Finite Group Databasebeta

Subgroups — #20274063

Order 6784.

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

normal subgroups only

This group has 172 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	#16810996	C₂	1	1	yes	maximal, characteristic
2	#16810908	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810823	C₂	1	1	yes	maximal, characteristic
4	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810370	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719913	—	2	1	no	—
9	C₂×#11719913	—	2	1	no	—
10	C₂×#11719924	—	2	1	no	—
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133214	C₂²	1	1	yes	characteristic
13	#13133148	—	2	1	no	—
14	C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₄⋊#88322	C₂²	1	1	yes	characteristic
16	#13133141	—	2	1	no	—
17	#13133137	—	2	1	no	—
18	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₅₃	—	2	1	no	—
20	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₅₃	—	2	1	no	—
22	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×D₂₁₂	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×Dic₁₀₆	—	2	1	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₄²×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719924	—	4	2	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	#11719913	—	4	2	no	—
41	#11719913	—	4	2	no	—
42	C₂²×D₅₃	—	4	1	no	—
43	Q₈×C₅₃	—	2	1	no	—
44	Q₈×C₅₃	SD₁₆	1	2	yes	complemented
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂×Dic₅₃	—	4	1	no	—
51	D₂₁₂	—	4	1	no	—
52	D₂₁₂	—	4	1	no	—
53	D₂₁₂	—	4	1	no	—
54	D₂₁₂	—	4	1	no	—
55	D₂₁₂	—	8	1	no	—
56	Dic₁₀₆	—	4	1	no	—
57	Dic₁₀₆	—	4	1	no	—
58	C₈×C₅₃	—	2	1	no	—
59	C₈×C₅₃	—	4	1	no	—
60	C₈×C₅₃	SD₁₆	1	2	yes	complemented
61	#88322	—	4	1	no	—
62	#88322	—	4	1	no	—
63	C₂²×C₅₃	#121	1	1	yes	characteristic
64	C₂×D₅₃	—	8	1	no	—
65	C₂×D₅₃	—	8	1	no	—
66	C₂×D₅₃	—	8	1	no	—
67	C₄×C₅₃	—	2	2	no	—
68	C₄×C₅₃	—	2	1	no	—
69	C₄×C₅₃	—	4	1	no	—
70	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₅₃	#135	1	1	yes	characteristic
73	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	Dic₅₃	—	8	1	no	—
75	#1666	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	#460	1	1	yes	characteristic
77	C₂×C₅₃	#465	1	1	yes	characteristic
78	C₂×C₅₃	#471	1	1	yes	characteristic
79	D₅₃	—	16	1	no	—
80	#492	—	53	1	no	—
81	#475	—	53	1	no	—
82	#460	—	53	1	no	—
83	C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
84	#334	—	53	1	no	—
85	#332	—	53	1	no	—
86	#331	—	53	1	no	—
87	C₅₃	#1666	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	106	1	no	—
89	C₂×SD₁₆	—	106	1	no	—
90	C₂×SD₁₆	—	106	1	no	—
91	C₄⋊Q₈	—	53	1	no	—
92	#127	—	53	1	no	—
93	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
94	#106	—	106	1	no	—
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	53	1	no	—
97	C₄⋊C₈	—	53	1	no	—
98	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
99	#103	—	106	1	no	—
100	#102	—	106	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×Q₈	—	106	1	no	—
104	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
105	C₂×D₄	—	106	1	no	—
106	C₂×D₄	—	106	1	no	—
107	C₂×D₄	—	106	1	no	—
108	SD₁₆	—	212	2	no	—
109	SD₁₆	—	212	2	no	—
110	SD₁₆	—	212	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	106	1	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	106	1	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
120	C₄²	—	2	1	no	—
121	C₄²	C₂²×D₅₃	1	1	yes	characteristic
122	C₂³	—	212	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	212	1	no	—
125	Q₈	—	212	1	no	—
126	Q₈	#11719913	1	2	yes	complemented
127	D₄	—	424	1	no	—
128	D₄	—	212	1	no	—
129	D₄	—	212	1	no	—
130	D₄	—	212	1	no	—
131	D₄	—	212	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	212	1	no	—
135	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
136	C₂×C₄	#11719945	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	4	1	no	—
140	C₈	—	212	1	no	—
141	C₈	—	212	1	no	—
142	C₈	#11719924	1	2	yes	complemented
143	C₂²	—	424	1	no	—
144	C₂²	—	424	1	no	—
145	C₂²	—	424	1	no	—
146	C₂²	#13133191	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	424	1	no	—
151	C₄	C₂×#11719924	1	1	yes	characteristic
152	C₄	C₂×#11719913	1	1	yes	characteristic
153	C₄	#13133239	1	1	yes	characteristic
154	C₄	#13133242	1	1	yes	characteristic
155	C₂	—	848	1	no	—
156	C₂	#16810823	1	1	yes	characteristic
157	C₂	#16810852	1	1	yes	characteristic
158	C₂	#16810886	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133191
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₈×Q₈	D₅₃
hypocenter	C₅₃	#1666
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #16810908	C₂	semidirect
4. #16810374	C₂	semidirect
5. #16810370	C₂	semidirect
7. C₈×Q₈×C₅₃	C₂	semidirect
44. Q₈×C₅₃	SD₁₆	semidirect
60. C₈×C₅₃	SD₁₆	semidirect
83. C₈×Q₈	D₅₃	semidirect
87. C₅₃	#1666	semidirect
126. Q₈	#11719913	semidirect
142. C₈	#11719924	semidirect