Finite Group Databasebeta

← C17×#12273 #16870017

Subgroups — C17×#12273 #16870017

Order 4352.

364 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2586C211yesmaximal, characteristic
2C17×#2028C211yesmaximal, characteristic
3C17×#1611C211yesmaximal, characteristic
4C17×#1605C211yesmaximal, characteristic
5C17×#1354C211yesmaximal, characteristic
6C17×#1346C211yesmaximal, characteristic
7C17×#1319C211yesmaximal, characteristic
8C17×#544C2211yescharacteristic
9C17×#483—21no—
10C17×#451—21no—
11C17×#419—21no—
12C17×#398C2211yescharacteristic
13C17×#395—21no—
14C17×#395C2211yescharacteristic
15C17×#391—21no—
16C17×#391C2211yescharacteristic
17C17×#389—21no—
18C17×#387—21no—
19C17×#387—21no—
20C4×C4⋊C4×C17—21no—
21C17×#343C2211yescharacteristic
22C17×#339—21no—
23C17×#328—21no—
24C17×#328—21no—
25C17×#324C2211yescharacteristic
26C17×#324C2211yescharacteristic
27C22×Q8×C17D411yescharacteristic
28C22×Q8×C17D411yescharacteristic
29C2×Q16×C17—41no—
30C2×M16×C17—21no—
31C17×C4⋊Q8—41no—
32C17×#125—41no—
33C17×C22⋊Q8—21no—
34C17×C22⋊Q8—21no—
35C17×#117—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17D411yescharacteristic
42C2×C4⋊C4×C17C2311yescharacteristic
43C2×C42×C17—21no—
44C2×C42×C17—21no—
45C2×C42×C17D411yescharacteristic
46C17×#106—41no—
47C17×#103—41no—
48C17×#103—41no—
49C17×#103—41no—
50C17×#103—41no—
51C17×C22⋊C8—21no—
52C17×C22⋊C8—21no—
53C17×#95—21no—
54C17×#95—21no—
55C17×#95—21no—
56C17×#95—21no—
57C17×#95—21no—
58C17×#95—21no—
59C17×#95—21no—
60C17×#95D411yescharacteristic
61C17×#95D411yescharacteristic
62C2×Q8×C17—21no—
63C2×Q8×C17—21no—
64C2×Q8×C17—21no—
65C2×Q8×C17—21no—
66C2×Q8×C17—42no—
67C2×Q8×C17—41no—
68C2×Q8×C17—41no—
69C2×Q8×C17—41no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79Q16×C17—82no—
80M16×C17—41no—
81M16×C17—41no—
82C2×C8×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17—41no—
85C4⋊C4×C17—24no—
86C4⋊C4×C17—21no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17—21no—
89C4⋊C4×C17—21no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—42no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—81no—
98C4⋊C4×C17—81no—
99C22⋊C4×C17—41no—
100C42×C17—41no—
101C42×C17—41no—
102C42×C17—42no—
103C42×C17—81no—
104#12273C1711yesmaximal, characteristic, direct product factor, Sylow
105C23×C17#12011yescharacteristic
106Q8×C17—42no—
107Q8×C17—42no—
108Q8×C17—81no—
109Q8×C17—81no—
110Q8×C17—81no—
111Q8×C17—81no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17#13511yescharacteristic
139C2×C4×C17#13711yescharacteristic
140C8×C17—81no—
141C8×C17—81no—
142C8×C17—81no—
143#2586C2×C1711yescharacteristic
144#2028C2×C1711yescharacteristic
145#1611C2×C1711yescharacteristic
146#1605C2×C1711yescharacteristic
147#1354C2×C1711yescharacteristic
148#1346C2×C1711yescharacteristic
149#1319C2×C1711yescharacteristic
150C22×C17—21no—
151C22×C17—21no—
152C22×C17#45211yescharacteristic
153C22×C17#45511yescharacteristic
154C22×C17B(4,𝔽2)11yescharacteristic
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—42no—
158C4×C17—42no—
159C4×C17—42no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—42no—
163C4×C17—81no—
164C4×C17—81no—
165C4×C17—81no—
166C4×C17—81no—
167#544C22×C1711yescharacteristic
168#483—21no—
169#451—21no—
170#419—21no—
171#398C22×C1711yescharacteristic
172#395—21no—
173#395C22×C1711yescharacteristic
174#391—21no—
175#391C22×C1711yescharacteristic
176#389—21no—
177#387—21no—
178#387—21no—
179C4×C4⋊C4—21no—
180#343C22×C1711yescharacteristic
181#339—21no—
182#328—21no—
183#328—21no—
184#324C22×C1711yescharacteristic
185#324C22×C1711yescharacteristic
186C2×C17—21no—
187C2×C17—21no—
188C2×C17#160611yescharacteristic
189C2×C17#219711yescharacteristic
190C2×C17#220511yescharacteristic
191C22×Q8D4×C1711yescharacteristic
192C22×Q8D4×C1711yescharacteristic
193C2×Q16—41no—
194C2×M16—21no—
195C4⋊Q8—41no—
196#125—41no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199#117—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4D4×C1711yescharacteristic
206C2×C4⋊C4C23×C1711yescharacteristic
207C2×C42—21no—
208C2×C42—21no—
209C2×C42D4×C1711yescharacteristic
210#106—41no—
211#103—41no—
212#103—41no—
213#103—41no—
214#103—41no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95D4×C1711yescharacteristic
225#95D4×C1711yescharacteristic
226C17#1227311yescharacteristic, direct product factor, Sylow
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C2×D4×C1711yescharacteristic
242C22×C4C2×D4×C1711yescharacteristic
243C22×C4C2×D4×C1711yescharacteristic
244Q16—82no—
245M16—41no—
246M16—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C42—41no—
266C42—42no—
267C42—41no—
268C42—81no—
269C23C17×#12011yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—81no—
273Q8—81no—
274Q8—81no—
275Q8—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4C17×#13511yescharacteristic
303C2×C4C17×#13711yescharacteristic
304C8—81no—
305C8—81no—
306C8—81no—
307C22—21no—
308C22—21no—
309C22C17×#45211yescharacteristic
310C22C17×#45511yescharacteristic
311C22C17×B(4,𝔽2)11yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C2—21no—
325C2—21no—
326C2C17×#160611yescharacteristic
327C2C17×#219711yescharacteristic
328C2C17×#220511yescharacteristic
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C17
prime radicalC23×C17#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C2×C4⋊C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
104. #12273C17direct (N × H)
226. C17#12273direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#25862
C17×#20282
C17×#16112
C17×#16052
C17×#13542
C17×#13462
C17×#13192
#1227317