Finite Group Databasebeta

← #16952254

Subgroups — #16952254

Order 4352.

386 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762611C211yesmaximal, characteristic
2C2×#11762611C211yesmaximal, characteristic
3#14529650C211yesmaximal, characteristic
4#14529517C211yesmaximal, characteristic, complemented
5#14526876C211yesmaximal, characteristic, complemented
6#14526876C211yesmaximal, characteristic, complemented
7C2×C17×#346C211yesmaximal, characteristic, complemented
8C22×D136C2211yescharacteristic
9C2×#10587489C2211yescharacteristic
10C2×#10587489C2211yescharacteristic
11C2×#10587489C2211yescharacteristic
12C2×#10587492—21no—
13#11762711—21no—
14#11762707—21no—
15#11762611—22no—
16#11762611C414yescomplemented
17#11762572—21no—
18C22×C16×C17C2211yescharacteristic
19C22×C16×C17C2211yescharacteristic
20C2×C4×C8×C17C2211yescharacteristic
21C17×#346—22no—
22C22×D68—21no—
23C22×C8×C17D411yescharacteristic
24C22×C8×C17C2311yescharacteristic
25C2×C42×C17D411yescharacteristic
26C2×#63227—41no—
27C2×#63227—41no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×#63225—41no—
32C2×D136—21no—
33C2×D136—21no—
34C2×D136—21no—
35C2×D136—21no—
36C2×D136—21no—
37C2×D136C2×C412yes—
38C2×C16×C17—21no—
39C2×C16×C17—21no—
40C2×C16×C17—21no—
41C2×C16×C17—21no—
42C2×C16×C17C2×C412yes—
43C2×C16×C17D412yes—
44C4×C8×C17—21no—
45C4×C8×C17—21no—
46#10587502—41no—
47#10587492—42no—
48#10587489—21no—
49#10587489—21no—
50#10587489—21no—
51#10587489—21no—
52#10587489—21no—
53#10587489C2×C412yes—
54C23×D17—41no—
55C22×C4×C17D811yescharacteristic
56C22×C4×C17D811yescharacteristic
57C22×C4×C17C2×D411yescharacteristic
58C22×Dic17—41no—
59C22×Dic17—41no—
60C22×Dic17—41no—
61C2×D68—22no—
62C2×D68—41no—
63C2×D68—41no—
64C2×D68—41no—
65C2×D68—41no—
66C2×D68—41no—
67C2×C8×C17—21no—
68C2×C8×C17—21no—
69C2×C8×C17SD1612yes—
70C2×C8×C17C22×C411yescharacteristic
71C2×C8×C17C2×D411yescharacteristic
72C2×C8×C17#4111yescharacteristic
73C2×C8×C17#4111yescharacteristic
74C2×C8×C17#4111yescharacteristic
75C2×C8×C17#4111yescharacteristic
76C42×C17—21no—
77C42×C17—21no—
78#63227—82no—
79#63227—82no—
80C4⋊Dic17—22no—
81C4⋊Dic17—41no—
82C4⋊Dic17—41no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85C4⋊Dic17—41no—
86#63225—82no—
87D136—22no—
88D136—22no—
89D136—41no—
90D136—41no—
91C16×C17—22no—
92C16×C17—22no—
93#13149—171nomaximal, Sylow
94C23×C17C2×D811yescharacteristic
95C22×D17—42no—
96C22×D17—81no—
97C22×D17—81no—
98C22×D17—81no—
99C22×D17—81no—
100C22×D17—81no—
101C22×D17—81no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17C4×D411yescharacteristic
109C2×C4×C17#12111yescharacteristic
110C2×C4×C17#12411yescharacteristic
111C2×C4×C17C2×SD1611yescharacteristic
112C2×C4×C17#13511yescharacteristic
113C2×C4×C17#13511yescharacteristic
114C2×Dic17—42no—
115C2×Dic17—81no—
116C2×Dic17—81no—
117C2×Dic17—81no—
118C2×Dic17—81no—
119C2×Dic17—81no—
120C2×Dic17—81no—
121C2×Dic17—81no—
122C2×Dic17—81no—
123D68—42no—
124D68—42no—
125D68—81no—
126D68—81no—
127C8×C17—22no—
128C8×C17#11711yescharacteristic
129C8×C17C4×D411yescharacteristic
130C8×C17#12311yescharacteristic
131C8×C17#12611yescharacteristic
132C2×#357—171no—
133C2×#357—171no—
134C2×#346D1711yescharacteristic, complemented
135#1970—171no—
136#1942—171no—
137#1380—171no—
138#1380—171no—
139C22×C17C4×D811yescharacteristic
140C22×C17#46611yescharacteristic
141C22×C17#48611yescharacteristic
142C22×C17#50911yescharacteristic
143C22×C17#50911yescharacteristic
144C22×C17#50911yescharacteristic
145C22×C17#51011yescharacteristic
146C2×D17—82no—
147C2×D17—82no—
148C2×D17—82no—
149C2×D17—161no—
150C2×D17—161no—
151C2×D17—161no—
152C2×D17—161no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17#38811yescharacteristic
156C4×C17C4×SD1611yescharacteristic
157C4×C17#46511yescharacteristic
158C4×C17#48811yescharacteristic
159Dic17—82no—
160Dic17—161no—
161Dic17—161no—
162C22×D8—171no—
163C22×C16C2×D1711yescharacteristic
164C22×C16C2×D1711yescharacteristic
165C2×#107—171no—
166C2×#107—171no—
167C2×#107—171no—
168C2×#102—341no—
169C2×C4×C8C2×D1711yescharacteristic
