Finite Group Databasebeta

← C17×#12095 #16869839

Subgroups — C17×#12095 #16869839

Order 4352.

458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2817C211yesmaximal, characteristic, complemented
2C17×#2022C211yesmaximal, characteristic
3C17×#1607C212yesmaximal
4C17×#1344C212yesmaximal, complemented
5C17×#1317C211yesmaximal, characteristic, complemented
6C17×#534C2211yescharacteristic
7C2×C17×#123—21no—
8C2×C17×C22⋊Q8—21no—
9C17×#480—22no—
10C2×C17×#99—21no—
11C17×#400—22no—
12C17×#397—21no—
13C17×#397C2212yes—
14C17×#395—22no—
15C17×#381C2211yescharacteristic
16C17×#342C2211yescharacteristic
17C17×#327—22no—
18C17×#323C2212yes—
19C22×Q8×C17—21no—
20C22×D4×C17—21no—
21C23×C4×C17D411yescharacteristic
22C17×#123—42no—
23C17×#123—41no—
24C17×#123—42no—
25C17×C22⋊Q8—42no—
26C17×C22⋊Q8—42no—
27C17×#121—22no—
28C17×#120—41no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—41no—
33C2×C4⋊C4×C17D412yes—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—41no—
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17C2311yescharacteristic
39C17×#107—42no—
40C17×#102—42no—
41C17×#99—42no—
42C17×C22⋊C8—22no—
43C17×#95—22no—
44C17×#95—21no—
45C17×#95—22no—
46C17×#95—22no—
47C17×#95D412yes—
48C24×C17—21no—
49C24×C17C2×D411yescharacteristic
50C2×Q8×C17—42no—
51C2×Q8×C17—81no—
52C2×Q8×C17—81no—
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17—41no—
59C2×D4×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—41no—
67C22×C4×C17—41no—
68C22×C4×C17—41no—
69C22×C4×C17D812yes—
70C22×C4×C17C2×D412yes—
71C2×C8×C17—42no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—44no—
76C4⋊C4×C17—44no—
77C4⋊C4×C17—81no—
78C22⋊C4×C17—22no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—44no—
82C22⋊C4×C17—42no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—41no—
86#12095C1711yesmaximal, characteristic, direct product factor, Sylow
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—42no—
90C23×C17—41no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—82no—
94C23×C17#12011yescharacteristic
95C23×C17C2×D811yescharacteristic
96C23×C17AGL(1,ℤ/8)11yescharacteristic
97Q8×C17—84no—
98D4×C17—82no—
99D4×C17—81no—
100D4×C17—81no—
101D4×C17—81no—
102D4×C17—81no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—44no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—42no—
120C2×C4×C17—81no—
121C2×C4×C17—82no—
122C8×C17—82no—
123#2817C2×C1711yescharacteristic, complemented
124#2022C2×C1711yescharacteristic
125#1607C2×C1712yes—
126#1344C2×C1712yescomplemented
127#1317C2×C1711yescharacteristic, complemented
128C22×C17—22no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—42no—
134C22×C17—81no—
135C22×C17—82no—
136C22×C17—81no—
137C22×C17—82no—
138C22×C17—161no—
139C22×C17#44711yescharacteristic
140C22×C17#45311yescharacteristic
141C22×C17B(4,𝔽2)11yescharacteristic
142C4×C17—42no—
143C4×C17—42no—
144C4×C17—82no—
145C4×C17—81no—
146C4×C17—82no—
147C4×C17—81no—
148C4×C17—82no—
149#534C22×C1711yescharacteristic
150C2×#123—21no—
151C2×C22⋊Q8—21no—
152#480—22no—
153C2×#99—21no—
154#400—22no—
155#397—21no—
156#397C22×C1712yes—
157#395—22no—
158#381C22×C1711yescharacteristic
159#342C22×C1711yescharacteristic
160#327—22no—
161#323C22×C1712yes—
162C2×C17—21no—
163C2×C17—21no—
164C2×C17—41no—
165C2×C17—41no—
166C2×C17—161no—
167C2×C17#159511yescharacteristic
168C2×C17#220112yes—
169C22×Q8—21no—
170C22×D4—21no—
171C23×C4D4×C1711yescharacteristic
172#123—42no—
173#123—41no—
174#123—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177#121—22no—
178#120—41no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4D4×C1712yes—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4D4×C1711yescharacteristic
188C2×C22⋊C4C23×C1711yescharacteristic
189#107—42no—
190#102—42no—
191#99—42no—
192C22⋊C8—22no—
193#95—22no—
194#95—21no—
195#95—22no—
196#95—22no—
197#95D4×C1712yes—
198C17#1209511yescharacteristic, direct product factor, Sylow
199C24—21no—
200C24C2×D4×C1711yescharacteristic
201C2×Q8—42no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×D4—22no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4D8×C1712yes—
221C22×C4C2×D4×C1712yes—
222C2×C8—42no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—81no—
229C22⋊C4—22no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—41no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—82no—
244C23C17×#12011yescharacteristic
245C23C2×D8×C1711yescharacteristic
246C23C17×AGL(1,ℤ/8)11yescharacteristic
247Q8—84no—
248D4—82no—
249D4—81no—
250D4—81no—
251D4—81no—
252D4—81no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—82no—
272C8—82no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—82no—
281C22—82no—
282C22—81no—
283C22—161no—
284C22C17×#44711yescharacteristic
285C22C17×#45311yescharacteristic
286C22C17×B(4,𝔽2)11yescharacteristic
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4—82no—
292C4—81no—
293C4—82no—
294C2—21no—
295C2—21no—
296C2—41no—
297C2—41no—
298C2—161no—
299C2C17×#159511yescharacteristic
300C2C17×#220112yes—
subgroupG/H
center, socleC22×C17B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C17
prime radicalC17×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C17 ◁ C23×C17 ◁ C2×C22⋊C4×C17 ◁ G

Complements

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

NH ≅ G/Ntype
1. C17×#2817C2semidirect
4. C17×#1344C2semidirect
5. C17×#1317C2semidirect
86. #12095C17direct (N × H)
123. #2817C2×C17semidirect
126. #1344C2×C17semidirect
127. #1317C2×C17semidirect
198. C17#12095direct (N × H)

Maximal subgroups

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

subgroupindex
C17×#28172
C17×#20222
C17×#16072
C17×#13442
C17×#13172
#1209517