Finite Group Databasebeta

← #17053215

Subgroups — #17053215

Order 4352.

658 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1M16×Dic34C211yesmaximal, characteristic
2#14537735C211yesmaximal, characteristic
3C2×#11762933C211yesmaximal, characteristic
4#14529560C211yesmaximal, characteristic
5#14529558C212yesmaximal
6#14529556C212yesmaximal
7#14528955C211yesmaximal, characteristic
8#14528954C212yesmaximal
9C4×#10587493C211yesmaximal, characteristic
10#14528685C212yesmaximal
11C17×#1950C211yesmaximal, characteristic
12C2×C4×Dic34C2211yescharacteristic
13#11762939—21no—
14#11762938—22no—
15#11762937—22no—
16#11762936—21no—
17C8×Dic34—21no—
18#11762933—21no—
19#11762933C414yes—
20#11762933C412yes—
21#11762933C412yes—
22C2×#10587490C2212yes—
23C2×#10587493C2212yes—
24C2×#10587493C2211yescharacteristic
25C2×#10587493C2212yes—
26C2×#10587493C2211yescharacteristic
27#11762833C2211yescharacteristic
28#11762832C2212yes—
29#11762832C2212yes—
30C2×#10587485C2212yes—
31C4×#63223C2211yescharacteristic
32M16×Dic17C2212yes—
33M16×Dic17C2212yes—
34C2×C4×#3622C2211yescharacteristic
35C2×C8×Dic17C2212yes—
36#11762574C2212yes—
37#11762573C2211yescharacteristic
38#11762573C2211yescharacteristic
39C2×C17×#108C2212yes—
40C17×#423C2211yescharacteristic
41C2×C17×C4⋊C8C2211yescharacteristic
42C4×M16×C17C2211yescharacteristic
43C17×#341C2212yes—
44C22×Dic34Q811yescharacteristic
45C2×M16×C17C2312yes—
46C2×M16×C17C2311yescharacteristic
47C2×M16×C17C2311yescharacteristic
48C22×C8×C17C2312yes—
49C2×C42×C17C2311yescharacteristic
50C2×C4⋊Dic17Q811yescharacteristic
51C2×#63225Q812yes—
52C2×C4×Dic17C2312yes—
53C2×#63223C2312yes—
54C2×#63223C2311yescharacteristic
55C2×#63223C2311yescharacteristic
56C22×#3622C2311yescharacteristic
57C22×#3622C2311yescharacteristic
58C4×Dic34C2×C412yes—
59C4×Dic34C2×C411yescharacteristic
60C4×Dic34C2×C411yescharacteristic
61C4×Dic34D411yescharacteristic
62C4×Dic34D411yescharacteristic
63C4×Dic34D412yes—
64C17×#108—42no—
65C17×C4⋊C8—21no—
66C17×C4⋊C8—21no—
67C17×C4⋊C8C2×C412yes—
68C17×C4⋊C8C2×C412yes—
69C17×#97—21no—
70C4×C8×C17—21no—
71#10587493—24no—
72#10587493—22no—
73#10587493—42no—
74#10587493Q814yescomplemented
75#10587490—42no—
76#10587486—22no—
77#10587486—22no—
78#10587485—22no—
79#10587485—22no—
80#10587485—22no—
81#10587485—22no—
82#10587485C2×C414yes—
83#10587485C2×C414yes—
84C8×Dic17—22no—
85C8×Dic17—22no—
86C8×Dic17C2×C414yes—
87C8×Dic17C2×C414yes—
88C4⋊#3622—21no—
89C4⋊#3622—21no—
90#10587474—21no—
91C4×#3622—21no—
92C4×#3622C2×C412yes—
93C4×#3622C2×C412yes—
94C22×C4×C17C2×Q811yescharacteristic
95C22×C4×C17C2×Q811yescharacteristic
96C22×C4×C17C2411yescharacteristic
97C22×Dic17C2×Q812yes—
98C22×Dic17C2×Q812yes—
99C2×Dic34—24no—
100C2×Dic34C4⋊C411yescharacteristic
101C2×Dic34C4⋊C412yes—
102C2×Dic34C4⋊C411yescharacteristic
103M16×C17—21no—
104M16×C17—22no—
105M16×C17—21no—
106M16×C17—22no—
107M16×C17—21no—
108M16×C17C2×Q812yes—
109C2×C8×C17—21no—
110C2×C8×C17—22no—
111C2×C8×C17—22no—
112C2×C8×C17—22no—
113C2×C8×C17—21no—
114C2×C8×C17C22×C414yes—
115C42×C17C22×C411yescharacteristic
116C42×C17C22×C411yescharacteristic
