Finite Group Databasebeta

← #16961096

Subgroups — #16961096

Order 4352.

1830 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
1C22×Q8×Dic17C211yesmaximal, characteristic, complemented
2C2×#11762702C213yesmaximal
3#14527055C211yesmaximal, characteristic, complemented
4#14527049C213yesmaximal, complemented
5#14527029C213yesmaximal
6#14527009C213yesmaximal, complemented
7C2×C17×#393C211yesmaximal, characteristic
8C2×Q8×Dic17—212no—
9C22×C4⋊Dic17C2213yes—
10C22×C4⋊Dic17C2213yes—
11C22×C22⋊Dic17C2211yescharacteristic
12C22×C22⋊Dic17C2213yes—
13C22×C4×Dic17C2213yes—
14#11762702—212no—
15C2×#10587502C2213yes—
16C2×#10587502C2213yes—
17C2×#10587502C2213yes—
18C2×#10587502C2216yes—
19C23×Q8×C17C2211yescharacteristic
20C22×C22⋊C4×C17C2213yes—
21C17×#393C418yescomplemented
22C2×C17×#95C2213yes—
23C25×C17C2311yescharacteristic
24C22×Q8×C17—26no—
25C22×Q8×C17—26no—
26C22×Q8×C17C2×C414yes—
27C23×C4×C17C2313yes—
28C23×C4×C17C2313yes—
29C23×Dic17C2311yescharacteristic
30C23×Dic17C2313yes—
31C23×Dic17C2313yes—
32C23×Dic17C2311yescharacteristic
33C2×C22⋊C4×C17—26no—
34C2×C22⋊C4×C17—26no—
35C2×C22⋊C4×C17C2×C4112yes—
36C2×C22⋊Dic17—26no—
37C2×C22⋊Dic17—26no—
38C2×C22⋊Dic17—46no—
39Q8×Dic17—232no—
40C2×C4⋊Dic17—26no—
41C2×C4⋊Dic17—212no—
42C2×C4⋊Dic17—26no—
43C2×C4⋊Dic17—46no—
44C2×C4×Dic17—212no—
45C2×C4×Dic17D4112yes—
46C17×#95C2×C4112yes—
47#10587502—26no—
48#10587502—26no—
49#10587502—26no—
50#10587502—46no—
51C24×C17—23no—
52C24×C17—23no—
53C24×C17—23no—
54C24×C17—21no—
55C24×C17—21no—
56C24×C17—23no—
57C24×C17C22×C412yes—
58C24×C17C2411yescharacteristic
59C2×Q8×C17—224no—
60C2×Q8×C17—28no—
61C2×Q8×C17—224no—
62C22×C4×C17—26no—
63C22×C4×C17—26no—
64C22×C4×C17—23no—
65C22×C4×C17—23no—
66C22×C4×C17—26no—
67C22×C4×C17—26no—
68C22×C4×C17C22×C416yes—
69C22×C4×C17C22×C416yes—
70C22×C4×C17C2×D416yes—
71C22×C4×C17#4116yes—
72C22×Dic17—21no—
73C22×Dic17—23no—
74C22×Dic17—26no—
75C22×Dic17—23no—
76C22×Dic17—26no—
77C22×Dic17—23no—
78C22×Dic17—23no—
79C22×Dic17—23no—
80C22×Dic17—26no—
81C22×Dic17—26no—
82C22×Dic17—23no—
83C22×Dic17—23no—
84C22×Dic17—21no—
85C22×Dic17—23no—
86C22×Dic17C2×D416yes—
87C22×Dic17C2×D416yes—
88C22⋊C4×C17—224no—
89C22⋊Dic17—412no—
90C22⋊Dic17—44no—
91C4⋊Dic17—224no—
92C4⋊Dic17—412no—
93C4×Dic17—224no—
94#14301—171nomaximal, Sylow
95C23×C17—26no—
96C23×C17—22no—
97C23×C17—26no—
98C23×C17—41no—
99C23×C17—43no—
100C23×C17—43no—
101C23×C17—43no—
102C23×C17—43no—
103C23×C17—43no—
104C23×C17—43no—
105C23×C17—46no—
106C23×C17—43no—
107C23×C17C23×C411yescharacteristic
108C23×C17C22×D413yes—
109C23×C17C2×#4113yes—
110C23×C17H(5,𝔽2)13yes—
111C23×C172−1+411yescharacteristic
112C23×C172−1+411yescharacteristic
113C23×C172−1+413yes—
114Q8×C17—232no—
115C2×C4×C17—212no—
116C2×C4×C17—26no—
117C2×C4×C17—26no—
118C2×C4×C17—212no—
119C2×C4×C17—26no—
120C2×C4×C17—212no—
121C2×C4×C17—43no—
122C2×C4×C17—43no—
123C2×C4×C17—46no—
124C2×C4×C17C4×D4112yes—
125C2×Dic17—26no—
126C2×Dic17—26no—
127C2×Dic17—26no—
128C2×Dic17—26no—
129C2×Dic17—26no—
130C2×Dic17—46no—
131C2×Dic17—46no—
132C2×Dic17—41no—
133C2×Dic17—43no—
134C2×Dic17—43no—
135C2×Dic17—43no—
136C2×Dic17—43no—
137C2×Dic17—43no—
138C2×Dic17—46no—
139C2×Dic17—46no—
140C2×Dic17#12014yes—
141C22×C4×Q8—171no—
142C2×#393D1711yescharacteristic
143C2×#388—173no—
144#1446—171no—
145#1444—173no—
146#1439—173no—
147#1434—173no—
148C22×C17—46no—
149C22×C17—46no—
150C22×C17—42no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17—83no—
154C22×C17—83no—
155C22×C17C2×C4×D413yes—
156C22×C17#51911yescharacteristic
157C22×C17#52013yes—
158C22×C17C2×#12011yescharacteristic
159C22×C17#52513yes—
160C22×C17#53513yes—
161C22×C17#53613yes—
162C22×C17#53613yes—
163C22×C17#53713yes—
164C22×C17#54113yes—
165C22×C17#54216yes—
166C22×C17#54213yes—
