Finite Group Databasebeta

← #16950321

Subgroups — #16950321

Order 4352.

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

normal subgroups only

This group has 500 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#124×Dic17C211yesmaximal, characteristic
2C2×#11762606C211yesmaximal, characteristic
3#14528778C212yesmaximal, complemented
4#14526910C212yesmaximal, complemented
5C2×C17×#355C211yesmaximal, characteristic
6C2×Q8×Dic17C2211yescharacteristic
7C2×D4×Dic17C411yescharacteristic
8#11762832—22no—
9C2×#10587507C2212yes—
10#11762716C2211yescharacteristic
11#11762712C411yescharacteristic
12#11762702—22no—
13#11762675C411yescharacteristic
14C22⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic
16#11762606—22no—
17#11762577—22no—
18C2×C17×#124C2211yescharacteristic
19C2×C17×#101C2212yes—
20C17×#355—22no—
21C22×Q8×C17C2311yescharacteristic
22C22×D4×C17C2×C411yescharacteristic
23C23×Dic17—21no—
24C2×M16×C17—22no—
25C17×#124—22no—
26C17×#124D412yes—
27C17×#124Q812yes—
28C2×C22⋊C4×C17—22no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—22no—
32Q8×Dic17—41no—
33Q8×Dic17—41no—
34D4×Dic17—41no—
35D4×Dic17—41no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17D412yes—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43C2×C4×Dic17D412yes—
44C2×#63223—22no—
45C17×#101—22no—
46C17×#101—22no—
47#10587507—22no—
48#10587507—22no—
49#10587502—22no—
50#10587502—22no—
51#10587485—42no—
52#10587485—42no—
53C24×C17—21no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17C2×D411yescharacteristic
57C2×Q8×C17C2×Q811yescharacteristic
58C2×Q8×C17#4112yes—
59C2×D4×C17—21no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17C4⋊C412yes—
63C22×C4×C17—21no—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D412yes—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—44no—
69C22×Dic17—41no—
70C22×Dic17—41no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C412yes—
73C22×Dic17C22⋊C411yescharacteristic
74M16×C17—42no—
75M16×C17—42no—
76C2×C8×C17—42no—
77C22⋊C4×C17—24no—
78C22⋊C4×C17—24no—
79C42×C17—21no—
80C42×C17C4⋊C412yes—
81C22⋊Dic17—42no—
82C22⋊Dic17—44no—
83C4⋊Dic17—41no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4×Dic17—22no—
87C4×Dic17—22no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90C4×Dic17—42no—
91C4×Dic17—42no—
92C4×Dic17—41no—
93#63223—42no—
94#63223—42no—
95C2×#3622—42no—
96#12801—171nomaximal, Sylow
97C23×C17—22no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17C2×C22⋊C411yescharacteristic
103Q8×C17—41no—
104Q8×C17—41no—
105Q8×C17—42no—
106D4×C17—42no—
107D4×C17—42no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17C2×C4⋊C411yescharacteristic
116C2×C4×C17#11711yescharacteristic
117C2×C4×C17#12112yes—
118C2×C4×C17C22⋊Q812yes—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—21no—
122C2×Dic17—22no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—42no—
126C2×Dic17—41no—
127C2×Dic17—42no—
128C2×Dic17—41no—
129C2×Dic17—82no—
130C2×Dic17—82no—
131C2×Dic17#9911yescharacteristic
132C2×Dic17#9911yescharacteristic
133C8×C17—82no—
134#3622—82no—
135C4×#124—171no—
136C2×#355—171no—
137C2×#355D1711yescharacteristic
138#1801—172no—
139#1394—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—44no—
143C22×C17—42no—
144C22×C17—82no—
145C22×C17—81no—
146C22×C17—81no—
147C22×C17#38011yescharacteristic
148C22×C17C2×#9911yescharacteristic
149C22×C17#41011yescharacteristic
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—42no—
