Finite Group Databasebeta

← #16953491

Subgroups — #16953491

Order 4352.

446 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763047C211yesmaximal, characteristic
2C2×#11762636C211yesmaximal, characteristic, complemented
3C2×#11762613C211yesmaximal, characteristic, complemented
4C2×#11762588C211yesmaximal, characteristic
5#14528810C211yesmaximal, characteristic
6#14526876C211yesmaximal, characteristic, complemented
7C2×C17×#358C211yesmaximal, characteristic, complemented
8C22×C8×D17C2211yescharacteristic
9C2×#10587632—21no—
10C22×#63214C2211yescharacteristic
11#11763047—22no—
12#11763047—22no—
13C2×#10587489C2211yescharacteristic
14C2×#10587489C2211yescharacteristic
15C2×#10587506—21no—
16#11762666—21no—
17#11762636—22no—
18#11762613—22no—
19#11762588—22no—
20#11762571—21no—
21C22×Q16×C17C2211yescharacteristic
22C22×C16×C17C2211yescharacteristic
23C2×C17×#107C2211yescharacteristic
24C17×#358—22no—
25C22×Q8×C17—21no—
26C22×C4×D17D411yescharacteristic
27C2×Q16×C17—21no—
28C2×Q16×C17—21no—
29C2×Q16×C17—21no—
30C2×Q16×C17—21no—
31C2×Q16×C17—21no—
32C2×Q16×C17D412yes—
33C22×C8×C17C2311yescharacteristic
34C2×C4⋊C4×C17—21no—
35#10587632—24no—
36#10587632—42no—
37C2×#63227—41no—
38C2×#63227—41no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×#63225—41no—
42C22×#3622D411yescharacteristic
43C2×C8×D17—21no—
44C2×C8×D17—21no—
45C2×C8×D17—21no—
46C2×C8×D17—21no—
47C2×C8×D17—21no—
48C2×C8×D17—21no—
49C2×C16×C17—21no—
50C2×C16×C17—21no—
51C2×C16×C17—21no—
52C17×#107—21no—
53C17×#107—21no—
54#10587506—24no—
55#10587502—41no—
56#10587489—21no—
57#10587489—21no—
58#10587489—21no—
59#10587489D412yes—
60C2×#63214—21no—
61C2×#63214—21no—
62C2×#63214D412yes—
63C23×D17D811yescharacteristic
64C2×Q8×C17—22no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C2×Q8×C17—41no—
68C2×Q8×C17—41no—
69C2×Q8×C17—41no—
70C22×C4×C17—41no—
71C22×C4×C17—41no—
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17D811yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82Q16×C17—22no—
83Q16×C17—22no—
84Q16×C17—41no—
85Q16×C17—41no—
86C2×C8×C17C2×D411yescharacteristic
87C2×C8×C17C2×D411yescharacteristic
88C2×C8×C17C2×D411yescharacteristic
89C2×C8×C17#4111yescharacteristic
90C2×C8×C17#4111yescharacteristic
91C2×C8×C17#4111yescharacteristic
92C4⋊C4×C17—41no—
93C4⋊C4×C17—41no—
94#63227—44no—
95#63227—82no—
96C4⋊Dic17—22no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100#63225—44no—
101C2×#3622—21no—
102C2×#3622—21no—
103C2×#3622—21no—
104C8×D17—41no—
105C8×D17—41no—
106C8×D17—41no—
107C8×D17—41no—
108C16×C17—41no—
109#63214—22no—
110#13382—171nomaximal, Sylow
111C23×C17C2×D811yescharacteristic
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17SD3212yes—
119Q8×C17—42no—
120Q8×C17—42no—
121Q8×C17—81no—
122Q8×C17—81no—
123C2×C4×C17—42no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17—81no—
127C2×C4×C17—81no—
128C2×C4×C17—81no—
129C2×C4×C17#12011yescharacteristic
130C2×C4×C17#12111yescharacteristic
131C2×C4×C17#12311yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C17#13711yescharacteristic
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—42no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17—81no—
142C2×Dic17—81no—
143C2×Dic17—81no—
144C4×D17—41no—
145C4×D17—41no—
146C4×D17—41no—
147C4×D17—41no—
148C8×C17#12111yescharacteristic
149C8×C17#12111yescharacteristic
150C8×C17#12311yescharacteristic
151C8×C17#12411yescharacteristic
152#3622—41no—
153C2×#467—171no—
154C2×#367—171no—
155C2×#358—171no—
156C2×#358D1711yescharacteristic, complemented
157C2×C22⋊C16—171no—
158#1808—171no—
159#1380—171no—
160C22×C17#44711yescharacteristic
161C22×C17#46611yescharacteristic
162C22×C17#48011yescharacteristic
163C22×C17C2×SD3211yescharacteristic
164C22×C17#50811yescharacteristic
165C22×C17#51011yescharacteristic
166C22×C17#51011yescharacteristic
167C2×D17—22no—
168C2×D17—22no—
169C2×D17—22no—
170C2×D17—41no—
171C2×D17—41no—
172C2×D17—41no—
173C2×D17—41no—
174C4×C17—82no—
175C4×C17—161no—
176C4×C17#39411yescharacteristic
177C4×C17#45211yescharacteristic
