Finite Group Databasebeta

← #15776802

Subgroups — #15776802

Order 3328.

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

normal subgroups only

This group has 294 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#13127825C211yesmaximal, characteristic
2#13117065C211yesmaximal, characteristic, complemented
3#13116970C211yesmaximal, characteristic
4#13112884C211yesmaximal, characteristic, complemented
5Q8⋊#6090C211yesmaximal, characteristic, complemented
6#13112796C211yesmaximal, characteristic
7C13×#1674C211yesmaximal, characteristic, complemented
8#11719289—21no—
9C13×#500C2211yescharacteristic
10C13×#462—21no—
11C8×Q8×C13C2211yescharacteristic
12#11718506—22no—
13#11718502—21no—
14#11718360C2211yescharacteristic
15C4×Dic52—21no—
16#11718357—21no—
17C4×D104C2211yescharacteristic
18C13×#335C2211yescharacteristic
19C13×#332—21no—
20#11718084—22no—
21#11718083—21no—
22#11718081—21no—
23C4⋊#6090C2211yescharacteristic
24C4×#6090C2211yescharacteristic
25C2×C13×Q16—41no—
26C13×C4⋊Q8—21no—
27C4×Q8×C13D411yescharacteristic
28#88200—41no—
29C2×Dic52—21no—
30C2×Dic52D412yes—
31C2×D104D411yescharacteristic
32C2×#6095—22no—
33#88152—42no—
34#88146—41no—
35#88129—22no—
36C4×D52—21no—
37#88123—21no—
38C4×Dic26—22no—
39C13×#107—21no—
40C13×#107—21no—
41C13×C4⋊C8—21no—
42C13×C4⋊C8—21no—
43C13×C4⋊C8D411yescharacteristic
44C13×#103—41no—
45C4×C8×C13—21no—
46C4×C8×C13C2311yescharacteristic
47#88061—21no—
48#88058D411yescharacteristic
49#88056—22no—
50C2×#6090—21no—
51C2×#6090—21no—
52#12383—131nomaximal, Sylow
53C2×Q8×C13—41no—
54C2×Q8×C13D811yescharacteristic
55C2×D52—21no—
56C2×C4×D13—41no—
57C2×Dic26—22no—
58C13×Q16—82no—
59C2×C8×C13—21no—
60C2×C8×C13—41no—
61C2×C8×C13C2×D411yescharacteristic
62C2×C8×C13C2×D411yescharacteristic
63C13×C4⋊C4—21no—
64C13×C4⋊C4—41no—
65C13×C4⋊C4—41no—
66C13×C4⋊C4D811yescharacteristic
67C42×C13—21no—
68C42×C13C2×D411yescharacteristic
69#6103—42no—
70#6103—42no—
71#6103—41no—
72C4⋊Dic13—21no—
73C4⋊Dic13—41no—
74C4⋊Dic13—41no—
75#6101—42no—
76C4×Dic13—42no—
77Dic52—42no—
78Dic52—41no—
79Dic52—41no—
80D104—21no—
81D104—21no—
82#6095—82no—
83#6090—41no—
84#6090—41no—
85#3393—131no—
86#2241—131no—
87#2224—131no—
88#1674D1311yescharacteristic, complemented
89#1362—131no—
90Q8⋊C16—131no—
91#1331—131no—
92C22×D13—41no—
93Q8×C13—21no—
94Q8×C13—21no—
95Q8×C13—81no—
96Q8×C13—81no—
97C2×C4×C13—21no—
98C2×C4×C13—21no—
99C2×C4×C13—81no—
100C2×C4×C13#12011yescharacteristic
101C2×C4×C13C2×D811yescharacteristic
102C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
103C2×Dic13—41no—
104C2×Dic13—42no—
105D52—41no—
106D52—41no—
107C4×D13—81no—
108C4×D13—81no—
109Dic26—42no—
110Dic26—42no—
111C8×C13—21no—
112C8×C13—41no—
113C8×C13—81no—
114C8×C13#13511yescharacteristic
115C8×C13#13711yescharacteristic
116#556—261no—
117#500C2×D1311yescharacteristic
118#495—261no—
119#494—131no—
120#464—262no—
121#462—21no—
122#461—261no—
123C8×Q8C2×D1311yescharacteristic
124C4×Q16—261no—
125C4×D8—131no—
126#366—261no—
127C4⋊C16—131no—
128#358—262no—
129#357—261no—
130C4×C16—131no—
131#335C2×D1311yescharacteristic
