Finite Group Databasebeta

← #20275201

Subgroups — #20275201

Order 6784.

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

normal subgroups only

This group has 346 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
1C2×#13133245C211yesmaximal, characteristic
2#16811097C211yesmaximal, characteristic, complemented
3#16810639C211yesmaximal, characteristic, complemented
4#16810629C211yesmaximal, characteristic, complemented
5#16810596C211yesmaximal, characteristic, complemented
6#16810380C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×D212C2211yescharacteristic
10#13133245—22no—
11#13133245—22no—
12C2×#11719913C412yescomplemented
13C2×D424C412yescomplemented
14C4×D212—22no—
15#13133179—22no—
16#13133176C2211yescharacteristic
17#13133176C2211yescharacteristic
18C22⋊C4×D53—22no—
19#13133143—22no—
20#13133139—22no—
21#13133131—22no—
22C2×M16×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24#117×C53C2211yescharacteristic
25C23×D53—21no—
26C22×C4×C53C2311yescharacteristic
27C2×C22⋊D53—21no—
28#11719945—21no—
29#11719945—21no—
30#11719945—21no—
31#11719945—21no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212C2×C412yes—
36C2×D212C2×C411yescharacteristic, complemented
37C2×D212D412yes—
38C2×D212D412yes—
39C2×C4×D53—21no—
40C2×C4×D53—24no—
41C2×Dic106C2×C411yescharacteristic, complemented
42M16×C53—21no—
43M16×C53—21no—
44M16×C53—21no—
45M16×C53—21no—
46C2×C8×C53—21no—
47C2×C8×C53C2×C412yescomplemented
48C4⋊C4×C53—21no—
49C22⋊C4×C53D412yes—
50C42×C53—21no—
51C22⋊Dic53—21no—
52C22⋊Dic53—21no—
53#11719921—24no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4×Dic53—21no—
57C4×Dic53D412yes—
58D424—24no—
59#11719913—24no—
60C23×C53C2×D411yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—21no—
65C22×D53—82no—
66C2×C4×C53—22no—
67C2×C4×C53C22×C411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C2×C4×C53#4111yescharacteristic
73C22⋊D53—41no—
74C22⋊D53—41no—
75C2×Dic53—21no—
76C2×Dic53—22no—
77C2×Dic53—41no—
78D212—21no—
79D212—22no—
80D212—21no—
81D212—22no—
82D212—22no—
83D212C22⋊C414yes—
84C4×D53—41no—
85C4×D53—41no—
86C4×D53—84no—
87Dic106—21no—
88Dic106—21no—
89C8×C53—22no—
90C8×C53—41no—
91C22×C53—41no—
92C22×C53C4×D411yescharacteristic
93C22×C53#12011yescharacteristic
94C22×C53#12111yescharacteristic
95C2×D53—24no—
96C2×D53—44no—
97C2×D53—41no—
98C2×D53—82no—
99C2×D53—82no—
100C2×D53—81no—
101C4×C53—42no—
102C4×C53C2×C22⋊C411yescharacteristic
103C4×C53C4×D411yescharacteristic
104C4×C53#12111yescharacteristic
105C4×C53#12311yescharacteristic
106Dic53—42no—
107Dic53—41no—
108Dic53—81no—
109#1904—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—21no—
113C2×C53#38611yescharacteristic
114D53—44no—
115D53—81no—
116C2×AGL(1,ℤ/8)—531no—
117#518—531no—
118#421—531no—
119#418—531no—
120#410—531no—
121#344D5311yescharacteristic, complemented
122H(3,ℤ/4)—531no—
123C53#190411yescharacteristic, complemented, Sylow
124C2×#41—531no—
125C22×D4—531no—
126AGL(1,ℤ/8)—1062no—
127AGL(1,ℤ/8)—1062no—
128C2×SD16—532no—
129C2×D8—532no—
130C2×M16C2×D5311yescharacteristic
131C2×M16C2×D5311yescharacteristic
132C4×D4—1062no—
133C4×D4—1062no—
134#117—531no—
135#117—531no—
136#117C2×D5311yescharacteristic
137C2×C22⋊C4—1062no—
138#104—1062no—
139#102—1062no—
140#99—1062no—
141C24—1061no—
142#41—1061no—
143#41—1061no—
144#41—1061no—
145#41—1061no—
146C2×Q8—531no—
147C2×D4—532no—
148C2×D4—531no—
149C2×D4—532no—
150C2×D4—532no—
151C2×D4—1061no—
152C2×D4—1061no—
153C2×D4—1061no—
154C2×D4—1061no—
155C22×C4—1064no—
156C22×C4—1061no—
157C22×C4C22×D5311yescharacteristic
158SD16—1064no—
159D8—1064no—
160M16—21no—
161M16—21no—
162M16—21no—
163M16—21no—
164C2×C8—21no—
165C2×C8C4×D5312yescomplemented
166C4⋊C4—21no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C22⋊C4—1064no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4D21212yes—
173C42—21no—
174C42—532no—
175C42—1061no—
176C23—1062no—
177C23—1062no—
178C23—1062no—
179C23—1061no—
180C23—4242no—
181C23C2×D21211yescharacteristic
182Q8—1061no—
183Q8—1061no—
184D4—534no—
185D4—1061no—
186D4—1062no—
187D4—1061no—
188D4—1062no—
189D4—1062no—
190D4—2121no—
191D4—2121no—
192C2×C4—22no—
193C2×C4—1061no—
194C2×C4—1062no—
195C2×C4—4244no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4—2121no—
199C2×C4C2×C4×D5311yescharacteristic
200C2×C4#1171994511yescharacteristic
201C2×C4D4×D5311yescharacteristic
202C2×C4D4×D5311yescharacteristic
203C2×C4D4×D5311yescharacteristic
204C2×C4#1171994911yescharacteristic
205C8—22no—
206C8—41no—
207C22—41no—
208C22—1064no—
209C22—4241no—
210C22—4242no—
211C22—4242no—
212C22—2124no—
213C22—2121no—
214C22C4×D21211yescharacteristic
215C22#1313318711yescharacteristic
216C22#1313319211yescharacteristic
217C4—42no—
218C4—4241no—
219C4—2122no—
220C4—2121no—
221C4C22⋊C4×D5311yescharacteristic
222C4#1313317911yescharacteristic
223C4#1313319011yescharacteristic
224C4#1313320211yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—4241no—
229C2—2124no—
230C2#1681050111yescharacteristic
subgroupG/H
centerC2#16810501
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#386
hypercenter#344D53
hypocenterC53#1904
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #16811097C2semidirect
3. #16810639C2semidirect
4. #16810629C2semidirect
5. #16810596C2semidirect
6. #16810380C2semidirect
7. C53×#344C2semidirect
12. C2×#11719913C4semidirect
13. C2×D424C4semidirect
36. C2×D212C2×C4semidirect
41. C2×Dic106C2×C4semidirect
47. C2×C8×C53C2×C4semidirect
121. #344D53semidirect
123. C53#1904semidirect
165. C2×C8C4×D53semidirect