Finite Group Databasebeta

← #16930967

Subgroups — #16930967

Order 4352.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539909C211yesmaximal, characteristic
2#14529757C211yesmaximal, characteristic, complemented
3#14527466C211yesmaximal, characteristic, complemented
4#14527344C211yesmaximal, characteristic
5#14526644C211yesmaximal, characteristic, complemented
6#14526615C211yesmaximal, characteristic, complemented
7C17×#2932C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9#11763205C412yescomplemented
10#11763155C412yescomplemented
11#11763125—22no—
12#11763099—22no—
13#11762862C2211yescharacteristic
14#11762862C2211yescharacteristic
15#11762675C2211yescharacteristic
16#11762572—21no—
17#11762572—21no—
18#11762557—22no—
19#11762543—22no—
20C2×C17×#131—21no—
21C17×#532C2211yescharacteristic
22C17×#435—22no—
23C17×#435—22no—
24C17×#423C2211yescharacteristic
25C17×#408—21no—
26C2×C17×#97C2211yescharacteristic
27C2×#41×C17—21no—
28C22×Dic34—21no—
29C17×#131—42no—
30C17×#131—42no—
31C2×M16×C17—22no—
32C2×M16×C17—21no—
33C22×C8×C17—22no—
34C22×C8×C17D411yescharacteristic
35C22×C8×C17D411yescharacteristic
36C17×C4⋊Q8C2×C411yescharacteristic
37C17×#127C2×C411yescharacteristic
38C17×#124—21no—
39C17×#121—21no—
40C17×#121D412yes—
41C4×D4×C17—21no—
42C4×D4×C17D412yes—
43C2×C42×C17C2311yescharacteristic
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47#10587555—21no—
48#10587555—21no—
49#10587555—21no—
50#10587555C2×C412yes—
51C17×C4⋊C8—21no—
52C17×C4⋊C8D412yes—
53C17×C22⋊C8—24no—
54C17×#97—21no—
55C17×#97—21no—
56#10587503—24no—
57#10587502—21no—
58#10587502—21no—
59C22⋊#3622—21no—
60C22⋊#3622—21no—
61#10587479—24no—
62C4⋊#3622—21no—
63C4⋊#3622—21no—
64C4×#3622—21no—
65C4×#3622C2×C412yes—
66#41×C17—41no—
67#41×C17—41no—
68#41×C17—41no—
69#41×C17—41no—
70C2×Q8×C17—21no—
71C2×D4×C17—22no—
72C2×D4×C17—21no—
73C2×D4×C17—21no—
74C2×D4×C17C22⋊C412yes—
75C22×C4×C17—22no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17#4111yescharacteristic
79C22×Dic17—41no—
80C22×Dic17—41no—
81C2×Dic34—22no—
82C2×Dic34—41no—
83C2×Dic34—82no—
84M16×C17—44no—
85M16×C17—41no—
86M16×C17—41no—
87C2×C8×C17—21no—
88C2×C8×C17—21no—
89C2×C8×C17—22no—
90C2×C8×C17—21no—
91C2×C8×C17—21no—
92C2×C8×C17—21no—
93C2×C8×C17—44no—
94C2×C8×C17SD1612yes—
95C4⋊C4×C17—21no—
96C4⋊C4×C17C22⋊C412yes—
97C22⋊C4×C17—22no—
98C22⋊C4×C17—22no—
99C42×C17C22×C411yescharacteristic
100C42×C17C2×D411yescharacteristic
101C42×C17C2×D411yescharacteristic
102C42×C17#4111yescharacteristic
103C4⋊Dic17—22no—
104C4⋊Dic17—22no—
105C4⋊Dic17—41no—
106C4⋊Dic17—41no—
107C4⋊Dic17—41no—
108C4⋊Dic17—41no—
109C2×#3622—22no—
110C2×#3622—41no—
111#9287—171nomaximal, Sylow
112C23×C17—22no—
113C23×C17#12111yescharacteristic
114Q8×C17—41no—
115Q8×C17—41no—
116D4×C17—42no—
117D4×C17—42no—
118D4×C17—42no—
119D4×C17—41no—
120D4×C17—41no—
121D4×C17—81no—
122C2×C4×C17—22no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17C2×C22⋊C411yescharacteristic
128C2×C4×C17C4×D411yescharacteristic
129C2×C4×C17C4×D411yescharacteristic
130C2×C4×C17#12011yescharacteristic
131C2×C4×C17#12111yescharacteristic
132C2×C4×C17#12311yescharacteristic
133C2×C4×C17C2×SD1611yescharacteristic
134C2×C4×C17#13511yescharacteristic
135C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C17#13711yescharacteristic
137C2×Dic17—42no—
138C2×Dic17—82no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17—81no—
142Dic34—82no—
143Dic34—82no—
144C8×C17—22no—
145C8×C17—42no—
146C8×C17—41no—
147#3622—42no—
148#3622—81no—
149#3227—171no—
150#2932D1711yescharacteristic
151#1993—171no—
152#1548—171no—
153#1517—171no—
154#1286—171no—
155#1275—171no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—42no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17#38611yescharacteristic
162C22×C17#46111yescharacteristic
163C22×C17#46311yescharacteristic
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—42no—
