	—
129	C₂³	D₄₂₄	1	1	yes	characteristic
130	Q₈	—	106	2	no	—
131	Q₈	—	212	1	no	—
132	D₄	—	106	2	no	—
133	D₄	—	424	2	no	—
134	D₄	—	212	1	no	—
135	D₄	—	212	2	no	—
136	D₄	—	212	2	no	—
137	D₄	—	212	1	no	—
138	D₄	—	212	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	212	2	no	—
142	C₂×C₄	—	212	1	no	—
143	C₂×C₄	—	212	1	no	—
144	C₂×C₄	—	212	2	no	—
145	C₂×C₄	D₄₂₄	1	1	yes	characteristic
146	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₅₃	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	424	1	no	—
152	C₂²	—	424	1	no	—
153	C₂²	—	424	2	no	—
154	C₂²	—	424	1	no	—
155	C₂²	—	212	2	no	—
156	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	424	1	no	—
159	C₄	—	212	2	no	—
160	C₄	#13133187	1	1	yes	characteristic
161	C₄	#13133245	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	848	1	no	—
165	C₂	—	424	2	no	—
166	C₂	#16810759	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810759
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#349	C₂
socle	C₂×C₅₃	#447
hypercenter	#349	D₅₃
hypocenter	C₅₃	#2190
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
1. #16811423	C₂	semidirect
2. #16811061	C₂	semidirect
3. #16810876	C₂	semidirect
4. #16810434	C₂	semidirect
5. C₅₃×#349	C₂	semidirect
11. C₂×D₄₂₄	C₂²	semidirect
15. #13133149	C₂²	semidirect
18. M₃₂×C₅₃	C₂²	semidirect
36. Dic₂₁₂	D₄	semidirect
39. D₄₂₄	D₄	semidirect
88. #349	D₅₃	semidirect
89. C₅₃	#2190	semidirect
100. M₃₂	C₂×D₅₃	semidirect