Finite Group Databasebeta

← #17194385

Subgroups — #17194385

Order 4352.

2806 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531661C218yesmaximal, complemented
2#14531653C218yesmaximal, complemented
3#14531402C212yesmaximal, complemented
4#14531387C214yesmaximal, complemented
5#14531167C212yesmaximal
6#14531167C214yesmaximal
7#14531141C212yesmaximal, complemented
8C23×C4⋊C4×C17C211yesmaximal, characteristic
9C2×#10587612C2214yes—
10C2×#10587612C2218yes—
11#11762708C22116yescomplemented
12#11762695C22132yescomplemented
13#11762672C2214yes—
14#11762672C2218yes—
15#11762669C2218yes—
16#11762669C2214yes—
17#11762669C2214yes—
18#11762669C2218yes—
19#11762665C2214yes—
20C4⋊C4×Dic17C22116yescomplemented
21C4×C22⋊Dic17C2218yes—
22C4×C22⋊Dic17C2214yes—
23C4×C22⋊Dic17C2214yes—
24C4×C22⋊Dic17C2218yes—
25C24×C4×C17C2211yescharacteristic
26C24×C4×C17C2212yes—
27C22×C4⋊C4×C17C4116yescomplemented
28C22×C4⋊C4×C17C2212yes—
29C22×C4⋊C4×C17C2211yescharacteristic
30C22×C4⋊C4×C17C2214yes—
31C22×C4⋊C4×C17C2211yescharacteristic
32C22×C4⋊C4×C17C2214yes—
33C25×C17C2311yescharacteristic
34C23×C4×C17—28no—
35C23×C4×C17—48no—
36C23×C4×C17C2×C418yes—
37C23×C4×C17C2×C4116yes—
38C23×C4×C17C2314yes—
39C23×C4×C17C2312yes—
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2312yes—
43C23×C4×C17C2314yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2311yescharacteristic
46C2×C4⋊C4×C17C2×C4132yes—
47C2×C4⋊C4×C17C2×C4116yes—
48C2×C4⋊C4×C17C2×C4132yes—
49C2×C4⋊C4×C17C2×C418yes—
50C2×C4⋊C4×C17C2×C418yes—
51C2×C4⋊C4×C17C2318yes—
52C2×C4⋊C4×C17C2314yes—
53C2×C4⋊C4×C17C2314yes—
54C2×C22⋊Dic17C2318yes—
55C2×C22⋊Dic17C2318yes—
56C2×C22⋊Dic17C2318yes—
57#10587612—216no—
58#10587612—416no—
59C2×C4⋊Dic17C2318yes—
60C2×C4⋊Dic17C23116yes—
61C2×C4×Dic17C2318yes—
62C2×C4×Dic17C23116yes—
63#10587502C2318yes—
64#10587502C2318yes—
65#10587502C23116yes—
66#10587502C23116yes—
67C24×C17—24no—
68C24×C17—24no—
69C24×C17—24no—
70C24×C17C22×C414yes—
71C24×C17C2411yescharacteristic
72C24×C17C2411yescharacteristic
73C24×C17C2411yescharacteristic
74C22×C4×C17—24no—
75C22×C4×C17—216no—
76C22×C4×C17—232no—
77C22×C4×C17—28no—
78C22×C4×C17—28no—
79C22×C4×C17—432no—
80C22×C4×C17—44no—
81C22×C4×C17—48no—
82C22×C4×C17—44no—
83C22×C4×C17C22×C414yes—
84C22×C4×C17C22×C4116yes—
85C22×C4×C17C22×C418yes—
86C22×C4×C17C22×C418yes—
87C22×C4×C17C22×C418yes—
88C22×C4×C17C22×C418yes—
89C22×C4×C17C22×C4116yes—
90C22×C4×C17C22×C414yes—
91C22×C4×C17#4118yes—
92C22×C4×C17C2412yes—
93C22×C4×C17C2414yes—
94C22×C4×C17C2414yes—
95C22×C4×C17C2412yes—
96C22×Dic17C24116yes—
97C4⋊C4×C17C22×C4132yes—
98C4⋊C4×C17C22×C4116yes—
99C4⋊C4×C17C22×C4116yes—
100C22⋊Dic17—216no—
101C22⋊Dic17—216no—
102C22⋊Dic17—216no—
103C4⋊Dic17—216no—
104C4⋊Dic17—416no—
105C4×Dic17—216no—
106C4×Dic17—416no—
