Finite Group Databasebeta

← #16948145

Subgroups — #16948145

Order 4352.

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

normal subgroups only

This group has 344 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
1Q16×D68C211yesmaximal, characteristic
2#14531013C211yesmaximal, characteristic, complemented
3#14528180C211yesmaximal, characteristic
4#14526834C211yesmaximal, characteristic, complemented
5#14526748C211yesmaximal, characteristic, complemented
6#14526738C211yesmaximal, characteristic
7C17×#2227C211yesmaximal, characteristic, complemented
8C2×Q16×D17—21no—
9Q8×D68—21no—
10#11763189C2211yescharacteristic
11#11763096—21no—
12#11763051—21no—
13#11763019—21no—
14#11762966—22no—
15C8×D68C2211yescharacteristic
16#11762612—22no—
17C4×#63214C2211yescharacteristic
18#11762562C2211yescharacteristic
19#11762558—21no—
20C17×#500C2211yescharacteristic
21C4×Q16×C17C2211yescharacteristic
22C17×#366—21no—
23C17×C4⋊C16C2211yescharacteristic
24C17×#358—21no—
25C2×Q8×D17—22no—
26C2×Q16×C17D411yescharacteristic
27C17×C4⋊Q8—21no—
28C4×Q8×C17—21no—
29C2×#63231—22no—
30Q16×D17—42no—
31Q16×D17—42no—
32C2×Dic68—21no—
33C2×Dic68D412yes—
34C2×C8×D17—21no—
35#10587583—22no—
36#10587583—42no—
37C4×D68—41no—
38C4×D68D411yescharacteristic
39#10587555—21no—
40#10587555—41no—
41C2×C16×C17—21no—
42C17×#107—21no—
43C17×#107—21no—
44C17×#107D411yescharacteristic
45C17×#103—21no—
46C4×C8×C17C2311yescharacteristic
47#10587491—21no—
48C2×#63214—21no—
49#10587481—22no—
50#10587480—41no—
51#10587479—41no—
52C4⋊#3622—21no—
53C4⋊#3622D411yescharacteristic
54C2×Q8×C17—21no—
55C2×Q8×C17—41no—
56Q8×D17—44no—
57Q8×D17—82no—
58C2×D68D811yescharacteristic
59C2×C4×D17—21no—
60C2×C4×D17—42no—
61C2×Dic34—22no—
62C2×Dic34—42no—
63Q16×C17—21no—
64Q16×C17SD1612yes—
65C2×C8×C17C2×D411yescharacteristic
66C2×C8×C17C2×D411yescharacteristic
67C4⋊C4×C17—21no—
68C4⋊C4×C17—41no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C42×C17—41no—
72C42×C17C2×D411yescharacteristic
73#63231—44no—
74#63227—21no—
75#63227—42no—
76C4⋊Dic17—42no—
77C4⋊Dic17—41no—
78C4⋊Dic17—41no—
79C4⋊Dic17D811yescharacteristic
80C2×#3622—21no—
81C2×#3622—41no—
82Dic68—42no—
83Dic68—41no—
84Dic68—41no—
85C8×D17—41no—
86C8×D17—41no—
87C16×C17—41no—
88#63214—41no—
89#12374—171nomaximal, Sylow
90C22×D17—21no—
91Q8×C17—22no—
92Q8×C17—41no—
93Q8×C17—161no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—81no—
97C2×C4×C17#12011yescharacteristic
98C2×C4×C17C2×D811yescharacteristic
99C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
100C2×Dic17—21no—
101C2×Dic17—42no—
102D68—21no—
103D68Q3212yes—
104C4×D17—41no—
105C4×D17—41no—
106C4×D17—84no—
107Dic34—42no—
108Dic34—42no—
109Dic34—84no—
110C8×C17—21no—
111C8×C17C2×SD1611yescharacteristic
112C8×C17AGL(1,ℤ/8)11yescharacteristic
113#3622—41no—
114#3622—81no—
115D4×Q16—171no—
116#2241—171no—
117#2227D1711yescharacteristic, complemented
118#1675—171no—
119#1362—171no—
120#1332—171no—
121D4⋊C16—171no—
122C22×C17#44711yescharacteristic
123C2×D17—22no—
124C2×D17—41no—
125C2×D17—41no—
126C4×C17—21no—
127C4×C17—42no—
128C4×C17—81no—
129C4×C17—161no—
130C4×C17#45011yescharacteristic
131C4×C17#45511yescharacteristic
