Finite Group Databasebeta

← #16930734

Subgroups — #16930734

Order 4352.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763009C211yesmaximal, characteristic
2Q8×#63223C211yesmaximal, characteristic, complemented
3#14529623C211yesmaximal, characteristic
4#14527451C211yesmaximal, characteristic
5#14526658C211yesmaximal, characteristic, complemented
6#14526633C211yesmaximal, characteristic, complemented
7C17×#1487C211yesmaximal, characteristic, complemented
8C22×#63230C412yes—
9C2×#10587560C2211yescharacteristic
10#11763009—21no—
11#11763009—21no—
12#11763009—21no—
13#11763009—21no—
14#11762940—21no—
15#11762940—22no—
16Q8×#3622—21no—
17#11762833—21no—
18#11762833C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587480C412yes—
21C2×#10587494—21no—
22C2×#10587494—21no—
23C4×#63223—21no—
24C4×#63223C2211yescharacteristic
25#11762552—22no—
26C2×C4×Q8×C17C2211yescharacteristic
27C17×#431C2211yescharacteristic
28C17×#423C2211yescharacteristic
29C17×Q8⋊C8—21no—
30C17×Q8⋊C8—21no—
31C22×Q8×C17C2×C411yescharacteristic
32C22×D68—21no—
33C22×D68—21no—
34C22×D68C2×C412yes—
35C2×M16×C17—21no—
36C4×Q8×C17—21no—
37C4×Q8×C17—21no—
38C4×Q8×C17D411yescharacteristic
39C4×Q8×C17D411yescharacteristic
40C4×Q8×C17D411yescharacteristic
41C4×Q8×C17D411yescharacteristic
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17C2×C411yescharacteristic
44C2×C42×C17—21no—
45C2×C42×C17C2311yescharacteristic
46C2×#63230—22no—
47C2×#63230—22no—
48C2×#63230—44no—
49C2×#63223—21no—
50C2×#63223D411yescharacteristic
51C2×#63223D411yescharacteristic
52C22×#3622C2×C412yes—
53#10587560—21no—
54#10587560—21no—
55#10587560—21no—
56#10587560—21no—
57C17×C4⋊C8—21no—
58C17×C4⋊C8—21no—
59C17×C4⋊C8—21no—
60C17×C22⋊C8—21no—
61C4×C8×C17—21no—
62#10587494—42no—
63#10587494—42no—
64#10587480—22no—
65#10587480—22no—
66C4⋊#3622—22no—
67C4⋊#3622—22no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622D412yes—
72#10587474—22no—
73#10587474—21no—
74C4×#3622—22no—
75C4×#3622—21no—
76C23×D17—22no—
77C2×Q8×C17—21no—
78C2×Q8×C17—22no—
79C2×Q8×C17—41no—
80C2×Q8×C17C22⋊C411yescharacteristic
81C2×Q8×C17C22⋊C411yescharacteristic
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17C22×C411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17#4111yescharacteristic
87C2×D68—22no—
88C2×D68—22no—
89C2×D68—42no—
90C2×D68—42no—
91C2×D68—41no—
92C2×D68—41no—
93C2×D68—41no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—41no—
97C2×D68—41no—
98C2×D68—41no—
99C2×D68—82no—
100M16×C17—41no—
101M16×C17—41no—
102C2×C8×C17—41no—
103C2×C8×C17—41no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17C22⋊C411yescharacteristic
110C4⋊C4×C17C22⋊C411yescharacteristic
111C42×C17—21no—
112C42×C17—21no—
113C42×C17—21no—
114C42×C17—21no—
115C42×C17C2×D411yescharacteristic
116C42×C17C2×D411yescharacteristic
117C42×C17C2×D411yescharacteristic
118C42×C17#4111yescharacteristic
119#63230—44no—
120#63230—44no—
121#63223—21no—
122#63223—21no—
123#63223—21no—
124#63223—41no—
125#63223—41no—
126#63223SD1612yescomplemented
127C2×#3622—21no—
128C2×#3622—22no—
129C2×#3622—21no—
130C2×#3622—22no—
