Finite Group Databasebeta

← #17961426

Subgroups — #17961426

Order 4352.

500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#11763717C212yesmaximal
2#14544650C212yesmaximal, complemented
3#14543882C212yesmaximal, complemented
4C4×#10587622C211yesmaximal, characteristic
5#11763812—22no—
6C2×#10587545C2214yes—
7#11763717—24no—
8#11763689—22no—
9C2×#10587622C2211yescharacteristic
10C2×C4×Dic34C2211yescharacteristic
11#11762714C2211yescharacteristic
12#11762712C411yescharacteristic
13#11762703—22no—
14#11762674C411yescharacteristic
15C4×C22⋊Dic17—22no—
16C42×Dic17C411yescharacteristic
17C2×C4×D4×C17C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×C4×C17—21no—
20C22×Dic34C2311yescharacteristic
21C2×#63247—24no—
22C4×D4×C17—41no—
23C4×D4×C17—41no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C22⋊C4×C17—21no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—22no—
29#10587622—22no—
30#10587622D412yes—
31#10587622Q812yes—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×#63225D412yes—
34C2×C4×Dic17—21no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D412yes—
39C4×Dic34—41no—
40C4×Dic34—41no—
41#10587545—24no—
42#10587545—24no—
43#10587544—42no—
44C4⋊#3624—42no—
45#10587502—22no—
46#10587502—22no—
47C24×C17—21no—
48C2×D4×C17—21no—
49C2×D4×C17—41no—
50C2×D4×C17—41no—
51C2×D4×C17C4⋊C412yes—
52C22×C4×C17—21no—
53C22×C4×C17—44no—
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C22×Dic17—21no—
60C22×Dic17—21no—
61C22×Dic17C22⋊C412yes—
62C22×Dic17C2×D412yes—
63C2×Dic34—41no—
64C2×Dic34—41no—
65C2×Dic34C2×D411yescharacteristic
66C2×Dic34C2×Q811yescharacteristic
67C2×Dic34#4112yes—
68#63247—44no—
69#63247—44no—
70C2×#3624—44no—
71C4⋊C4×C17—41no—
72C22⋊C4×C17—42no—
73C42×C17—21no—
74C42×C17—21no—
75C22⋊Dic17—24no—
76C22⋊Dic17—24no—
77C22⋊Dic17—44no—
78C4⋊Dic17—41no—
79#63225—42no—
80C4×Dic17—22no—
81C4×Dic17—22no—
82C4×Dic17—21no—
83C4×Dic17—42no—
84C4×Dic17—42no—
85C4×Dic17—41no—
86C4×Dic17C4⋊C412yes—
87#12801—171nomaximal, Sylow
88C23×C17—22no—
89C23×C17—42no—
90C23×C17—42no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17C2×C22⋊C411yescharacteristic
94D4×C17—42no—
95D4×C17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—22no—
99C2×C4×C17—21no—
100C2×C4×C17—41no—
101C2×C4×C17—42no—
102C2×C4×C17—41no—
103C2×C4×C17—82no—
104C2×C4×C17—82no—
105C2×C4×C17#9911yescharacteristic
106C2×C4×C17#9911yescharacteristic
107C2×C4×C17C2×C4⋊C411yescharacteristic
108C2×C4×C17#11711yescharacteristic
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—42no—
119C2×Dic17#12112yes—
120C2×Dic17C22⋊Q812yes—
121Dic34—41no—
122Dic34—41no—
123Dic34—42no—
124#3624—84no—
125C4×#124—171no—
126C2×#355—172no—
127#1801—172no—
128#1394—172no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—44no—
132C22×C17—42no—
133C22×C17—82no—
134C22×C17—81no—
135C22×C17—81no—
136C22×C17#38011yescharacteristic
137C22×C17C2×#9911yescharacteristic
138C22×C17#41011yescharacteristic
139C4×C17—21no—
140C4×C17—21no—
141C4×C17—21no—
142C4×C17—41no—
143C4×C17—81no—
144C4×C17#35512yescomplemented
145Dic17—22no—
146Dic17—22no—
147Dic17—42no—
148Dic17—42no—
149Dic17—42no—
150C2×#124—171no—
151C2×C4×Q8—171no—
152C2×C4×D4#59511yescharacteristic
153#423—342no—
154C2×#101—174no—
155#391—171no—
156#390—171no—
157#388—342no—
158#382—171no—
159C4×C22⋊C4—342no—
160C43—171no—
161#355—344no—
162#343—342no—
163C2×C17—21no—
164C2×C17—21no—
165C2×C17—82no—
166C2×C17#179911yescharacteristic
167C2×C17C2×#35511yescharacteristic
168C2×C17#213211yescharacteristic
169C22×Q8—171no—
170C22×D4C2×#59511yescharacteristic
171C23×C4—21no—
172C2×M16—344no—
173#124—172no—
174#124—172no—
175#124—342no—
176C4×Q8—681no—
177C4×Q8—681no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—172no—
181C2×C4⋊C4—171no—
182C2×C4⋊C4C2×#59511yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—342no—
185C2×C22⋊C4—342no—
186C2×C42—171no—
187C2×C42—172no—
188C2×C42—171no—
189C2×C42—341no—
190C2×C42—341no—
191C2×C42C2×#59511yescharacteristic
192C4⋊C8—682no—
193C4⋊C8—682no—
194#101—344no—
195#101—344no—
196#95—342no—
197#95—342no—
198C17#1280111yescharacteristic, complemented, Sylow
199C24—21no—
200C2×Q8—171no—
201C2×Q8—171no—
202C2×Q8—172no—
203C2×Q8—681no—
204C2×Q8—681no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊#59512yes—
209C22×C4—21no—
210C22×C4—44no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—172no—
214C22×C4—172no—
215C22×C4—341no—
216C22×C4—341no—
217C22×C4C22⋊#59511yescharacteristic
218C22×C4C22⋊#59511yescharacteristic
219C22×C4C22×#59511yescharacteristic
220M16—684no—
221M16—684no—
222C2×C8—684no—
223C4⋊C4—41no—
224C4⋊C4—682no—
225C4⋊C4—681no—
226C22⋊C4—42no—
227C22⋊C4—344no—
228C22⋊C4—344no—
229C22⋊C4—684no—
230C42—21no—
231C42—21no—
232C42—172no—
233C42—341no—
234C42—342no—
235C42—342no—
236C42—681no—
237C42—682no—
238C42—682no—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C2×C22⋊#59511yescharacteristic
245Q8—681no—
246Q8—681no—
247Q8—682no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—172no—
260C2×C4—172no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—682no—
271C2×C4#1058755111yescharacteristic
272C2×C4#1058755111yescharacteristic
273C2×C4C2×C4⋊#59511yescharacteristic
274C2×C4#1058767711yescharacteristic
275C8—1364no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22#1176375111yescharacteristic
284C22C2×#1058755111yescharacteristic
285C22#1176378811yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—81no—
291C4—342no—
292C4—342no—
293C4—682no—
294C4—682no—
295C4—682no—
296C4#1176371712yescomplemented
297C2—21no—
298C2—21no—
299C2—82no—
300C2#1454464411yescharacteristic
301C2C2×#1176371711yescharacteristic
302C2#1454507511yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×D4×C17C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC22×Dic34C23
hypercenterC2×C4×D4#595
hypocenterC17#12801
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #14544650C2semidirect
3. #14543882C2semidirect
144. C4×C17#355semidirect
198. C17#12801semidirect
296. C4#11763717semidirect

Maximal subgroups

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

subgroupindex
C2×#117637172
#145446502
#145438822
C4×#105876222
#1280117