Finite Group Databasebeta

← #16950310

Subgroups — #16950310

Order 4352.

544 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 95 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#14531331C211yesmaximal, characteristic
2C2×#11762603C212yesmaximal
3#14528770C212yesmaximal
4#14526909C211yesmaximal, characteristic
5C17×#1393C211yesmaximal, characteristic
6C2×#10587622C2211yescharacteristic
7C2×#10587573C411yescharacteristic, complemented
8C2×D4×Dic17C2211yescharacteristic
9C2×#10587505C2212yes—
10#11762715C411yescharacteristic, complemented
11#11762697—22no—
12#11762674C411yescharacteristic, complemented
13#11762667—22no—
14C22⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic, complemented
16#11762603—24no—
17#11762575—22no—
18C2×C17×#99C2212yes—
19C17×#390C2211yescharacteristic
20C22×D4×C17C2311yescharacteristic
21C23×Dic17D412yes—
22C22×Dic34C2×C411yescharacteristic
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C22⋊C4×C17—22no—
25C2×C22⋊C4×C17—22no—
26C2×C42×C17C2×C411yescharacteristic
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—22no—
29C2×C22⋊Dic17D412yes—
30#10587622—22no—
31#10587622D412yes—
32#10587622Q812yes—
33D4×Dic17—41no—
34D4×Dic17—41no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—42no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43#10587573—41no—
44#10587573—41no—
45C17×#99—22no—
46C17×#99—22no—
47#10587505—22no—
48#10587505—22no—
49#10587502—22no—
50#10587502—22no—
51#10587502—42no—
52C24×C17C2×D412yes—
53C2×D4×C17—41no—
54C2×D4×C17—41no—
55C2×D4×C17C2×D411yescharacteristic
56C2×D4×C17C2×Q811yescharacteristic
57C2×D4×C17#4112yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—22no—
65C22×Dic17—22no—
66C22×Dic17—44no—
67C22×Dic17—42no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C412yes—
70C22×Dic17C22⋊C411yescharacteristic
71C2×Dic34—21no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×Dic34C4⋊C412yescomplemented
75C4⋊C4×C17—41no—
76C22⋊C4×C17—44no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C42×C17—41no—
80C22⋊Dic17—24no—
81C22⋊Dic17—24no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C22⋊Dic17—42no—
85C4⋊Dic17—41no—
86#63225—42no—
87#63225—84no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90C4×Dic17—21no—
91C4×Dic17—42no—
92C4×Dic17—42no—
93C4×Dic17—41no—
94C4×Dic17—41no—
95C4×Dic17C4⋊C412yescomplemented
96#12798—171nomaximal, Sylow
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17C2×C22⋊C411yescharacteristic
102C23×C17#12112yes—
103C23×C17C22⋊Q812yes—
104D4×C17—41no—
105D4×C17—41no—
106D4×C17—42no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—82no—
115C2×C4×C17—82no—
116C2×C4×C17—82no—
117C2×C4×C17C2×C4⋊C411yescharacteristic
118C2×C4×C17#11711yescharacteristic
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—22no—
122C2×Dic17—21no—
123C2×Dic17—22no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—44no—
129C2×Dic17—42no—
130C2×Dic17—41no—
131C2×Dic17—82no—
132C2×Dic17—82no—
133C2×Dic17—82no—
134C2×Dic17—82no—
135C2×Dic17#9911yescharacteristic
136C2×Dic17#9911yescharacteristic
137Dic34—42no—
138Dic34—42no—
139C4×#124—171no—
140C2×#354—172no—
141#1799—172no—
142#1393—171no—
143#1393D1711yescharacteristic
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—22no—
147C22×C17—22no—
148C22×C17—42no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—82no—
153C22×C17#38011yescharacteristic
154C22×C17C2×#9911yescharacteristic
155C22×C17#41011yescharacteristic
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—42no—
159C4×C17—81no—
160C4×C17—162no—
