Finite Group Databasebeta

← C17×#12966 #16870710

Subgroups — C17×#12966 #16870710

Order 4352.

580 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C17×#2380C211yesmaximal, characteristic
2C2×C17×#353C211yesmaximal, characteristic
3C17×#1887C211yesmaximal, characteristic
4C17×#1867C212yesmaximal, complemented
5C17×#1390C212yesmaximal, complemented
6C2×C17×#127C2211yescharacteristic
7C17×#518C2211yescharacteristic
8C17×#418—21no—
9C2×C17×#100C2211yescharacteristic
10C2×C17×#99C2212yes—
11C17×#386—22no—
12C17×#386—21no—
13C17×#386—22no—
14C17×#386C2212yes—
15C17×#385—22no—
16C17×#383—21no—
17C17×#353C414yescomplemented
18C17×#342—22no—
19C17×#339—21no—
20C22×D4×C17—22no—
21C22×D4×C17D411yescharacteristic
22C22×D4×C17C2311yescharacteristic
23C23×C4×C17—21no—
24C23×C4×C17D412yes—
25C2×M16×C17—21no—
26C17×#127—22no—
27C17×#127C2×C412yes—
28C17×#127Q812yes—
29C4×D4×C17—42no—
30C17×#117—21no—
31C2×C4⋊C4×C17—42no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17D412yes—
38C2×C42×C17D411yescharacteristic
39C17×#102—42no—
40C17×#100—22no—
41C17×#99—22no—
42C17×#99C2×C414yescomplemented
43C17×#95—21no—
44C17×#95—22no—
45C17×#95—21no—
46C17×#95—22no—
47C17×#95—42no—
48C24×C17—21no—
49C24×C17C2×D412yes—
50C2×D4×C17—24no—
51C2×D4×C17—24no—
52C2×D4×C17—21no—
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17—82no—
58C2×D4×C17—82no—
59C2×D4×C17C4⋊C412yescomplemented
60C2×D4×C17C22×C411yescharacteristic
61C2×D4×C17C2×Q811yescharacteristic
62C2×D4×C17#4112yes—
63C22×C4×C17—21no—
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17—82no—
71C22×C4×C17—82no—
72C22×C4×C17—81no—
73C22×C4×C17C2×D411yescharacteristic
74M16×C17—41no—
75M16×C17—41no—
76C2×C8×C17—41no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—84no—
79C22⋊C4×C17—24no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—82no—
85C22⋊C4×C17—82no—
86C22⋊C4×C17—81no—
87C42×C17—21no—
88C42×C17—41no—
89C42×C17C4⋊C412yescomplemented
90#12966C1711yesmaximal, characteristic, direct product factor, Sylow
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17C4×D412yes—
101C23×C17#12011yescharacteristic
102C23×C17C22⋊Q812yes—
103D4×C17—41no—
104D4×C17—41no—
105D4×C17—42no—
106D4×C17—42no—
107D4×C17—81no—
108D4×C17—84no—
109D4×C17—84no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—42no—
114C2×C4×C17—44no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—44no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—82no—
124C2×C4×C17—82no—
125C2×C4×C17—84no—
126C2×C4×C17—82no—
127C2×C4×C17—82no—
128C2×C4×C17—82no—
129C2×C4×C17C2×C4⋊C411yescharacteristic
130C2×C4×C17#12311yescharacteristic
131C8×C17—81no—
132#2380C2×C1711yescharacteristic
133C2×#353C2×C1711yescharacteristic
134#1887C2×C1711yescharacteristic
135#1867C2×C1712yescomplemented
136#1390C2×C1712yescomplemented
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17—44no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—82no—
147C22×C17—81no—
148C22×C17—81no—
149C22×C17—82no—
150C22×C17#38511yescharacteristic
151C22×C17B(4,𝔽2)11yescharacteristic
152C22×C17#45811yescharacteristic
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—42no—
156C4×C17—82no—
157C4×C17—84no—
158C4×C17—82no—
159C4×C17—82no—
160C2×#127C22×C1711yescharacteristic
161#518C22×C1711yescharacteristic
162#418—21no—
163C2×#100C22×C1711yescharacteristic
164C2×#99C22×C1712yes—
165#386—21no—
166#386—22no—
167#386—22no—
168#386C22×C1712yes—
169#385—22no—
170#383—21no—
171#353C4×C1714yescomplemented
172#342—22no—
173#339—21no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—42no—
177C2×C17—42no—
178C2×C17—82no—
179C2×C17#186711yescharacteristic
180C2×C17#220012yes—
181C22×D4—22no—
182C22×D4D4×C1711yescharacteristic
183C22×D4C23×C1711yescharacteristic
184C23×C4—21no—
185C23×C4D4×C1712yes—
186C2×M16—21no—
187#127—22no—
188#127C2×C4×C1712yes—
189#127Q8×C1712yes—
190C4×D4—42no—
191#117—21no—
192C2×C4⋊C4—42no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4D4×C1712yes—
199C2×C42D4×C1711yescharacteristic
200#102—42no—
201#100—22no—
202#99—22no—
203#99C2×C4×C1714yescomplemented
204#95—21no—
205#95—22no—
206#95—21no—
207#95—22no—
208#95—42no—
209C17#1296611yescharacteristic, direct product factor, Sylow
210C24—21no—
211C24C2×D4×C1712yes—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—21no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4C4⋊C4×C1712yescomplemented
222C2×D4C22×C4×C1711yescharacteristic
223C2×D4C2×Q8×C1711yescharacteristic
224C2×D4#41×C1712yes—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—81no—
235C22×C4C2×D4×C1711yescharacteristic
236M16—41no—
237M16—41no—
238C2×C8—41no—
239C4⋊C4—22no—
240C4⋊C4—84no—
241C22⋊C4—24no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—81no—
249C42—21no—
250C42—41no—
251C42C4⋊C4×C1712yescomplemented
252C23—22no—
253C23—22no—
254C23—22no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—41no—
261C23C4×D4×C1712yes—
262C23C17×#12011yescharacteristic
263C23C17×C22⋊Q812yes—
264D4—41no—
265D4—41no—
266D4—42no—
267D4—42no—
268D4—81no—
269D4—84no—
270D4—84no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4C2×C4⋊C4×C1711yescharacteristic
291C2×C4C17×#12311yescharacteristic
292C8—81no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—81no—
304C22—81no—
305C22—82no—
306C22C17×#38511yescharacteristic
307C22C17×B(4,𝔽2)11yescharacteristic
308C22C17×#45811yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—82no—
313C4—84no—
314C4—82no—
315C4—82no—
316C2—21no—
317C2—21no—
318C2—42no—
319C2—42no—
320C2—82no—
321C2C17×#186711yescharacteristic
322C2C17×#220012yes—
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutatorC2×D4C22×C4×C17
Frattini, cosocleC22×D4C23×C17
prime radicalC2×C17×#127C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C22×D4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
4. C17×#1867C2semidirect
5. C17×#1390C2semidirect
17. C17×#353C4semidirect
42. C17×#99C2×C4semidirect
59. C2×D4×C17C4⋊C4semidirect
89. C42×C17C4⋊C4semidirect
90. #12966C17direct (N × H)
135. #1867C2×C17semidirect
136. #1390C2×C17semidirect
171. #353C4×C17semidirect
203. #99C2×C4×C17semidirect
209. C17#12966direct (N × H)
221. C2×D4C4⋊C4×C17semidirect
251. C42C4⋊C4×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#23802
C2×C17×#3532
C17×#18872
C17×#18672
C17×#13902
#1296617