Finite Group Databasebeta

← #14534849

Subgroups — #14534849

Order 2176.

442 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17⋊#401C211yesmaximal, characteristic, complemented
2#11762743C211yesmaximal, characteristic, complemented
3#11762737C211yesmaximal, characteristic, complemented
4#11762734C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic
6#11762723C211yesmaximal, characteristic
7#11762721C211yesmaximal, characteristic, complemented
8#11762721C211yesmaximal, characteristic, complemented
9#11762713C211yesmaximal, characteristic, complemented
10#11762703C211yesmaximal, characteristic, complemented
11#11762702C211yesmaximal, characteristic, complemented
12#11762691C211yesmaximal, characteristic, complemented
13#11762682C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
15C17×#386C211yesmaximal, characteristic
16C22×D4×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C4⋊Dic17C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×C4×Dic17C2211yescharacteristic
36C2×C4×Dic17C2211yescharacteristic
37C17×#95C2211yescharacteristic
38C17×#95C2211yescharacteristic
39#10587502C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C24×C17C2311yescharacteristic
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C22×C4×C17—22no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C2×Dic34—22no—
73C2×Dic34—41no—
74C2×Dic34—41no—
75C22⋊C4×C17—22no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—41no—
78C22⋊Dic17—22no—
79C22⋊Dic17—22no—
80C22⋊Dic17—22no—
81C22⋊Dic17—22no—
82C22⋊Dic17—41no—
83C22⋊Dic17—41no—
84C22⋊Dic17—41no—
85C22⋊Dic17—41no—
86C4⋊Dic17—41no—
87#63225—22no—
88#63225—41no—
89#63225—41no—
90#63225—41no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17#4112yes—
107C23×C17C2411yescharacteristic
108D4×C17—44no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
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—44no—
124C2×C4×C17#4112yes—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17#4112yes—
149Dic34—44no—
150#2653—171nomaximal, Sylow
151C22×C17—22no—
152C22×C17—22no—
153C22×C17—22no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C2×#4111yescharacteristic
166C22×C17C2×#4111yescharacteristic
167C22×C17C2×#4111yescharacteristic
168C22×C17H(5,𝔽2)11yescharacteristic
169C22×C17H(5,𝔽2)11yescharacteristic
170C22×C17H(5,𝔽2)11yescharacteristic
171C22×C172−1+411yescharacteristic
172C4×C17—22no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—41no—
177Dic17—22no—
178Dic17—41no—
179Dic17—41no—
180Dic17—41no—
181Dic17—41no—
182Dic17—41no—
183Dic17—41no—
184Dic17—41no—
185#401—171no—
186#398—171no—
187#397—171no—
188#397—171no—
189#394—171no—
190#394—171no—
191#393—171no—
192#393—171no—
193#391—171no—
194#388—171no—
195#388—171no—
196#387—171no—
197#386D1711yescharacteristic
198#385—171no—
199C4×C22⋊C4—171no—
200C2×C17—42no—
201C2×C17—81no—
202C2×C17#52911yescharacteristic
203C2×C17#53811yescharacteristic
204C2×C17#53811yescharacteristic
205C2×C17#53811yescharacteristic
206C2×C17#53911yescharacteristic
207C2×C17#54211yescharacteristic
208C2×C17#54211yescharacteristic
209C22×Q8—171no—
210C22×D4C2×D1711yescharacteristic
211C23×C4C2×D1711yescharacteristic
212C2×C4⋊C4—171no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—171no—
217C2×C22⋊C4—171no—
218C2×C22⋊C4—171no—
219C2×C22⋊C4—171no—
220C2×C22⋊C4—171no—
221C2×C22⋊C4—171no—
222C2×C22⋊C4—171no—
223C2×C22⋊C4—171no—
224C2×C22⋊C4—171no—
225C2×C22⋊C4C2×D1711yescharacteristic
226C2×C22⋊C4C2×D1711yescharacteristic
227C2×C22⋊C4C2×D1711yescharacteristic
228C2×C42—171no—
229C2×C42—171no—
230#95—171no—
231#95—171no—
232#95—171no—
233#95—171no—
234#95—171no—
235#95—171no—
236#95—171no—
237#95—171no—
238#95—171no—
239#95—171no—
240#95—171no—
241#95—171no—
242#95C2×D1711yescharacteristic
243#95C2×D1711yescharacteristic
244C17#265311yescharacteristic, complemented, Sylow
245C24C22×D1711yescharacteristic
246C24C22×D1711yescharacteristic
247C2×Q8—342no—
248C2×Q8—681no—
249C2×Q8—681no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—171no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4C22×D1711yescharacteristic
265C22×C4C22×D1711yescharacteristic
266C22×C4C22×D1711yescharacteristic
267C22×C4C22×D1711yescharacteristic
268C22×C4C22×D1711yescharacteristic
269C4⋊C4—342no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—342no—
278C22⋊C4—342no—
279C22⋊C4—342no—
280C22⋊C4—342no—
281C22⋊C4—681no—
282C22⋊C4—681no—
283C22⋊C4—681no—
284C22⋊C4—681no—
285C42—681no—
286C42—681no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23#6325212yes—
301C23C23×D1711yescharacteristic
302Q8—684no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—172no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4#6325412yes—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C2×#6325211yescharacteristic
359C22C2×#6325411yescharacteristic
360C22#1058769111yescharacteristic
361C22#1058769111yescharacteristic
362C22#1058769111yescharacteristic
363C22#41×D1711yescharacteristic
364C22#1058769711yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—342no—
371C4—681no—
372C4—681no—
373C4—681no—
374C4—681no—
375C4—681no—
376C4—681no—
377C4—681no—
378C2—42no—
379C2—81no—
380C2#1176333911yescharacteristic
381C2#1176339111yescharacteristic
382C2#1176339411yescharacteristic
383C2#1176339711yescharacteristic
384C2#1176340111yescharacteristic
385C2#1176342611yescharacteristic
386C2#1176342711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#386C2
prime radicalC22×D4×C17C22
hypercenter#386D17
hypocenterC17#2653
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C17⋊#401C2semidirect
2. #11762743C2semidirect
3. #11762737C2semidirect
4. #11762734C2semidirect
7. #11762721C2semidirect
8. #11762721C2semidirect
9. #11762713C2semidirect
10. #11762703C2semidirect
11. #11762702C2semidirect
12. #11762691C2semidirect
13. #11762682C2semidirect
14. C4×C22⋊Dic17C2semidirect
244. C17#2653semidirect

Maximal subgroups

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

subgroupindex
C17⋊#4012
#117627432
#117627372
#117627342
#117627232
#117627212
#117627132
#117627032
#117627022
#117626912
#117626822
C4×C22⋊Dic172
C17×#3862
#265317