Finite Group Databasebeta

← #20275078

Subgroups — #20275078

Order 6784.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#16811423C211yesmaximal, characteristic
2#16811389C211yesmaximal, characteristic, complemented
3#16810639C211yesmaximal, characteristic, complemented
4#16810620C211yesmaximal, characteristic, complemented
5#16810589C211yesmaximal, characteristic, complemented
6#16810395C211yesmaximal, characteristic, complemented
7C53×#341C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C2×#11719945C2211yescharacteristic
10#13133250—21no—
11#13133250C412yescomplemented
12#13133245—21no—
13#13133245C412yescomplemented
14#13133239—22no—
15#13133225—22no—
16#13133223—22no—
17C2×#11719917C2211yescharacteristic
18M16×D53—22no—
19#13133176C2211yescharacteristic
20#13133143—22no—
21#13133142—21no—
22#13133139—21no—
23#13133129—22no—
24C2×M16×C53C2211yescharacteristic
25C2×M16×C53C2211yescharacteristic
26C22×C8×C53C2211yescharacteristic
27C22×C4×C53C2311yescharacteristic
28C2×C22⋊D53—21no—
29C2×C22⋊D53—21no—
30#11719945—21no—
31#11719945—21no—
32#11719945—21no—
33#11719945—21no—
34#11719945—21no—
35#11719945—21no—
36#11719945C2×C412yes—
37#11719945D412yes—
38C2×D212C2×C411yescharacteristic
39C2×D212D411yescharacteristic
40C2×C4×D53—21no—
41C2×C4×D53—21no—
42C2×Dic106C2×C411yescharacteristic
43C2×Dic106D411yescharacteristic
44M16×C53—21no—
45M16×C53—21no—
46M16×C53C2×C412yes—
47M16×C53D412yes—
48C2×C8×C53—21no—
49C2×C8×C53—21no—
50C2×C8×C53—21no—
51C2×C8×C53—21no—
52C2×C8×C53—21no—
53C22⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4×Dic53—21no—
56#11719917—21no—
57#11719917—21no—
58C2×#88322D412yes—
59Dic212—22no—
60D424—22no—
61#11719913—22no—
62#11719913—22no—
63#11719912—24no—
64C8×D53—24no—
65C23×C53#4111yescharacteristic
66C22×D53—21no—
67C22×D53—21no—
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53#4111yescharacteristic
74C22⋊D53—22no—
75C22⋊D53—22no—
76C22⋊D53—41no—
77C22⋊D53—41no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—41no—
81D212—21no—
82D212—21no—
83D212—21no—
84D212C22⋊C412yes—
85C4×D53—22no—
86C4×D53—22no—
87C4×D53—41no—
88C4×D53—41no—
89Dic106—21no—
90Dic106—21no—
91Dic106—21no—
92Dic106C22⋊C412yes—
93C8×C53—22no—
94C8×C53—22no—
95C8×C53—41no—
96#88322—22no—
97C22×C53—41no—
98C22×C53C4×D411yescharacteristic
99C22×C53#12111yescharacteristic
100C22×C53#12311yescharacteristic
101C2×D53—22no—
102C2×D53—42no—
103C2×D53—41no—
104C2×D53—81no—
105C4×C53C2×C22⋊C411yescharacteristic
106C4×C53C4×D411yescharacteristic
107C4×C53#12011yescharacteristic
108C4×C53#12111yescharacteristic
109Dic53—22no—
110Dic53—41no—
111Dic53—81no—
112#1881—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#38611yescharacteristic
117D53—42no—
118D53—81no—
119#575—531no—
120#568—531no—
121#421—531no—
122#416—531no—
123#408—531no—
124#344—531no—
125#341D5311yescharacteristic, complemented
126C53#188111yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C2×#41—531no—
129#137—532no—
130#137—1061no—
131AGL(1,ℤ/8)—532no—
132AGL(1,ℤ/8)—1061no—
133#135—1062no—
134#131—1062no—
135#131—1062no—
136C2×M16—531no—
137C2×M16—1062no—
138C2×M16C2×D5311yescharacteristic
139C2×M16C2×D5311yescharacteristic
140C22×C8C2×D5311yescharacteristic
141#117—531no—
142#104—1062no—
143#103—1061no—
144#102—1061no—
145C22⋊C8—1062no—
146#41—532no—
147#41—532no—
148#41—1061no—
149#41—1061no—
150#41—1061no—
151#41—1061no—
152#41—1061no—
153#41—1061no—
154C2×Q8—531no—
155C2×Q8—531no—
156C2×D4—531no—
157C2×D4—531no—
158C2×D4—1061no—
159C2×D4—1061no—
160C22×C4—1061no—
161C22×C4—1061no—
162C22×C4C22×D5311yescharacteristic
163Q16—1062no—
164SD16—1062no—
165SD16—1062no—
166D8—1062no—
167M16—21no—
168M16—21no—
169M16—1064no—
170M16—1061no—
171M16—1061no—
172M16C4×D5312yes—
173M16D21212yes—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—532no—
180C2×C8—1064no—
181C4⋊C4—1061no—
182C22⋊C4—1061no—
183C42—1061no—
184C23—1061no—
185C23—1061no—
186C23#1171994911yescharacteristic
187Q8—532no—
188Q8—1061no—
189Q8—1061no—
190Q8—1061no—
191D4—532no—
192D4—1062no—
193D4—1062no—
194D4—1061no—
195D4—1061no—
196D4—1061no—
197D4—2121no—
198D4—2121no—
199C2×C4—1061no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1061no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4C2×C4×D5311yescharacteristic
207C2×C4C2×D21211yescharacteristic
208C2×C4#1171994511yescharacteristic
209C2×C4D4×D5311yescharacteristic
210C2×C4D4×D5311yescharacteristic
211C2×C4D4×D5311yescharacteristic
212C8—22no—
213C8—22no—
214C8—41no—
215C8—1062no—
216C22—41no—
217C22—1062no—
218C22—4241no—
219C22—2122no—
220C22—2121no—
221C22#1313317911yescharacteristic
222C22#1313319211yescharacteristic
223C22#1313320211yescharacteristic
224C4—1062no—
225C4—4241no—
226C4—2121no—
227C4C22⋊C4×D5311yescharacteristic
228C4C4×D21211yescharacteristic
229C4#1313318711yescharacteristic
230C4#1313319011yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—4241no—
235C2—2122no—
236C2#1681050111yescharacteristic
subgroupG/H
centerC4#13133187
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#341C2
socleC2×C53#386
hypercenter#341D53
hypocenterC53#1881
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #341

Complements

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

NH ≅ G/Ntype
2. #16811389C2semidirect
3. #16810639C2semidirect
4. #16810620C2semidirect
5. #16810589C2semidirect
6. #16810395C2semidirect
7. C53×#341C2semidirect
11. #13133250C4semidirect
13. #13133245C4semidirect
125. #341D53semidirect
126. C53#1881semidirect