107#34178—171nomaximal, Sylow
108C23×C17—232no—
109C23×C17—28no—
110C23×C17—28no—
111C23×C17—28no—
112C23×C17—22no—
113C23×C17—22no—
114C23×C17—22no—
115C23×C17—22no—
116C23×C17—22no—
117C23×C17—22no—
118C23×C17C23×C412yes—
119C23×C17C23×C412yes—
120C23×C17C23×C412yes—
121C23×C17C2×#4114yes—
122C23×C17H(5,𝔽2)14yes—
123C23×C172−1+414yes—
124C23×C17C2511yescharacteristic
125C2×C4×C17—28no—
126C2×C4×C17—232no—
127C2×C4×C17—24no—
128C2×C4×C17—28no—
129C2×C4×C17—28no—
130C2×C4×C17—216no—
131C2×C4×C17—216no—
132C2×C4×C17—48no—
133C2×C4×C17—416no—
134C2×C4×C17—48no—
135C2×C4×C17#117116yescomplemented
136C2×C4×C17C23×C414yes—
137C2×C4×C17C23×C418yes—
138C2×C4×C17C23×C418yes—
139C2×C4×C17C23×C414yes—
140C2×C4×C17C2×#4118yes—
141C2×Dic17—216no—
142C2×Dic17—216no—
143C2×Dic17—216no—
144C23×C4⋊C4D1711yescharacteristic
145#2345—178no—
146#2344—178no—
147#2316—172no—
148#2313—174no—
149#2280—172no—
150#2280—174no—
151#2274—172no—
152C22×C17—232no—
153C22×C17—22no—
154C22×C17—22no—
155C22×C17—22no—
156C22×C17—22no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—28no—
160C22×C17—28no—
161C22×C17—28no—
162C22×C17C2×#11714yes—
163C22×C17#51814yes—
164C22×C17#51914yes—
165C22×C17C24×C411yescharacteristic
166C22×C17C22×#4112yes—
167C22×C17C2×H(5,𝔽2)12yes—
168C22×C17C2×2−1+412yes—
169C4×C17—24no—
170C4×C17—28no—
171C4×C17—28no—
172C4×C17C2×#11718yes—
173Dic17—416no—
174C24×C4C2×D1711yescharacteristic
175C24×C4C2×D1712yes—
176C2×#117—174no—
177C2×#117—178no—
178C22×C4⋊C4Dic17116yescomplemented
179C22×C4⋊C4C2×D1712yes—
180C22×C4⋊C4C2×D1711yescharacteristic
181C22×C4⋊C4C2×D1714yes—
182C22×C4⋊C4C2×D1714yes—
183C22×C4⋊C4C2×D1711yescharacteristic
184#389—1716no—
185#387—1732no—
186#381—174no—
187#381—178no—
188#380—178no—
189#380—174no—
190#380—178no—
191#380—174no—
192#379—174no—
193C4×C4⋊C4—1716no—
194C4×C22⋊C4—178no—
195C4×C22⋊C4—174no—
196C4×C22⋊C4—174no—
197C4×C22⋊C4—178no—
198C2×C17—24no—
199C2×C17—24no—
200C2×C17—24no—
201C2×C17C22×#11711yescharacteristic
202C2×C17C2×#51811yescharacteristic
203C2×C17C2×#51911yescharacteristic
204C2×C17#349214yes—
205C25C22×D1711yescharacteristic
206C23×C4—28no—
207C23×C4—48no—
208C23×C4C2×Dic1718yes—
209C23×C4C2×Dic17116yes—
210C23×C4C22×D1714yes—
211C23×C4C22×D1712yes—
212C23×C4C22×D1712yes—
213C23×C4C22×D1711yescharacteristic
214C23×C4C22×D1712yes—
215C23×C4C22×D1714yes—
216C23×C4C22×D1712yes—
217C23×C4C22×D1711yescharacteristic
218#117—3416no—
219#117—6816no—
220C2×C4⋊C4—178no—
221C2×C4⋊C4—1716no—
222C2×C4⋊C4C2×Dic17132yes—
223C2×C4⋊C4C2×Dic17116yes—
224C2×C4⋊C4C2×Dic17132yes—
225C2×C4⋊C4C2×Dic1718yes—
226C2×C4⋊C4C2×Dic1718yes—
227C2×C4⋊C4C22×D1714yes—
228C2×C4⋊C4C22×D1718yes—
229C2×C4⋊C4C22×D1714yes—
230C2×C22⋊C4—178no—
231C2×C22⋊C4—178no—
232C2×C22⋊C4—178no—
233C2×C42—178no—
