Finite Group Databasebeta

← #16952649

Subgroups — #16952649

Order 4352.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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#14533769C211yesmaximal, characteristic
2C2×#11762607C214yesmaximal
3C4×#10587507C211yesmaximal, characteristic
4C17×#1801C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6C2×#10587630C2212yes—
7C2×#10587507C2212yes—
8C4×#63223—21no—
9#11762739—21no—
10#11762739—21no—
11#11762714C412yes—
12#11762707—21no—
13#11762707—21no—
14C4×C4⋊Dic17C411yescharacteristic
15#11762607—28no—
16#11762578—21no—
17C2×C4×Q8×C17C2211yescharacteristic
18C17×#423—21no—
19C2×C17×#101C2212yes—
20C17×#343—21no—
21C22×Q8×C17C2311yescharacteristic
22C22×Dic34C2×C412yes—
23C2×M16×C17—22no—
24C4×Q8×C17—41no—
25C4×Q8×C17—41no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C42×C17D411yescharacteristic
32#10587630—24no—
33#10587630—24no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×#63225—22no—
38C2×#63225—22no—
39C2×C4×Dic17C2×C412yes—
40C2×#63223—22no—
41#10587555—41no—
42#10587555—41no—
43C17×C4⋊C8—41no—
44C17×C4⋊C8—41no—
45C17×#101—24no—
46#10587507—22no—
47#10587507D414yes—
48#10587502—22no—
49#10587502—22no—
50#10587502—21no—
51#10587474—41no—
52C4×#3622—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17—41no—
55C2×Q8×C17C2×D412yes—
56C2×Q8×C17#4112yes—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—22no—
65C22×Dic17—22no—
66C2×Dic34—24no—
67C2×Dic34—42no—
68C2×Dic34—42no—
69M16×C17—42no—
70M16×C17—42no—
71C2×C8×C17—42no—
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C42×C17—21no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—21no—
81C4⋊Dic17—24no—
82C4⋊Dic17—42no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85#63225—44no—
86#63225—42no—
87#63225—42no—
88C4×Dic17—24no—
89#63223—24no—
90#63223—42no—
91C2×#3622—42no—
92#13227—171nomaximal, Sylow
93C23×C17C2×C22⋊C411yescharacteristic
94Q8×C17—42no—
95Q8×C17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17#9911yescharacteristic
110C2×C4×C17#9911yescharacteristic
111C2×C4×C17C4×D412yes—
112C2×C4×C17#12112yes—
113C2×C4×C17#12411yescharacteristic
114C2×C4×C17#12711yescharacteristic
115C2×Dic17—24no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121Dic34—84no—
122C8×C17—82no—
123#3622—44no—
124#2546—171no—
125C2×#356—174no—
126#1801D1711yescharacteristic
127C4×#101—171no—
128C22×C17—21no—
129C22×C17—21no—
130C22×C17#39011yescharacteristic
131C22×C17C2×#9911yescharacteristic
132C22×C17#41011yescharacteristic
133C4×C17—21no—
134C4×C17—22no—
135C4×C17—21no—
136C4×C17—22no—
137C4×C17—21no—
138C4×C17—41no—
139C4×C17—41no—
140C4×C17—41no—
141C4×C17#35612yescomplemented
142Dic17—44no—
143Dic17—82no—
144C2×C4⋊Q8—172no—
145C2×C4⋊Q8—171no—
146C2×C4×Q8C2×D1711yescharacteristic
147#423—21no—
148C2×#101—172no—
149C2×#101C2×D1712yes—
150C4×M16—341no—
151#398—341no—
152#398—341no—
153#391—172no—
154#389—341no—
155#389—341no—
156C4×C4⋊C4—171no—
157#356—348no—
158#343—21no—
159#343—341no—
160C2×C17—21no—
161C2×C17—21no—
162C2×C17#196411yescharacteristic
163C2×C17C2×#35611yescharacteristic
164C2×C17#213511yescharacteristic
165C22×Q8—172no—
166C22×Q8C22×D1711yescharacteristic
167C2×M16—22no—
168C2×M16—342no—
169C4⋊Q8—344no—
170C4⋊Q8—344no—
171C4⋊Q8—681no—
172C4⋊Q8—681no—
173C4×Q8—41no—
174C4×Q8—41no—
175C2×C4⋊C4—341no—
176C2×C4⋊C4—342no—
177C2×C4⋊C4—342no—
178C2×C4⋊C4—341no—
179C2×C4⋊C4—341no—
180C2×C4⋊C4C4×D1711yescharacteristic
181C2×C4⋊C4D6811yescharacteristic
182C2×C4⋊C4C22⋊D1711yescharacteristic
183C2×C42—172no—
184C2×C42C4×D1711yescharacteristic
185C2×C42D6811yescharacteristic
186C2×C42C22⋊D1711yescharacteristic
187C4⋊C8—41no—
188C4⋊C8—41no—
189#101—24no—
190#101—174no—
191#101—342no—
192#97—681no—
193C4×C8—681no—
194#95—341no—
195#95—342no—
196#95—342no—
197C17#1322711yescharacteristic, complemented, Sylow
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—344no—
201C2×Q8—682no—
202C2×Q8—682no—
203C2×Q8D4×D1712yes—
204C2×Q8#6325612yes—
205C22×C4—342no—
206C22×C4—342no—
207C22×C4#6322711yescharacteristic
208C22×C4#6322711yescharacteristic
209C22×C4#6322711yescharacteristic
210C22×C4#6322711yescharacteristic
211C22×C4C2×C4×D1711yescharacteristic
212C22×C4C2×D6811yescharacteristic
213C22×C4C2×C22⋊D1711yescharacteristic
214M16—42no—
215M16—42no—
216M16—344no—
217M16—682no—
218C2×C8—42no—
219C2×C8—682no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—344no—
224C4⋊C4—682no—
225C4⋊C4—681no—
226C4⋊C4—681no—
227C4⋊C4—684no—
228C4⋊C4—682no—
229C4⋊C4—682no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—344no—
237C23C2×#6322711yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—1364no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—344no—
255C2×C4—682no—
256C2×C4—682no—
257C2×C4—682no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4#1058747711yescharacteristic
261C2×C4#1058747711yescharacteristic
262C2×C4#1058757912yes—
263C2×C4#1058758112yes—
264C2×C4#1058762611yescharacteristic
265C2×C4#1058763311yescharacteristic
266C8—82no—
267C8—684no—
268C22—21no—
269C22—21no—
270C22#1176271011yescharacteristic
271C22C2×#1058747711yescharacteristic
272C22#1176278211yescharacteristic
273C4—21no—
274C4—22no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—684no—
282C4—1362no—
283C4#1176260712yescomplemented
284C2—21no—
285C2—21no—
286C2#1452961911yescharacteristic
287C2C2×#1176260711yescharacteristic
288C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1801C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#1801D17
hypocenterC17#13227
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
141. C4×C17#356semidirect
197. C17#13227semidirect
283. C4#11762607semidirect

Maximal subgroups

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

subgroupindex
#145337692
C2×#117626072
C4×#105875072
C17×#18012
#1322717