155Dic17—21no—
156Dic17—42no—
157Dic17—41no—
158Dic17—42no—
159Dic17—81no—
160Dic17#35512yescomplemented
161C2×#124C2×D1711yescharacteristic
162C2×C4×Q8—171no—
163C2×C4×D4—171no—
164#423—342no—
165C2×#101—172no—
166C2×#101C2×D1712yes—
167#391—171no—
168#390—171no—
169#388—342no—
170#382—171no—
171C4×C22⋊C4—342no—
172C43—171no—
173#355—22no—
174#355—342no—
175#343—342no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—82no—
179C2×C17#179911yescharacteristic
180C2×C17C2×#35511yescharacteristic
181C2×C17#213211yescharacteristic
182C22×Q8C22×D1711yescharacteristic
183C22×D4C4×D1711yescharacteristic
184C23×C4—341no—
185C2×M16—22no—
186C2×M16—342no—
187#124—22no—
188#124Dic3412yes—
189#124C22⋊D1712yes—
190C4×Q8—681no—
191C4×Q8—681no—
192C4×D4—681no—
193C4×D4—681no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—172no—
196C2×C4⋊C4—171no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—341no—
199C2×C22⋊C4—342no—
200C2×C42—171no—
201C2×C42—172no—
202C2×C42—171no—
203C2×C42—341no—
204C2×C42—341no—
205C2×C42C4×D1711yescharacteristic
206C4⋊C8—682no—
207C4⋊C8—682no—
208#101—22no—
209#101—22no—
210#101—342no—
211#101—342no—
212#95—342no—
213#95—342no—
214C17#1280111yescharacteristic, complemented, Sylow
215C24—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8C2×Dic3411yescharacteristic
219C2×Q8#6325412yes—
220C2×Q8C2×C22⋊D1711yescharacteristic
221C2×D4—21no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#6322512yes—
225C22×C4—21no—
226C22×C4—171no—
227C22×C4—172no—
228C22×C4—171no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—681no—
232C22×C4—681no—
233C22×C4—684no—
234C22×C4C2×C4×D1711yescharacteristic
235C22×C4D4×D1712yes—
236M16—42no—
237M16—42no—
238M16—682no—
239M16—682no—
240C2×C8—42no—
241C2×C8—682no—
242C4⋊C4—681no—
243C4⋊C4—682no—
244C4⋊C4—681no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—684no—
248C22⋊C4—682no—
249C42—21no—
250C42—342no—
251C42—342no—
252C42—341no—
253C42—341no—
254C42—681no—
255C42—682no—
256C42—682no—
257C42#6322512yes—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23C22⋊C4×D1711yescharacteristic
264Q8—41no—
265Q8—42no—
266Q8—41no—
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—171no—
277C2×C4—171no—
278C2×C4—342no—
279C2×C4—342no—
280C2×C4—341no—
281C2×C4—342no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—682no—
286C2×C4—682no—
287C2×C4—681no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4#1058756311yescharacteristic
291C2×C4#1058756412yes—
292C2×C4C2×#6322511yescharacteristic
293C2×C4#1058762512yes—
294C8—82no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22#1176266711yescharacteristic
304C22#99×D1711yescharacteristic
305C22#1176278111yescharacteristic
306C4—22no—
307C4—21no—
308C4—22no—
309C4—21no—
310C4—42no—
311C4—172no—
312C4—341no—
313C4—682no—
314C4—681no—
315C4—682no—
316C4—1361no—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#1452877011yescharacteristic
321C2D17×#35511yescharacteristic
322C2#1453046611yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC2×C17×#355C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC22×Q8×C17C23
hypercenterC2×#355D17
hypocenterC17#12801
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #14528778C2semidirect
4. #14526910C2semidirect
160. Dic17#355semidirect
214. C17#12801semidirect

Maximal subgroups

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

subgroupindex
#124×Dic172
C2×#117626062
#145287782
#145269102
C2×C17×#3552
#1280117