167C4×C17—212no—
168C4×C17—46no—
169Dic17—24no—
170Dic17—46no—
171Dic17—81no—
172Dic17—83no—
173C23×Q8C2×D1711yescharacteristic
174C2×C4×Q8—3412no—
175C22×C4⋊C4—173no—
176C22×C4⋊C4—173no—
177C22×C22⋊C4—171no—
178C22×C22⋊C4—173no—
179C22×C22⋊C4C2×D1713yes—
180C22×C42—173no—
181#393Dic1718yescomplemented
182#388—3412no—
183C2×#95—173no—
184C2×#95—173no—
185C2×#95—173no—
186C2×#95—176no—
187C2×#95C2×D1713yes—
188C2×C17—82no—
189C2×C17C4×#12011yescharacteristic
190C2×C17#232313yes—
191C2×C17#232613yes—
192C2×C17#261411yescharacteristic
193C2×C17#261511yescharacteristic
194C2×C17#263213yes—
195C2×C17#263713yes—
196C25C22×D1711yescharacteristic
197C22×Q8—26no—
198C22×Q8—26no—
199C22×Q8C2×Dic1714yes—
200C23×C4—171no—
201C23×C4—173no—
202C23×C4—171no—
203C23×C4—173no—
204C23×C4C22×D1713yes—
205C23×C4C22×D1713yes—
206C4×Q8—3432no—
207C2×C4⋊C4—3412no—
208C2×C4⋊C4—346no—
209C2×C4⋊C4—346no—
210C2×C4⋊C4—686no—
211C2×C22⋊C4—26no—
212C2×C22⋊C4—26no—
213C2×C22⋊C4—346no—
214C2×C22⋊C4—346no—
215C2×C22⋊C4—686no—
216C2×C22⋊C4C2×Dic17112yes—
217C2×C42—1712no—
218C2×C42—3412no—
219#95—346no—
220#95—346no—
221#95—346no—
222#95—686no—
223#95C2×Dic17112yes—
224C17#1430111yescharacteristic, complemented, Sylow
225C24—23no—
226C24—23no—
227C24—21no—
228C24—21no—
229C24—23no—
230C24—23no—
231C24C22×Dic1712yes—
232C24C23×D1711yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—176no—
243C22×C4—176no—
244C22×C4—341no—
245C22×C4—343no—
246C22×C4—346no—
247C22×C4—343no—
248C22×C4—343no—
249C22×C4—346no—
250C22×C4—343no—
251C22×C4—343no—
252C22×C4—346no—
253C22×C4—343no—
254C22×C4—341no—
255C22×C4—346no—
256C22×C4—343no—
257C22×C4—343no—
258C22×C4D4×D1716yes—
259C22×C4#6325416yes—
260C22×C4C22×Dic1716yes—
261C22×C4C22×Dic1716yes—
262C4⋊C4—3424no—
263C4⋊C4—6812no—
264C22⋊C4—224no—
265C22⋊C4—684no—
266C22⋊C4—6812no—
267C42—3424no—
268C23—26no—
269C23—22no—
270C23—26no—
271C23—41no—
272C23—43no—
273C23—43no—
274C23—43no—
275C23—43no—
276C23—43no—
277C23—43no—
278C23—46no—
279C23—43no—
280C23C2×D4×D1713yes—
281C23C2×#6325413yes—
282C23#1058769411yescharacteristic
283C23#1058769411yescharacteristic
284C23#1058769613yes—
285C23#1058769713yes—
286C23C23×Dic1711yescharacteristic
287Q8—232no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—212no—
294C2×C4—43no—
295C2×C4—43no—
296C2×C4—46no—
297C2×C4—174no—
298C2×C4—346no—
299C2×C4—346no—
300C2×C4—346no—
301C2×C4—346no—
302C2×C4—346no—
303C2×C4—686no—
304C2×C4—686no—
305C2×C4—681no—
306C2×C4—683no—
307C2×C4—683no—
308C2×C4—683no—
309C2×C4—683no—
310C2×C4—683no—
311C2×C4—686no—
312C2×C4—686no—
313C2×C4D4×Dic17112yes—
314C22—46no—
315C22—46no—
316C22—42no—
317C22—83no—
318C22—81no—
319C22—81no—
320C22—83no—
321C22C2×D4×Dic1713yes—
322C22#1176328711yescharacteristic
323C22#1176329213yes—
324C22#120×D1711yescharacteristic
325C22#1176331513yes—
326C22#1176337713yes—
327C22#1176337913yes—
328C22#1176338213yes—
329C22#1176338913yes—
330C22#1176341913yes—
331C22#1176342816yes—
332C22#1176343213yes—
333C4—212no—
334C4—46no—
335C4—344no—
336C4—686no—
337C4—1361no—
338C4—1363no—
339C2—82no—
340C2#120×Dic1711yescharacteristic
341C2#1453146813yes—
342C2#1453149713yes—
343C2#1453450211yescharacteristic
344C2#1453451111yescharacteristic
345C2#1453464313yes—
346C2#1453468913yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#393C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×#393D17
hypocenterC17#14301
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic17C2semidirect
3. #14527055C2semidirect
4. #14527049C2semidirect
6. #14527009C2semidirect
21. C17×#393C4semidirect
181. #393Dic17semidirect
224. C17#14301semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic172
C2×#117627022
#145270552
#145270492
#145270292
#145270092
C2×C17×#3932
#1430117