Finite Group Databasebeta

← #16938086

Subgroups — #16938086

Order 4352.

398 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 93 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#14538581C211yesmaximal, characteristic
2#14527554C211yesmaximal, characteristic, complemented
3#14527442C211yesmaximal, characteristic, complemented
4#14527347C211yesmaximal, characteristic, complemented
5#14527105C211yesmaximal, characteristic
6#14526642C211yesmaximal, characteristic, complemented
7C17×#1804C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763109—22no—
10#11763099—21no—
11#11763089—21no—
12#11762862C2211yescharacteristic
13#11762862C2211yescharacteristic
14#11762849—21no—
15#11762833C2211yescharacteristic
16#11762814—21no—
17#11762559—22no—
18#11762557—21no—
19#11762554—21no—
20#11762545—22no—
21#11762533—22no—
22C2×C4×D4×C17C2211yescharacteristic
23C2×C17×C4⋊C8C2211yescharacteristic
24C2×C17×#102—21no—
25C17×#382C2211yescharacteristic
26C17×#340—21no—
27C22×D4×C17D411yescharacteristic
28C23×C4×C17—21no—
29C22×C8×C17—21no—
30C4×D4×C17—21no—
31C4×D4×C17—21no—
32C4×D4×C17—21no—
33C4×D4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C42×C17C2311yescharacteristic
38#10587612—21no—
39#10587610—21no—
40#10587610—21no—
41C2×#63223—21no—
42#10587556—21no—
43#10587555D411yescharacteristic
44#10587555D411yescharacteristic
45C4×Dic34—21no—
46C4×Dic34—21no—
47C17×C4⋊C8—21no—
48C17×C4⋊C8—21no—
49C17×#102—42no—
50C17×#95—21no—
51C17×#95—21no—
52#10587503—22no—
53#10587503—22no—
54C22⋊#3622—21no—
55C22⋊#3622D412yes—
56#10587479—22no—
57#10587478—22no—
58C4⋊#3622—21no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622D412yes—
62C4×#3622—21no—
63C4×#3622—21no—
64C24×C17—21no—
65C2×D4×C17—21no—
66C2×D4×C17—21no—
67C2×D4×C17—21no—
68C2×D4×C17—21no—
69C2×D4×C17—21no—
70C2×D4×C17SD1612yes—
71C22×C4×C17—21no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17—82no—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17#4111yescharacteristic
79C2×Dic34—21no—
80C2×Dic34—21no—
81C2×C8×C17—41no—
82C2×C8×C17—41no—
83C2×C8×C17—41no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C42×C17C2×D411yescharacteristic
91C42×C17C2×D411yescharacteristic
92C42×C17#4111yescharacteristic
93C42×C17#4111yescharacteristic
94C22⋊Dic17—41no—
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17—21no—
99#63225—41no—
100#63225—41no—
101C4×Dic17—41no—
102#63223—41no—
103#63223—41no—
104C2×#3622—22no—
105C2×#3622—22no—
106C2×#3622—41no—
107#10468—171nomaximal, Sylow
108C23×C17—22no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—81no—
112C23×C17—81no—
113C23×C17#12111yescharacteristic
114D4×C17—22no—
115D4×C17—22no—
116D4×C17—41no—
117D4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—82no—
122C2×C4×C17—82no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17—81no—
127C2×C4×C17#12011yescharacteristic
128C2×C4×C17#12111yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12311yescharacteristic
131C2×C4×C17#12311yescharacteristic
132C2×C4×C17#12411yescharacteristic
133C2×C4×C17C2×SD1611yescharacteristic
134C2×C4×C17#13511yescharacteristic
135C2×C4×C17#13711yescharacteristic
136C2×C4×C17#13711yescharacteristic
137C2×Dic17—41no—
138C2×Dic17—41no—
139Dic34—81no—
140Dic34—81no—
141C8×C17—81no—
142#3622—42no—
143#3622—42no—
144#3622—81no—
145#3077—171no—
146#1804D1711yescharacteristic
147#1570—171no—
148#1542—171no—
149#1518—171no—
150#1460—171no—
151#1286—171no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—42no—
155C22×C17—82no—
156C22×C17—82no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17#39411yescharacteristic
160C22×C17#46511yescharacteristic
161C22×C17#47011yescharacteristic
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—81no—
167C4×C17—161no—
168C4×C17#44811yescharacteristic
