Finite Group Databasebeta

← C17×#2515 #14525521

Subgroups — C17×#2515 #14525521

Order 2176.

412 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#400C211yesmaximal, characteristic, complemented
2C17×#398C211yesmaximal, characteristic, complemented
3C17×#397C211yesmaximal, characteristic
4C17×#396C211yesmaximal, characteristic
5C17×#388C211yesmaximal, characteristic
6C17×#388C211yesmaximal, characteristic
7C17×#387C211yesmaximal, characteristic, complemented
8C17×#387C211yesmaximal, characteristic, complemented
9C17×#387C211yesmaximal, characteristic, complemented
10C17×#385C211yesmaximal, characteristic
11C17×#383C211yesmaximal, characteristic, complemented
12C17×#382C211yesmaximal, characteristic, complemented
13C17×#381C211yesmaximal, characteristic
14C4×C4⋊C4×C17C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C17C211yesmaximal, characteristic
16C23×C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C4⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊C4×C17C2211yescharacteristic
29C2×C22⋊C4×C17C2211yescharacteristic
30C2×C22⋊C4×C17C2211yescharacteristic
31C2×C42×C17C2211yescharacteristic
32C2×C42×C17C2211yescharacteristic
33C2×C42×C17C2211yescharacteristic
34C2×C42×C17C2211yescharacteristic
35C17×#95C2211yescharacteristic
36C17×#95C2211yescharacteristic
37C17×#95C2211yescharacteristic
38C17×#95C2211yescharacteristic
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
41C17×#95C2211yescharacteristic
42C17×#95C2211yescharacteristic
43C17×#95C2211yescharacteristic
44C17×#95C2211yescharacteristic
45C17×#95C2211yescharacteristic
46C17×#95C2211yescharacteristic
47C17×#95C2211yescharacteristic
48C17×#95C2211yescharacteristic
49C17×#95C2211yescharacteristic
50C17×#95C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—41no—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C4⋊C4×C17—22no—
70C4⋊C4×C17—22no—
71C4⋊C4×C17—22no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—22no—
78C22⋊C4×C17—22no—
79C22⋊C4×C17—22no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—41no—
83C42×C17—22no—
84C42×C17—22no—
85C42×C17—22no—
86C42×C17—41no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17#4112yes—
94C23×C17C2411yescharacteristic
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
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—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—44no—
134C2×C4×C17#4112yes—
135C2×C4×C17#4112yes—
136C2×C4×C17#4112yes—
137C2×C4×C17#4112yes—
138#2515C1711yesmaximal, characteristic, direct product factor, Sylow
139C22×C17—22no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17C2×#4111yescharacteristic
147C22×C17C2×#4111yescharacteristic
148C22×C17C2×#4111yescharacteristic
149C22×C17C2×#4111yescharacteristic
150C22×C17C2×#4111yescharacteristic
151C22×C17H(5,𝔽2)11yescharacteristic
152C22×C172−1+411yescharacteristic
153C4×C17—22no—
154C4×C17—22no—
155C4×C17—22no—
156C4×C17—22no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—41no—
167#400C2×C1711yescharacteristic, complemented
168#398C2×C1711yescharacteristic, complemented
169#397C2×C1711yescharacteristic
170#396C2×C1711yescharacteristic
171#388C2×C1711yescharacteristic
172#388C2×C1711yescharacteristic
173#387C2×C1711yescharacteristic, complemented
174#387C2×C1711yescharacteristic, complemented
175#387C2×C1711yescharacteristic, complemented
176#385C2×C1711yescharacteristic
177#383C2×C1711yescharacteristic, complemented
178#382C2×C1711yescharacteristic, complemented
179#381C2×C1711yescharacteristic
180C4×C4⋊C4C2×C1711yescharacteristic, complemented
181C4×C22⋊C4C2×C1711yescharacteristic
182C2×C17—42no—
183C2×C17#52911yescharacteristic
184C2×C17#52911yescharacteristic
185C2×C17#54211yescharacteristic
186C2×C17#55111yescharacteristic
