Finite Group Databasebeta

← #17052637

Subgroups — #17052637

Order 4352.

914 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762916C211yesmaximal, characteristic
2C2×Q16×Dic17C211yesmaximal, characteristic
3Q8×#63225C212yesmaximal
4#14529522C211yesmaximal, characteristic
5#14529520C211yesmaximal, characteristic
6#14529052C212yesmaximal
7#14529050C212yesmaximal
8#14528939C211yesmaximal, characteristic
9#14528787C212yesmaximal
10#14528652C211yesmaximal, characteristic
11C2×C17×#441C211yesmaximal, characteristic
12C2×Q8×Dic17C2212yes—
13C2×Q8×Dic17C2212yes—
14#11762916—24no—
15Q16×Dic17C418yescomplemented
16C2×#10587489C2211yescharacteristic
17C2×#10587488C2211yescharacteristic
18C2×#10587479C2212yes—
19C2×#10587485C2211yescharacteristic
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587486C2211yescharacteristic
23C2×C8×Dic17C2211yescharacteristic
24#11762707—22no—
25#11762707C2212yes—
26#11762705—22no—
27#11762705C2212yes—
28#11762678—22no—
29#11762678C2212yes—
30C4×#63225—22no—
31C4×#63225C2211yescharacteristic
32#11762572C2212yes—
33#11762571C2212yes—
34#11762564C2211yescharacteristic
35#11762563C2211yescharacteristic
36C22×Q16×C17C2211yescharacteristic
37C2×C4×Q8×C17C2212yes—
38C17×#441—24no—
39C2×C17×#106C2211yescharacteristic
40C2×C17×#103C2212yes—
41C2×C17×#97C2211yescharacteristic
42C22×Q8×C17C2312yes—
43C2×Q16×C17C2×C414yes—
44C2×Q16×C17D414yes—
45C2×Q16×C17Q814yes—
46C22×C8×C17C2311yescharacteristic
47C22×C8×C17C2311yescharacteristic
48C4×Q8×C17—24no—
49C4×Q8×C17—42no—
50C2×C4⋊C4×C17—22no—
51C2×C4⋊C4×C17C2312yes—
52C2×C42×C17—22no—
53C2×C42×C17C2311yescharacteristic
54Q8×Dic17—24no—
55Q8×Dic17—24no—
56Q8×Dic17—42no—
57Q8×Dic17C2×C418yes—
58C2×C4⋊Dic17—22no—
59C2×C4⋊Dic17—22no—
60C2×C4⋊Dic17C2312yes—
61C2×C4⋊Dic17C2312yes—
62C2×#63225—22no—
63C2×#63225—22no—
64C2×#63225—22no—
65C2×#63225—22no—
66C2×#63225D412yes—
67C2×C4×Dic17—22no—
68C2×C4×Dic17—22no—
69C2×C4×Dic17C2311yescharacteristic
70C2×C4×Dic17C2311yescharacteristic
71C22×#3622C2311yescharacteristic
72C22×#3622C2311yescharacteristic
73C17×#106—22no—
74C17×#103—24no—
75C17×#97—22no—
76#10587506—24no—
77#10587506C2×C418yes—
78#10587502—22no—
79#10587502—22no—
80#10587502D412yes—
81#10587489C2×C414yes—
82#10587488—22no—
83#10587486—22no—
84#10587485—22no—
85C8×Dic17C2×C414yes—
86#10587479—42no—
87C2×Q8×C17—22no—
88C2×Q8×C17—22no—
89C2×Q8×C17—22no—
90C2×Q8×C17C22×C414yes—
91C2×Q8×C17C2×D414yes—
92C2×Q8×C17C2×Q814yes—
93C22×C4×C17—22no—
94C22×C4×C17—22no—
95C22×C4×C17C2×D412yes—
96C22×C4×C17C2411yescharacteristic
97C22×Dic17—22no—
98C22×Dic17—22no—
99C22×Dic17C2×D412yes—
100C22×Dic17C2×D411yescharacteristic
101C22×Dic17C2×D411yescharacteristic
102Q16×C17C4⋊C4116yescomplemented
103C2×C8×C17—21no—
104C2×C8×C17—21no—
105C2×C8×C17—21no—
106C2×C8×C17C22×C412yes—
107C2×C8×C17C2×D412yes—
108C2×C8×C17C2×Q812yes—
109C4⋊C4×C17—22no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—44no—
112C42×C17—21no—
113C42×C17—21no—
114C42×C17—44no—
115C4⋊Dic17—22no—
116C4⋊Dic17—28no—
117C4⋊Dic17—22no—
118C4⋊Dic17—22no—