117C42×C17C2×D411yescharacteristic
118C42×C17C2×D411yescharacteristic
119C4⋊Dic17C4⋊C412yes—
120C4⋊Dic17C4⋊C411yescharacteristic
121C4⋊Dic17C4⋊C411yescharacteristic
122#63225C4⋊C412yes—
123#63225C4⋊C412yes—
124#63225C4⋊C412yes—
125#63225C4⋊C412yes—
126C4×Dic17C22×C412yes—
127C4×Dic17C22×C412yes—
128C4×Dic17C2×D412yes—
129C4×Dic17C2×D412yes—
130#63223—21no—
131#63223—22no—
132#63223—21no—
133#63223—21no—
134#63223—22no—
135#63223C2×Q812yes—
136C2×#3622—21no—
137C2×#3622—21no—
138C2×#3622—21no—
139C2×#3622—21no—
140C2×#3622—22no—
141C2×#3622C22×C412yes—
142C2×#3622C22×C412yes—
143C2×#3622C2×Q812yes—
144C2×#3622#4112yes—
145Q8×#108—171nomaximal, Sylow
146C23×C17C22×Q811yescharacteristic
147C2×C4×C17—22no—
148C2×C4×C17—41no—
149C2×C4×C17C2×C4⋊C411yescharacteristic
150C2×C4×C17C2×C4⋊C411yescharacteristic
151C2×C4×C17C2×C4⋊C411yescharacteristic
152C2×C4×C17C2×C4⋊C411yescharacteristic
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—24no—
160C2×Dic17—42no—
161C2×Dic17C2×C4⋊C412yes—
162C2×Dic17C2×C4⋊C412yes—
163C2×Dic17C2×C4⋊C412yes—
164C2×Dic17C2×C4⋊C412yes—
165Dic34—22no—
166Dic34—24no—
167Dic34#10814yescomplemented
168C8×C17—24no—
169C8×C17—22no—
170C8×C17—42no—
171C8×C17—41no—
172#3622—22no—
173#3622—22no—
174#3622—41no—
175#3622—42no—
176#3622C4×Q814yes—
177Q8×M16—171no—
178Q8×M16—171no—
179C2×C8×Q8—171no—
180#1950—172no—
181#1950—171no—
182#1950—172no—
183#1950D1711yescharacteristic
184#1848—172no—
185#1848—171no—
186C4×#108—171no—
187C4×#108—172no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17C22×C4⋊C411yescharacteristic
191C22×C17#55711yescharacteristic
192C22×C17Q8211yescharacteristic
193C4×C17—21no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17C2×#10812yes—
197C4×C17C2×C4×Q811yescharacteristic
198C4×C17#52011yescharacteristic
199C4×C17D4×Q811yescharacteristic
200C4×C17#55011yescharacteristic
201Dic17—22no—
202Dic17—22no—
203Dic17—22no—
204Dic17C2×#10814yes—
205C2×C4×Q8—171no—
206#446—341no—
207#446—342no—
208#446—341no—
209#446—342no—
210C8×Q8—172no—
211C8×Q8—174no—
212C8×Q8—172no—
213C8×Q8—341no—
214C8×Q8—341no—
215C2×#108—171no—
216C2×#108—171no—
217C2×#108—172no—
218C2×#108—172no—
219C2×#108—172no—
220C2×#108C2×D1712yes—
221#423—172no—
222#423—172no—
223#423—171no—
224#423C2×D1711yescharacteristic
225C2×C4⋊C8—172no—
226C2×C4⋊C8C2×D1711yescharacteristic
227C4×M16—172no—
228C4×M16—172no—
229C4×M16—171no—
230C4×M16C2×D1711yescharacteristic
231C2×C4×C8—172no—
232C2×C4×C8—171no—
233#341—171no—
234#341—171no—
235#341—172no—
236#341C2×D1712yes—
237C2×C17—21no—
238C2×C17—21no—
239C2×C17Q8×C4⋊C411yescharacteristic
240C2×C17C22×#10811yescharacteristic
241C2×C17#291611yescharacteristic
242C22×Q8—171no—
243C2×M16—171no—
244C2×M16—171no—
245C2×M16—172no—
246C2×M16C22×D1711yescharacteristic
247C2×M16C22×D1712yes—
248C2×M16C22×D1711yescharacteristic
249C22×C8—171no—
250C22×C8—171no—
251C22×C8C22×D1712yes—
252C4×Q8—171no—
253C4×Q8—172no—
254C4×Q8—171no—
255C4×Q8—171no—
256C4×Q8—172no—
257C4×Q8—171no—
258C2×C4⋊C4—172no—
259C2×C4⋊C4—171no—
260C2×C42—172no—
261C2×C42C22×D1711yescharacteristic
262#108—42no—
263#108—174no—
264#108—342no—
