Finite Group Databasebeta

← C17×#13388 #16871132

Subgroups — C17×#13388 #16871132

Order 4352.

396 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×#3059C211yesmaximal, characteristic
2C17×#2159C211yesmaximal, characteristic, complemented
3C2×C17×#362C211yesmaximal, characteristic, complemented
4C2×C17×#358C211yesmaximal, characteristic, complemented
5C2×C17×#350C211yesmaximal, characteristic
6C17×#1816C211yesmaximal, characteristic
7C17×#1382C211yesmaximal, characteristic, complemented
8C22×Q16×C17C2211yescharacteristic
9C2×C17×#131C2211yescharacteristic
10C17×#536—21no—
11C2×C17×M32C2211yescharacteristic
12C22×C16×C17C2211yescharacteristic
13C17×#470—22no—
14C17×#467—21no—
15C17×#467—21no—
16C2×C17×#108C2211yescharacteristic
17C17×#427C2211yescharacteristic
18C2×C17×#107C2211yescharacteristic
19C17×#419—21no—
20C17×#417—21no—
21C17×#367—21no—
22C17×#366—21no—
23C17×#362—22no—
24C17×#358—22no—
25C17×#350—22no—
26C17×#343—21no—
27C2×#41×C17D411yescharacteristic
28C22×Q8×C17—21no—
29C2×Q16×C17—21no—
30C2×Q16×C17—21no—
31C2×Q16×C17—21no—
32C2×Q16×C17—21no—
33C2×Q16×C17—21no—
34C2×Q16×C17D412yes—
35C17×#131—21no—
36C17×#131—21no—
37C17×#131—21no—
38C17×#131—21no—
39C2×M16×C17—21no—
40C2×M16×C17—21no—
41C2×M16×C17D411yescharacteristic
42C22×C8×C17—21no—
43C22×C8×C17C2311yescharacteristic
44C17×C4⋊Q8—41no—
45C17×#124—41no—
46C17×#123—42no—
47C17×C22⋊Q8—22no—
48C17×C22⋊Q8—22no—
49C17×#117—21no—
50C2×C4⋊C4×C17—21no—
51C17×M32—21no—
52C17×M32D412yes—
53C2×C16×C17—21no—
54C2×C16×C17—21no—
55C2×C16×C17—21no—
56C2×C16×C17—21no—
57C17×#108—21no—
58C17×#108—21no—
59C17×#107—21no—
60C17×#107—21no—
61C17×#107D411yescharacteristic
62C17×#107D411yescharacteristic
63C17×#106—21no—
64C17×#103—22no—
65C17×#103—22no—
66C17×#103—41no—
67C17×#102—41no—
68#41×C17—21no—
69#41×C17—21no—
70#41×C17—21no—
71#41×C17—21no—
72C2×Q8×C17—22no—
73C2×Q8×C17—41no—
74C2×Q8×C17—41no—
75C2×Q8×C17—41no—
76C2×Q8×C17—41no—
77C2×Q8×C17—41no—
78C2×Q8×C17D811yescharacteristic
79C2×D4×C17—21no—
80C2×D4×C17D811yescharacteristic
81C22×C4×C17—21no—
82C22×C4×C17—41no—
83C22×C4×C17—41no—
84C22×C4×C17C2×D411yescharacteristic
85Q16×C17—22no—
86Q16×C17—22no—
87Q16×C17—41no—
88Q16×C17—41no—
89M16×C17—21no—
90M16×C17—21no—
91M16×C17—41no—
92M16×C17—41no—
93M16×C17—41no—
94M16×C17—41no—
95C2×C8×C17—21no—
96C2×C8×C17—41no—
97C2×C8×C17—41no—
98C2×C8×C17—41no—
99C2×C8×C17C2×D411yescharacteristic
100C2×C8×C17C2×D411yescharacteristic
101C2×C8×C17C2×D411yescharacteristic
102C2×C8×C17#4111yescharacteristic
103C2×C8×C17#4111yescharacteristic
104C2×C8×C17#4111yescharacteristic
105C4⋊C4×C17—21no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—42no—
108C4⋊C4×C17—42no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C22⋊C4×C17—42no—
112C22⋊C4×C17—42no—
113C22⋊C4×C17—41no—
114C42×C17—41no—
115C16×C17—22no—
116C16×C17—41no—
117#13388C1711yesmaximal, characteristic, direct product factor, Sylow
118C23×C17—21no—
119C23×C17AGL(1,ℤ/8)11yescharacteristic
120Q8×C17—21no—
121Q8×C17—21no—
122Q8×C17—42no—
123Q8×C17—42no—
124Q8×C17—81no—
125Q8×C17—81no—
126D4×C17—21no—
127D4×C17—21no—
128D4×C17—41no—
129D4×C17—41no—
130C2×C4×C17—21no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—42no—
136C2×C4×C17—81no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17—161no—
140C2×C4×C17#12011yescharacteristic
141C2×C4×C17#12111yescharacteristic
142C2×C4×C17#12311yescharacteristic
143C2×C4×C17C2×D811yescharacteristic
144C2×C4×C17#13511yescharacteristic
145C2×C4×C17#13711yescharacteristic
146C8×C17—41no—
147C8×C17—81no—
148C8×C17#12111yescharacteristic
149C8×C17#12111yescharacteristic
150C8×C17#12311yescharacteristic
151C8×C17#12411yescharacteristic
152#3059C2×C1711yescharacteristic
153#2159C2×C1711yescharacteristic, complemented
154C2×#362C2×C1711yescharacteristic, complemented
155C2×#358C2×C1711yescharacteristic, complemented
156C2×#350C2×C1711yescharacteristic
157#1816C2×C1711yescharacteristic
158#1382C2×C1711yescharacteristic, complemented
