Finite Group Databasebeta

← #27296532

Subgroups — #27296532

Order 13568.

434 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 67 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#20286025C211yesmaximal, characteristic, complemented
2#20276627C212yesmaximal, complemented
3#20272723C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#2054C211yesmaximal, characteristic, complemented
6#16811173—21no—
7C2×#13133158—22no—
8#16810904—22no—
9#16810859—21no—
10#16810859C2212yes—
11#16810819—22no—
12#16810424—22no—
13C22⋊#11719908C2212yes—
14C22×D8×C53C2211yescharacteristic
15C2×C4⋊C8×C53C2211yescharacteristic
16C2×#102×C53—21no—
17C53×#390—21no—
18C53×#340C2211yescharacteristic
19C2×#11719945—41no—
20C2×D424—21no—
21C2×D424D412yes—
22#13133217—21no—
23#13133214—21no—
24#13133205—41no—
25#13133191—22no—
26#13133191—41no—
27C4×D212—22no—
28C4×D212—41no—
29#13133158—44no—
30C2×#11719908—22no—
31#13133151—21no—
32#13133151D412yes—
33#13133139—22no—
34#13133139—22no—
35C22×D4×C53—21no—
36C2×D8×C53—22no—
37C2×D8×C53—42no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53C2311yescharacteristic
40C2×C4⋊C4×C53—21no—
41C2×C22⋊C4×C53—41no—
42C2×C22⋊C4×C53—41no—
43C2×C42×C53D411yescharacteristic
44C4⋊C8×C53—22no—
45#102×C53—42no—
46C24×C53—41no—
47C2×D4×C53—22no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53D811yescharacteristic
55C22×C4×C53D811yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C2×C22⋊D53—42no—
58#11719945—82no—
59#11719945—82no—
60C2×D212—22no—
61C2×D212—41no—
62C2×D212—41no—
63C2×C4×D53—42no—
64C2×Dic106—41no—
65D8×C53—24no—
66D8×C53—42no—
67C2×C8×C53—21no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53C2×D412yes—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—82no—
74C22⋊C4×C53—82no—
75C42×C53—21no—
76C42×C53—21no—
77#11719921—42no—
78#11719921—42no—
79C4⋊Dic53—22no—
80C4⋊Dic53—41no—
81D424—42no—
82D424—81no—
83#11719908—42no—
84C23×C53—42no—
85C23×C53—81no—
86C23×C53—81no—
87C23×C53—81no—
88C23×C53—81no—
89C23×C53—81no—
90C23×C53—81no—
91C23×C53C2×D811yescharacteristic
92C22×D53—42no—
93D4×C53—42no—
94D4×C53—42no—
95D4×C53—81no—
96D4×C53—81no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—81no—
104C2×C4×C53—81no—
105C2×C4×C53—81no—
106C2×C4×C53D1612yescomplemented
107C2×C4×C53#12011yescharacteristic
108C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
109C22⋊D53—82no—
110C22⋊D53—82no—
111C2×Dic53—42no—
112D212—42no—
113D212—42no—
114D212—82no—
115D212—82no—
116C4×D53—84no—
117Dic106—82no—
118C8×C53—22no—
119C8×C53—41no—
120#12158—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—82no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53—161no—
127C22×C53—161no—
128C22×C53—161no—
129C22×C53—161no—
130C22×C53#44711yescharacteristic
131C22×C53C2×D1611yescharacteristic
132C22×C53#50911yescharacteristic
133C2×D53—82no—
134C2×D53—82no—
135C2×D53—82no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—161no—
140C4×C53#45411yescharacteristic
141C4×C53B(4,𝔽2)11yescharacteristic
142Dic53—82no—
143#3245—531no—
144#2184—532no—
145#2054D5311yescharacteristic, complemented
146#1347—532no—
147#1322—531no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—162no—
151C2×C53#159811yescharacteristic
152C2×C53#218411yescharacteristic
153C2×C53#219011yescharacteristic
154D53—162no—
155C22×D8C2×D5311yescharacteristic
156#532—1061no—
157C2×D16—1062no—
158#474—1062no—
159#466—532no—
160#466—1061no—
161#459—1062no—
162C2×C4⋊C8C2×D5311yescharacteristic
163C2×#102—21no—
164#390—21no—
165#357—1062no—
166C22⋊C16—532no—
167#340C2×D5311yescharacteristic
168C53#1215811yescharacteristic, complemented, Sylow
169C2×#41—2121no—
170C22×D4—21no—
171C2×D8—22no—
172C2×D8—42no—
173C2×D8—532no—
174C2×D8—1061no—
175C22×C8C22⋊D5311yescharacteristic
176C22×C8C22×D5311yescharacteristic
177C4⋊Q8—1061no—
178#127—1061no—
179#124—2121no—
180#121—1062no—
181#121—2121no—
182C4×D4—1062no—
183C4×D4—2121no—
184C2×C4⋊C4—21no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C42C22⋊D5311yescharacteristic
188D16—2124no—
189C2×C16—1062no—
190#107—532no—
191#107—1061no—
192C4⋊C8—22no—
193#102—42no—
194#102—1062no—
195#102—1062no—
196C24—41no—
197#41—4242no—
198#41—4242no—
199C2×Q8—2121no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—1062no—
207C2×D4—2121no—
208C2×D4—2121no—
209C2×D4—2122no—
210C22×C4—41no—
211C22×C4—2122no—
212C22×C4#1171992211yescharacteristic
213C22×C4#1171992211yescharacteristic
214C22×C4C2×C22⋊D5311yescharacteristic
215D8—24no—
216D8—42no—
217D8—4241no—
218D8—2122no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8D4×D5312yes—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—1062no—
226C4⋊C4—2121no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C42—21no—
232C42—21no—
233C16—2122no—
234C23—42no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—2122no—
242C23C2×#1171992211yescharacteristic
243Q8—4242no—
244D4—42no—
245D4—42no—
246D4—81no—
247D4—81no—
248D4—4242no—
249D4—4242no—
250D4—4242no—
251D4—4242no—
252D4—2122no—
253D4—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—4244no—
264C2×C4—2122no—
265C2×C4#1313315812yescomplemented
266C2×C4#1313318511yescharacteristic
267C2×C4#1313324211yescharacteristic
268C8—22no—
269C8—41no—
270C22—21no—
271C22—21no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22—4242no—
280C22—4242no—
281C22—4242no—
282C22#1681075611yescharacteristic
283C22C2×#1313315811yescharacteristic
284C22#1681105811yescharacteristic
285C4—22no—
286C4—21no—
287C4—41no—
288C4—161no—
289C4—4242no—
290C4#1681079511yescharacteristic
291C4#1681080911yescharacteristic
292C2—21no—
293C2—21no—
294C2—162no—
295C2—8482no—
296C2#2027364911yescharacteristic
297C2#2027662611yescharacteristic
298C2#2027666011yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2054C2
socleC22×C53#447
hypercenter#2054D53
hypocenterC53#12158
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. #20286025C2semidirect
2. #20276627C2semidirect
3. #20272723C2semidirect
4. #20272655C2semidirect
5. C53×#2054C2semidirect
106. C2×C4×C53D16semidirect
145. #2054D53semidirect
168. C53#12158semidirect
265. C2×C4#13133158semidirect

Maximal subgroups

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

subgroupindex
#202860252
#202766272
#202727232
#202726552
C53×#20542
#1215853