Finite Group Databasebeta

← #20275751

Subgroups — #20275751

Order 6784.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#13133248C211yesmaximal, characteristic
2#16811197C211yesmaximal, characteristic
3C2×#13133145C211yesmaximal, characteristic
4#16810632C211yesmaximal, characteristic, complemented
5C2×#13133136C211yesmaximal, characteristic, complemented
6#16810383C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C53C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133248—22no—
10#13133248—22no—
11C2×#11719925—21no—
12C2×#11719924—21no—
13C2×#11719917C2211yescharacteristic
14C2×#11719917C2211yescharacteristic
15#13133215—21no—
16#13133206—21no—
17#13133200—22no—
18#13133193—22no—
19#13133176C2211yescharacteristic
20#13133145—22no—
21#13133141—22no—
22#13133136—22no—
23C22×Q8×C53C2211yescharacteristic
24C22×Q8×C53C2211yescharacteristic
25C4⋊Q8×C53—22no—
26C4⋊Q8×C53—22no—
27C2×C4⋊C4×C53—22no—
28C2×C42×C53C2211yescharacteristic
29C2×Q8×C53—21no—
30C2×Q8×C53—21no—
31C2×Q8×C53—21no—
32C2×Q8×C53—21no—
33C2×Q8×C53—21no—
34C2×Q8×C53—21no—
35C2×Q8×C53—21no—
36C2×Q8×C53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—21no—
39C2×Q8×C53—21no—
40C2×Q8×C53D412yes—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53D412yes—
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—21no—
46#11719945—21no—
47#11719945—21no—
48#11719945—21no—
49#11719945—21no—
50C2×D212D411yescharacteristic
51C2×C4×D53—21no—
52C2×Dic106D411yescharacteristic
53C4⋊C4×C53—28no—
54C42×C53—21no—
55C42×C53D412yes—
56C22⋊Dic53—21no—
57#11719925—24no—
58#11719924—24no—
59#11719921—24no—
60C4⋊Dic53D412yes—
61#11719919—24no—
62C4×Dic53—21no—
63#11719917—21no—
64#11719917—21no—
65#11719917—21no—
66#11719917D412yes—
67C2×#88322—21no—
68C2×#88322—21no—
69C23×C53C2×D411yescharacteristic
70C22×D53—21no—
71Q8×C53—22no—
72Q8×C53—22no—
73Q8×C53—41no—
74Q8×C53—41no—
75Q8×C53—41no—
76Q8×C53—41no—
77Q8×C53—41no—
78Q8×C53—41no—
79C2×C4×C53—24no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53#4111yescharacteristic
93C22⋊D53—41no—
94C22⋊D53—41no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97D212—21no—
98D212—21no—
99C4×D53—41no—
100C4×D53—41no—
101Dic106—21no—
102Dic106—21no—
103#88322—22no—
104#88322—41no—
105C22×C53—22no—
106C22×C53#12011yescharacteristic
107C22×C53#12011yescharacteristic
108C22×C53#12111yescharacteristic
109C2×D53—41no—
110C2×D53—42no—
111C4×C53—24no—
112C4×C53—42no—
113C4×C53—42no—
114C4×C53#12011yescharacteristic
115C4×C53#12111yescharacteristic
116C4×C53#12111yescharacteristic
117C4×C53#12711yescharacteristic
118Dic53—41no—
119Dic53—42no—
120#2005—531nomaximal, Sylow
121C2×C53—21no—
122C2×C53—21no—
123C2×C53#39211yescharacteristic
124C2×C53#45612yes—
125D53—81no—
126C2×#137—531no—
127#536—531no—
128C2×C4⋊Q8D5311yescharacteristic, complemented
129C2×#104—531no—
130#419—531no—
131C2×#101—531no—
132#339—531no—
133C53#200511yescharacteristic, complemented, Sylow
134C2×#41—531no—
135C22×Q8C2×D5311yescharacteristic
136C22×Q8C2×D5311yescharacteristic
137#137—1062no—
138#137—1062no—
139C2×Q16—1061no—
140C2×SD16—1061no—
141C2×M16—531no—
142C2×M16—531no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—1061no—
146#124—1061no—
147#123—1062no—
148C22⋊Q8—1062no—
149#117—531no—
150C2×C4⋊C4—22no—
151C2×C42C2×D5311yescharacteristic
152#104—1062no—
153#103—1062no—
154#101—1062no—
155#41—1061no—
156#41—1061no—
157#41—1061no—
158#41—1061no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—531no—
171C2×Q8C22⋊D5312yes—
172C2×D4—531no—
173C2×D4—1061no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1061no—
177C22×C4C22⋊D5312yes—
178C22×C4C22×D5311yescharacteristic
179Q16—1064no—
180SD16—1064no—
181M16—532no—
182M16—1061no—
183M16—1061no—
184M16—1061no—
185C2×C8—1061no—
186C2×C8—1061no—
187C4⋊C4—28no—
188C4⋊C4—532no—
189C4⋊C4—1064no—
190C22⋊C4—1064no—
191C22⋊C4—1061no—
192C42—21no—
193C42—1061no—
194C42C22⋊D5312yes—
195C23—1061no—
196C23C2×C22⋊D5311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—1061no—
206Q8—1061no—
207D4—1061no—
208D4—1061no—
209D4—2121no—
210D4—2121no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—1062no—
220C2×C4—1061no—
221C2×C4—2121no—
222C2×C4—2121no—
223C2×C4D4×D5311yescharacteristic
224C2×C4D4×D5311yescharacteristic
225C2×C4D4×D5311yescharacteristic
226C2×C4#1171994711yescharacteristic
227C2×C4C2×C22⋊D5311yescharacteristic
228C2×C4C2×C22⋊D5311yescharacteristic
229C8—1062no—
230C8—2121no—
231C22—22no—
232C22—2121no—
233C22—2122no—
234C22#1313318511yescharacteristic
235C22#1313318611yescharacteristic
236C22#1313318911yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—2121no—
241C4—2122no—
242C4#1313318511yescharacteristic
243C4#1313318811yescharacteristic
244C4#1313318911yescharacteristic
245C4#1313321311yescharacteristic
246C2—21no—
247C2—21no—
248C2—4241no—
249C2#1681053111yescharacteristic
250C2#1681080512yes—
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC2×C4⋊Q8×C53C2
socleC22×C53#120
hypercenterC2×C4⋊Q8D53
hypocenterC53#2005
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #16810632C2semidirect
5. C2×#13133136C2semidirect
6. #16810383C2semidirect
7. C2×C4⋊Q8×C53C2semidirect
128. C2×C4⋊Q8D53semidirect
133. C53#2005semidirect