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17#44711yescharacteristic
165C22×C17#46811yescharacteristic
166C22×C17#48211yescharacteristic
167C4×C17—41no—
168C4×C17—81no—
169C4×C17—81no—
170C4×C17—82no—
171C4×C17—161no—
172C4×C17#39411yescharacteristic
173C4×C17#45211yescharacteristic
174C4×C17#46611yescharacteristic
175C4×C17#48011yescharacteristic
176C22×Q16C22×C1711yescharacteristic
177C2×#131C22×C1711yescharacteristic
178#536—21no—
179C2×M32C22×C1711yescharacteristic
180C22×C16C22×C1711yescharacteristic
181#470—22no—
182#467—21no—
183#467—21no—
184C2×#108C22×C1711yescharacteristic
185#427C22×C1711yescharacteristic
186C2×#107C22×C1711yescharacteristic
187#419—21no—
188#417—21no—
189#367—21no—
190#366—21no—
191#362—22no—
192#358—22no—
193#350—22no—
194#343—21no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17—81no—
198C2×C17#203311yescharacteristic
199C2×C17#219311yescharacteristic
200C2×C17#219511yescharacteristic
201C2×#41D4×C1711yescharacteristic
202C22×Q8—21no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16—21no—
208C2×Q16D4×C1712yes—
209#131—21no—
210#131—21no—
211#131—21no—
212#131—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16D4×C1711yescharacteristic
216C22×C8—21no—
217C22×C8C23×C1711yescharacteristic
218C4⋊Q8—41no—
219#124—41no—
220#123—42no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223#117—21no—
224C2×C4⋊C4—21no—
225M32—21no—
226M32D4×C1712yes—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231#108—21no—
232#108—21no—
233#107—21no—
234#107—21no—
235#107D4×C1711yescharacteristic
236#107D4×C1711yescharacteristic
237#106—21no—
238#103—22no—
239#103—22no—
240#103—41no—
241#102—41no—
242C17#1338811yescharacteristic, direct product factor, Sylow
243#41—21no—
244#41—21no—
245#41—21no—
246#41—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8D8×C1711yescharacteristic
254C2×D4—21no—
255C2×D4D8×C1711yescharacteristic
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C2×D4×C1711yescharacteristic
260Q16—22no—
261Q16—22no—
262Q16—41no—
263Q16—41no—
264M16—21no—
265M16—21no—
266M16—41no—
267M16—41no—
268M16—41no—
269M16—41no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8C2×D4×C1711yescharacteristic
275C2×C8C2×D4×C1711yescharacteristic
276C2×C8C2×D4×C1711yescharacteristic
277C2×C8#41×C1711yescharacteristic
278C2×C8#41×C1711yescharacteristic
279C2×C8#41×C1711yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C42—41no—
290C16—22no—
291C16—41no—
292C23—21no—
293C23C17×AGL(1,ℤ/8)11yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—42no—
298Q8—81no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—161no—
314C2×C4C17×#12011yescharacteristic
315C2×C4C17×#12111yescharacteristic
316C2×C4C17×#12311yescharacteristic
317C2×C4C2×D8×C1711yescharacteristic
318C2×C4C17×#13511yescharacteristic
319C2×C4C17×#13711yescharacteristic
320C8—41no—
321C8—81no—
322C8C17×#12111yescharacteristic
323C8C17×#12111yescharacteristic
324C8C17×#12311yescharacteristic
325C8C17×#12411yescharacteristic
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22C17×#44711yescharacteristic
332C22C17×#46811yescharacteristic
333C22C17×#48211yescharacteristic
334C4—41no—
335C4—81no—
336C4—81no—
337C4—82no—
338C4—161no—
339C4C17×#39411yescharacteristic
340C4C17×#45211yescharacteristic
341C4C17×#46611yescharacteristic
342C4C17×#48011yescharacteristic
343C2—21no—
344C2—21no—
345C2—81no—
346C2C17×#203311yescharacteristic
347C2C17×#219311yescharacteristic
348C2C17×#219511yescharacteristic
subgroupG/H
center, socleC22×C17#447
commutator, Frattini, cosocleC22×C8C23×C17
prime radicalC2×#41×C17D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2159C2semidirect
3. C2×C17×#362C2semidirect
4. C2×C17×#358C2semidirect
7. C17×#1382C2semidirect
117. #13388C17direct (N × H)
153. #2159C2×C17semidirect
154. C2×#362C2×C17semidirect
155. C2×#358C2×C17semidirect
158. #1382C2×C17semidirect
242. C17#13388direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#30592
C17×#21592
C2×C17×#3622
C2×C17×#3582
C2×C17×#3502
C17×#18162
C17×#13822
#1338817