Finite Group Databasebeta

← C17×#7754 #16865498

Subgroups — C17×#7754 #16865498

Order 4352.

536 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2683C211yesmaximal, characteristic
2C17×#2058C212yesmaximal, complemented
3C17×#2013C211yesmaximal, characteristic
4C17×#1271C212yesmaximal, complemented
5C17×#1270C211yesmaximal, characteristic
6C22×SD16×C17—21no—
7C2×C17×C4⋊Q8C2211yescharacteristic
8C2×C17×C22⋊Q8—22no—
9C2×C17×#106—22no—
10C2×C17×#103—21no—
11C17×#396—22no—
12C17×#395—21no—
13C17×#390C2211yescharacteristic
14C17×#389C2212yes—
15C17×#386—21no—
16C17×#340—22no—
17C17×#336C2211yescharacteristic
18C17×#336C2212yes—
19C22×Q8×C17—21no—
20C22×Q8×C17—21no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17—21no—
23C2×SD16×C17—42no—
24C2×SD16×C17—42no—
25C2×SD16×C17—42no—
26C22×C8×C17—21no—
27C22×C8×C17—22no—
28C17×C4⋊Q8—42no—
29C17×C22⋊Q8—44no—
30C17×C22⋊Q8—44no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17D412yes—
35C2×C4⋊C4×C17D412yes—
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17—21no—
40C2×C42×C17C2311yescharacteristic
41C17×#106—44no—
42C17×#103—42no—
43C17×#95—21no—
44C17×#95—22no—
45C24×C17—21no—
46C2×Q8×C17—22no—
47C2×Q8×C17—44no—
48C2×Q8×C17—41no—
49C2×Q8×C17—41no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×Q8×C17—41no—
53C2×Q8×C17—81no—
54C2×D4×C17—21no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17SD1612yes—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—44no—
62C22×C4×C17—41no—
63C22×C4×C17—81no—
64C22×C4×C17C2×D412yes—
65C22×C4×C17C2×D411yescharacteristic
66SD16×C17—48no—
67C2×C8×C17—24no—
68C2×C8×C17—22no—
69C2×C8×C17—41no—
70C2×C8×C17—42no—
71C2×C8×C17—41no—
72C2×C8×C17—42no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—24no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—44no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17SD1614yescomplemented
82C22⋊C4×C17—44no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—81no—
86C42×C17—41no—
87#7754C1711yesmaximal, characteristic, direct product factor, Sylow
88C23×C17—22no—
89C23×C17—42no—
90C23×C17—41no—
91C23×C17—42no—
92C23×C17—41no—
93C23×C17#12011yescharacteristic
94Q8×C17—42no—
95Q8×C17—42no—
96Q8×C17—81no—
97Q8×C17—81no—
98Q8×C17—84no—
99D4×C17—42no—
100D4×C17—81no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—21no—
105C2×C4×C17—24no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—84no—
115C2×C4×C17—81no—
116C2×C4×C17—81no—
117C2×C4×C17C2×SD1611yescharacteristic
118C2×C4×C17C2×SD1612yes—
119C2×C4×C17AGL(1,ℤ/8)12yes—
120C2×C4×C17#13711yescharacteristic
121C8×C17—44no—
122C8×C17—42no—
123#2683C2×C1711yescharacteristic
124#2058C2×C1712yescomplemented
125#2013C2×C1711yescharacteristic
126#1271C2×C1712yescomplemented
127#1270C2×C1711yescharacteristic
128C22×C17—42no—
129C22×C17—44no—
130C22×C17—82no—
131C22×C17—81no—
132C22×C17—81no—
133C22×C17#44811yescharacteristic
134C22×C17#45012yes—
135C22×C17#45311yescharacteristic
136C22×C17#45612yes—
137C22×C17B(4,𝔽2)11yescharacteristic
138C4×C17—21no—
139C4×C17—21no—
140C4×C17—22no—
141C4×C17—22no—
142C4×C17—44no—
143C4×C17—82no—