131C2×#3622—42no—
132#9254—171nomaximal, Sylow
133C23×C17C4×D411yescharacteristic
134C22×D17—42no—
135C22×D17—42no—
136C22×D17—42no—
137C22×D17—82no—
138C22×D17—82no—
139Q8×C17—21no—
140Q8×C17—21no—
141Q8×C17—21no—
142Q8×C17—42no—
143Q8×C17#10012yescomplemented
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17—81no—
153C2×C4×C17C2×C22⋊C411yescharacteristic
154C2×C4×C17C4×D411yescharacteristic
155C2×C4×C17#12011yescharacteristic
156C2×C4×C17#12111yescharacteristic
157C2×C4×C17#12111yescharacteristic
158C2×C4×C17#12311yescharacteristic
159C2×C4×C17C2×SD1611yescharacteristic
160C2×C4×C17#13511yescharacteristic
161C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C17#13711yescharacteristic
163D68—42no—
164D68—42no—
165D68—82no—
166D68—82no—
167D68—82no—
168D68—81no—
169D68—81no—
170D68—81no—
171D68—81no—
172C8×C17—81no—
173C8×C17—81no—
174#3622—22no—
175#3622—42no—
176#3622—41no—
177#3622—42no—
178C2×#460—171no—
179Q8×M16—171no—
180#1965—171no—
181#1545—171no—
182#1487D1711yescharacteristic, complemented
183#1292—171no—
184#1283—171no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17#38611yescharacteristic
188C22×C17C4×SD1611yescharacteristic
189C22×C17#44011yescharacteristic
190C2×D17—82no—
191C2×D17—82no—
192C2×D17—82no—
193C2×D17—162no—
194C2×D17—162no—
195C4×C17—21no—
196C4×C17—22no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—41no—
200C4×C17—41no—
201C4×C17—41no—
202C4×C17C2×#10011yescharacteristic
203C4×C17#41311yescharacteristic
204C4×C17#44811yescharacteristic
205C4×C17#44911yescharacteristic
206C4×C17#46011yescharacteristic
207C4×C17#46411yescharacteristic
208C22×SD16—172no—
209C2×#127—171no—
210C2×C4×Q8C2×D1711yescharacteristic
211#460—341no—
212#460—341no—
213#460—341no—
214#460—341no—
215#446—342no—
216#446—341no—
217C8×Q8—341no—
218#431C2×D1711yescharacteristic
219#423—171no—
220#423—341no—
221#423C2×D1711yescharacteristic
222C2×C4⋊C8—171no—
223C2×#102—172no—
224C2×#100—341no—
225C2×#100—341no—
226C4×M16—171no—
227C4×M16—341no—
228#331—342no—
229Q8⋊C8—21no—
230Q8⋊C8—21no—
231C2×C17—21no—
232C2×C17—21no—
233C2×C17#188011yescharacteristic
234C2×C17#190011yescharacteristic
235C2×C17#190311yescharacteristic
236D17—162no—
237C22×Q8C4×D1711yescharacteristic
238C22×D4—172no—
239C22×D4—341no—
240C22×D4—341no—
241C2×SD16—342no—
242C2×SD16—342no—
243C2×SD16—684no—
244C2×M16—21no—
245C2×M16—171no—
246C2×M16—171no—
247C2×M16—341no—
248C22×C8—172no—
249#127—341no—
250#127—341no—
251#127—341no—
252#127—341no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8D6811yescharacteristic
256C4×Q8D6811yescharacteristic
257C4×Q8C22⋊D1711yescharacteristic
258C4×Q8C22⋊D1711yescharacteristic
259C2×C4⋊C4—21no—
260C2×C4⋊C4C4×D1711yescharacteristic
261C2×C42—21no—
262C2×C42C22×D1711yescharacteristic
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—172no—
267C4⋊C8—342no—
268C4⋊C8—342no—
269C4⋊C8—341no—
270C4⋊C8—341no—
271C4⋊C8—341no—
272#102—342no—
273#102—342no—
274#100—682no—
275#100—682no—
276C22⋊C8—21no—
277#97—342no—
278#97—341no—
279C4×C8—21no—
280C4×C8—342no—
281C4×C8—341no—
282C17#925411yescharacteristic, complemented, Sylow
