Finite Group Databasebeta

← #27297607

Subgroups — #27297607

Order 13568.

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

normal subgroups only

This group has 340 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#20276964C211yesmaximal, characteristic
2#20276755C211yesmaximal, characteristic, complemented
3#20274075C211yesmaximal, characteristic, complemented
4#20272761C211yesmaximal, characteristic, complemented
5Q8⋊#11719908C211yesmaximal, characteristic, complemented
6#20272679C211yesmaximal, characteristic, complemented
7C53×#3396C211yesmaximal, characteristic, complemented
8C2×#13133160—21no—
9C2×#13133158—22no—
10#16811000C2211yescharacteristic
11#16810823—21no—
12C4⋊#11719908C2211yescharacteristic
13#16810423—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810366—21no—
17C53×#559—21no—
18C53×#495—21no—
19C53×#493C2211yescharacteristic
20C53×#463—22no—
21C53×#460—21no—
22C8×Q8×C53C2211yescharacteristic
23C4×Q16×C53C2211yescharacteristic
24C4×D8×C53—21no—
25C2×#11719924—41no—
26C2×D424—21no—
27C2×D424—21no—
28#13133214—21no—
29#13133160—42no—
30#13133158—44no—
31C2×#11719908—21no—
32C2×#11719908—21no—
33C4⋊#88322—21no—
34#13133137—41no—
35C2×Q16×C53D411yescharacteristic
36C2×SD16×C53—22no—
37C2×D8×C53—21no—
38C2×D8×C53D412yes—
39#127×C53—21no—
40#124×C53—22no—
41#124×C53—41no—
42#123×C53—42no—
43C4×Q8×C53—21no—
44C4×Q8×C53D411yescharacteristic
45C4×D4×C53—22no—
46C4×D4×C53—41no—
47#107×C53D411yescharacteristic
48C4⋊C8×C53—21no—
49C4⋊C8×C53D411yescharacteristic
50#103×C53—21no—
51#102×C53—22no—
52C4×C8×C53—21no—
53C4×C8×C53C2311yescharacteristic
54C2×Q8×C53—21no—
55C2×Q8×C53D811yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C22×C4×C53—42no—
60C2×D212—41no—
61C2×D212—41no—
62C2×D212—41no—
63Q16×C53—21no—
64Q16×C53—21no—
65SD16×C53—82no—
66D8×C53—42no—
67D8×C53—41no—
68D8×C53—41no—
69C2×C8×C53—21no—
70C2×C8×C53C2×D411yescharacteristic
71C2×C8×C53C2×D411yescharacteristic
72C4⋊C4×C53—21no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53D811yescharacteristic
76C22⋊C4×C53—42no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C42×C53—21no—
80C42×C53—41no—
81C42×C53C2×D411yescharacteristic
82#11719924—82no—
83C2×#88322—41no—
84D424—42no—
85D424—41no—
86D424—41no—
87#11719908—22no—
88#11719908—41no—
89C23×C53—42no—
90C22×D53—81no—
91Q8×C53—21no—
92Q8×C53—21no—
93Q8×C53—41no—
94Q8×C53—41no—
95D4×C53—42no—
96D4×C53—42no—
97D4×C53—81no—
98D4×C53—81no—
99D4×C53—161no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—162no—
105C2×C4×C53#12011yescharacteristic
106C2×C4×C53C2×D811yescharacteristic
107C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
108D212—81no—
109D212—81no—
110D212—81no—
111D212—81no—
112D212—161no—
113C8×C53—21no—
114C8×C53—41no—
115C8×C53#13511yescharacteristic
116C8×C53AGL(1,ℤ/8)11yescharacteristic
117#88322—81no—
118#12329—531nomaximal, Sylow
119C22×C53—82no—
120C22×C53—82no—
121C22×C53—82no—
122C22×C53#44711yescharacteristic
123C2×D53—161no—
124C2×D53—161no—
125C2×D53—161no—
126C4×C53—21no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—81no—
130C4×C53—81no—
131C4×C53#44911yescharacteristic
132C4×C53#45311yescharacteristic
133C4×C53#50811yescharacteristic
134C4×C53#50911yescharacteristic
135#3396D5311yescharacteristic, complemented
136#2246—531no—