169C4×C17#45211yescharacteristic
170C4×C17#48311yescharacteristic
171C4×C17#48511yescharacteristic
172Dic17—81no—
173Dic17—81no—
174#533—171no—
175C2×C4×D4C2×D1711yescharacteristic
176#478—342no—
177#476—341no—
178#474—341no—
179#431—171no—
180#431—171no—
181#428—341no—
182#423—171no—
183C2×C4⋊C8C2×D1711yescharacteristic
184#418—341no—
185C2×#102—21no—
186#382C2×D1711yescharacteristic
187#340—21no—
188#334—342no—
189#333—341no—
190#332—341no—
191#328—342no—
192#323—342no—
193C2×C17—21no—
194C2×C17—21no—
195C2×C17—82no—
196C2×C17#203511yescharacteristic
197C2×C17#204811yescharacteristic
198C2×C17#205311yescharacteristic
199C22×D4C22⋊D1711yescharacteristic
200C23×C4—21no—
201C2×M16—341no—
202C22×C8—21no—
203C4⋊Q8—171no—
204C4⋊Q8—171no—
205#125—341no—
206C22⋊Q8—341no—
207C22⋊Q8—341no—
208C4×Q8—341no—
209C4×Q8—341no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214#117—341no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C22⋊D1711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C42C22×D1711yescharacteristic
219#107—342no—
220#106—342no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—172no—
224C4⋊C8—341no—
225C4⋊C8—341no—
226C4⋊C8—341no—
227#102—42no—
228#102—342no—
229#102—342no—
230C22⋊C8—172no—
231C22⋊C8—341no—
232C4×C8—341no—
233C4×C8—341no—
234#95—21no—
235#95—21no—
236C17#1046811yescharacteristic, complemented, Sylow
237C24—21no—
238C2×Q8—341no—
239C2×Q8—341no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4#6322912yes—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4D4×D1711yescharacteristic
252C22×C4#6325411yescharacteristic
253C22×C4C2×C22⋊D1711yescharacteristic
254M16—681no—
255M16—681no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—342no—
260C2×C8—342no—
261C2×C8—681no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—341no—
267C4⋊C4—341no—
268C4⋊C4—341no—
269C4⋊C4—341no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—681no—
275C42—681no—
276C42#6325211yescharacteristic
277C42#6325211yescharacteristic
278C42D4×D1711yescharacteristic
279C42D4×D1711yescharacteristic
280C23—22no—
281C23—41no—
282C23—41no—
283C23—81no—
284C23—81no—
285C23#1058762511yescharacteristic
286Q8—1361no—
287Q8—1361no—
288D4—22no—
289D4—22no—
290D4—41no—
291D4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4#1058756911yescharacteristic
304C2×C4#1058757011yescharacteristic
305C2×C4#1058757011yescharacteristic
306C2×C4#1058757111yescharacteristic
307C2×C4#1058761511yescharacteristic
308C2×C4C2×#6322911yescharacteristic
309C2×C4#1058762311yescharacteristic
310C2×C4#1058762811yescharacteristic
311C2×C4#1058763611yescharacteristic
312C2×C4#1058763711yescharacteristic
313C8—81no—
314C8—682no—
315C8—682no—
316C8—1361no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—82no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#1176272411yescharacteristic
325C22#1176303611yescharacteristic
326C22#1176306511yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—161no—
333C4—1361no—
334C4—1361no—
335C4#1176294711yescharacteristic
336C4#1176296911yescharacteristic
337C4#1176313911yescharacteristic
338C4#1176315011yescharacteristic
339C2—21no—
340C2—21no—
341C2—82no—
342C2#1452998011yescharacteristic
343C2#1453005511yescharacteristic
344C2#1453008811yescharacteristic
subgroupG/H
centerC22#11762724
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1804C2
socleC22×C17#394
prime radicalC22×D4×C17D4
hypercenter#1804D17
hypocenterC17#10468
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. #14527554C2semidirect
3. #14527442C2semidirect
4. #14527347C2semidirect
6. #14526642C2semidirect
236. C17#10468semidirect

Maximal subgroups

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

subgroupindex
#145385812
#145275542
#145274422
#145273472
#145271052
#145266422
C17×#18042
#1046817