Finite Group Databasebeta

← #16922634

Subgroups — #16922634

Order 4352.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528852C211yesmaximal, characteristic
2#14528764C211yesmaximal, characteristic
3#14526610C212yesmaximal, complemented
4#14526607C212yesmaximal, complemented
5C4×C17×C22⋊Q8C211yesmaximal, characteristic
6C2×C4⋊#3622—21no—
7C2×C22⋊#3622—21no—
8C2×C22⋊#3622—21no—
9#11762564—21no—
10#11762564C2212yes—
11#11762564C2212yes—
12C2×C17×C22⋊Q8C411yescharacteristic
13C2×C4×Q8×C17C411yescharacteristic
14C22×C42×C17C2211yescharacteristic
15C17×#391C2211yescharacteristic
16C17×#389C2211yescharacteristic
17C17×#387—21no—
18C17×#385—21no—
19C17×#380C411yescharacteristic
20C4×C4⋊C4×C17—21no—
21C4×C4⋊C4×C17C411yescharacteristic
22C4×C22⋊C4×C17—21no—
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C17×C22⋊Q8—22no—
28C17×C22⋊Q8—22no—
29C4×Q8×C17—42no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C42×C17—21no—
39C2×C42×C17—21no—
40C2×C42×C17—42no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×C42×C17C2311yescharacteristic
45C22×#3622—22no—
46C22×#3622—22no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C22⋊#3622—42no—
53C22⋊#3622—42no—
54C4⋊#3622—42no—
55C24×C17C22⋊C411yescharacteristic
56C2×Q8×C17—21no—
57C2×Q8×C17—41no—
58C2×Q8×C17—41no—
59C2×Q8×C17M1612yes—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—22no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—41no—
69C22×C4×C17—41no—
70C22×C4×C17—41no—
71C22×C4×C17—41no—
72C22×C4×C17—41no—
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17SD1612yes—
77C22×C4×C17Q1612yes—
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C4⋊C4×C17—24no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17M1612yes—
88C22⋊C4×C17—24no—
89C22⋊C4×C17—42no—
90C42×C17—22no—
91C42×C17—22no—
92C42×C17—41no—
93C42×C17—41no—
94C42×C17—42no—
95C42×C17—42no—
96C2×#3622—42no—
97C2×#3622—42no—
98C2×#3622—42no—
99C2×#3622—42no—
100C2×#3622—42no—
101C2×#3622—42no—
102#7648—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17#10312yes—
110C23×C17C2×C22⋊C411yescharacteristic
111Q8×C17—42no—
112Q8×C17—42no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—44no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17#9912yes—
144C2×C4×C17#10312yes—
145C2×C4×C17C2×M1611yescharacteristic
146C2×C4×C17C2×M1611yescharacteristic
147C2×C4×C17C2×SD1611yescharacteristic
148C2×C4×C17C2×Q1611yescharacteristic
149C2×C4×C17#13512yes—
150#3622—82no—
151#3622—82no—
152C4×C22⋊Q8D1711yescharacteristic
153#1817—171no—
154#1797—171no—
155#1272—172no—
156#1271—172no—
157C22×C17—22no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17#40711yescharacteristic
165C22×C17C2×#9911yescharacteristic
166C22×C17#41311yescharacteristic
167C22×C17C2×#10311yescharacteristic
168C22×C17#41611yescharacteristic
169C22×C17#42111yescharacteristic
170C22×C17#42111yescharacteristic
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—22no—
174C4×C17—22no—
175C4×C17—21no—
176C4×C17—22no—
177C4×C17—21no—
178C4×C17—41no—
179C4×C17—42no—
180C4×C17—42no—
181C4×C17—81no—
182C4×C17—81no—
183C2×C22⋊Q8Dic1711yescharacteristic
184C2×C4×Q8Dic1711yescharacteristic
185C22×C42C2×D1711yescharacteristic
186C2×C4⋊C8—341no—
187C2×C22⋊C8—341no—
188C2×C22⋊C8—341no—
189#391C2×D1711yescharacteristic
190#389C2×D1711yescharacteristic
191#387—21no—
192#385—21no—
193#380Dic1711yescharacteristic
194C4×C4⋊C4—21no—
195C4×C4⋊C4Dic1711yescharacteristic
196C4×C22⋊C4—21no—
197#336—172no—
198#336—172no—
199#336—341no—
200C2×C17—42no—
201C2×C17#146511yescharacteristic
202C2×C17#148711yescharacteristic
203C2×C17#148811yescharacteristic
204C2×C17#151611yescharacteristic
205C2×C17#152011yescharacteristic
206C2×C17#153511yescharacteristic
207C2×C17#154211yescharacteristic
208C22×Q8C2×Dic1711yescharacteristic
209C23×C4C2×Dic1711yescharacteristic
210C23×C4C22⋊D1711yescharacteristic
211C23×C4C22⋊D1711yescharacteristic
212C22×C8—342no—
213C22×C8—342no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C4×Q8—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic1711yescharacteristic
221C2×C4⋊C4C2×Dic1711yescharacteristic
222C2×C4⋊C4C2×Dic1711yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42C2×Dic1711yescharacteristic
229C2×C42C22⋊D1711yescharacteristic
230C2×C42C22⋊D1711yescharacteristic
231C2×C42C22×D1711yescharacteristic
232C4⋊C8—682no—
233C22⋊C8—682no—
234C22⋊C8—682no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240C17#764811yescharacteristic, complemented, Sylow
241C24C22⋊Dic1711yescharacteristic
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8#6322312yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4#6323012yes—
260C22×C4#6323112yes—
261C22×C4C22⋊Dic1711yescharacteristic
262C22×C4C22⋊Dic1711yescharacteristic
263C22×C4C22⋊Dic1711yescharacteristic
264C22×C4C22×Dic1711yescharacteristic
265C22×C4C2×C22⋊D1711yescharacteristic
266C22×C4C2×C22⋊D1711yescharacteristic
267C2×C8—682no—
268C2×C8—682no—
269C2×C8—682no—
270C2×C8—682no—
271C2×C8—682no—
272C2×C8—682no—
273C4⋊C4—24no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4#6322312yes—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—42no—
287C42—42no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1058750612yes—
295C23C2×C22⋊Dic1711yescharacteristic
296Q8—42no—
297Q8—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4#1058750512yes—
329C2×C4#1058750612yes—
330C2×C4C2×#6322311yescharacteristic
331C2×C4C2×#6322311yescharacteristic
332C2×C4C2×#6323011yescharacteristic
333C2×C4C2×#6323111yescharacteristic
334C2×C4#1058763612yes—
335C8—1362no—
336C8—1362no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1176277311yescharacteristic
345C22C2×#1058750511yescharacteristic
346C22#1176279311yescharacteristic
347C22C2×#1058750611yescharacteristic
348C22#1176280511yescharacteristic
349C22#1176282611yescharacteristic
350C22#1176282611yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C2—42no—
364C2#1452712211yescharacteristic
365C2#1452721311yescharacteristic
366C2#1452721811yescharacteristic
367C2#1452734011yescharacteristic
368C2#1452735611yescharacteristic
369C2#1452741511yescharacteristic
370C2#1452744111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC4×C17×C22⋊Q8C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenterC4×C22⋊Q8D17
hypocenterC17#7648
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14526610C2semidirect
4. #14526607C2semidirect
240. C17#7648semidirect

Maximal subgroups

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

subgroupindex
#145288522
#145287642
#145266102
#145266072
C4×C17×C22⋊Q82
#764817