132#332—21no—
133C22×C13#44711yescharacteristic
134C2×D13—81no—
135C2×D13—81no—
136C2×D13—81no—
137C4×C13—21no—
138C4×C13—41no—
139C4×C13—41no—
140C4×C13—161no—
141C4×C13#45211yescharacteristic
142C4×C13#45511yescharacteristic
143C4×C13#50811yescharacteristic
144C4×C13#50911yescharacteristic
145Dic13—82no—
146Dic13—81no—
147C2×Q16—41no—
148C2×Q16—132no—
149C2×Q16—261no—
150C2×SD16—262no—
151C2×D8—131no—
152C4⋊Q8—21no—
153C4⋊Q8—261no—
154#126—522no—
155#124—262no—
156C22⋊Q8—521no—
157C4×Q8—262no—
158C4×Q8C22⋊D1311yescharacteristic
159C4×D4—261no—
160#117—521no—
161C2×C16—261no—
162C2×C16—261no—
163#107—21no—
164#107—21no—
165#107—131no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8C22⋊D1311yescharacteristic
169#103—41no—
170#103—262no—
171#102—261no—
172C4×C8—21no—
173C4×C8C22×D1311yescharacteristic
174C2×C13#164111yescharacteristic
175C2×C13#218611yescharacteristic
176C2×C13#219511yescharacteristic
177D13—161no—
178C2×Q8—41no—
179C2×Q8—262no—
180C2×Q8#610411yescharacteristic
181C2×D4—261no—
182C22×C4—521no—
183Q16—82no—
184Q16—522no—
185Q16—521no—
186Q16—521no—
187SD16—1042no—
188D8—261no—
189D8—261no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8D4×D1311yescharacteristic
193C2×C8D4×D1311yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—261no—
198C4⋊C4—522no—
199C4⋊C4—521no—
200C4⋊C4—521no—
201C4⋊C4#610411yescharacteristic
202C22⋊C4—522no—
203C22⋊C4—522no—
204C22⋊C4—521no—
205C42—21no—
206C42—522no—
207C42C2×C22⋊D1311yescharacteristic
208C16—521no—
209C16—521no—
210C13#1238311yescharacteristic, complemented, Sylow
211C23—521no—
212Q8—21no—
213Q8—21no—
214Q8—81no—
215Q8—81no—
216Q8—522no—
217Q8—522no—
218D4—521no—
219D4—521no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—81no—
223C2×C4—521no—
224C2×C4—522no—
225C2×C4—1041no—
226C2×C4—1041no—
227C2×C4C2×#610411yescharacteristic
228C2×C4#8819111yescharacteristic
229C2×C4#8820311yescharacteristic
230C8—21no—
231C8—41no—
232C8—81no—
233C8#8817211yescharacteristic
234C8#8817311yescharacteristic
235C22—1041no—
236C22—1041no—
237C22—1041no—
238C22#1171869711yescharacteristic
239C4—21no—
240C4—41no—
241C4—41no—
242C4—161no—
243C4—1041no—
244C4—1042no—
245C4#1171880811yescharacteristic
246C4#1171884811yescharacteristic
247C4#1171885511yescharacteristic
248C4#1171886311yescharacteristic
249C2—2081no—
250C2#1311402711yescharacteristic
251C2#1311675711yescharacteristic
252C2#1311681211yescharacteristic
subgroupG/H
centerC22#11718697
commutator, cosocleC4×C8×C13C23
FrattiniC4×C8C22×D13
FittingC13×#1674C2
socleC22×C13#447
prime radicalC2×D104D4
hypercenter#1674D13
hypocenterC13#12383
LCSG ▷ C4×C8×C13 ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1674

Complements

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

NH ≅ G/Ntype
2. #13117065C2semidirect
4. #13112884C2semidirect
5. Q8⋊#6090C2semidirect
7. C13×#1674C2semidirect
88. #1674D13semidirect
210. C13#12383semidirect

Maximal subgroups

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

subgroupindex
#131278252
#131170652
#131169702
#131128842
Q8⋊#60902
#131127962
C13×#16742
#1238313