Finite Group Databasebeta

← #20275197

Subgroups — #20275197

Order 6784.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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#16811390C211yesmaximal, characteristic, complemented
2#16810640C211yesmaximal, characteristic, complemented
3#16810628C211yesmaximal, characteristic, complemented
4C2×#13133134C211yesmaximal, characteristic, complemented
5#16810394C211yesmaximal, characteristic, complemented
6#16810381C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic
8#13133224—22no—
9#13133224—22no—
10C2×#11719917—22no—
11C2×#11719917C2211yescharacteristic
12C2×#11719917C2211yescharacteristic
13C2×#11719917C2211yescharacteristic
14#13133176C2211yescharacteristic
15#13133144—22no—
16#13133138—22no—
17#13133134—22no—
18C2×#41×C53C2211yescharacteristic
19C22×D4×C53C2211yescharacteristic
20AGL(1,ℤ/8)×C53—22no—
21AGL(1,ℤ/8)×C53—22no—
22C2×SD16×C53C412yescomplemented
23C2×D8×C53C412yescomplemented
24C2×M16×C53C2211yescharacteristic
25C24×C53—21no—
26#41×C53—21no—
27#41×C53—21no—
28#41×C53D412yes—
29#41×C53D412yes—
30C2×Q8×C53C2×C411yescharacteristic
31C2×D4×C53—21no—
32C2×D4×C53—21no—
33C2×D4×C53—21no—
34C2×D4×C53—21no—
35C2×D4×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53C2×C412yes—
38C2×D4×C53C2×C411yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53C2311yescharacteristic
41SD16×C53—24no—
42D8×C53—24no—
43M16×C53—21no—
44M16×C53—21no—
45C2×C8×C53C2×C412yes—
46C22⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4×Dic53—21no—
49#11719917—24no—
50#11719917—21no—
51#11719917—21no—
52#11719917—21no—
53#11719917—21no—
54#11719917—21no—
55#11719917D412yes—
56C2×#88322—24no—
57C2×#88322—21no—
58C2×#88322—21no—
59C2×#88322D412yes—
60C23×C53—21no—
61C23×C53—22no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53—82no—
65C23×C53#4111yescharacteristic
66Q8×C53—21no—
67Q8×C53C22⋊C412yes—
68D4×C53—22no—
69D4×C53—22no—
70D4×C53—22no—
71D4×C53—21no—
72D4×C53—22no—
73D4×C53—41no—
74D4×C53—41no—
75D4×C53C22⋊C412yes—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53C22×C411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53#4111yescharacteristic
85C2×Dic53—41no—
86C8×C53—22no—
87#88322—22no—
88#88322—22no—
89#88322—41no—
90C22×C53—22no—
91C22×C53—41no—
92C22×C53—42no—
93C22×C53—81no—
94C22×C53—82no—
95C22×C53—82no—
96C22×C53—82no—
97C22×C53C4×D411yescharacteristic
98C22×C53#12111yescharacteristic
99C22×C53#12311yescharacteristic
100C4×C53—22no—
101C4×C53C2×C22⋊C411yescharacteristic
102C4×C53C4×D411yescharacteristic
103C4×C53#12011yescharacteristic
104C4×C53#12111yescharacteristic
105Dic53—81no—
106#1903—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53—21no—
110C2×C53—42no—
111C2×C53—82no—
112C2×C53#38611yescharacteristic
113C2×AGL(1,ℤ/8)D5311yescharacteristic
114#568—531no—
115#421—531no—
116#418—531no—
117C2×#100—531no—
118#344—531no—
119#338—531no—
120C53#190311yescharacteristic, complemented, Sylow
121C2×#41C2×D5311yescharacteristic
122C22×D4C2×D5311yescharacteristic
123AGL(1,ℤ/8)—22no—
124AGL(1,ℤ/8)—22no—
125C2×SD16Dic5312yescomplemented
126C2×D8Dic5312yescomplemented
127#131—1062no—
128#131—1062no—
129C2×M16—531no—
130C2×M16—531no—
131C2×M16—531no—
132C2×M16—1062no—
133C2×M16C2×D5311yescharacteristic
134#117—531no—
135#104—1062no—
136#102—1062no—
137#100—1062no—
138C24—21no—
139#41—21no—
140#41—21no—
141#41C22⋊D5312yes—
142#41C22⋊D5312yes—
143C2×Q8C2×Dic5311yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×Dic5312yes—
151C2×D4C2×Dic5311yescharacteristic
152C22×C4—21no—
153C22×C4C22×D5311yescharacteristic
154SD16—24no—
155D8—24no—
156M16—21no—
157M16—21no—
158M16—532no—
159M16—1064no—
160M16—1061no—
161M16—1061no—
162M16—1061no—
163M16—1061no—
164M16—1061no—
165C2×C8—532no—
166C2×C8—1064no—
167C2×C8—1061no—
168C2×C8—1061no—
169C2×C8C2×Dic5312yes—
170C4⋊C4—1061no—
171C22⋊C4—1061no—
172C42—1061no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—82no—
178C23#1171994711yescharacteristic
179Q8—21no—
180Q8C22⋊Dic5312yes—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—21no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊Dic5312yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—2121no—
193C2×C4D4×D5311yescharacteristic
194C2×C4D4×D5311yescharacteristic
195C2×C4#1171994711yescharacteristic
196C2×C4C22×Dic5311yescharacteristic
197C2×C4C2×C22⋊D5311yescharacteristic
198C2×C4C2×C22⋊D5311yescharacteristic
199C8—22no—
200C8—1062no—
201C8—1062no—
202C8—2121no—
203C22—22no—
204C22—41no—
205C22—42no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22D4×Dic5311yescharacteristic
211C22#1313318911yescharacteristic
212C22#1313319811yescharacteristic
213C4—22no—
214C4—4241no—
215C4C2×C22⋊Dic5311yescharacteristic
216C4D4×Dic5311yescharacteristic
217C4#1313318511yescharacteristic
218C4#1313318911yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—82no—
224C2#1681050311yescharacteristic
subgroupG/H
centerC2#16810503
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×AGL(1,ℤ/8)×C53C2
socleC2×C53#386
hypercenterC2×AGL(1,ℤ/8)D53
hypocenterC53#1903
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #16811390C2semidirect
2. #16810640C2semidirect
3. #16810628C2semidirect
4. C2×#13133134C2semidirect
5. #16810394C2semidirect
6. #16810381C2semidirect
22. C2×SD16×C53C4semidirect
23. C2×D8×C53C4semidirect
120. C53#1903semidirect
125. C2×SD16Dic53semidirect
126. C2×D8Dic53semidirect