Finite Group Databasebeta

← #16938674

Subgroups — #16938674

Order 4352.

448 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 81 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
1#14529253C212yesmaximal
2#14528966C211yesmaximal, characteristic
3#14527304C212yesmaximal
4#14526661C211yesmaximal, characteristic
5C17×#3189C211yesmaximal, characteristic
6C22×#63223C2211yescharacteristic
7C2×#10587485—21no—
8C2×#10587507—22no—
9#11762773C2212yes—
10C4×C22⋊Dic17C2211yescharacteristic
11#11762578—22no—
12#11762563—21no—
13#11762546—22no—
14#11762546—22no—
15C2×C17×C22⋊Q8C2212yes—
16C17×#484—22no—
17C17×#467—21no—
18C17×#467—21no—
19C17×#451—22no—
20C2×C17×C22⋊C8C2211yescharacteristic
21C22×Q8×C17D412yes—
22C23×C4×C17C2311yescharacteristic
23C2×Q16×C17—21no—
24C2×Q16×C17—21no—
25C22×C8×C17D411yescharacteristic
26C22×C8×C17D411yescharacteristic
27C17×C22⋊Q8—22no—
28C17×C22⋊Q8—22no—
29C17×C22⋊Q8—22no—
30C17×C22⋊Q8—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—22no—
34C2×C22⋊Dic17—21no—
35C2×C4×Dic17—21no—
36C2×#63223—22no—
37C2×#63223—22no—
38C2×#63223—42no—
39C22×#3622—21no—
40C17×#107—21no—
41C17×#107—21no—
42C17×#103—22no—
43C17×#103—22no—
44C17×C22⋊C8—22no—
45#10587507—44no—
46#10587502D411yescharacteristic
47#10587502D411yescharacteristic
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50#10587485—42no—
51C24×C17#4111yescharacteristic
52C2×Q8×C17—22no—
53C2×Q8×C17—22no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—41no—
59C22×C4×C17—41no—
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D412yes—
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17#4112yes—
64C22×Dic17—21no—
65C22×Dic17—21no—
66Q16×C17—81no—
67Q16×C17—81no—
68C2×C8×C17—21no—
69C2×C8×C17—21no—
70C2×C8×C17—41no—
71C2×C8×C17—41no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—42no—
76C22⋊C4×C17—42no—
77C22⋊C4×C17—42no—
78C22⋊Dic17—42no—
79C4×Dic17—41no—
80C4×Dic17—41no—
81#63223—24no—
82#63223—42no—
83#63223—42no—
84#63223—42no—
85C2×#3622—22no—
86C2×#3622—41no—
87C2×#3622—41no—
88C2×#3622—42no—
89#10559—171nomaximal, Sylow
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17#12111yescharacteristic
95C23×C17#12312yes—
96Q8×C17—42no—
97Q8×C17—42no—
98Q8×C17—82no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—41no—
104C2×C4×C17—42no—
105C2×C4×C17—44no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—82no—
109C2×C4×C17#12011yescharacteristic
110C2×C4×C17#12112yes—
111C2×C4×C17#12411yescharacteristic
112C2×Dic17—22no—
113C2×Dic17—41no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C8×C17—41no—
119C8×C17—41no—
120#3622—42no—
121#3622—82no—
122#3189D1711yescharacteristic
123#1901—172no—
124#1851—171no—
125#1508—172no—
126#1293—171no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—22no—
130C22×C17—22no—
131C22×C17—22no—
132C22×C17—81no—
133C22×C17—81no—
134C22×C17#39411yescharacteristic
135C22×C17#47311yescharacteristic
136C22×C17#47311yescharacteristic
137C4×C17—22no—
138C4×C17—21no—
139C4×C17—82no—
140C4×C17—82no—
141C4×C17B(4,𝔽2)11yescharacteristic
142C4×C17#45811yescharacteristic
143Dic17—42no—
144Dic17—81no—
145C22×M16—171no—
146C2×C22⋊Q8C2×D1712yes—
147#484—22no—
148#467—21no—
149#467—21no—
150#451—22no—
151C2×C4⋊C8—341no—
152C2×#101—342no—
153#407—172no—
154C2×C22⋊C8C2×D1711yescharacteristic
155C4×C22⋊C4—171no—
156#343—342no—
157#336—341no—
158#328—342no—
159#328—342no—
160C2×C17—21no—
161C2×C17—21no—
162C2×C17—42no—
163C2×C17#204411yescharacteristic
164C2×C17#204711yescharacteristic
165C2×C17#205011yescharacteristic
166C22×Q8C22⋊D1712yes—
167C23×C4C22×D1711yescharacteristic
168C2×Q16—21no—
169C2×Q16—21no—
170C2×M16—342no—
171C2×M16—342no—
172C2×M16—682no—
173C22×C8—341no—
174C22×C8C22⋊D1711yescharacteristic
175C22×C8C22⋊D1711yescharacteristic
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C22⋊D1712yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—341no—
184C2×C42—341no—
185#107—21no—
186#107—21no—
187C4⋊C8—682no—
188#103—22no—
189#103—22no—
190#101—684no—
191C22⋊C8—22no—
192C22⋊C8—342no—
193C22⋊C8—342no—
194#95—171no—
195#95—171no—
196C17#1055911yescharacteristic, complemented, Sylow
197C24#6325411yescharacteristic
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—42no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—341no—
207C22×C4—341no—
208C22×C4D4×D1711yescharacteristic
209C22×C4#6325412yes—
210C22×C4C2×C22⋊D1711yescharacteristic
211C22×C4C2×C22⋊D1712yes—
212Q16—81no—
213Q16—81no—
214M16—344no—
215M16—682no—
216M16—682no—
217M16—682no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—342no—
223C2×C8—681no—
224C2×C8—681no—
225C2×C8—682no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—682no—
233C42—681no—
234C42—681no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23#1058762112yes—
240C23#1058762411yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—82no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4—342no—
255C2×C4—681no—
256C2×C4—681no—
257C2×C4—681no—
258C2×C4—681no—
259C2×C4—681no—
260C2×C4#1058762211yescharacteristic
261C2×C4#1058762512yes—
262C2×C4#1058763911yescharacteristic
263C8—41no—
264C8—41no—
265C8—682no—
266C8—1362no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—81no—
273C22—81no—
274C22#1176272311yescharacteristic
275C22#1176308111yescharacteristic
276C22#1176308111yescharacteristic
277C4—22no—
278C4—21no—
279C4—82no—
280C4—82no—
281C4—682no—
282C4—1361no—
283C4#1176299611yescharacteristic
284C4#1176299911yescharacteristic
285C2—21no—
286C2—21no—
287C2—42no—
288C2#1453003211yescharacteristic
289C2#1453004911yescharacteristic
290C2#1453006811yescharacteristic
subgroupG/H
centerC22#11762723
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#3189C2
socleC22×C17#394
prime radicalC24×C17#41
hypercenter#3189D17
hypocenterC17#10559
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
196. C17#10559semidirect

Maximal subgroups

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

subgroupindex
#145292532
#145289662
#145273042
#145266612
C17×#31892
#1055917