170#390—341no—
171#389—341no—
172#357—174no—
173#357—342no—
174#346—22no—
175#340—341no—
176C2×C17#193411yescharacteristic
177C2×C17#217511yescharacteristic
178C2×C17#217711yescharacteristic
179C2×C17#221811yescharacteristic
180C2×C17#222011yescharacteristic
181C2×C17#224311yescharacteristic
182C2×C17#224411yescharacteristic
183D17—162no—
184C22×D4—341no—
185C2×D8—172no—
186C2×D8—341no—
187C2×D8—341no—
188C2×D8—341no—
189C2×D8—341no—
190C2×D8—341no—
191C22×C8D6811yescharacteristic
192C22×C8C22×D1711yescharacteristic
193C2×C4⋊C4—341no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—681no—
197C2×C22⋊C4—681no—
198C2×C22⋊C4—681no—
199C2×C42D6811yescharacteristic
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16C4×D1712yes—
205C2×C16C22⋊D1712yes—
206#107—172no—
207#107—341no—
208#107—341no—
209#107—341no—
210#107—341no—
211#107—341no—
212#102—682no—
213C4×C8—21no—
214C4×C8—21no—
215#95—681no—
216C17#1314911yescharacteristic, complemented, Sylow
217C24—681no—
218C2×D4—342no—
219C2×D4—681no—
220C2×D4—681no—
221C2×D4—681no—
222C2×D4—681no—
223C2×D4—681no—
224C22×C4—681no—
225C22×C4—681no—
226C22×C4—681no—
227C22×C4D13611yescharacteristic
228C22×C4D13611yescharacteristic
229C22×C4C2×D6811yescharacteristic
230D8—342no—
231D8—342no—
232D8—681no—
233D8—681no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8#6321912yes—
237C2×C8C2×C4×D1711yescharacteristic
238C2×C8#6325211yescharacteristic
239C2×C8#6325211yescharacteristic
240C2×C8#6325211yescharacteristic
241C2×C8#6325211yescharacteristic
242C2×C8C2×C22⋊D1711yescharacteristic
243C4⋊C4—342no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C4⋊C4—681no—
247C4⋊C4—681no—
248C4⋊C4—681no—
249C4⋊C4—1362no—
250C22⋊C4—1362no—
251C22⋊C4—1362no—
252C42—21no—
253C42—21no—
254C16—22no—
255C16—22no—
256C23—682no—
257C23—1361no—
258C23—1361no—
259C23—1361no—
260C23—1361no—
261C23—1361no—
262C23—1361no—
263C23C2×D13611yescharacteristic
264D4—682no—
265D4—682no—
266D4—1361no—
267D4—1361no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—682no—
275C2×C4—1361no—
276C2×C4—1361no—
277C2×C4—1361no—
278C2×C4—1361no—
279C2×C4—1361no—
280C2×C4—1361no—
281C2×C4—1361no—
282C2×C4—1361no—
283C2×C4C4×D6811yescharacteristic
284C2×C4#1058756111yescharacteristic
285C2×C4C2×#6321911yescharacteristic
286C2×C4#1058759011yescharacteristic
287C2×C4#1058759011yescharacteristic
288C2×C4#1058761611yescharacteristic
289C8—22no—
290C8#1058755811yescharacteristic
291C8#1058756211yescharacteristic
292C8C4×C22⋊D1711yescharacteristic
293C8#1058761511yescharacteristic
294C22—2721no—
295C22—2721no—
296C22—2721no—
297C22—2721no—
298C22—1362no—
299C22—1362no—
300C22—1362no—
301C22C4×D13611yescharacteristic
302C22#1176304511yescharacteristic
303C22#1176315611yescharacteristic
304C22#1176324711yescharacteristic
305C22#1176324711yescharacteristic
306C22#1176324711yescharacteristic
307C22#1176325211yescharacteristic
308C4—41no—
309C4—41no—
310C4—2721no—
311C4—2721no—
312C4—1362no—
313C4#1176270011yescharacteristic
314C4C4×#6321911yescharacteristic
315C4#1176303811yescharacteristic
316C4#1176316411yescharacteristic
317C2—2722no—
318C2#1452946711yescharacteristic
319C2#1453065011yescharacteristic
320C2#1453066111yescharacteristic
321C2#1453088711yescharacteristic
322C2#1453089711yescharacteristic
323C2#1453102111yescharacteristic
324C2#1453102611yescharacteristic
subgroupG/H
centerC23C2×D136
commutatorC2×C8×C17C22×C4
FrattiniC22×C8C22×D17
FittingC2×C17×#346C2
socleC23×C17C2×D8
prime radicalC22×D136C22
cosocleC22×C8×C17C23
hypercenterC2×#346D17
hypocenterC17#13149
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×#346

Complements

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

NH ≅ G/Ntype
4. #14529517C2semidirect
5. #14526876C2semidirect
6. #14526876C2semidirect
7. C2×C17×#346C2semidirect
16. #11762611C4semidirect
134. C2×#346D17semidirect
216. C17#13149semidirect

Maximal subgroups

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

subgroupindex
C2×#117626112
#145296502
#145295172
#145268762
C2×C17×#3462
#1314917