Finite Group Databasebeta

← C17×#12986 #16870730

Subgroups — C17×#12986 #16870730

Order 4352.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#3407C211yesmaximal, characteristic, complemented
2C2×C17×#444C211yesmaximal, characteristic
3C17×#2117C211yesmaximal, characteristic, complemented
4C17×#2117C211yesmaximal, characteristic, complemented
5C17×#2115C211yesmaximal, characteristic, complemented
6C17×#1377C211yesmaximal, characteristic, complemented
7C17×#1375C211yesmaximal, characteristic, complemented
8C2×C17×#135C411yescharacteristic
9C2×C17×#131C2211yescharacteristic
10C2×C17×#131C2211yescharacteristic
11C17×#504—22no—
12C17×#504—22no—
13C2×C17×M32—22no—
14C2×C17×M32C2211yescharacteristic
15C2×C17×M32C2211yescharacteristic
16C2×C17×M32C2211yescharacteristic
17C2×C17×M32C2211yescharacteristic
18C17×#444—22no—
19C17×#444—22no—
20C2×C17×#108C411yescharacteristic
21C2×C17×#104C411yescharacteristic
22C2×C17×#104C411yescharacteristic
23C2×C17×#97C2211yescharacteristic
24C17×#350—22no—
25C17×#350—22no—
26C17×#349—22no—
27C2×#41×C17C2×C411yescharacteristic
28C2×#41×C17C2×C411yescharacteristic
29C17×#135—22no—
30C17×#135—22no—
31C2×Q16×C17C811yescharacteristic
32C2×SD16×C17C811yescharacteristic
33C2×SD16×C17C811yescharacteristic
34C2×D8×C17C811yescharacteristic
35C17×#131—21no—
36C17×#131—21no—
37C17×#131—21no—
38C17×#131—21no—
39C17×#131—21no—
40C17×#131—21no—
41C17×#131D412yes—
42C17×#131D412yes—
43C2×M16×C17—21no—
44C2×M16×C17—21no—
45C2×M16×C17C2×C411yescharacteristic
46C2×M16×C17C2×C411yescharacteristic
47C22×C8×C17—21no—
48C22×C8×C17—21no—
49C22×C8×C17C2×C411yescharacteristic
50C22×C8×C17C2311yescharacteristic
51C2×C42×C17C2×C411yescharacteristic
52C17×M32—24no—
53C17×M32—21no—
54C17×M32—21no—
55C17×M32—21no—
56C17×M32—21no—
57C17×M32—21no—
58C17×M32—21no—
59C17×M32—21no—
60C17×M32D412yes—
61C2×C16×C17—24no—
62C2×C16×C17—21no—
63C2×C16×C17—21no—
64C2×C16×C17—21no—
65C2×C16×C17D412yes—
66C17×#108—22no—
67C17×#104—22no—
68C17×#104—22no—
69C17×#97—21no—
70C17×#97—21no—
71#41×C17—21no—
72#41×C17—21no—
73#41×C17—21no—
74#41×C17—21no—
75#41×C17—21no—
76#41×C17—21no—
77#41×C17C22⋊C412yes—
78#41×C17M1612yes—
79C2×Q8×C17C2×C811yescharacteristic
80C2×Q8×C17C2×C811yescharacteristic
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17C2×C811yescharacteristic
84C2×D4×C17C2×C811yescharacteristic
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17C22×C411yescharacteristic
89Q16×C17—22no—
90SD16×C17—22no—
91SD16×C17—22no—
92D8×C17—22no—
93M16×C17—21no—
94M16×C17—22no—
95M16×C17—22no—
96M16×C17—21no—
97M16×C17—21no—
98M16×C17—41no—
99M16×C17—41no—
100M16×C17C22⋊C412yes—
101C2×C8×C17—21no—
102C2×C8×C17—21no—
103C2×C8×C17—21no—
104C2×C8×C17—22no—
105C2×C8×C17—22no—
106C2×C8×C17—21no—
107C2×C8×C17—41no—
108C2×C8×C17—41no—
109C2×C8×C17C2×C811yescharacteristic
110C2×C8×C17C2×C811yescharacteristic
111C2×C8×C17C2×D411yescharacteristic
112C2×C8×C17C2×D411yescharacteristic
113C2×C8×C17C2×D411yescharacteristic
114C2×C8×C17C2×D411yescharacteristic