132C4×C17C2×Q3211yescharacteristic
133C4×C17#51011yescharacteristic
134Dic17—41no—
135Dic17—82no—
136C22×Q16—341no—
137D4×Q8—341no—
138#500C2×D1711yescharacteristic
139#494—171no—
140#476—341no—
141#467—341no—
142#462—341no—
143#451—342no—
144C4×Q16C2×D1711yescharacteristic
145C8×D4—171no—
146#366—21no—
147C4⋊C16C2×D1711yescharacteristic
148#358—21no—
149#358—342no—
150C4×C16—171no—
151#335—171no—
152#333—341no—
153C2×C17#163911yescharacteristic
154C2×C17#218811yescharacteristic
155C2×C17#219311yescharacteristic
156D17—42no—
157C22×Q8—342no—
158C2×Q16—172no—
159C2×Q16—342no—
160C2×Q16—341no—
161C2×Q16—682no—
162C2×Q16—682no—
163C2×Q16C22⋊D1711yescharacteristic
164C22×C8—341no—
165C4⋊Q8—21no—
166C4⋊Q8—341no—
167C4⋊Q8—681no—
168C22⋊Q8—342no—
169C22⋊Q8—682no—
170C4×Q8—21no—
171C4×D4—171no—
172C4×D4—681no—
173C2×C16—21no—
174C2×C16—341no—
175#107—21no—
176#107—21no—
177#107C22⋊D1711yescharacteristic
178#106—681no—
179C4⋊C8—171no—
180C4⋊C8—341no—
181#103—21no—
182#103—342no—
183#102—681no—
184C22⋊C8—341no—
185C4×C8C22×D1711yescharacteristic
186C17#1237411yescharacteristic, complemented, Sylow
187C2×Q8—21no—
188C2×Q8—41no—
189C2×Q8—342no—
190C2×Q8—682no—
191C2×Q8—684no—
192C2×Q8—1362no—
193C2×D4—171no—
194C22×C4—341no—
195C22×C4—682no—
196Q16—21no—
197Q16—682no—
198Q16—684no—
199Q16—681no—
200Q16—681no—
201Q16#6323012yes—
202C2×C8—341no—
203C2×C8—681no—
204C2×C8—681no—
205C2×C8—681no—
206C2×C8D4×D1711yescharacteristic
207C2×C8C2×C22⋊D1711yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—171no—
213C4⋊C4—682no—
214C4⋊C4—681no—
215C4⋊C4—681no—
216C22⋊C4—341no—
217C22⋊C4—682no—
218C42—41no—
219C42D4×D1711yescharacteristic
220C16—41no—
221C16—681no—
222C23—341no—
223Q8—22no—
224Q8—41no—
225Q8—161no—
226Q8—682no—
227Q8—682no—
228Q8—1364no—
229D4—172no—
230D4—341no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—81no—
234C2×C4—341no—
235C2×C4—681no—
236C2×C4—681no—
237C2×C4—682no—
238C2×C4—1364no—
239C2×C4D8×D1711yescharacteristic
240C2×C4#1058759711yescharacteristic
241C2×C4#1058762311yescharacteristic
242C8—21no—
243C8—681no—
244C8—1361no—
245C8C2×#6323011yescharacteristic
246C8#1058763511yescharacteristic
247C22—342no—
248C22—681no—
249C22—681no—
250C22#1176294111yescharacteristic
251C4—21no—
252C4—42no—
253C4—81no—
254C4—161no—
255C4—681no—
256C4—1362no—
257C4#1176295911yescharacteristic
258C4#1176298511yescharacteristic
259C4Q32×D1711yescharacteristic
260C4#1176325111yescharacteristic
261C2—682no—
262C2#1452796211yescharacteristic
263C2#1453071611yescharacteristic
264C2#1453074511yescharacteristic
subgroupG/H
centerC22#11762941
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#2227C2
socleC22×C17#447
prime radicalC2×D68D8
hypercenter#2227D17
hypocenterC17#12374
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2227

Complements

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

NH ≅ G/Ntype
2. #14531013C2semidirect
4. #14526834C2semidirect
5. #14526748C2semidirect
7. C17×#2227C2semidirect
117. #2227D17semidirect
186. C17#12374semidirect

Maximal subgroups

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

subgroupindex
Q16×D682
#145310132
#145281802
#145268342
#145267482
#145267382
C17×#22272
#1237417