144C4×C17—82no—
145C4×C17—81no—
146C22×SD16—21no—
147C2×C4⋊Q8C22×C1711yescharacteristic
148C2×C22⋊Q8—22no—
149C2×#106—22no—
150C2×#103—21no—
151#396—22no—
152#395—21no—
153#390C22×C1711yescharacteristic
154#389C22×C1712yes—
155#386—21no—
156#340—22no—
157#336C22×C1712yes—
158#336C22×C1711yescharacteristic
159C2×C17—82no—
160C2×C17#159611yescharacteristic
161C2×C17#160412yes—
162C2×C17#162911yescharacteristic
163C2×C17#165112yes—
164C2×C17#166011yescharacteristic
165C22×Q8—21no—
166C22×Q8—21no—
167C22×D4D4×C1711yescharacteristic
168C23×C4—21no—
169C2×SD16—42no—
170C2×SD16—42no—
171C2×SD16—42no—
172C22×C8—21no—
173C22×C8—22no—
174C4⋊Q8—42no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4D4×C1712yes—
181C2×C4⋊C4D4×C1712yes—
182C2×C4⋊C4D4×C1711yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C42C23×C1711yescharacteristic
187#106—44no—
188#103—42no—
189#95—21no—
190#95—22no—
191C17#775411yescharacteristic, direct product factor, Sylow
192C24—21no—
193C2×Q8—22no—
194C2×Q8—44no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—81no—
201C2×D4—21no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4SD16×C1712yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—44no—
209C22×C4—41no—
210C22×C4—81no—
211C22×C4C2×D4×C1712yes—
212C22×C4C2×D4×C1711yescharacteristic
213SD16—48no—
214C2×C8—22no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—42no—
220C4⋊C4—22no—
221C4⋊C4—24no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4SD16×C1714yescomplemented
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—81no—
233C42—41no—
234C23—22no—
235C23—42no—
236C23—41no—
237C23—42no—
238C23—41no—
239C23C17×#12011yescharacteristic
240Q8—42no—
241Q8—42no—
242Q8—81no—
243Q8—81no—
244Q8—84no—
245D4—42no—
246D4—81no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—84no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×SD16×C1711yescharacteristic
264C2×C4C2×SD16×C1712yes—
265C2×C4C17×AGL(1,ℤ/8)12yes—
266C2×C4C17×#13711yescharacteristic
267C8—42no—
268C8—44no—
269C22—42no—
270C22—44no—
271C22—82no—
272C22—81no—
273C22—81no—
274C22C17×#44811yescharacteristic
275C22C17×#45012yes—
276C22C17×#45311yescharacteristic
277C22C17×#45612yes—
278C22C17×B(4,𝔽2)11yescharacteristic
279C4—21no—
280C4—21no—
281C4—22no—
282C4—22no—
283C4—44no—
284C4—82no—
285C4—82no—
286C4—81no—
287C2—82no—
288C2C17×#159611yescharacteristic
289C2C17×#160412yes—
290C2C17×#162911yescharacteristic
291C2C17×#165112yes—
292C2C17×#166011yescharacteristic
subgroupG/H
center, socleC23×C17#120
commutator, Frattini, cosocleC2×C42C23×C17
prime radicalC22×D4×C17D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C17 ◁ C2×C42×C17 ◁ G

Complements

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

NH ≅ G/Ntype
2. C17×#2058C2semidirect
4. C17×#1271C2semidirect
81. C4⋊C4×C17SD16semidirect
87. #7754C17direct (N × H)
124. #2058C2×C17semidirect
126. #1271C2×C17semidirect
191. C17#7754direct (N × H)
228. C4⋊C4SD16×C17semidirect

Maximal subgroups

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

subgroupindex
C17×#26832
C17×#20582
C17×#20132
C17×#12712
C17×#12702
#775417