283C24—342no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8#6322711yescharacteristic
288C2×Q8#6322711yescharacteristic
289C2×D4—342no—
290C2×D4—342no—
291C2×D4—682no—
292C2×D4—682no—
293C2×D4—681no—
294C2×D4—681no—
295C2×D4—681no—
296C2×D4—681no—
297C2×D4—681no—
298C2×D4—681no—
299C2×D4—681no—
300C2×D4—681no—
301C2×D4—1362no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×C4×D1711yescharacteristic
305C22×C4D4×D1711yescharacteristic
306C22×C4#6325411yescharacteristic
307SD16—684no—
308SD16—684no—
309M16—41no—
310M16—41no—
311M16—172no—
312M16—341no—
313M16—341no—
314M16—341no—
315M16—681no—
316M16—681no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—341no—
320C2×C8—342no—
321C2×C8—341no—
322C2×C8—342no—
323C2×C8—682no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4#6322711yescharacteristic
330C4⋊C4#6322711yescharacteristic
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C2×D6811yescharacteristic
336C42D4×D1711yescharacteristic
337C42#6325411yescharacteristic
338C42C2×C22⋊D1711yescharacteristic
339C23—682no—
340C23—682no—
341C23—682no—
342C23—1362no—
343C23—1362no—
344C23#1058756711yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—21no—
348Q8—42no—
349Q8#1058749412yescomplemented
350D4—682no—
351D4—682no—
352D4—1362no—
353D4—1362no—
354D4—1362no—
355D4—1361no—
356D4—1361no—
357D4—1361no—
358D4—1361no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4#1058756711yescharacteristic
369C2×C4#1058756811yescharacteristic
370C2×C4#1058757211yescharacteristic
371C2×C4SD16×D1711yescharacteristic
372C2×C4#1058760011yescharacteristic
373C2×C4#1058760411yescharacteristic
374C2×C4#1058760511yescharacteristic
375C2×C4C2×#6322711yescharacteristic
376C2×C4#1058762511yescharacteristic
377C2×C4#1058762511yescharacteristic
378C8—81no—
379C8—81no—
380C8—342no—
381C8—681no—
382C8—682no—
383C8—682no—
384C22—21no—
385C22—21no—
386C22—2722no—
387C22—2722no—
388C22—1362no—
389C22—1362no—
390C22—1362no—
391C22#1176268611yescharacteristic
392C22#1176289411yescharacteristic
393C22#1176291211yescharacteristic
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4C2×#1058749411yescharacteristic
402C4#1176279111yescharacteristic
403C4#1176294911yescharacteristic
404C4#1176295711yescharacteristic
405C4#1176300711yescharacteristic
406C4#1176303311yescharacteristic
407C2—21no—
408C2—21no—
409C2—2722no—
410C2#1452913711yescharacteristic
411C2#1452924511yescharacteristic
412C2#1452926211yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1487C2
socleC22×C17#386
prime radicalC2×#10587560C22
cosocleC2×C42×C17C23
hypercenter#1487D17
hypocenterC17#9254
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. Q8×#63223C2semidirect
5. #14526658C2semidirect
6. #14526633C2semidirect
7. C17×#1487C2semidirect
126. #63223SD16semidirect
143. Q8×C17#100semidirect
182. #1487D17semidirect
282. C17#9254semidirect
349. Q8#10587494semidirect

Maximal subgroups

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

subgroupindex
C2×#117630092
Q8×#632232
#145296232
#145274512
#145266582
#145266332
C17×#14872
#925417