Finite Group Databasebeta

← #16938662

Subgroups — #16938662

Order 4352.

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

normal subgroups only

This group has 500 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#14529248C212yesmaximal, complemented
2#14528966C211yesmaximal, characteristic, complemented
3#14527300C212yesmaximal, complemented
4#14526661C211yesmaximal, characteristic, complemented
5C17×#3186C211yesmaximal, characteristic
6C22×#63223C2211yescharacteristic
7C2×#10587485—21no—
8C2×#10587504—22no—
9#11762773C2212yes—
10C4×C22⋊Dic17C2211yescharacteristic
11#11762578—22no—
12#11762563—21no—
13#11762543—22no—
14#11762543—22no—
15C2×C17×#121C2212yes—
16C17×#480—22no—
17C17×#466—21no—
18C17×#466—21no—
19C17×#447—22no—
20C2×C17×C22⋊C8C2211yescharacteristic
21C22×D4×C17—22no—
22C22×D4×C17D412yes—
23C23×C4×C17C2311yescharacteristic
24C2×D8×C17—21no—
25C2×D8×C17—21no—
26C22×C8×C17D411yescharacteristic
27C22×C8×C17D411yescharacteristic
28C17×#121—22no—
29C17×#121—22no—
30C17×#121—22no—
31C17×#121—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×#102—22no—
43C17×#102—22no—
44C17×C22⋊C8—22no—
45#10587504—44no—
46#10587502D411yescharacteristic
47#10587502D411yescharacteristic
48C22⋊#3622—22no—
49C22⋊#3622—22no—
50#10587485—42no—
51C24×C17—22no—
52C24×C17#4111yescharacteristic
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—42no—
56C2×D4×C17—42no—
57C2×D4×C17—42no—
58C2×D4×C17—42no—
59C2×D4×C17—82no—
60C2×D4×C17—82no—
61C22×C4×C17—22no—
62C22×C4×C17—41no—
63C22×C4×C17—41no—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17#4112yes—
68C22×Dic17—21no—
69C22×Dic17—21no—
70D8×C17—81no—
71D8×C17—81no—
72C2×C8×C17—21no—
73C2×C8×C17—21no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
76C4⋊C4×C17—22no—
77C4⋊C4×C17—22no—
78C22⋊C4×C17—42no—
79C22⋊C4×C17—42no—
80C22⋊Dic17—42no—
81C4×Dic17—41no—
82C4×Dic17—41no—
83#63223—24no—
84#63223—42no—
85#63223—42no—
86#63223—42no—
87C2×#3622—22no—
88C2×#3622—41no—
89C2×#3622—41no—
90C2×#3622—42no—
91#10557—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17—44no—
97C23×C17—42no—
98C23×C17—82no—
99C23×C17—82no—
100C23×C17#12111yescharacteristic
101C23×C17#12312yes—
102D4×C17—42no—
103D4×C17—42no—
104D4×C17—82no—
105D4×C17—84no—
106D4×C17—82no—
107D4×C17—82no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—82no—
116C2×C4×C17#12011yescharacteristic
117C2×C4×C17#12112yes—
118C2×C4×C17#12411yescharacteristic
119C2×Dic17—22no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C8×C17—41no—
126C8×C17—41no—
127#3622—42no—
128#3622—82no—
129#3186D1711yescharacteristic
130#1900—172no—
131#1851—171no—
132#1507—172no—
133#1293—171no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—81no—
140C22×C17—81no—
141C22×C17—82no—
142C22×C17—82no—
143C22×C17—82no—
144C22×C17—164no—
145C22×C17#39411yescharacteristic
146C22×C17#47211yescharacteristic
147C22×C17#47211yescharacteristic
148C4×C17—22no—
149C4×C17—21no—
150C4×C17—82no—
151C4×C17B(4,𝔽2)11yescharacteristic
152C4×C17#45811yescharacteristic
153Dic17—42no—
154Dic17—81no—
155C22×M16—171no—
156C2×#121C2×D1712yes—
157#480—22no—
158#466—21no—
159#466—21no—
160#447—22no—
161C2×C4⋊C8—341no—
162C2×#100—342no—
163#407—172no—
164C2×C22⋊C8C2×D1711yescharacteristic
165C4×C22⋊C4—171no—
166#343—342no—
167#336—341no—
168#327—342no—
169#327—342no—
170C2×C17—21no—
171C2×C17—21no—
172C2×C17—42no—
173C2×C17—162no—
174C2×C17#204411yescharacteristic
175C2×C17#204511yescharacteristic
176C2×C17#205011yescharacteristic
177C22×D4—22no—
178C22×D4C22⋊D1712yes—
179C23×C4C22×D1711yescharacteristic
180C2×D8—21no—
181C2×D8—21no—
182C2×M16—342no—
183C2×M16—342no—
184C2×M16—682no—
185C22×C8—341no—
186C22×C8C22⋊D1711yescharacteristic
187C22×C8C22⋊D1711yescharacteristic
188#121—22no—
189#121—22no—
190#121—22no—
191#121—22no—
192C2×C4⋊C4C22⋊D1712yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—341no—
195C2×C42—341no—
196#107—21no—
197#107—21no—
198C4⋊C8—682no—
199#102—22no—
200#102—22no—
201#100—684no—
202C22⋊C8—22no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205#95—171no—
206#95—171no—
207C17#1055711yescharacteristic, complemented, Sylow
208C24—22no—
209C24#6325411yescharacteristic
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—82no—
217C2×D4—82no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4D4×D1711yescharacteristic
224C22×C4#6325412yes—
225C22×C4C2×C22⋊D1711yescharacteristic
226C22×C4C2×C22⋊D1712yes—
227D8—81no—
228D8—81no—
229M16—344no—
230M16—682no—
231M16—682no—
232M16—682no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—342no—
238C2×C8—681no—
239C2×C8—681no—
240C2×C8—682no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—682no—
246C42—681no—
247C42—681no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—44no—
253C23—42no—
254C23—82no—
255C23—82no—
256C23#1058762112yes—
257C23#1058762411yescharacteristic
258D4—42no—
259D4—42no—
260D4—82no—
261D4—84no—
262D4—82no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—82no—
272C2×C4—342no—
273C2×C4—681no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4#1058762211yescharacteristic
279C2×C4#1058762512yes—
280C2×C4#1058763911yescharacteristic
281C8—41no—
282C8—41no—
283C8—682no—
284C8—1362no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22#1176272311yescharacteristic
297C22#1176307611yescharacteristic
298C22#1176307611yescharacteristic
299C4—22no—
300C4—21no—
301C4—82no—
302C4—682no—
303C4—1361no—
304C4#1176299611yescharacteristic
305C4#1176299911yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—162no—
310C2#1453003211yescharacteristic
311C2#1453003711yescharacteristic
312C2#1453006811yescharacteristic
subgroupG/H
centerC22#11762723
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
Fitting, prime radicalC17×#3186C2
socleC22×C17#394
hypercenter#3186D17
hypocenterC17#10557
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #14529248C2semidirect
2. #14528966C2semidirect
3. #14527300C2semidirect
4. #14526661C2semidirect
207. C17#10557semidirect

Maximal subgroups

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

subgroupindex
#145292482
#145289662
#145273002
#145266612
C17×#31862
#1055717