119C4⋊Dic17—44no—
120C4⋊Dic17C22×C414yes—
121#63225—24no—
122#63225—44no—
123#63225—82no—
124#63225—82no—
125#63225—82no—
126C4×Dic17—21no—
127C4×Dic17—21no—
128C4×Dic17—28no—
129C4×Dic17—44no—
130C4×Dic17C22×C412yes—
131C4×Dic17#4112yes—
132C2×#3622—21no—
133C2×#3622—21no—
134C2×#3622—21no—
135C2×#3622—21no—
136C2×#3622—21no—
137C2×#3622C22×C412yes—
138#22674—171nomaximal, Sylow
139C23×C17C22×D411yescharacteristic
140Q8×C17—24no—
141Q8×C17—24no—
142Q8×C17—24no—
143Q8×C17C2×C4⋊C418yes—
144C2×C4×C17—24no—
145C2×C4×C17—24no—
146C2×C4×C17—24no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17C23×C411yescharacteristic
154C2×C4×C17C22×D411yescharacteristic
155C2×C4×C17C22×Q811yescharacteristic
156C2×C4×C17C2×#4111yescharacteristic
157C2×C4×C17H(5,𝔽2)11yescharacteristic
158C2×C4×C172−1+411yescharacteristic
159C2×Dic17—22no—
160C2×Dic17—24no—
161C2×Dic17—22no—
162C2×Dic17—22no—
163C2×Dic17—22no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17—42no—
169C2×Dic17C4×D412yes—
170C2×Dic17C4×D412yes—
171C2×Dic17#13714yes—
172C8×C17—41no—
173C8×C17C2×C4⋊C414yes—
174#3622—22no—
175#3622—41no—
176C2×#441—171no—
177C2×#441D1711yescharacteristic
178C2×C4×Q16—171no—
179Q8×C4⋊C4—172no—
180#1943—171no—
181#1943—171no—
182#1865—172no—
183#1865—172no—
184#1844—171no—
185#1803—172no—
186#1770—171no—
187C22×C17C2×C4×D411yescharacteristic
188C22×C17D4211yescharacteristic
189C22×C17D4×Q811yescharacteristic
190C22×C17C2×#13712yes—
191C22×C17#57712yes—
192C4×C17—28no—
193C4×C17—42no—
194C4×C17—82no—
195C4×C17C22×C4⋊C411yescharacteristic
196C4×C17#52011yescharacteristic
197C4×C17#54711yescharacteristic
198C4×C17#55411yescharacteristic
199Dic17—22no—
200Dic17—44no—
201Dic17—42no—
202Dic17—82no—
203Dic17#44114yescomplemented
204C22×Q16C2×D1711yescharacteristic
205C2×C4×Q8—172no—
206C2×C4×Q8—172no—
207C2×C4×Q8C2×D1712yes—
208#441—24no—
209#441—344no—
210C4×Q16—178no—
211C2×#107—171no—
212C2×#106—171no—
213C2×#106—172no—
214C2×#106C2×D1711yescharacteristic
215C2×C4⋊C8—171no—
216C2×#103—172no—
217C2×#103—172no—
218C2×#103C2×D1712yes—
219C2×#97—171no—
220C2×#97C2×D1711yescharacteristic
221C2×C4×C8—171no—
222#389—172no—
223#389—172no—
224#389—342no—
225#389—342no—
226#384—172no—
227#384—342no—
228C4×C4⋊C4—171no—
229C4×C4⋊C4—342no—
230#340—172no—
231#340—172no—
232#336—171no—
233#336—171no—
234C2×C17D4×C4⋊C411yescharacteristic
235C2×C17C2×#44111yescharacteristic
236C2×C17#295211yescharacteristic
237C2×C17#327812yes—
238C2×C17#338712yes—
239C22×Q8C22×D1712yes—
240C2×Q16Dic3414yes—
241C2×Q16C4×D1714yes—
242C2×Q16C22⋊D1714yes—
243C22×C8—171no—
244C22×C8—171no—
245C22×C8C22×D1711yescharacteristic
246C22×C8C22×D1711yescharacteristic
247C4×Q8—24no—
248C4×Q8—42no—
249C4×Q8—178no—
250C4×Q8—344no—
251C4×Q8—344no—
252C4×Q8—682no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4—172no—
256C2×C4⋊C4—172no—
257C2×C4⋊C4—342no—
258C2×C4⋊C4—342no—
259C2×C4⋊C4—342no—
260C2×C4⋊C4—342no—
261C2×C4⋊C4—342no—
262C2×C4⋊C4—342no—
263C2×C4⋊C4C22×D1712yes—
264C2×C42—22no—
265C2×C42—171no—
266C2×C42—171no—
267C2×C42—342no—
