Finite Group Databasebeta

← #16952971

Subgroups — #16952971

Order 4352.

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

normal subgroups only

This group has 612 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#14532149C211yesmaximal, characteristic
2C4×#10587504C211yesmaximal, characteristic
3#14526901C214yesmaximal
4C17×#1799C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic, complemented
6C2×#10587610—21no—
7C2×#10587504C2212yes—
8C4×#63223—21no—
9#11762729—21no—
10#11762721—21no—
11#11762715—21no—
12#11762689C2212yes—
13#11762681C412yescomplemented
14#11762669C411yescharacteristic, complemented
15#11762578—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×#99C2212yes—
18C17×#380—21no—
19C17×#342—21no—
20C22×D4×C17C2311yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17C2×C412yes—
24C22×Dic34—21no—
25C22×Dic34—21no—
26C22×Dic34—44no—
27C22×Dic34—41no—
28C22×Dic34—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17—22no—
34C2×C22⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17C2×C412yes—
39C2×C4⋊Dic17—21no—
40#10587610—44no—
41C2×#63225—22no—
42C2×#63223—22no—
43C17×#99—24no—
44C17×#95—41no—
45#10587504—22no—
46#10587504Q814yescomplemented
47#10587502—22no—
48#10587502—21no—
49#10587502—22no—
50#10587474—41no—
51C4×#3622—41no—
52C24×C17C22×C411yescharacteristic
53C24×C17C2×D411yescharacteristic
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17C2×Q812yes—
57C2×D4×C17#4112yes—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—22no—
71C22×Dic17—22no—
72C22×Dic17—44no—
73C22×Dic17—42no—
74C2×Dic34—24no—
75C2×Dic34—44no—
76C2×Dic34—42no—
77C2×Dic34—42no—
78C2×Dic34—48no—
79C2×Dic34—44no—
80C2×Dic34—41no—
81C2×Dic34—41no—
82C2×Dic34—44no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—82no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C42×C17—21no—
90C42×C17—21no—
91C22⋊Dic17—44no—
92C22⋊Dic17—42no—
93C4⋊Dic17—42no—
94#63225—44no—
95#63223—24no—
96#63223—42no—
97C2×#3622—42no—
98#13293—171nomaximal, Sylow
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17C2×C22⋊C411yescharacteristic
108C23×C17C4×Q812yes—
109C23×C17#12411yescharacteristic
110C23×C17C4⋊Q811yescharacteristic
111D4×C17—42no—
112D4×C17—42no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—22no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—81no—
126C2×C4×C17—82no—
127C2×C4×C17—82no—
128C2×C4×C17—82no—
129C2×C4×C17#9911yescharacteristic
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17C22⋊Q812yes—
132C2×Dic17—24no—
133C2×Dic17—42no—
134C2×Dic17—44no—
135C2×Dic17—42no—
136C2×Dic17—44no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140Dic34—44no—
141Dic34—48no—
142Dic34—44no—
143#3622—44no—
144#2409—171no—
145#1799D1711yescharacteristic
146C4×#100—171no—
147#1390—174no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—22no—
152C22×C17—21no—
153C22×C17—42no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—81no—
162C22×C17#39111yescharacteristic
163C22×C17C2×#9911yescharacteristic
164C22×C17#41011yescharacteristic
165C4×C17—21no—
166C4×C17—22no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—162no—
171C4×C17AΣL(1,𝔽16)12yes—
172Dic17—44no—
173Dic17—82no—
174C23×Q8—171no—
175C2×C22⋊Q8—341no—
176C2×C4×D4C2×D1711yescharacteristic
177C2×#100—172no—
178C2×#99C2×D1712yes—
179C4×M16—341no—
180#395—341no—
181#393—341no—
182#391—341no—
183#386—172no—
184#385—172no—
185#380—21no—
186#380—171no—
187#343—341no—
188#342—21no—
189C2×C17—21no—
190C2×C17—21no—
191C2×C17—41no—
192C2×C17—42no—
193C2×C17—41no—
194C2×C17#199611yescharacteristic
195C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
196C2×C17#212311yescharacteristic
197C22×Q8—341no—
198C22×Q8—341no—
199C22×Q8—681no—
200C22×Q8—681no—
201C22×Q8—684no—
202C22×D4C22×D1711yescharacteristic
203C23×C4—172no—
204C23×C4C4×D1711yescharacteristic
205C23×C4C22⋊D1711yescharacteristic
206C2×M16—342no—
207C22⋊Q8—684no—
208C4×D4—41no—
209C4×D4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—342no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4D6811yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—172no—
216C2×C22⋊C4—342no—
217C2×C22⋊C4C4×D1711yescharacteristic
218C2×C22⋊C4C22⋊D1711yescharacteristic
219C2×C42D6811yescharacteristic
220#100—174no—
221#100—342no—
222#99—24no—
223#97—681no—
224C4×C8—681no—
225#95—41no—
226#95—342no—
227#95—341no—
228#95—342no—
229C17#1329311yescharacteristic, complemented, Sylow
230C24C2×C4×D1711yescharacteristic
231C24C2×C22⋊D1711yescharacteristic
232C2×Q8—344no—
233C2×Q8—682no—
234C2×Q8—682no—
235C2×Q8—684no—
236C2×Q8—684no—
237C2×Q8—688no—
238C2×Q8—684no—
239C2×Q8—681no—
240C2×Q8—681no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#6325412yes—
244C2×D4Q8×D1712yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—342no—
253C22×C4—342no—
254C22×C4—684no—
255C22×C4—682no—
256C22×C4#6322711yescharacteristic
257C22×C4#6322711yescharacteristic
258C22×C4#6322711yescharacteristic
259C22×C4#6322711yescharacteristic
260C22×C4C2×D6811yescharacteristic
261M16—344no—
262M16—682no—
263C2×C8—682no—
264C4⋊C4—41no—
265C4⋊C4—82no—
266C4⋊C4—682no—
267C4⋊C4—684no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—684no—
273C22⋊C4—682no—
274C42—21no—
275C42—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23#1058757312yes—
285C23C2×#6322711yescharacteristic
286C23#1058762211yescharacteristic
287C23#1058763011yescharacteristic
288Q8—684no—
289Q8—688no—
290Q8—684no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—344no—
310C2×C4—682no—
311C2×C4—684no—
312C2×C4—682no—
313C2×C4—684no—
314C2×C4—682no—
315C2×C4—682no—
316C2×C4—682no—
317C2×C4#1058747711yescharacteristic
318C2×C4#1058747711yescharacteristic
319C2×C4#1058758312yes—
320C8—684no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#1176271411yescharacteristic
336C22C2×#1058747711yescharacteristic
337C22#1176278211yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—162no—
344C4—684no—
345C4—1362no—
346C4#1176259512yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—42no—
351C2—41no—
352C2#1452976811yescharacteristic
353C2C2×#1176259511yescharacteristic
354C2#1453043711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1799D17
hypocenterC17#13293
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C23×Dic34C4semidirect
13. #11762681C4semidirect
14. #11762669C4semidirect
46. #10587504Q8semidirect
229. C17#13293semidirect

Maximal subgroups

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

subgroupindex
#145321492
C4×#105875042
#145269012
C17×#17992
#1329317