Finite Group Databasebeta

Subgroups — #20274126

Order 6784.

176 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16810971	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810818	C₂	1	1	yes	maximal, characteristic
3	D₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810373	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810372	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719922	—	2	1	no	—
9	C₂²×#88322	—	2	1	no	—
10	#13133214	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₂²	1	1	yes	characteristic
13	#13133137	—	2	1	no	—
14	#13133137	—	2	1	no	—
15	#13133137	—	2	1	no	—
16	C₂²⋊#88322	—	2	1	no	—
17	C₄×#88322	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	#106×C₅₃	—	2	1	no	—
21	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#102×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂²⋊C₄×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719922	—	4	2	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	—	4	2	no	—
39	C₂×#88322	D₄	1	1	yes	characteristic
40	C₂×#88322	D₄	1	1	yes	characteristic
41	C₂³×C₅₃	—	2	1	no	—
42	C₂²×D₅₃	—	4	1	no	—
43	Q₈×C₅₃	—	8	1	no	—
44	D₄×C₅₃	—	2	1	no	—
45	D₄×C₅₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	4	1	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	—	4	1	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	#41	1	1	yes	characteristic
53	D₂₁₂	—	4	1	no	—
54	D₂₁₂	—	4	1	no	—
55	D₂₁₂	—	4	1	no	—
56	D₂₁₂	—	4	1	no	—
57	D₂₁₂	—	8	1	no	—
58	C₈×C₅₃	—	4	1	no	—
59	#88322	—	2	1	no	—
60	#88322	—	2	1	no	—
61	#88322	—	4	1	no	—
62	#88322	—	4	1	no	—
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	—	4	1	no	—
65	C₂²×C₅₃	—	4	1	no	—
66	C₂²×C₅₃	#121	1	1	yes	characteristic
67	C₂×D₅₃	—	8	1	no	—
68	C₂×D₅₃	—	8	1	no	—
69	C₂×D₅₃	—	8	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	4	1	no	—
72	C₄×C₅₃	—	8	1	no	—
73	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₅₃	#135	1	1	yes	characteristic
75	C₄×C₅₃	#135	1	1	yes	characteristic
76	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	#1677	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	—	4	2	no	—
79	C₂×C₅₃	#463	1	1	yes	characteristic
80	C₂×C₅₃	#465	1	1	yes	characteristic
81	C₂×C₅₃	#472	1	1	yes	characteristic
82	D₅₃	—	16	1	no	—
83	#486	—	53	1	no	—
84	#476	D₅₃	1	1	yes	characteristic, complemented
85	#459	—	53	1	no	—
86	C₈×D₄	—	53	1	no	—
87	#334	—	53	1	no	—
88	#333	—	53	1	no	—
89	#331	—	53	1	no	—
90	C₅₃	#1677	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	106	1	no	—
92	C₂²×C₈	—	106	1	no	—
93	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
94	#127	—	53	1	no	—
95	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
96	#106	—	2	1	no	—
97	C₄⋊C₈	—	53	1	no	—
98	C₄⋊C₈	—	53	1	no	—
99	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	106	1	no	—
102	#102	—	106	1	no	—
103	#102	—	106	1	no	—
104	C₂²⋊C₈	—	106	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	106	1	no	—
108	C₂×D₄	—	106	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	212	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	—	53	1	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	—	106	1	no	—
118	C₂×C₈	—	212	2	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₅₃	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	212	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	424	1	no	—
129	D₄	—	212	1	no	—
130	D₄	—	212	1	no	—
131	D₄	—	212	1	no	—
132	D₄	—	212	1	no	—
133	D₄	#11719924	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	#11719947	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	106	1	no	—
143	C₈	—	106	1	no	—
144	C₈	—	212	1	no	—
145	C₈	—	212	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	424	1	no	—
150	C₂²	—	424	1	no	—
151	C₂²	—	424	1	no	—
152	C₂²	#13133189	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	1	no	—
156	C₄	C₂×#11719924	1	1	yes	characteristic
157	C₄	#13133237	1	1	yes	characteristic
158	C₄	#13133238	1	1	yes	characteristic
159	C₄	#13133244	1	1	yes	characteristic
160	C₂	—	4	2	no	—
161	C₂	—	848	1	no	—
162	C₂	#16810840	1	1	yes	characteristic
163	C₂	#16810854	1	1	yes	characteristic
164	C₂	#16810891	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133189
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#476	C₂
socle	C₂²×C₅₃	#121
hypercenter	#476	D₅₃
hypocenter	C₅₃	#1677
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #16810971	C₂	semidirect
3. D₄×#88322	C₂	semidirect
4. #16810373	C₂	semidirect
5. #16810372	C₂	semidirect
6. #16810366	C₂	semidirect
7. C₅₃×#476	C₂	semidirect
84. #476	D₅₃	semidirect
90. C₅₃	#1677	semidirect