Finite Group Databasebeta

← #27296533

Subgroups — #27296533

Order 13568.

434 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276626C212yesmaximal, complemented
2#20276026C211yesmaximal, characteristic, complemented
3#20272724C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#3245C211yesmaximal, characteristic, complemented
6C22×D424C2211yescharacteristic
7C2×#13133158—22no—
8C2×#13133139—21no—
9#16810525—21no—
10#16810423—22no—
11C22⋊#11719908C2212yes—
12#16810386C2211yescharacteristic
13C53×#532—21no—
14C53×#474—22no—
15C53×#466—21no—
16C53×#466C2212yes—
17C53×#459—22no—
18C2×C4⋊C8×C53C2211yescharacteristic
19C22×D212—21no—
20C2×D424—22no—
21C2×D424—42no—
22C2×C4⋊Dic53—21no—
23C2×#11719921—41no—
24C2×#11719921—41no—
25#13133158—44no—
26C2×#11719908—22no—
27#13133139—42no—
28C2×#41×C53—41no—
29C2×D8×C53—21no—
30C2×D8×C53D412yes—
31C22×C8×C53D411yescharacteristic
32C22×C8×C53C2311yescharacteristic
33C4⋊Q8×C53—21no—
34#127×C53—21no—
35#124×C53—41no—
36#121×C53—22no—
37#121×C53—41no—
38C4×D4×C53—22no—
39C4×D4×C53—41no—
40C2×C42×C53D411yescharacteristic
41#107×C53—21no—
42#107×C53D412yes—
43C4⋊C8×C53—22no—
44#102×C53—22no—
45#102×C53—22no—
46C23×D53—41no—
47#41×C53—82no—
48#41×C53—82no—
49C2×Q8×C53—41no—
50C2×D4×C53—22no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53D811yescharacteristic
56C22×C4×C53D811yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—41no—
59C2×D212—22no—
60C2×D212—41no—
61C2×D212—41no—
62C2×D212—41no—
63C2×D212—41no—
64C2×D212—41no—
65D8×C53—42no—
66D8×C53—81no—
67C2×C8×C53—21no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53C2×D412yes—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—41no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C42×C53—21no—
76C42×C53—21no—
77#11719921—82no—
78#11719921—82no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81D424—24no—
82D424—42no—
83#11719908—42no—
84C23×C53—42no—
85C23×C53C2×D811yescharacteristic
86C22×D53—42no—
87C22×D53—81no—
88C22×D53—81no—
89C22×D53—81no—
90C22×D53—81no—
91C22×D53—81no—
92C22×D53—81no—
93Q8×C53—82no—
94D4×C53—42no—
95D4×C53—42no—
96D4×C53—82no—
97D4×C53—82no—
98D4×C53—82no—
99D4×C53—82no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—84no—
108C2×C4×C53D1612yescomplemented
109C2×C4×C53#12011yescharacteristic
110C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
111C2×Dic53—81no—
112C2×Dic53—81no—
113C2×Dic53—81no—
114D212—42no—
115D212—42no—
116D212—81no—
117D212—81no—
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#44711yescharacteristic
127C22×C53C2×D1611yescharacteristic
128C22×C53#50911yescharacteristic
129C2×D53—82no—
130C2×D53—82no—
131C2×D53—82no—
132C2×D53—161no—
133C2×D53—161no—
134C2×D53—161no—
135C2×D53—161no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—82no—
140C4×C53#45411yescharacteristic
141C4×C53B(4,𝔽2)11yescharacteristic
142Dic53—161no—
143#3245D5311yescharacteristic, complemented
144#2184—532no—
145#2054—531no—
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×D8—531no—
156#532—21no—
157C2×D16—1062no—
158#474—22no—
159#466—21no—
160#466C2×D5312yes—
161#459—22no—
162C2×C4⋊C8C2×D5311yescharacteristic
163C2×#102—1061no—
164#390—1061no—
165#357—1062no—
166C22⋊C16—532no—
167#340—531no—
168C53#1215811yescharacteristic, complemented, Sylow
169C2×#41—41no—
170C22×D4—1061no—
171C2×D8—21no—
172C2×D8—1062no—
173C2×D8—2122no—
174C2×D8C22⋊D5312yes—
175C22×C8C22⋊D5311yescharacteristic
176C22×C8C22×D5311yescharacteristic
177C4⋊Q8—21no—
178#127—21no—
179#124—41no—
180#121—22no—
181#121—41no—
182C4×D4—22no—
183C4×D4—41no—
184C2×C4⋊C4—1061no—
185C2×C22⋊C4—2121no—
186C2×C22⋊C4—2121no—
187C2×C42C22⋊D5311yescharacteristic
188D16—2124no—
189C2×C16—1062no—
190#107—21no—
191#107C22⋊D5312yes—
192C4⋊C8—22no—
193#102—22no—
194#102—22no—
195#102—2122no—
196C24—2121no—
197#41—82no—
198#41—82no—
199C2×Q8—41no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—1062no—
205C2×D4—2121no—
206C2×D4—2121no—
207C2×D4—2121no—
208C2×D4—2121no—
209C2×D4—2121no—
210C22×C4—42no—
211C22×C4—2121no—
212C22×C4#1171992211yescharacteristic
213C22×C4#1171992211yescharacteristic
214C22×C4C2×C22⋊D5311yescharacteristic
215D8—42no—
216D8—81no—
217D8—1064no—
218D8—2122no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8C2×C22⋊D5312yes—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—2121no—
226C4⋊C4—2121no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—4242no—
230C22⋊C4—4242no—
231C42—21no—
232C42—21no—
233C16—2122no—
234C23—42no—
235C23—4241no—
236C23—4241no—
237C23—4241no—
238C23—4241no—
239C23—4241no—
240C23—4241no—
241C23—2122no—
242C23C2×#1171992211yescharacteristic
243Q8—82no—
244D4—42no—
245D4—42no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—4241no—
251D4—4241no—
252D4—2122no—
253D4—2122no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4—4241no—
263C2×C4—4241no—
264C2×C4—4241no—
265C2×C4#1313315812yescomplemented
266C2×C4#1313318611yescharacteristic
267C2×C4#1313324311yescharacteristic
268C8—22no—
269C8—41no—
270C22—21no—
271C22—21no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—8481no—
276C22—8481no—
277C22—8481no—
278C22—8481no—
279C22—4242no—
280C22—4242no—
281C22—4242no—
282C22#1681075711yescharacteristic
283C22C2×#1313315811yescharacteristic
284C22#1681105911yescharacteristic
285C4—22no—
286C4—21no—
287C4—41no—
288C4—82no—
289C4—8481no—
290C4#1681079611yescharacteristic
291C4#1681081011yescharacteristic
292C2—21no—
293C2—21no—
294C2—162no—
295C2—8482no—
296C2#2027365011yescharacteristic
297C2#2027662711yescharacteristic
298C2#2027666111yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#3245C2
socleC22×C53#447
hypercenter#3245D53
hypocenterC53#12158
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #20276626C2semidirect
2. #20276026C2semidirect
3. #20272724C2semidirect
4. #20272655C2semidirect
5. C53×#3245C2semidirect
108. C2×C4×C53D16semidirect
143. #3245D53semidirect
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
#202766262
#202760262
#202727242
#202726552
C53×#32452
#1215853