115C2×C8×C17#4111yescharacteristic
116C2×C8×C17#4111yescharacteristic
117C42×C17—21no—
118C42×C17—21no—
119C16×C17—22no—
120C16×C17—22no—
121C16×C17—41no—
122C16×C17—41no—
123#12986C1711yesmaximal, characteristic, direct product factor, Sylow
124C23×C17—21no—
125C23×C17—21no—
126C23×C17#13111yescharacteristic
127Q8×C17—21no—
128Q8×C17—21no—
129Q8×C17—21no—
130Q8×C17C22⋊C812yes—
131D4×C17—22no—
132D4×C17—22no—
133D4×C17—21no—
134D4×C17—21no—
135D4×C17—21no—
136D4×C17—41no—
137D4×C17—41no—
138D4×C17C22⋊C812yes—
139C2×C4×C17—21no—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—21no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—81no—
147C2×C4×C17C2×C22⋊C411yescharacteristic
148C2×C4×C17C4×D411yescharacteristic
149C2×C4×C17C4×D411yescharacteristic
150C2×C4×C17C22×C811yescharacteristic
151C2×C4×C17C2×M1611yescharacteristic
152C2×C4×C17#13111yescharacteristic
153C8×C17—22no—
154C8×C17—21no—
155C8×C17—21no—
156C8×C17—41no—
157C8×C17#12011yescharacteristic
158C8×C17#12111yescharacteristic
159C8×C17#12111yescharacteristic
160C8×C17#12311yescharacteristic
161#3407C2×C1711yescharacteristic, complemented
162C2×#444C2×C1711yescharacteristic
163#2117C2×C1711yescharacteristic, complemented
164#2117C2×C1711yescharacteristic, complemented
165#2115C2×C1711yescharacteristic, complemented
166#1377C2×C1711yescharacteristic, complemented
167#1375C2×C1711yescharacteristic, complemented
168C22×C17—22no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—81no—
172C22×C17—81no—
173C22×C17#40811yescharacteristic
174C22×C17C8×D411yescharacteristic
175C22×C17#43511yescharacteristic
176C4×C17—22no—
177C4×C17—41no—
178C4×C17—81no—
179C4×C17#38611yescharacteristic
180C4×C17C2×C22⋊C811yescharacteristic
181C4×C17C8×D411yescharacteristic
182C4×C17#43511yescharacteristic
183C2×#135C4×C1711yescharacteristic
184C2×#131C22×C1711yescharacteristic
185C2×#131C22×C1711yescharacteristic
186#504—22no—
187#504—22no—
188C2×M32—22no—
189C2×M32C22×C1711yescharacteristic
190C2×M32C22×C1711yescharacteristic
191C2×M32C22×C1711yescharacteristic
192C2×M32C22×C1711yescharacteristic
193#444—22no—
194#444—22no—
195C2×#108C4×C1711yescharacteristic
196C2×#104C4×C1711yescharacteristic
197C2×#104C4×C1711yescharacteristic
198C2×#97C22×C1711yescharacteristic
199#350—22no—
200#350—22no—
201#349—22no—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17—21no—
205C2×C17—42no—
206C2×C17—81no—
207C2×C17#186911yescharacteristic
208C2×#41C2×C4×C1711yescharacteristic
209C2×#41C2×C4×C1711yescharacteristic
210#135—22no—
211#135—22no—
212C2×Q16C8×C1711yescharacteristic
213C2×SD16C8×C1711yescharacteristic
214C2×SD16C8×C1711yescharacteristic
215C2×D8C8×C1711yescharacteristic
216#131—21no—
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131D4×C1712yes—
223#131D4×C1712yes—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16C2×C4×C1711yescharacteristic
227C2×M16C2×C4×C1711yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8C2×C4×C1711yescharacteristic
231C22×C8C23×C1711yescharacteristic
