Finite Group Databasebeta

Subgroups — #20276677

Order 6784.

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

normal subgroups only

This group has 210 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	#16811440	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811059	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133161	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810438	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810429	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810407	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133264	—	2	1	no	—
9	#13133250	—	2	1	no	—
10	#13133250	—	2	2	no	—
11	#13133239	—	2	1	no	—
12	#13133239	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₂	—	2	1	no	—
14	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
15	#13133161	—	2	2	no	—
16	#13133161	—	2	1	no	—
17	#13133158	—	2	1	no	—
18	#13133157	C₂²	1	1	yes	characteristic
19	C₂×#11719908	C₂²	1	1	yes	characteristic
20	AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
21	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
22	#131×C₅₃	C₂²	1	1	yes	characteristic
23	#108×C₅₃	C₂²	1	1	yes	characteristic
24	#100×C₅₃	—	2	1	no	—
25	#41×C₅₃	D₄	1	1	yes	characteristic
26	C₂×D₄×C₅₃	—	2	1	no	—
27	Q₈×D₅₃	—	4	1	no	—
28	#11719947	—	4	2	no	—
29	#11719947	—	4	1	no	—
30	#11719945	—	2	2	no	—
31	#11719945	—	2	1	no	—
32	C₂×Dic₁₀₆	—	2	2	no	—
33	C₂×Dic₁₀₆	—	2	1	no	—
34	SD₁₆×C₅₃	—	4	1	no	—
35	D₈×C₅₃	—	2	1	no	—
36	D₈×C₅₃	—	2	1	no	—
37	D₈×C₅₃	—	4	1	no	—
38	M₁₆×C₅₃	—	2	1	no	—
39	M₁₆×C₅₃	—	2	1	no	—
40	M₁₆×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	Dic₂₁₂	—	2	2	no	—
44	Dic₂₁₂	—	2	1	no	—
45	Dic₂₁₂	D₄	1	1	yes	characteristic
46	Dic₂₁₂	D₄	1	2	yes	—
47	D₄₂₄	D₄	1	1	yes	characteristic
48	#11719913	—	2	2	no	—
49	#11719913	—	2	1	no	—
50	#11719908	—	2	1	no	—
51	#11719908	—	2	1	no	—
52	C₂³×C₅₃	—	4	1	no	—
53	Q₈×C₅₃	D₈	1	1	yes	characteristic
54	D₄×C₅₃	—	2	1	no	—
55	D₄×C₅₃	—	4	1	no	—
56	D₄×C₅₃	—	4	1	no	—
57	D₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₅₃	—	4	2	no	—
61	C₂²⋊D₅₃	—	4	1	no	—
62	C₂×Dic₅₃	—	4	2	no	—
63	C₂×Dic₅₃	—	4	1	no	—
64	D₂₁₂	—	2	1	no	—
65	C₄×D₅₃	—	4	2	no	—
66	C₄×D₅₃	—	4	1	no	—
67	Dic₁₀₆	—	2	1	no	—
68	Dic₁₀₆	—	2	2	no	—
69	Dic₁₀₆	—	4	2	no	—
70	Dic₁₀₆	—	4	1	no	—
71	C₈×C₅₃	—	2	1	no	—
72	C₈×C₅₃	—	4	1	no	—
73	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	—	8	1	no	—
77	C₂²×C₅₃	—	16	1	no	—
78	C₂²×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₅₃	—	4	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	#120	1	1	yes	characteristic
82	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
83	Dic₅₃	—	4	1	no	—
84	Dic₅₃	—	4	2	no	—
85	#2193	—	53	1	no	maximal, Sylow
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	—	4	1	no	—
88	C₂×C₅₃	—	16	1	no	—
89	C₂×C₅₃	#447	1	1	yes	characteristic
90	D₅₃	—	8	1	no	—
91	#578	—	53	1	no	—
92	#509	—	53	1	no	—
93	C₂×SD₃₂	—	53	1	no	—
94	#472	D₅₃	1	1	yes	characteristic, complemented
95	#362	—	53	1	no	—
96	#359	—	53	1	no	—
97	#350	—	53	1	no	—
98	C₅₃	#2193	1	1	yes	characteristic, complemented, Sylow
99	2₋¹⁺⁴	—	106	1	no	—
100	#137	—	106	2	no	—
101	#137	—	106	1	no	—
102	AGL(1,ℤ/8)	—	2	1	no	—
103	#135	—	53	1	no	—
104	#135	—	106	1	no	—
105	C₂×Q₁₆	—	53	1	no	—
106	C₂×Q₁₆	—	106	1	no	—
107	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
108	#131	C₂×D₅₃	1	1	yes	characteristic
109	SD₃₂	—	106	2	no	—
110	SD₃₂	—	106	1	no	—
111	D₁₆	—	106	1	no	—
112	M₃₂	—	53	1	no	—
113	C₂×C₁₆	—	53	1	no	—
114	#108	C₂×D₅₃	1	1	yes	characteristic
115	#100	—	2	1	no	—
116	#41	—	106	2	no	—
117	#41	—	106	1	no	—
118	#41	—	212	2	no	—
119	#41	—	212	1	no	—
120	#41	C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂×Q₈	—	106	2	no	—
122	C₂×Q₈	—	106	1	no	—
123	C₂×Q₈	—	212	1	no	—
124	C₂×D₄	—	2	1	no	—
125	Q₁₆	—	53	2	no	—
126	Q₁₆	—	53	1	no	—
127	Q₁₆	—	106	2	no	—
128	Q₁₆	—	106	1	no	—
129	SD₁₆	—	4	1	no	—
130	SD₁₆	—	106	2	no	—
131	SD₁₆	—	106	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	4	1	no	—
135	D₈	—	53	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	2	1	no	—
138	M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₁₆	—	106	1	no	—
142	C₁₆	—	106	1	no	—
143	C₂³	—	4	1	no	—
144	Q₈	—	106	2	no	—
145	Q₈	—	106	1	no	—
146	Q₈	—	212	2	no	—
147	Q₈	—	212	1	no	—
148	Q₈	#11719922	1	1	yes	characteristic
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	106	1	no	—
153	D₄	—	212	2	no	—
154	D₄	—	212	1	no	—
155	D₄	#11719922	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	212	2	no	—
158	C₂×C₄	—	212	2	no	—
159	C₂×C₄	—	212	1	no	—
160	C₂×C₄	—	212	1	no	—
161	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	4	1	no	—
164	C₈	D₄×D₅₃	1	1	yes	characteristic
165	C₈	D₄×D₅₃	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	212	1	no	—
170	C₂²	#13133242	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	212	2	no	—
173	C₄	—	212	1	no	—
174	C₄	#13133185	1	1	yes	characteristic
175	C₄	C₂×#11719922	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	4	1	no	—
178	C₂	—	16	1	no	—
179	C₂	—	424	1	no	—
180	C₂	#16810756	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810756
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#472	C₂
socle	C₂×C₅₃	#447
hypercenter	#472	D₅₃
hypocenter	C₅₃	#2193
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #472

Complements

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

N	H ≅ G/N	type
1. #16811440	C₂	semidirect
2. #16811059	C₂	semidirect
3. C₂×#13133161	C₂	semidirect
4. #16810438	C₂	semidirect
5. #16810429	C₂	semidirect
6. #16810407	C₂	semidirect
7. C₅₃×#472	C₂	semidirect
94. #472	D₅₃	semidirect
98. C₅₃	#2193	semidirect