178C4×C17#46811yescharacteristic
179C4×C17#48211yescharacteristic
180Dic17—41no—
181Dic17—82no—
182Dic17—161no—
183C22×Q16C2×D1711yescharacteristic
184C23×C8—171no—
185C2×C22⋊Q8—341no—
186C22×C16—171no—
187C22×C16C2×D1711yescharacteristic
188#467—342no—
189#467—342no—
190C2×#107—171no—
191C2×#107—171no—
192C2×#107C2×D1711yescharacteristic
193C2×#103—341no—
194#380—341no—
195#367—342no—
196#358—22no—
197#358—342no—
198C22⋊C16—342no—
199#340—341no—
200C2×C17#203311yescharacteristic
201C2×C17#218511yescharacteristic
202C2×C17#218911yescharacteristic
203C2×C17#221711yescharacteristic
204C2×C17#221911yescharacteristic
205C2×C17#223511yescharacteristic
206C2×C17#223711yescharacteristic
207D17—42no—
208C22×Q8—21no—
209C23×C4—171no—
210C2×Q16—21no—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16C22⋊D1712yes—
216C22×C8—171no—
217C22×C8—341no—
218C22×C8—341no—
219C22×C8—341no—
220C22×C8—341no—
221C22×C8—341no—
222C22×C8—341no—
223C22×C8C22×D1711yescharacteristic
224C22⋊Q8—344no—
225C22⋊Q8—682no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4—681no—
230C2×C22⋊C4—681no—
231C2×C22⋊C4—681no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—172no—
236C2×C16—341no—
237C2×C16—341no—
238#107—21no—
239#107—21no—
240#107—172no—
241#107—341no—
242#107—341no—
243#107—341no—
244#103—344no—
245#95—681no—
246C17#1338211yescharacteristic, complemented, Sylow
247C24—171no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—171no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4D4×D1711yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—41no—
269Q16—41no—
270C2×C8—341no—
271C2×C8—341no—
272C2×C8—341no—
273C2×C8—681no—
274C2×C8—681no—
275C2×C8—681no—
276C2×C8—681no—
277C2×C8#6325211yescharacteristic
278C2×C8#6325211yescharacteristic
279C2×C8D4×D1711yescharacteristic
280C2×C8D4×D1711yescharacteristic
281C2×C8#6325411yescharacteristic
282C2×C8C2×C22⋊D1711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—342no—
286C4⋊C4—684no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C22⋊C4—684no—
291C22⋊C4—1362no—
292C16—41no—
293C16—342no—
294C23—172no—
295C23—341no—
296C23—341no—
297C23—341no—
298C23—341no—
299C23—341no—
300C23—341no—
301C23D8×D1711yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—682no—
320C2×C4—1361no—
321C2×C4—1361no—
322C2×C4—1361no—
323C2×C4—1361no—
324C2×C4—1361no—
325C2×C4#1058756911yescharacteristic
326C2×C4#1058757011yescharacteristic
327C2×C4#1058759711yescharacteristic
328C2×C4#1058760111yescharacteristic
329C2×C4#1058760211yescharacteristic
330C2×C4#1058762311yescharacteristic
331C8—681no—
332C8#1058757011yescharacteristic
333C8#1058757111yescharacteristic
334C8#1058761511yescharacteristic
335C8#1058762511yescharacteristic
336C22—342no—
337C22—342no—
338C22—342no—
339C22—681no—
340C22—681no—
341C22—681no—
342C22—681no—
343C22#1176294111yescharacteristic
344C22#1176304411yescharacteristic
345C22#1176312111yescharacteristic
346C22SD32×D1711yescharacteristic
347C22#1176324211yescharacteristic
348C22#1176324811yescharacteristic
349C22#1176325111yescharacteristic
350C4—82no—
351C4—161no—
352C4—2721no—
353C4—681no—
354C4—1362no—
355C4#1176272411yescharacteristic
356C4#1176296811yescharacteristic
357C4#1176305411yescharacteristic
358C4#1176313211yescharacteristic
359C2—682no—
360C2#1452996411yescharacteristic
361C2#1453069911yescharacteristic
362C2#1453072311yescharacteristic
363C2#1453088311yescharacteristic
364C2#1453089111yescharacteristic
365C2#1453098011yescharacteristic
366C2#1453099111yescharacteristic
subgroupG/H
centerC23D8×D17
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC2×C17×#358C2
socleC23×C17C2×D8
prime radicalC23×D17D8
hypercenterC2×#358D17
hypocenterC17#13382
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×#11762636C2semidirect
3. C2×#11762613C2semidirect
6. #14526876C2semidirect
7. C2×C17×#358C2semidirect
156. C2×#358D17semidirect
246. C17#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#117630472
C2×#117626362
C2×#117626132
C2×#117625882
#145288102
#145268762
C2×C17×#3582
#1338217