Finite Group Databasebeta

← #16952984

Subgroups — #16952984

Order 4352.

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

normal subgroups only

This group has 616 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×#10587505C211yesmaximal, characteristic
3#14526903C214yesmaximal
4C17×#1800C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic, complemented
6C2×#10587610—21no—
7C2×#10587505C2212yes—
8#11762729—21no—
9#11762721—21no—
10#11762715—21no—
11#11762689C2212yes—
12#11762681C412yescomplemented
13#11762669C411yescharacteristic, complemented
14C4×C22⋊Dic17—21no—
15#11762576—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#423—21no—
18C2×C17×#100C2212yes—
19C17×#343—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—
29C2×M16×C17—22no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C22⋊Dic17—22no—
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17C2×C412yes—
39C2×C4⋊Dic17—21no—
40#10587610—44no—
41C2×#63225—22no—
42C2×C4×Dic17—41no—
43C17×C4⋊C8—41no—
44C17×C4⋊C8—41no—
45C17×#100—24no—
46#10587505—22no—
47#10587505Q814yescomplemented
48#10587502—22no—
49#10587502—21no—
50#10587502—22no—
51#10587502—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×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—44no—
72C22×Dic17—42no—
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—
83M16×C17—42no—
84M16×C17—42no—
85C2×C8×C17—42no—
86C4⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C42×C17—21no—
90C42×C17—21no—
91C22⋊Dic17—24no—
92C22⋊Dic17—44no—
93C22⋊Dic17—42no—
94C22⋊Dic17—42no—
95C4⋊Dic17—42no—
96#63225—44no—
97C4×Dic17—82no—
98#13296—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×C17C22⋊Q812yes—
110D4×C17—42no—
111D4×C17—42no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—81no—
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17#12411yescharacteristic
128C2×C4×C17C4⋊Q811yescharacteristic
129C2×Dic17—24no—
130C2×Dic17—42no—
131C2×Dic17—44no—
132C2×Dic17—42no—
133C2×Dic17—44no—
134C2×Dic17—44no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—82no—
139C2×Dic17—82no—
140Dic34—44no—
141Dic34—48no—
142Dic34—44no—
143C8×C17—82no—
144#2409—171no—
145#1800D1711yescharacteristic
146C4×#99—171no—
147#1390—174no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—81no—
160C22×C17#39111yescharacteristic
161C22×C17C2×#9911yescharacteristic
162C22×C17#41011yescharacteristic
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—41no—
169C4×C17AΣL(1,𝔽16)12yes—
170Dic17—44no—
171Dic17—82no—
172Dic17—84no—
173C23×Q8—171no—
174C2×C22⋊Q8—341no—
175C2×C4×D4C2×D1711yescharacteristic
176#423—21no—
177C2×#100C2×D1712yes—
178C2×#99—172no—
179#395—341no—
180#393—341no—
181#391—341no—
182#386—172no—
183#385—172no—
184#380—171no—
185C4×C22⋊C4—341no—
186#343—21no—
187#342—341no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17—42no—
191C2×C17—42no—
192C2×C17#199611yescharacteristic
193C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
194C2×C17#212311yescharacteristic
195C22×Q8—341no—
196C22×Q8—341no—
197C22×Q8—681no—
198C22×Q8—681no—
199C22×Q8—684no—
200C22×D4C22×D1711yescharacteristic
201C23×C4—172no—
202C23×C4C4×D1711yescharacteristic
203C23×C4D6811yescharacteristic
204C2×M16—22no—
205C22⋊Q8—684no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—342no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4C22⋊D1711yescharacteristic
211C2×C22⋊C4—172no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4C4×D1711yescharacteristic
215C2×C22⋊C4D6811yescharacteristic
216C2×C42—681no—
217C2×C42C22⋊D1711yescharacteristic
218C4⋊C8—41no—
219C4⋊C8—41no—
220#100—24no—
221#99—174no—
222#99—342no—
223#95—342no—
224#95—341no—
225#95—342no—
226#95—681no—
227C17#1329611yescharacteristic, complemented, Sylow
228C24C2×C4×D1711yescharacteristic
229C24C2×D6811yescharacteristic
230C2×Q8—344no—
231C2×Q8—682no—
232C2×Q8—682no—
233C2×Q8—684no—
234C2×Q8—684no—
235C2×Q8—688no—
236C2×Q8—684no—
237C2×Q8—681no—
238C2×Q8—681no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4#6325412yes—
242C2×D4Q8×D1712yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—684no—
252C22×C4—682no—
253C22×C4—682no—
254C22×C4#6322711yescharacteristic
255C22×C4#6322711yescharacteristic
256C22×C4#6322711yescharacteristic
257C22×C4#6322711yescharacteristic
258C22×C4C2×C22⋊D1711yescharacteristic
259M16—42no—
260M16—42no—
261C2×C8—42no—
262C4⋊C4—41no—
263C4⋊C4—682no—
264C4⋊C4—684no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—344no—
268C22⋊C4—684no—
269C22⋊C4—682no—
270C22⋊C4—682no—
271C42—21no—
272C42—21no—
273C42—1362no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#1058757312yes—
283C23#1058758312yes—
284C23C2×#6322711yescharacteristic
285Q8—684no—
286Q8—688no—
287Q8—684no—
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—344no—
304C2×C4—682no—
305C2×C4—684no—
306C2×C4—682no—
307C2×C4—684no—
308C2×C4—684no—
309C2×C4—682no—
310C2×C4—682no—
311C2×C4—682no—
312C2×C4—1362no—
313C2×C4—1362no—
314C2×C4#1058747711yescharacteristic
315C2×C4#1058747711yescharacteristic
316C2×C4#1058762211yescharacteristic
317C2×C4#1058763011yescharacteristic
318C8—82no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1176271411yescharacteristic
332C22C2×#1058747711yescharacteristic
333C22#1176278211yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—684no—
341C4—1362no—
342C4—1364no—
343C4#1176259712yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2#1452976811yescharacteristic
349C2C2×#1176259711yescharacteristic
350C2#1453043911yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1800C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1800D17
hypocenterC17#13296
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
5. C23×Dic34C4semidirect
12. #11762681C4semidirect
13. #11762669C4semidirect
47. #10587505Q8semidirect
227. C17#13296semidirect

Maximal subgroups

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

subgroupindex
#145321492
C4×#105875052
#145269032
C17×#18002
#1329617