Finite Group Databasebeta

← #20275196

Subgroups — #20275196

Order 6784.

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

normal subgroups only

This group has 324 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#16811389C211yesmaximal, characteristic, complemented
3#16810639C211yesmaximal, characteristic, complemented
4#16810629C211yesmaximal, characteristic, complemented
5C2×#13133133C211yesmaximal, characteristic, complemented
6#16810394C211yesmaximal, characteristic, complemented
7C53×#338C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×D212C2211yescharacteristic
10#13133245—22no—
11#13133245—22no—
12C2×#11719913C412yescomplemented
13C2×D424C412yescomplemented
14#13133225—22no—
15#13133223—22no—
16C2×#11719917C2211yescharacteristic
17M16×D53—22no—
18#13133176C2211yescharacteristic
19#13133143—22no—
20#13133139—22no—
21#13133133—22no—
22C2×M16×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24C2×M16×C53C2211yescharacteristic
25C23×D53—21no—
26C22×C4×C53C2311yescharacteristic
27C2×C22⋊D53—21no—
28#11719945—21no—
29#11719945—21no—
30#11719945D412yes—
31#11719945D412yes—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—21no—
37C2×D212C2×C412yes—
38C2×D212C2×C411yescharacteristic
39C2×C4×D53—21no—
40C2×Dic106C2×C411yescharacteristic
41M16×C53—21no—
42M16×C53—21no—
43M16×C53—21no—
44M16×C53—21no—
45M16×C53—21no—
46M16×C53—21no—
47C2×C8×C53—21no—
48C2×C8×C53C2×C412yes—
49C2×C8×C53D412yes—
50C22⋊Dic53—21no—
51C4⋊Dic53—21no—
52C4×Dic53—21no—
53#11719917—21no—
54#11719917D412yes—
55C2×#88322—21no—
56D424—24no—
57#11719913—24no—
58#11719912—24no—
59C8×D53—24no—
60C23×C53#4111yescharacteristic
61C22×D53—21no—
62C22×D53—22no—
63C22×D53—41no—
64C22×D53—41no—
65C22×D53—82no—
66C2×C4×C53C22×C411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C22⋊D53—22no—
73C22⋊D53—22no—
74C2×Dic53—21no—
75C2×Dic53—41no—
76D212—22no—
77D212—21no—
78D212—22no—
79D212—41no—
80D212—41no—
81D212C22⋊C412yes—
82C4×D53—22no—
83C4×D53—22no—
84Dic106—21no—
85Dic106C22⋊C412yes—
86C8×C53—22no—
87C8×C53—22no—
88C8×C53—41no—
89#88322—22no—
90C22×C53—41no—
91C22×C53C4×D411yescharacteristic
92C22×C53#12111yescharacteristic
93C22×C53#12311yescharacteristic
94C2×D53—22no—
95C2×D53—42no—
96C2×D53—81no—
97C2×D53—82no—
98C2×D53—82no—
99C2×D53—82no—
100C4×C53C2×C22⋊C411yescharacteristic
101C4×C53C4×D411yescharacteristic
102C4×C53#12011yescharacteristic
103C4×C53#12111yescharacteristic
104Dic53—22no—
105Dic53—81no—
106#1903—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—21no—
110C2×C53#38611yescharacteristic
111D53—42no—
112D53—82no—
113C2×AGL(1,ℤ/8)—531no—
114#568—531no—
115#421—531no—
116#418—531no—
117C2×#100—531no—
118#344—531no—
119#338D5311yescharacteristic, complemented
120C53#190311yescharacteristic, complemented, Sylow
121C2×#41—531no—
122C22×D4—531no—
123AGL(1,ℤ/8)—1062no—
124AGL(1,ℤ/8)—1062no—
125C2×SD16—532no—
126C2×D8—532no—
127#131—1062no—
128#131—1062no—
129C2×M16—531no—
130C2×M16—1062no—
131C2×M16C2×D5311yescharacteristic
132C2×M16C2×D5311yescharacteristic
133C2×M16C2×D5311yescharacteristic
134#117—531no—
135#104—1062no—
136#102—1062no—
137#100—1062no—
138C24—1061no—
139#41—532no—
140#41—532no—
141#41—1061no—
142#41—1061no—
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—532no—
146C2×D4—1061no—
147C2×D4—1061no—
148C2×D4—1061no—
149C2×D4—1061no—
150C2×D4—1061no—
151C2×D4—1061no—
152C22×C4—1061no—
153C22×C4C22×D5311yescharacteristic
154SD16—1064no—
155D8—1064no—
156M16—21no—
157M16—21no—
158M16—21no—
159M16—21no—
160M16—21no—
161M16—21no—
162M16—532no—
163M16—1064no—
164M16—1061no—
165C2×C8—21no—
166C2×C8—1064no—
167C2×C8—1061no—
168C2×C8C4×D5312yes—
169C2×C8D21212yes—
170C4⋊C4—1061no—
171C22⋊C4—1061no—
172C42—1061no—
173C23—1061no—
174C23—1062no—
175C23—4242no—
176C23—2121no—
177C23—2121no—
178C23#1171994911yescharacteristic
179Q8—532no—
180Q8—1061no—
181D4—532no—
182D4—1062no—
183D4—1062no—
184D4—1062no—
185D4—1061no—
186D4—1062no—
187D4—2121no—
188D4—2121no—
189C2×C4—1061no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—2121no—
193C2×C4C2×C4×D5311yescharacteristic
194C2×C4C2×D21211yescharacteristic
195C2×C4#1171994511yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C2×C4D4×D5311yescharacteristic
198C2×C4D4×D5311yescharacteristic
199C8—22no—
200C8—22no—
201C8—41no—
202C8—1062no—
203C22—41no—
204C22—1062no—
205C22—4242no—
206C22—4242no—
207C22—4242no—
208C22—4241no—
209C22—2122no—
210C22#1313317911yescharacteristic
211C22#1313319211yescharacteristic
212C22#1313320211yescharacteristic
213C4—1062no—
214C4—4241no—
215C4C22⋊C4×D5311yescharacteristic
216C4C4×D21211yescharacteristic
217C4#1313318711yescharacteristic
218C4#1313319011yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—4242no—
223C2—2122no—
224C2#1681050111yescharacteristic
subgroupG/H
centerC2#16810501
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#338C2
socleC2×C53#386
hypercenter#338D53
hypocenterC53#1903
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #338

Complements

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

NH ≅ G/Ntype
2. #16811389C2semidirect
3. #16810639C2semidirect
4. #16810629C2semidirect
5. C2×#13133133C2semidirect
6. #16810394C2semidirect
7. C53×#338C2semidirect
12. C2×#11719913C4semidirect
13. C2×D424C4semidirect
119. #338D53semidirect
120. C53#1903semidirect