Finite Group Databasebeta

← #20275077

Subgroups — #20275077

Order 6784.

310 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 89 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#16811421C211yesmaximal, characteristic
2#16810639C211yesmaximal, characteristic, complemented
3#16810619C211yesmaximal, characteristic, complemented
4#16810591C211yesmaximal, characteristic, complemented
5#16810394C211yesmaximal, characteristic, complemented
6#16810387C211yesmaximal, characteristic, complemented
7C53×#568C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133248—21no—
10#13133247C412yescomplemented
11#13133243—21no—
12#13133242C412yescomplemented
13#13133237—22no—
14C2×#11719917C2211yescharacteristic
15C22×#88322C2211yescharacteristic
16#13133176C2211yescharacteristic
17#13133143—22no—
18#13133141—21no—
19#13133138—21no—
20#13133129—22no—
21C2×#41×C53C2211yescharacteristic
22#131×C53—22no—
23#131×C53—22no—
24C2×M16×C53—22no—
25C2×M16×C53C2211yescharacteristic
26C2×M16×C53C2211yescharacteristic
27#41×C53—21no—
28#41×C53—21no—
29#41×C53C2×C412yes—
30#41×C53D412yes—
31C2×Q8×C53D411yescharacteristic
32C2×D4×C53—21no—
33C2×D4×C53D411yescharacteristic
34C22×C4×C53—21no—
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—21no—
37#11719945—21no—
38#11719945—21no—
39#11719945—21no—
40#11719945—21no—
41C2×D212C2×C411yescharacteristic
42C2×C4×D53—21no—
43C2×Dic106C2×C411yescharacteristic
44M16×C53—24no—
45M16×C53—21no—
46M16×C53—21no—
47M16×C53—21no—
48M16×C53D412yes—
49C2×C8×C53—24no—
50C2×C8×C53—21no—
51C2×C8×C53D412yes—
52C22⋊Dic53—21no—
53#11719925—22no—
54#11719924—22no—
55#11719923—22no—
56#11719922—22no—
57C4⋊Dic53—21no—
58C4×Dic53—21no—
59#11719917—21no—
60#11719917C2×C412yes—
61C2×#88322—21no—
62C2×#88322—21no—
63C2×#88322—21no—
64C2×#88322—21no—
65C23×C53—21no—
66C23×C53#4111yescharacteristic
67C22×D53—21no—
68Q8×C53—21no—
69Q8×C53C22⋊C412yes—
70D4×C53—22no—
71D4×C53—22no—
72D4×C53—21no—
73D4×C53C22⋊C412yes—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53C22×C411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53#4111yescharacteristic
83C22⋊D53—41no—
84C22⋊D53—41no—
85C2×Dic53—21no—
86C2×Dic53—41no—
87D212—21no—
88D212—21no—
89C4×D53—41no—
90C4×D53—41no—
91Dic106—21no—
92Dic106—21no—
93C8×C53—22no—
94C8×C53—22no—
95#88322—22no—
96#88322—41no—
97C22×C53—22no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53C4×D411yescharacteristic
101C22×C53#12111yescharacteristic
102C22×C53#12311yescharacteristic
103C2×D53—42no—
104C2×D53—41no—
105C4×C53—22no—
106C4×C53C2×C22⋊C411yescharacteristic
107C4×C53C4×D411yescharacteristic
108C4×C53#12011yescharacteristic
109C4×C53#12111yescharacteristic
110Dic53—41no—
111Dic53—81no—
112#1881—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53—42no—
117C2×C53#38611yescharacteristic
118D53—81no—
119#575—531no—
120#568D5311yescharacteristic, complemented
121#421—531no—
122#416—531no—
123#408—531no—
124#344—531no—
125#341—531no—
126C53#188111yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C2×#41C2×D5311yescharacteristic
129#137—532no—
130#137—1061no—
131AGL(1,ℤ/8)—532no—
132AGL(1,ℤ/8)—1061no—
133#135—1062no—
134#131—22no—
135#131—22no—
136C2×M16—22no—
137C2×M16—531no—
138C2×M16C2×D5311yescharacteristic
139C2×M16C2×D5311yescharacteristic
140C22×C8—531no—
141#117—531no—
142#104—1062no—
143#103—1061no—
144#102—1061no—
145C22⋊C8—1062no—
146#41—21no—
147#41—21no—
148#41—1061no—
149#41—1061no—
150#41—1061no—
151#41—1061no—
152#41C4×D5312yes—
153#41C22⋊D5312yes—
154C2×Q8—531no—
155C2×Q8D21211yescharacteristic
156C2×D4—21no—
157C2×D4—531no—
158C2×D4—1061no—
159C2×D4D21211yescharacteristic
160C22×C4—21no—
161C22×C4—1061no—
162C22×C4C22×D5311yescharacteristic
163Q16—1062no—
164SD16—1062no—
165SD16—1062no—
166D8—1062no—
167M16—24no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—532no—
172M16—1061no—
173M16C22⋊D5312yes—
174C2×C8—24no—
175C2×C8—21no—
176C2×C8—1061no—
177C2×C8—1061no—
178C2×C8—1061no—
179C2×C8—1061no—
180C2×C8C22⋊D5312yes—
181C4⋊C4—1061no—
182C22⋊C4—1061no—
183C42—1061no—
184C23—21no—
185C23—1061no—
186C23#1171994511yescharacteristic
187Q8—21no—
188Q8—1061no—
189Q8—1061no—
190Q8#1171992112yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—1061no—
195D4—1061no—
196D4—2121no—
197D4—2121no—
198D4#1171992112yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—1061no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4C2×C4×D5311yescharacteristic
207C2×C4C2×D21211yescharacteristic
208C2×C4#1171994511yescharacteristic
209C2×C4C2×C22⋊D5311yescharacteristic
210C2×C4C2×C22⋊D5311yescharacteristic
211C2×C4C2×C22⋊D5311yescharacteristic
212C8—22no—
213C8—22no—
214C8—1062no—
215C8—2121no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—2122no—
220C22—2121no—
221C22C4×C22⋊D5311yescharacteristic
222C22#1313319111yescharacteristic
223C22#1313320011yescharacteristic
224C4—22no—
225C4—4241no—
226C4—2121no—
227C4C2×#1171992111yescharacteristic
228C4C4×C22⋊D5311yescharacteristic
229C4#1313318611yescharacteristic
230C4#1313319111yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—4241no—
236C2#1681050211yescharacteristic
subgroupG/H
centerC4#13133186
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#568C2
socleC2×C53#386
hypercenter#568D53
hypocenterC53#1881
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #16810639C2semidirect
3. #16810619C2semidirect
4. #16810591C2semidirect
5. #16810394C2semidirect
6. #16810387C2semidirect
7. C53×#568C2semidirect
10. #13133247C4semidirect
12. #13133242C4semidirect
120. #568D53semidirect
126. C53#1881semidirect