234C2×C42—1716no—
235#95—1716no—
236#95—178no—
237#95—178no—
238#95—1716no—
239C17#3417811yescharacteristic, complemented, Sylow
240C24—24no—
241C24—24no—
242C24—24no—
243C24C22×Dic1714yes—
244C24C23×D1711yescharacteristic
245C24C23×D1711yescharacteristic
246C24C23×D1711yescharacteristic
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—232no—
252C22×C4—432no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—1716no—
257C22×C4#6325218yes—
258C22×C4C22×Dic1714yes—
259C22×C4C22×Dic17116yes—
260C22×C4C22×Dic1718yes—
261C22×C4C22×Dic1718yes—
262C22×C4C22×Dic1718yes—
263C22×C4C22×Dic1718yes—
264C22×C4C22×Dic17116yes—
265C22×C4C22×Dic1714yes—
266C22×C4C23×D1712yes—
267C22×C4C23×D1714yes—
268C22×C4C23×D1714yes—
269C22×C4C23×D1712yes—
270C4⋊C4—3416no—
271C4⋊C4—6816no—
272C4⋊C4C22×Dic17132yes—
273C4⋊C4C22×Dic17116yes—
274C4⋊C4C22×Dic17116yes—
275C22⋊C4—3416no—
276C22⋊C4—3416no—
277C22⋊C4—3416no—
278C42—3416no—
279C42—6816no—
280C23—232no—
281C23—28no—
282C23—28no—
283C23—28no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23C2×#6325214yes—
291C23#1058769114yes—
292C23#1058769414yes—
293C23C23×Dic1712yes—
294C23C23×Dic1712yes—
295C23C23×Dic1712yes—
296C23C24×D1711yescharacteristic
297C2×C4—28no—
298C2×C4—232no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—3416no—
308C2×C4—3416no—
309C2×C4—3416no—
310C2×C4#10587612116yescomplemented
311C2×C4C2×#6325218yes—
312C2×C4C23×Dic1714yes—
313C2×C4C23×Dic1718yes—
314C2×C4C23×Dic1718yes—
315C2×C4C23×Dic1714yes—
316C22—232no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—28no—
324C22—28no—
325C22—28no—
326C22C2×#1058761214yes—
327C22#1176328414yes—
328C22#1176328714yes—
329C22C24×Dic1711yescharacteristic
330C22C22×#6325212yes—
331C22C2×#1058769112yes—
332C22C2×#1058769412yes—
333C4—24no—
334C4—28no—
335C4—28no—
336C4—6816no—
337C4C2×#1058761218yes—
338C2—24no—
339C2—24no—
340C2—24no—
341C2C22×#1058761211yescharacteristic
342C2C2×#1176328411yescharacteristic
343C2C2×#1176328711yescharacteristic
344C2#1454249314yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC23×C4⋊C4×C17C2
socle, cosocleC23×C17C25
prime radicalC25×C17C23
hypercenterC23×C4⋊C4D17
hypocenterC17#34178
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #14531661C2semidirect
2. #14531653C2semidirect
3. #14531402C2semidirect
4. #14531387C2semidirect
7. #14531141C2semidirect
11. #11762708C22semidirect
12. #11762695C22semidirect
20. C4⋊C4×Dic17C22semidirect
27. C22×C4⋊C4×C17C4semidirect
135. C2×C4×C17#117semidirect
178. C22×C4⋊C4Dic17semidirect
239. C17#34178semidirect
310. C2×C4#10587612semidirect

Maximal subgroups

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

subgroupindex
#145316612
#145316532
#145314022
#145313872
#145311672
#145311412
C23×C4⋊C4×C172
#3417817