232C2×C42C2×C4×C1711yescharacteristic
233M32—24no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—21no—
241M32D4×C1712yes—
242C2×C16—24no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16D4×C1712yes—
247#108—22no—
248#104—22no—
249#104—22no—
250#97—21no—
251#97—21no—
252C17#1298611yescharacteristic, direct product factor, Sylow
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41C22⋊C4×C1712yes—
260#41M16×C1712yes—
261C2×Q8C2×C8×C1711yescharacteristic
262C2×Q8C2×C8×C1711yescharacteristic
263C2×D4—21no—
264C2×D4—21no—
265C2×D4C2×C8×C1711yescharacteristic
266C2×D4C2×C8×C1711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C22×C4×C1711yescharacteristic
271Q16—22no—
272SD16—22no—
273SD16—22no—
274D8—22no—
275M16—21no—
276M16—22no—
277M16—22no—
278M16—21no—
279M16—21no—
280M16—41no—
281M16—41no—
282M16C22⋊C4×C1712yes—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8C2×C8×C1711yescharacteristic
292C2×C8C2×C8×C1711yescharacteristic
293C2×C8C2×D4×C1711yescharacteristic
294C2×C8C2×D4×C1711yescharacteristic
295C2×C8C2×D4×C1711yescharacteristic
296C2×C8C2×D4×C1711yescharacteristic
297C2×C8#41×C1711yescharacteristic
298C2×C8#41×C1711yescharacteristic
299C42—21no—
300C42—21no—
301C16—22no—
302C16—22no—
303C16—41no—
304C16—41no—
305C23—21no—
306C23—21no—
307C23C17×#13111yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8C17×C22⋊C812yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319D4C17×C22⋊C812yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4C2×C22⋊C4×C1711yescharacteristic
329C2×C4C4×D4×C1711yescharacteristic
330C2×C4C4×D4×C1711yescharacteristic
331C2×C4C22×C8×C1711yescharacteristic
332C2×C4C2×M16×C1711yescharacteristic
333C2×C4C17×#13111yescharacteristic
334C8—22no—
335C8—21no—
336C8—21no—
337C8—41no—
338C8C17×#12011yescharacteristic
339C8C17×#12111yescharacteristic
340C8C17×#12111yescharacteristic
341C8C17×#12311yescharacteristic
342C22—22no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22C17×#40811yescharacteristic
348C22C8×D4×C1711yescharacteristic
349C22C17×#43511yescharacteristic
350C4—22no—
351C4—41no—
352C4—81no—
353C4C17×#38611yescharacteristic
354C4C2×C17×C22⋊C811yescharacteristic
355C4C8×D4×C1711yescharacteristic
356C4C17×#43511yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2C17×#186911yescharacteristic
subgroupG/H
centerC4×C17#386
commutatorC2×C4C22×C8×C17
Frattini, cosocleC22×C8C23×C17
socleC2×C17#1869
prime radicalC2×C17×#135C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C17 ◁ C22×C8×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#3407C2semidirect
3. C17×#2117C2semidirect
4. C17×#2117C2semidirect
5. C17×#2115C2semidirect
6. C17×#1377C2semidirect
7. C17×#1375C2semidirect
123. #12986C17direct (N × H)
161. #3407C2×C17semidirect
163. #2117C2×C17semidirect
164. #2117C2×C17semidirect
165. #2115C2×C17semidirect
166. #1377C2×C17semidirect
167. #1375C2×C17semidirect
252. C17#12986direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#34072
C2×C17×#4442
C17×#21172
C17×#21152
C17×#13772
C17×#13752
#1298617