Finite Group Databasebeta

← #28299437

Subgroups — #28299437

Order 13568.

528 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290429C212yesmaximal, complemented
2#20289535C212yesmaximal, complemented
3#20289531C212yesmaximal
4#124×Dic53C211yesmaximal, characteristic
5C2×C4×#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×Q8×Dic53C411yescharacteristic
9C2×D4×Dic53C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810526C2211yescharacteristic
12#16810516—22no—
13#16810489C411yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C2211yescharacteristic
16C2×#124×C53C411yescharacteristic
17C22×#88324—24no—
18C4×#88324—28no—
19C23×Dic53—21no—
20Q8×Dic53—42no—
21D4×Dic53—42no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C22⋊Dic53—21no—
26C2×C22⋊Dic53—22no—
27C2×C4×Dic53—21no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C2×C4×Dic53D412yes—
31C2×C4×Dic53C2311yescharacteristic
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53C2×C411yescharacteristic
36#124×C53—22no—
37#124×C53C814yescomplemented
38C2×C22⋊C4×C53—22no—
39C2×C42×C53C2×C411yescharacteristic
40C24×C53—21no—
41C2×Q8×C53—21no—
42C2×Q8×C53—41no—
43C2×Q8×C53—41no—
44C2×Q8×C53C2×C812yes—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53C2×C812yes—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53—42no—
55C22×Dic53—44no—
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D412yes—
59C2×#88324—28no—
60C2×#88324—44no—
61C2×#88324—44no—
62C22⋊C4×C53—28no—
63C42×C53C2×C812yes—
64C42×C53M1612yes—
65C22⋊Dic53—42no—
66C22⋊Dic53—44no—
67C4⋊Dic53—42no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4×Dic53—22no—
71C4×Dic53—24no—
72C4×Dic53—24no—
73C4×Dic53—41no—
74C4×Dic53—42no—
75C23×C53—22no—
76C23×C53—41no—
77C23×C53—41no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—42no—
82Q8×C53—42no—
83D4×C53—42no—
84D4×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53C22⋊C814yes—
93C2×C4×C53C22×C811yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—84no—
104C2×Dic53#10012yes—
105C2×Dic53#10414yes—
106C2×Dic53#13514yes—
107#88324—48no—
108#7567—531nomaximal, Sylow
109C22×C53—42no—
110C22×C53—44no—
111C22×C53—81no—
112C22×C53—82no—
113C22×C53—81no—
114C22×C53C2×C22⋊C811yescharacteristic
115C22×C53#41011yescharacteristic
116C22×C53C2×#10011yescharacteristic
117C22×C53#41612yes—
118C22×C53C2×#10412yes—
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—24no—
122C4×C53—42no—
123Dic53—24no—
124Dic53—41no—
125Dic53—81no—
126Dic53—81no—
127Dic53#32914yes—
128C4×#124—531no—
129#1744—532no—
130#1272—532no—
131#1270—532no—
132C2×C53—82no—
133C2×C53#146011yescharacteristic
134C2×C53#147612yes—
135C2×C53#150012yes—
136C2×C53C2×#32911yescharacteristic
137C2×C53#154311yescharacteristic
138C2×#124#615711yescharacteristic
139C2×C4×Q8—531no—
140C2×C4×D4—531no—
141C2×C4×C8—1062no—
142#391—531no—
143#390—531no—
144#388—1062no—
145#382—531no—
146C4×C22⋊C4—1062no—
147C43—531no—
148#336—534no—
149#336—1062no—
150C53#756711yescharacteristic, complemented, Sylow
151C22×Q8C2×#615711yescharacteristic
152C22×D4C2×#615711yescharacteristic
153C23×C4—1061no—
154C22×C8—1064no—
155#124—22no—
156#124#8832414yescomplemented
157C4×Q8—2122no—
158C4×D4—2122no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—1061no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—1061no—
164C2×C22⋊C4—1062no—
165C2×C42—532no—
166C2×C42—532no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C4×C8—1068no—
172#95—1062no—
173#95—1062no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8C2×#8832412yes—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4C2×#8832412yes—
183C22×C4—21no—
184C22×C4—531no—
185C22×C4—532no—
186C22×C4—531no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4—2122no—
190C22×C4—2124no—
191C22×C4C22⋊#615712yes—
192C22×C4C22×#615711yescharacteristic
193C2×C8—1068no—
194C2×C8—2124no—
195C2×C8—2124no—
196C4⋊C4—2121no—
197C4⋊C4—2121no—
198C4⋊C4—2122no—
199C22⋊C4—28no—
200C22⋊C4—2124no—
201C22⋊C4—2122no—
202C42—1064no—
203C42—1062no—
204C42—1064no—
205C42—2121no—
206C42—2122no—
207C42C2×#8832412yes—
208C42#1171994012yes—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊#615711yescharacteristic
215Q8—42no—
216Q8—42no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—532no—
227C2×C4—534no—
228C2×C4—534no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—4244no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4C22⋊#8832414yes—
239C2×C4C22×#8832411yescharacteristic
240C2×C4C2×#1171994011yescharacteristic
241C8—2128no—
242C22—42no—
243C22—44no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22C2×C22⋊#8832411yescharacteristic
248C22#1681160311yescharacteristic
249C22D53⋊#10011yescharacteristic
250C22#1681161312yes—
251C22D53⋊#10412yes—
252C4—21no—
253C4—21no—
254C4—24no—
255C4—42no—
256C4—534no—
257C4—1064no—
258C4—4241no—
259C4—4241no—
260C4—2121no—
261C2—82no—
262C2#2028995511yescharacteristic
263C2#2029000612yes—
264C2#2029008712yes—
265C2D53⋊#32911yescharacteristic
266C2#2029021911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×#124#6157
hypocenterC53#7567
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20290429C2semidirect
2. #20289535C2semidirect
37. #124×C53C8semidirect
150. C53#7567semidirect
156. #124#88324semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202904292
#202895352
#202895312
#124×Dic532
#756753