Finite Group Databasebeta

← #20276367

Subgroups — #20276367

Order 6784.

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

normal subgroups only

This group has 234 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

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811195C211yesmaximal, characteristic
2#16810604C211yesmaximal, characteristic
3#16810595C211yesmaximal, characteristic
4#16810414C211yesmaximal, characteristic, complemented
5#16810413C211yesmaximal, characteristic, complemented
6#16810412C211yesmaximal, characteristic, complemented
7C53×#352C211yesmaximal, characteristic, complemented
8C2×#11719947C2211yescharacteristic
9C22×Dic106C411yescharacteristic, complemented
10M16×D53—21no—
11#13133216—21no—
12#13133203—21no—
13#13133201C411yescharacteristic, complemented
14#13133200C411yescharacteristic, complemented
15#13133198C2211yescharacteristic
16#13133195—21no—
17#13133194—21no—
18#13133173—21no—
19#13133173C411yescharacteristic, complemented
20#13133135—21no—
21#13133134C2211yescharacteristic
22#13133132C2211yescharacteristic
23#13133131—21no—
24#123×C53C2211yescharacteristic
25#100×C53C2211yescharacteristic
26#99×C53C2211yescharacteristic
27C2×D4×C53C2311yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C2×C22⋊D53C2×C411yescharacteristic
30C2×C22⋊D53D411yescharacteristic
31C22×Dic53C2×C411yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C22×Dic53D411yescharacteristic
34#11719947—41no—
35#11719947—41no—
36C2×C4×D53D411yescharacteristic
37C2×Dic106—21no—
38C2×Dic106—21no—
39C2×Dic106—41no—
40C2×Dic106—41no—
41C2×Dic106D411yescharacteristic
42M16×C53—21no—
43C4⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53C2×C411yescharacteristic
47C22⋊Dic53—21no—
48C22⋊Dic53—21no—
49C22⋊Dic53C2×C411yescharacteristic
50#11719921—21no—
51#11719921—21no—
52C4⋊Dic53—21no—
53#11719919—21no—
54#11719919—21no—
55#11719919—21no—
56#11719919—41no—
57C4×Dic53—21no—
58C4×Dic53—41no—
59#11719917—21no—
60#11719912—41no—
61C8×D53—41no—
62C23×C53C22×C411yescharacteristic
63C23×C53C2×D411yescharacteristic
64C22×D53C22⋊C411yescharacteristic
65D4×C53—41no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53C2×D411yescharacteristic
71C22⋊D53—41no—
72C22⋊D53—41no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53C22⋊C411yescharacteristic
82C2×Dic53C22⋊C411yescharacteristic
83C2×Dic53C22⋊C411yescharacteristic
84C4×D53—21no—
85C4×D53—21no—
86Dic106—41no—
87Dic106—42no—
88Dic106—41no—
89Dic106—41no—
90C8×C53—41no—
91#88322—41no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—41no—
95C22×C53—41no—
96C22×C53C2×C22⋊C411yescharacteristic
97C2×D53—21no—
98C2×D53—41no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—41no—
102C4×C53—81no—
103Dic53—21no—
104Dic53—21no—
105Dic53—41no—
106Dic53—41no—
107Dic53—81no—
108Dic53#9911yescharacteristic
109Dic53#9911yescharacteristic
110#2123—531nomaximal, Sylow
111C2×C53—21no—
112C2×C53—41no—
113C2×C53—41no—
114C2×C53C2×#9911yescharacteristic
115D53—41no—
116#536—531no—
117#413—531no—
118#410—531no—
119#352—531no—
120#352—531no—
121#352—531no—
122#352D5311yescharacteristic, complemented
123C53#212311yescharacteristic, complemented, Sylow
124C2×#41—531no—
125C22×Q8—531no—
126C2×M16—1061no—
127C4⋊Q8—1061no—
128#124—1061no—
129#123—531no—
130#123—531no—
131#123—531no—
132#123C2×D5311yescharacteristic
133C22⋊Q8—1061no—
134C22⋊Q8—1061no—
135#117—531no—
136#117—1061no—
137#101—1061no—
138#100—531no—
139#100C2×D5311yescharacteristic
140#99—531no—
141#99—1061no—
142#99C2×D5311yescharacteristic
143#41—2121no—
144#41—2121no—
145C2×Q8—531no—
146C2×Q8—1061no—
147C2×Q8—1061no—
148C2×Q8—2121no—
149C2×Q8—2121no—
150C2×D4—531no—
151C2×D4—531no—
152C2×D4C22×D5311yescharacteristic
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4C4×D5311yescharacteristic
158M16—21no—
159M16—1061no—
160M16—2121no—
161C2×C8—2121no—
162C4⋊C4—21no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4—1061no—
167C4⋊C4—2121no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—531no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C22⋊C4—1061no—
174C22⋊C4—1061no—
175C22⋊C4C4×D5311yescharacteristic
176C42—1061no—
177C42—2121no—
178C23—531no—
179C23C2×C4×D5311yescharacteristic
180C23D4×D5311yescharacteristic
181Q8—2121no—
182Q8—2121no—
183Q8—2121no—
184Q8—2122no—
185D4—41no—
186D4—2121no—
187D4—2121no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—531no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—1061no—
196C2×C4—1061no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1061no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4D4×D5311yescharacteristic
206C8—41no—
207C8—2121no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—1061no—
213C22—2121no—
214C22C22⋊C4×D5311yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—531no—
220C4—531no—
221C4—1061no—
222C4—1061no—
223C4—4241no—
224C4—2121no—
225C4—2121no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—2121no—
230C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#352C2
socleC2×C53C2×#99
hypercenter#352D53
hypocenterC53#2123
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #16810414C2semidirect
5. #16810413C2semidirect
6. #16810412C2semidirect
7. C53×#352C2semidirect
9. C22×Dic106C4semidirect
13. #13133201C4semidirect
14. #13133200C4semidirect
19. #13133173C4semidirect
122. #352D53semidirect
123. C53#2123semidirect