Finite Group Databasebeta

← #27271302

Subgroups — #27271302

Order 13568.

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

normal subgroups only

This group has 388 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
1#20275013C211yesmaximal, characteristic
2C4×#13133134C211yesmaximal, characteristic
3C4×#99×C53C211yesmaximal, characteristic
4C22×#11719917C411yescharacteristic, complemented
5C2×#13133134—21no—
6C2×C22⋊#88322—21no—
7C2×C22⋊#88322C411yescharacteristic, complemented
8C4×#11719917—21no—
9#16810392C412yescomplemented
10#16810378—21no—
11C2×C4×D4×C53C2211yescharacteristic
12C2×#99×C53—21no—
13C4×C22⋊C4×C53—21no—
14C53×#342C412yes—
15C2×#11719917—22no—
16C2×#11719917—41no—
17C2×#11719917—42no—
18C22×#88322—21no—
19C22×#88322—21no—
20#13133134—24no—
21C22⋊#88322—41no—
22C22⋊#88322—42no—
23#13133128—41no—
24C4×#88322—41no—
25C22×D4×C53D411yescharacteristic
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C4×D4×C53—41no—
29C4×D4×C53—41no—
30C2×C4⋊C4×C53C2×C411yescharacteristic
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53C2×C411yescharacteristic
33C2×C22⋊C4×C53Q811yescharacteristic
34C2×C42×C53—41no—
35C2×C42×C53D411yescharacteristic
36#99×C53—24no—
37#95×C53—41no—
38C24×C53C22⋊C411yescharacteristic
39C24×C53C4⋊C411yescharacteristic
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53—41no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53—42no—
52C22×C4×C53C4211yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C4⋊C411yescharacteristic
56C22×C4×C53C4⋊C411yescharacteristic
57C4⋊C4×C53—41no—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C22⋊C4×C53—42no—
62C42×C53—21no—
63C42×C53—21no—
64C42×C53—81no—
65C42×C53—81no—
66#11719917—24no—
67#11719917—42no—
68#11719917—42no—
69#11719917—42no—
70C2×#88322—22no—
71C2×#88322—42no—
72C2×#88322—41no—
73C2×#88322—41no—
74C2×#88322—41no—
75C2×#88322—41no—
76C2×#88322—41no—
77C23×C53—21no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53—41no—
86C23×C53#9511yescharacteristic
87C23×C53#10412yes—
88D4×C53—44no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—44no—
105C2×C4×C53—81no—
106C2×C4×C53—82no—
107C2×C4×C53—82no—
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#10412yes—
111#88322—44no—
112#88322—42no—
113#88322—81no—
114#7282—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—22no—
118C22×C53—42no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—81no—
125C22×C53—81no—
126C22×C53—81no—
127C22×C53#33911yescharacteristic
128C22×C53#33911yescharacteristic
129C22×C53#34211yescharacteristic
130C4×C53—22no—
131C4×C53—21no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—84no—
136C4×C53AΣL(1,𝔽16)11yescharacteristic
137C4×C53#35211yescharacteristic
138#1870—531no—
139C4×#100—531no—
140C4×#99D5311yescharacteristic
141C2×C53—21no—
142C2×C53—21no—
143C2×C53—42no—
144C2×C53—81no—
145C2×C53#129711yescharacteristic
146C2×C53#139011yescharacteristic
147C2×C53#139511yescharacteristic
148C22×M16—531no—
149C2×C4×D4C2×D5311yescharacteristic
150C2×#100—1061no—
151C2×#99—21no—
152C2×C22⋊C8—531no—
153C2×C22⋊C8—1061no—
154C4×M16—1061no—
155C4×C22⋊C4—21no—
156#343—532no—
157#342Dic5312yes—
158#336—1061no—
159C53#728211yescharacteristic, complemented, Sylow
160C22×D4C22⋊D5311yescharacteristic
161C23×C4C4×D5311yescharacteristic
162C23×C4C22⋊D5311yescharacteristic
163C2×M16—1062no—
164C2×M16—2122no—
165C2×M16—2121no—
166C22×C8—1061no—
167C22×C8—1061no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4C4×D5311yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4Dic10611yescharacteristic
173C2×C22⋊C4C2×Dic5311yescharacteristic
174C2×C42—41no—
175C2×C42D21211yescharacteristic
176#100—1064no—
177#99—24no—
178C22⋊C8—2122no—
179C22⋊C8—2121no—
180#97—2121no—
181C4×C8—2121no—
182#95—41no—
183C24#1171991911yescharacteristic
184C24C22⋊Dic5311yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—41no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4C4×Dic5311yescharacteristic
198C22×C4#1171991911yescharacteristic
199C22×C4C4⋊Dic5311yescharacteristic
200C22×C4#1171992111yescharacteristic
201C22×C4#1171992111yescharacteristic
202M16—1064no—
203M16—2122no—
204M16—2122no—
205M16—2122no—
206C2×C8—1062no—
207C2×C8—2121no—
208C2×C8—2121no—
209C2×C8—2121no—
210C2×C8—2122no—
211C2×C8—2121no—
212C2×C8—2121no—
213C4⋊C4—41no—
214C22⋊C4—24no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C42—21no—
219C42—21no—
220C42—81no—
221C42—81no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23#1313312411yescharacteristic
232C23#1313314412yes—
233D4—44no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4#1313313111yescharacteristic
254C2×C4#1313313111yescharacteristic
255C2×C4#1313314512yes—
256C8—4241no—
257C8—2124no—
258C8—2122no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22#1681038211yescharacteristic
272C22#1681038311yescharacteristic
273C22#1681038911yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—84no—
280C4#1681040911yescharacteristic
281C4#1681041211yescharacteristic
282C2—21no—
283C2—21no—
284C2—42no—
285C2—81no—
286C2#2027260111yescharacteristic
287C2#2027283211yescharacteristic
288C2#2027284311yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC22×C4×C53C42
FrattiniC2×C4×D4C2×D53
FittingC4×#99×C53C2
socleC22×C53#342
prime radicalC22×D4×C53D4
cosocleC2×C4×D4×C53C22
hypercenterC4×#99D53
hypocenterC53#7282
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
4. C22×#11719917C4semidirect
7. C2×C22⋊#88322C4semidirect
9. #16810392C4semidirect
159. C53#7282semidirect

Maximal subgroups

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

subgroupindex
#202750132
C4×#131331342
C4×#99×C532
#728253