268C2×C42—342no—
269C2×C42C22×D1711yescharacteristic
270#107—174no—
271#106—22no—
272#106—342no—
273#106—682no—
274C4⋊C8—342no—
275#103—24no—
276#103—178no—
277#103—344no—
278#97—22no—
279#97—342no—
280C4×C8—174no—
281#95—172no—
282#95—342no—
283#95—342no—
284C17#2267411yescharacteristic, complemented, Sylow
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8C2×Dic3414yes—
289C2×Q8C2×C4×D1714yes—
290C2×Q8C2×C22⋊D1714yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—171no—
294C22×C4—171no—
295C22×C4—172no—
296C22×C4—342no—
297C22×C4—342no—
298C22×C4D4×D1712yes—
299C22×C4C23×D1711yescharacteristic
300Q16#63225116yescomplemented
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—172no—
305C2×C8—341no—
306C2×C8—341no—
307C2×C8—341no—
308C2×C8—341no—
309C2×C8—341no—
310C2×C8C2×Dic3412yes—
311C2×C8C2×C4×D1712yes—
312C2×C8C2×C22⋊D1712yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—44no—
316C4⋊C4—174no—
317C4⋊C4—344no—
318C4⋊C4—348no—
319C4⋊C4—342no—
320C4⋊C4—342no—
321C4⋊C4—342no—
322C4⋊C4—684no—
323C4⋊C4—684no—
324C4⋊C4—1362no—
325C4⋊C4—1362no—
326C4⋊C4—1362no—
327C42—21no—
328C42—21no—
329C42—44no—
330C42—172no—
331C42—172no—
332C42—348no—
333C42—341no—
334C42—341no—
335C42—684no—
336C23C2×D4×D1711yescharacteristic
337Q8—24no—
338Q8—24no—
339Q8—24no—
340Q8C2×#6322518yes—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—172no—
351C2×C4—174no—
352C2×C4—172no—
353C2×C4—342no—
354C2×C4—344no—
355C2×C4—342no—
356C2×C4—342no—
357C2×C4—342no—
358C2×C4—682no—
359C2×C4—682no—
360C2×C4—682no—
361C2×C4—682no—
362C2×C4—682no—
363C2×C4C22×Dic3411yescharacteristic
364C2×C4C22×C4×D1711yescharacteristic
365C2×C4#1058769111yescharacteristic
366C2×C4#41×D1711yescharacteristic
367C2×C4#1058769711yescharacteristic
368C2×C4C22×C22⋊D1711yescharacteristic
369C8—41no—
370C8—342no—
371C8—681no—
372C8C2×#6322514yes—
373C22C4×D4×D1711yescharacteristic
374C22D4×C22⋊D1711yescharacteristic
375C22D4×Dic3411yescharacteristic
376C22#137×D1712yes—
377C22#1176361812yes—
378C4—28no—
379C4—42no—
380C4—82no—
381C4—174no—
382C4—342no—
383C4—682no—
384C4—684no—
385C4—1362no—
386C4C22×#6322511yescharacteristic
387C4#1176328811yescharacteristic
388C4#1176345811yescharacteristic
389C4#1176351811yescharacteristic
390C2D4×#6322511yescharacteristic
391C2D17×#44111yescharacteristic
392C2#1453766411yescharacteristic
393C2#1454044412yes—
394C2#1454171712yes—
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#441C2
socle, prime radicalC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenterC2×#441D17
hypocenterC17#22674
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

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

NH ≅ G/Ntype
15. Q16×Dic17C4semidirect
102. Q16×C17C4⋊C4semidirect
203. Dic17#441semidirect
284. C17#22674semidirect
300. Q16#63225semidirect

Maximal subgroups

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

subgroupindex
C2×#117629162
C2×Q16×Dic172
Q8×#632252
#145295222
#145295202
#145290522
#145290502
#145289392
#145287872
#145286522
C2×C17×#4412
#2267417