Finite Group Databasebeta

← #28299436

Subgroups — #28299436

Order 13568.

528 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 122 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#20290430C212yesmaximal, complemented
2#20289536C212yesmaximal, complemented
3#20289531C212yesmaximal
4C4×#13133204C211yesmaximal, characteristic
5C2×C4×#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133204C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10#16810528C2211yescharacteristic
11#16810526C2211yescharacteristic
12#16810517—22no—
13#16810488C411yescharacteristic
14C4×C22⋊Dic53—22no—
15C42×Dic53C2211yescharacteristic
16C2×C4×D4×C53C411yescharacteristic
17C22×#88324—24no—
18C4×#88324—28no—
19C22×Dic106C2×C411yescharacteristic
20#13133204—22no—
21#13133204C814yescomplemented
22C4×Dic106—42no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×#11719919—21no—
25C2×C22⋊Dic53—22no—
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×D4×C53C2×C411yescharacteristic
35C23×C4×C53—21no—
36C4×D4×C53—42no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C42×C53C2×C411yescharacteristic
40C24×C53—21no—
41C2×D4×C53—21no—
42C2×D4×C53—41no—
43C2×D4×C53—41no—
44C2×D4×C53C2×C812yes—
45C22×C4×C53—21no—
46C22×C4×C53—42no—
47C22×C4×C53—44no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C2×D412yes—
55C2×Dic106—21no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106C2×C812yes—
59C2×#88324—28no—
60C2×#88324—44no—
61C2×#88324—44no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—42no—
64C42×C53—22no—
65C22⋊Dic53—28no—
66C22⋊Dic53—44no—
67C4⋊Dic53—41no—
68#11719919—42no—
69C4×Dic53—24no—
70C4×Dic53—24no—
71C4×Dic53—41no—
72C4×Dic53—42no—
73C4×Dic53C2×C812yes—
74C4×Dic53M1612yes—
75C23×C53—22no—
76C23×C53—41no—
77C23×C53—41no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81D4×C53—42no—
82D4×C53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—84no—
90C2×C4×C53#10012yes—
91C2×C4×C53#10414yes—
92C2×C4×C53C22×C811yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53C22⋊C814yes—
104C2×Dic53#13514yes—
105Dic106—42no—
106Dic106—42no—
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—41no—
122C4×C53—81no—
123C4×C53#32914yes—
124Dic53—24no—
125Dic53—24no—
126Dic53—42no—
127Dic53—81no—
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—531no—
139C2×C4×Q8—531no—
140C2×C4×D4#615711yescharacteristic
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×Q8—531no—
152C22×D4C2×#615711yescharacteristic
153C23×C4—21no—
154C22×C8—1064no—
155#124—534no—
156#124—1062no—
157C4×Q8—2122no—
158C4×D4—42no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4C2×#615711yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—1062no—
164C2×C22⋊C4—1062no—
165C2×C42—531no—
166C2×C42—532no—
167C2×C42—532no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C4×C8—1068no—
172#95—1062no—
173#95—1062no—
174C24—21no—
175C2×Q8—532no—
176C2×Q8—1061no—
177C2×Q8—2121no—
178C2×Q8—2121no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4D53⋊C812yes—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—532no—
187C22×C4—532no—
188C22×C4—1061no—
189C22×C4—1061no—
190C22×C4C22⋊#615711yescharacteristic
191C22×C4C22⋊#615711yescharacteristic
192C22×C4C22×#615711yescharacteristic
193C2×C8—1068no—
194C2×C8—2124no—
195C2×C8—2124no—
196C4⋊C4—41no—
197C4⋊C4—2121no—
198C4⋊C4—2122no—
199C22⋊C4—42no—
200C22⋊C4—1068no—
201C22⋊C4—2124no—
202C42—22no—
203C42—532no—
204C42—532no—
205C42—1064no—
206C42—1064no—
207C42—2121no—
208C42—2122no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C2×C22⋊#615711yescharacteristic
215Q8—2122no—
216Q8—2122no—
217D4—42no—
218D4—42no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—84no—
226C2×C4—534no—
227C2×C4—534no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—2121no—
234C2×C4—2122no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4#1313327712yes—
238C2×C4#1313328114yes—
239C2×C4C2×D53⋊C811yescharacteristic
240C2×C4D53⋊M1611yescharacteristic
241C8—2128no—
242C22—42no—
243C22—44no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22D53⋊(C22⋊C8)11yescharacteristic
248C22#1681160211yescharacteristic
249C22C2×#1313327711yescharacteristic
250C22#1681161312yes—
251C22C2×#1313328112yes—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—81no—
256C4—1064no—
257C4—1064no—
258C4—4241no—
259C4—2122no—
260C4#1681147814yes—
261C2—82no—
262C2#2028995411yescharacteristic
263C2#2029000712yes—
264C2#2029008612yes—
265C2C2×#1681147811yescharacteristic
266C2#2029021811yescharacteristic
subgroupG/H
centerC22×C4C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×D4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×D4#6157
hypocenterC53#7567
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20290430C2semidirect
2. #20289536C2semidirect
21. #13133204C8semidirect
150. C53#7567semidirect

Maximal subgroups

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

subgroupindex
#202904302
#202895362
#202895312
C4×#131332042
#756753