137#2207—531no—
138#1668—531no—
139#1361—531no—
140Q8⋊C16—531no—
141#1332—531no—
142C2×C53—162no—
143C2×C53#162111yescharacteristic
144C2×C53#218611yescharacteristic
145C2×C53#219211yescharacteristic
146D53—321no—
147#559—21no—
148C2×SD32—1061no—
149C2×D16—1062no—
150#495—21no—
151#493—531no—
152#493C2×D5311yescharacteristic
153#463—22no—
154#460—21no—
155#460—1061no—
156C8×Q8C2×D5311yescharacteristic
157C4×Q16C2×D5311yescharacteristic
158C4×D8—21no—
159C4⋊C16—531no—
160#357—1061no—
161C4×C16—531no—
162#335—531no—
163#331—1061no—
164C53#1232911yescharacteristic, complemented, Sylow
165C2×Q16C22⋊D5311yescharacteristic
166C2×SD16—22no—
167C2×SD16—2121no—
168C2×D8—21no—
169C2×D8—1061no—
170C2×D8—1061no—
171C2×D8C22⋊D5312yes—
172#127—21no—
173#127—1061no—
174#124—22no—
175#124—41no—
176#123—42no—
177C4×Q8—21no—
178C4×Q8C22⋊D5311yescharacteristic
179C4×D4—22no—
180C4×D4—41no—
181SD32—2122no—
182D16—2124no—
183C2×C16—1061no—
184C2×C16—1061no—
185#107C22⋊D5311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—1061no—
188C4⋊C8C22⋊D5311yescharacteristic
189#103—21no—
190#102—22no—
191#102—2121no—
192C4×C8—21no—
193C4×C8C22×D5311yescharacteristic
194C2×Q8—21no—
195C2×Q8#1171992211yescharacteristic
196C2×D4—22no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—2121no—
200C2×D4—2121no—
201C2×D4—2121no—
202C22×C4—42no—
203Q16—21no—
204Q16—21no—
205SD16—82no—
206SD16—4242no—
207D8—42no—
208D8—41no—
209D8—41no—
210D8—2122no—
211D8—2121no—
212D8—2121no—
213C2×C8—21no—
214C2×C8—2121no—
215C2×C8C2×C22⋊D5311yescharacteristic
216C2×C8C2×C22⋊D5311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4#1171992211yescharacteristic
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—21no—
225C42—41no—
226C42C2×C22⋊D5311yescharacteristic
227C16—1062no—
228C16—2121no—
229C23—42no—
230C23—4241no—
231Q8—21no—
232Q8—21no—
233Q8—41no—
234Q8—41no—
235D4—42no—
236D4—42no—
237D4—81no—
238D4—81no—
239D4—161no—
240D4—8481no—
241D4—4241no—
242D4—4241no—
243D4—4241no—
244D4—4241no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—162no—
250C2×C4#1313318611yescharacteristic
251C2×C4C2×#1171992211yescharacteristic
252C2×C4#1313324311yescharacteristic
253C8—21no—
254C8—41no—
255C8—4241no—
256C8#1313323711yescharacteristic
257C8#1313324211yescharacteristic
258C22—82no—
259C22—82no—
260C22—82no—
261C22—8481no—
262C22—8481no—
263C22—8481no—
264C22#1681075711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—81no—
269C4—81no—
270C4#1681076811yescharacteristic
271C4#1681079111yescharacteristic
272C4#1681105311yescharacteristic
273C4#1681105811yescharacteristic
274C2—162no—
275C2—16961no—
276C2#2027378011yescharacteristic
277C2#2027663811yescharacteristic
278C2#2027667211yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3396C2
socleC22×C53#447
prime radical#20276964C2
hypercenter#3396D53
hypocenterC53#12329
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
2. #20276755C2semidirect
3. #20274075C2semidirect
4. #20272761C2semidirect
5. Q8⋊#11719908C2semidirect
6. #20272679C2semidirect
7. C53×#3396C2semidirect
135. #3396D53semidirect
164. C53#12329semidirect

Maximal subgroups

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

subgroupindex
#202769642
#202767552
#202740752
#202727612
Q8⋊#117199082
#202726792
C53×#33962
#1232953