169C4×C17C4×SD1611yescharacteristic
170C4×C17#44011yescharacteristic
171C4×C17#45011yescharacteristic
172C4×C17#45211yescharacteristic
173Dic17—82no—
174Dic17—161no—
175C2×#131—21no—
176#532C2×D1711yescharacteristic
177C2×C4⋊Q8—171no—
178#498—172no—
179#486—172no—
180#481—342no—
181#476—342no—
182#435—22no—
183#435—22no—
184#431—171no—
185#431—171no—
186#423C2×D1711yescharacteristic
187#408—21no—
188C2×#97C2×D1711yescharacteristic
189#382—171no—
190#340—341no—
191#340—341no—
192#333—342no—
193#327—342no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17—82no—
197C2×C17#187711yescharacteristic
198C2×C17#190311yescharacteristic
199C2×C17#190911yescharacteristic
200C2×#41—21no—
201C22×Q8—341no—
202#131—42no—
203#131—42no—
204C2×M16—22no—
205C2×M16—21no—
206C22×C8—22no—
207C22×C8C22⋊D1711yescharacteristic
208C22×C8C22⋊D1711yescharacteristic
209C4⋊Q8—172no—
210C4⋊Q8—341no—
211C4⋊Q8—341no—
212C4⋊Q8—341no—
213C4⋊Q8C4×D1711yescharacteristic
214#127C4×D1711yescharacteristic
215#124—21no—
216#121—21no—
217#121D6812yes—
218C4×D4—21no—
219C4×D4C22⋊D1712yes—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C42C22×D1711yescharacteristic
224#106—344no—
225C4⋊C8—21no—
226C4⋊C8—341no—
227C4⋊C8—341no—
228C4⋊C8C22⋊D1712yes—
229#102—344no—
230C22⋊C8—24no—
231C22⋊C8—341no—
232C22⋊C8—341no—
233#97—21no—
234#97—21no—
235C4×C8—172no—
236C4×C8—341no—
237#95—341no—
238#95—341no—
239C17#928711yescharacteristic, complemented, Sylow
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×Q8—342no—
246C2×Q8—681no—
247C2×Q8—1362no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4#6322712yes—
252C22×C4—22no—
253C22×C4—681no—
254C22×C4—681no—
255C22×C4C2×D6811yescharacteristic
256C22×C4#6325211yescharacteristic
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—44no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—44no—
268C2×C8—342no—
269C2×C8—681no—
270C2×C8#6322912yes—
271C4⋊C4—21no—
272C4⋊C4—342no—
273C4⋊C4—342no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4#6322712yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C2×C4×D1711yescharacteristic
282C42#6325211yescharacteristic
283C42C2×C22⋊D1711yescharacteristic
284C42C2×C22⋊D1711yescharacteristic
285C23—22no—
286C23#1058761611yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—1362no—
290Q8—1362no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—682no—
303C2×C4—1361no—
304C2×C4—1361no—
305C2×C4—1361no—
306C2×C4—1362no—
307C2×C4C2×#6322711yescharacteristic
308C2×C4C4×C22⋊D1711yescharacteristic
309C2×C4C4×C22⋊D1711yescharacteristic
310C2×C4#1058761511yescharacteristic
311C2×C4#1058761611yescharacteristic
312C2×C4#1058761811yescharacteristic
313C2×C4C2×#6322911yescharacteristic
314C2×C4#1058763611yescharacteristic
315C2×C4#1058763711yescharacteristic
316C2×C4#1058763911yescharacteristic
317C8—22no—
318C8—42no—
319C8—41no—
320C8—682no—
321C8—1361no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22#1176268811yescharacteristic
328C22#1176301411yescharacteristic
329C22#1176302411yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—2721no—
336C4—1362no—
337C4C4×#6322911yescharacteristic
338C4#1176290911yescharacteristic
339C4#1176296011yescharacteristic
340C4#1176297111yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2#1452911811yescharacteristic
345C2#1452926411yescharacteristic
346C2#1452930611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2932C2
socleC22×C17#386
prime radicalC17×#532C22
cosocleC2×C42×C17C23
hypercenter#2932D17
hypocenterC17#9287
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #14529757C2semidirect
3. #14527466C2semidirect
5. #14526644C2semidirect
6. #14526615C2semidirect
9. #11763205C4semidirect
10. #11763155C4semidirect
239. C17#9287semidirect

Maximal subgroups

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

subgroupindex
#145399092
#145297572
#145274662
#145273442
#145266442
#145266152
C17×#29322
#928717