265#108—344no—
266#108—682no—
267#108—682no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—174no—
271C4⋊C8—174no—
272C4⋊C8—342no—
273C4⋊C8—342no—
274C4⋊C8—342no—
275C4⋊C8—342no—
276C4⋊C8—341no—
277C4⋊C8—341no—
278C4⋊C8C4×D1712yes—
279C4⋊C8C4×D1712yes—
280#97—21no—
281#97—342no—
282#97—342no—
283#97—341no—
284C4×C8—21no—
285C4×C8—174no—
286C4×C8—174no—
287C4×C8—172no—
288C4×C8—172no—
289C4×C8—342no—
290C4×C8—342no—
291C4×C8—341no—
292C17Q8×#10811yescharacteristic, complemented, Sylow
293C2×Q8—171no—
294C2×Q8—172no—
295C2×Q8—171no—
296C2×Q8—344no—
297C22×C4—172no—
298C22×C4—172no—
299C22×C4Q8×D1711yescharacteristic
300C22×C4Q8×D1711yescharacteristic
301C22×C4C23×D1711yescharacteristic
302M16—21no—
303M16—21no—
304M16—22no—
305M16—21no—
306M16—22no—
307M16—172no—
308M16—341no—
309M16—341no—
310M16—342no—
311M16—341no—
312M16—342no—
313M16Q8×D1712yes—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—172no—
320C2×C8—172no—
321C2×C8—172no—
322C2×C8—172no—
323C2×C8—341no—
324C2×C8—341no—
325C2×C8—341no—
326C2×C8—342no—
327C2×C8—341no—
328C2×C8C2×C4×D1714yes—
329C4⋊C4—172no—
330C4⋊C4—172no—
331C4⋊C4—172no—
332C4⋊C4—172no—
333C4⋊C4—172no—
334C4⋊C4—171no—
335C4⋊C4—171no—
336C42—172no—
337C42—172no—
338C42—172no—
339C42—172no—
340C42C2×C4×D1711yescharacteristic
341C42C2×C4×D1711yescharacteristic
342C42D4×D1711yescharacteristic
343C42D4×D1711yescharacteristic
344C23C2×Q8×D1711yescharacteristic
345Q8—174no—
346Q8—344no—
347Q8—342no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—172no—
351C2×C4—172no—
352C2×C4—172no—
353C2×C4—172no—
354C2×C4—344no—
355C2×C4—682no—
356C2×C4C4⋊C4×D1711yescharacteristic
357C2×C4C4⋊C4×D1711yescharacteristic
358C2×C4C4⋊C4×D1711yescharacteristic
359C2×C4C4⋊C4×D1711yescharacteristic
360C2×C4C22×C4×D1711yescharacteristic
361C2×C4C2×D4×D1711yescharacteristic
362C2×C4#1058769111yescharacteristic
363C2×C4C2×Q8×D1711yescharacteristic
364C2×C4#41×D1711yescharacteristic
365C2×C4#1058769711yescharacteristic
366C8—22no—
367C8—24no—
368C8—41no—
369C8—42no—
370C8—174no—
371C8—342no—
372C8—342no—
373C8—682no—
374C8—681no—
375C22—21no—
376C22—21no—
377C22C2×C4⋊C4×D1711yescharacteristic
378C22#1176353611yescharacteristic
379C22#1176353911yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—174no—
384C4—342no—
385C4—342no—
386C4—342no—
387C4#108×D1712yes—
388C4C4×Q8×D1711yescharacteristic
389C4#1176328811yescharacteristic
390C4#1176347211yescharacteristic
391C4#1176347711yescharacteristic
392C2—21no—
393C2—21no—
394C2#1453169811yescharacteristic
395C2C2×#108×D1711yescharacteristic
396C2#1453739411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1950C2
socleC22×C17C22×C4⋊C4
prime radicalC23×C17C22×Q8
cosocleC22×C4×C17C24
hypercenter#1950D17
hypocenterC17Q8×#108
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
74. #10587493Q8semidirect
167. Dic34#108semidirect
292. C17Q8×#108semidirect

Maximal subgroups

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

subgroupindex
M16×Dic342
#145377352
C2×#117629332
#145295602
#145295582
#145295562
#145289552
#145289542
C4×#105874932
#145286852
C17×#19502
Q8×#10817