161Dic17—21no—
162Dic17—42no—
163Dic17—42no—
164Dic17—42no—
165Dic17—41no—
166Dic17—162no—
167Dic17#35412yescomplemented
168C2×#124—171no—
169C2×C4×Q8—171no—
170C2×C4×D4—171no—
171C2×#99—172no—
172C2×#99C2×D1712yes—
173#391—171no—
174#390C2×D1711yescharacteristic
175#388—342no—
176#382—171no—
177#380—342no—
178C4×C22⋊C4—342no—
179C43—171no—
180#354—344no—
181#342—342no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17—42no—
185C2×C17—42no—
186C2×C17#179911yescharacteristic
187C2×C17C2×#35411yescharacteristic
188C2×C17#212611yescharacteristic
189C22×Q8—171no—
190C22×D4C22×D1711yescharacteristic
191C23×C4—172no—
192#124—172no—
193#124—172no—
194#124—342no—
195C4×Q8—681no—
196C4×Q8—681no—
197C4×D4—681no—
198C4×D4—681no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4—682no—
202C2×C4⋊C4C4×D1711yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—172no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4—342no—
208C2×C42—171no—
209C2×C42—171no—
210C2×C42—341no—
211C2×C42—341no—
212C2×C42—341no—
213C2×C42C4×D1711yescharacteristic
214#99—22no—
215#99—22no—
216#99—342no—
217#99—342no—
218#95—342no—
219#95—342no—
220#95—682no—
221C17#1279811yescharacteristic, complemented, Sylow
222C24D4×D1712yes—
223C2×Q8—172no—
224C2×Q8—341no—
225C2×Q8—681no—
226C2×Q8—681no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4#6325212yes—
230C2×D4D4×D1711yescharacteristic
231C2×D4Q8×D1711yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—171no—
236C22×C4—172no—
237C22×C4—171no—
238C22×C4—342no—
239C22×C4—342no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4—684no—
243C22×C4—682no—
244C22×C4C2×C4×D1711yescharacteristic
245C4⋊C4—41no—
246C4⋊C4—681no—
247C4⋊C4—682no—
248C4⋊C4—1364no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—344no—
253C22⋊C4—344no—
254C22⋊C4—682no—
255C22⋊C4—682no—
256C22⋊C4—682no—
257C42—41no—
258C42—172no—
259C42—341no—
260C42—341no—
261C42—341no—
262C42—682no—
263C42—681no—
264C42—681no—
265C42—682no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23C22⋊C4×D1711yescharacteristic
271C23#1058757012yes—
272C23#1058758212yes—
273Q8—682no—
274Q8—682no—
275D4—42no—
276D4—41no—
277D4—41no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—171no—
289C2×C4—171no—
290C2×C4—342no—
291C2×C4—341no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—342no—
295C2×C4—684no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—682no—
302C2×C4—1362no—
303C2×C4—1362no—
304C2×C4—1362no—
305C2×C4—1362no—
306C2×C4#1058755811yescharacteristic
307C2×C4C4⋊C4×D1711yescharacteristic
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—82no—
317C22#1176266811yescharacteristic
318C22#99×D1711yescharacteristic
319C22#1176278111yescharacteristic
320C4—21no—
321C4—21no—
322C4—42no—
323C4—81no—
324C4—162no—
325C4—2722no—
326C4—172no—
327C4—341no—
328C4—682no—
329C4—682no—
330C4—682no—
331C4—681no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2#1452877111yescharacteristic
337C2D17×#35411yescharacteristic
338C2#1453044611yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1393C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1393D17
hypocenterC17#12798
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
7. C2×#10587573C4semidirect
10. #11762715C4semidirect
12. #11762674C4semidirect
15. C42×Dic17C4semidirect
74. C2×Dic34C4⋊C4semidirect
95. C4×Dic17C4⋊C4semidirect
167. Dic17#354semidirect
221. C17#12798semidirect

Maximal subgroups

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

subgroupindex
#145313312
C2×#117626032
#145287702
#145269092
C17×#13932
#1279817