187C2×C17#55211yescharacteristic
188C2×C17#55211yescharacteristic
189C2×C17#55311yescharacteristic
190C23×C4C22×C1711yescharacteristic
191C2×C4⋊C4C22×C1711yescharacteristic
192C2×C4⋊C4C22×C1711yescharacteristic
193C2×C4⋊C4C22×C1711yescharacteristic
194C2×C4⋊C4C22×C1711yescharacteristic
195C2×C4⋊C4C22×C1711yescharacteristic
196C2×C4⋊C4C22×C1711yescharacteristic
197C2×C4⋊C4C22×C1711yescharacteristic
198C2×C4⋊C4C22×C1711yescharacteristic
199C2×C22⋊C4C22×C1711yescharacteristic
200C2×C22⋊C4C22×C1711yescharacteristic
201C2×C22⋊C4C22×C1711yescharacteristic
202C2×C22⋊C4C22×C1711yescharacteristic
203C2×C22⋊C4C22×C1711yescharacteristic
204C2×C22⋊C4C22×C1711yescharacteristic
205C2×C42C22×C1711yescharacteristic
206C2×C42C22×C1711yescharacteristic
207C2×C42C22×C1711yescharacteristic
208C2×C42C22×C1711yescharacteristic
209#95C22×C1711yescharacteristic
210#95C22×C1711yescharacteristic
211#95C22×C1711yescharacteristic
212#95C22×C1711yescharacteristic
213#95C22×C1711yescharacteristic
214#95C22×C1711yescharacteristic
215#95C22×C1711yescharacteristic
216#95C22×C1711yescharacteristic
217#95C22×C1711yescharacteristic
218#95C22×C1711yescharacteristic
219#95C22×C1711yescharacteristic
220#95C22×C1711yescharacteristic
221#95C22×C1711yescharacteristic
222#95C22×C1711yescharacteristic
223#95C22×C1711yescharacteristic
224#95C22×C1711yescharacteristic
225C17#251511yescharacteristic, direct product factor, Sylow
226C24C23×C1711yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4C23×C1711yescharacteristic
231C22×C4C23×C1711yescharacteristic
232C22×C4C23×C1711yescharacteristic
233C22×C4C23×C1711yescharacteristic
234C22×C4C23×C1711yescharacteristic
235C22×C4C23×C1711yescharacteristic
236C22×C4C23×C1711yescharacteristic
237C22×C4C23×C1711yescharacteristic
238C22×C4C23×C1711yescharacteristic
239C22×C4C23×C1711yescharacteristic
240C22×C4C23×C1711yescharacteristic
241C22×C4C23×C1711yescharacteristic
242C22×C4C23×C1711yescharacteristic
243C22×C4C23×C1711yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—41no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—41no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#41×C1712yes—
269C23C24×C1711yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
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—44no—
309C2×C4#41×C1712yes—
310C2×C4#41×C1712yes—
311C2×C4#41×C1712yes—
312C2×C4#41×C1712yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#41×C1711yescharacteristic
321C22C2×#41×C1711yescharacteristic
322C22C2×#41×C1711yescharacteristic
323C22C2×#41×C1711yescharacteristic
324C22C2×#41×C1711yescharacteristic
325C22C17×H(5,𝔽2)11yescharacteristic
326C22C17×2−1+411yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C2—42no—
342C2C17×#52911yescharacteristic
343C2C17×#52911yescharacteristic
344C2C17×#54211yescharacteristic
345C2C17×#55111yescharacteristic
346C2C17×#55211yescharacteristic
347C2C17×#55211yescharacteristic
348C2C17×#55311yescharacteristic
subgroupG/H
center, socleC23×C17C24
commutator, Frattini, cosocleC23C24×C17
prime radicalC24×C17C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#400C2semidirect
2. C17×#398C2semidirect
7. C17×#387C2semidirect
8. C17×#387C2semidirect
9. C17×#387C2semidirect
11. C17×#383C2semidirect
12. C17×#382C2semidirect
14. C4×C4⋊C4×C17C2semidirect
138. #2515C17direct (N × H)
167. #400C2×C17semidirect
168. #398C2×C17semidirect
173. #387C2×C17semidirect
174. #387C2×C17semidirect
175. #387C2×C17semidirect
177. #383C2×C17semidirect
178. #382C2×C17semidirect
180. C4×C4⋊C4C2×C17semidirect
225. C17#2515direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#4002
C17×#3982
C17×#3972
C17×#3962
C17×#3882
C17×#3872
C17×#3852
C17×#3832
C17×#3822
C17×#3812
C4×C4⋊C4×C172
C4×C22⋊C4×C172
#251517