Finite Group Databasebeta

← C17×#12964 #16870708

Subgroups — C17×#12964 #16870708

Order 4352.

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

normal subgroups only

This group has 672 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×#2366C211yesmaximal, characteristic
2C2×C17×#353C211yesmaximal, characteristic
3C17×#1887C211yesmaximal, characteristic
4C17×#1867C212yesmaximal, complemented
5C17×#1390C212yesmaximal, complemented
6C2×C17×#127C2211yescharacteristic
7C17×#518C2211yescharacteristic
8C2×C4×D4×C17—22no—
9C17×#418—21no—
10C2×C17×#100C2211yescharacteristic
11C2×C17×#99C2212yes—
12C17×#390—22no—
13C17×#386—21no—
14C17×#386C2212yes—
15C17×#385—22no—
16C4×C4⋊C4×C17—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—41no—
30C4×D4×C17—41no—
31C4×D4×C17—48no—
32C17×#117—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—42no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17D412yes—
42C2×C42×C17—22no—
43C2×C42×C17D411yescharacteristic
44C17×#102—41no—
45C17×#102—41no—
46C17×#100—22no—
47C17×#99—22no—
48C17×#99C2×C414yescomplemented
49C17×#95—22no—
50C17×#95—42no—
51C24×C17—21no—
52C24×C17C2×D412yes—
53C2×D4×C17—24no—
54C2×D4×C17—24no—
55C2×D4×C17—21no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17—44no—
59C2×D4×C17—44no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17C4⋊C412yescomplemented
63C2×D4×C17C22×C411yescharacteristic
64C2×D4×C17C2×Q811yescharacteristic
65C2×D4×C17#4112yes—
66C22×C4×C17—21no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—44no—
72C22×C4×C17—44no—
73C22×C4×C17—42no—
74C22×C4×C17—42no—
75C22×C4×C17—81no—
76C22×C4×C17C2×D411yescharacteristic
77M16×C17—41no—
78M16×C17—41no—
79C2×C8×C17—41no—
80C4⋊C4×C17—24no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—84no—
85C22⋊C4×C17—24no—
86C22⋊C4×C17—44no—
87C22⋊C4×C17—44no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—42no—
92C22⋊C4×C17—81no—
93C42×C17—21no—
94C42×C17—44no—
95C42×C17—41no—
96C42×C17C4⋊C412yescomplemented
97#12964C1711yesmaximal, characteristic, direct product factor, Sylow
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—22no—
101C23×C17—42no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—42no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17C4×D412yes—
108C23×C17#12011yescharacteristic
109C23×C17C22⋊Q812yes—
110D4×C17—41no—
111D4×C17—41no—
112D4×C17—42no—
113D4×C17—42no—
114D4×C17—48no—
115D4×C17—48no—
116D4×C17—81no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—24no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—44no—
126C2×C4×C17—42no—
127C2×C4×C17—44no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—42no—
132C2×C4×C17—82no—
133C2×C4×C17—84no—
134C2×C4×C17—82no—
135C2×C4×C17—82no—
136C2×C4×C17—82no—
137C2×C4×C17—82no—
138C2×C4×C17C2×C4⋊C411yescharacteristic
139C2×C4×C17#12311yescharacteristic
140C8×C17—81no—
141#2366C2×C1711yescharacteristic
142C2×#353C2×C1711yescharacteristic
143#1887C2×C1711yescharacteristic
144#1867C2×C1712yescomplemented
145#1390C2×C1712yescomplemented
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—22no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—44no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—82no—
157C22×C17—82no—
158C22×C17—81no—
159C22×C17—81no—
160C22×C17#38511yescharacteristic
161C22×C17B(4,𝔽2)11yescharacteristic
162C22×C17#45811yescharacteristic
163C4×C17—24no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—44no—
167C4×C17—82no—
168C4×C17—84no—
169C4×C17—81no—
170C4×C17—81no—
171C2×#127C22×C1711yescharacteristic
172#518C22×C1711yescharacteristic
173C2×C4×D4—22no—
174#418—21no—
175C2×#100C22×C1711yescharacteristic
176C2×#99C22×C1712yes—
177#390—22no—
178#386—21no—
179#386C22×C1712yes—
180#385—22no—
181C4×C4⋊C4—21no—
182#353C4×C1714yescomplemented
183#342—22no—
184#339—21no—
185C2×C17—21no—
186C2×C17—21no—
187C2×C17—42no—
188C2×C17—42no—
189C2×C17—82no—
190C2×C17#186711yescharacteristic
191C2×C17#219611yescharacteristic
192C2×C17#219811yescharacteristic
193C22×D4—22no—
194C22×D4D4×C1711yescharacteristic
195C22×D4C23×C1711yescharacteristic
196C23×C4—21no—
197C23×C4D4×C1712yes—
198C2×M16—21no—
199#127—22no—
200#127C2×C4×C1712yes—
201#127Q8×C1712yes—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—48no—
205#117—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—42no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4D4×C1712yes—
215C2×C42—22no—
216C2×C42D4×C1711yescharacteristic
217#102—41no—
218#102—41no—
219#100—22no—
220#99—22no—
221#99C2×C4×C1714yescomplemented
222#95—22no—
223#95—42no—
224C17#1296411yescharacteristic, direct product factor, Sylow
225C24—21no—
226C24C2×D4×C1712yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4C4⋊C4×C1712yescomplemented
237C2×D4C22×C4×C1711yescharacteristic
238C2×D4C2×Q8×C1711yescharacteristic
239C2×D4#41×C1712yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4C2×D4×C1711yescharacteristic
251M16—41no—
252M16—41no—
253C2×C8—41no—
254C4⋊C4—24no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—42no—
258C4⋊C4—84no—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—81no—
267C42—21no—
268C42—44no—
269C42—41no—
270C42C4⋊C4×C1712yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23C4×D4×C1712yes—
281C23C17×#12011yescharacteristic
282C23C17×C22⋊Q812yes—
283D4—41no—
284D4—41no—
285D4—42no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4C2×C4⋊C4×C1711yescharacteristic
312C2×C4C17×#12311yescharacteristic
313C8—81no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22C17×#38511yescharacteristic
329C22C17×B(4,𝔽2)11yescharacteristic
330C22C17×#45811yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—82no—
336C4—84no—
337C4—81no—
338C4—81no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2C17×#186711yescharacteristic
345C2C17×#219611yescharacteristic
346C2C17×#219811yescharacteristic
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
48. C17×#99C2×C4semidirect
62. C2×D4×C17C4⋊C4semidirect
96. C42×C17C4⋊C4semidirect
97. #12964C17direct (N × H)
144. #1867C2×C17semidirect
145. #1390C2×C17semidirect
182. #353C4×C17semidirect
221. #99C2×C4×C17semidirect
224. C17#12964direct (N × H)
236. C2×D4C4⋊C4×C17semidirect
270. C42C4⋊C4×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#23662
C2×C17×#3532
C17×#18872
C17×#18672
C17×#13902
#1296417