Finite Group Databasebeta

← #16945118

Subgroups — #16945118

Order 4352.

418 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 79 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#14534652C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526807C212yesmaximal, complemented
4#14526783C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic, complemented
7#11762773—21no—
8#11762745C411yescharacteristic, complemented
9#11762745C2211yescharacteristic
10#11762740—21no—
11#11762734C411yescharacteristic, complemented
12#11762734C2211yescharacteristic
13#11762723—21no—
14#11762682—21no—
15#11762682—21no—
16#11762677C411yescharacteristic, complemented
17#11762535—21no—
18#11762533C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×#99—21no—
21C17×#380—21no—
22C17×#342C2212yes—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C4×D4×C17—41no—
27C4×D4×C17—41no—
28C2×C4⋊C4×C17—41no—
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17C2311yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63223—41no—
45#10587556—41no—
46#10587556—41no—
47C17×#99—42no—
48C17×#95—41no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55C22⋊#3622—22no—
56C22⋊#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C2×D411yescharacteristic
59C2×D4×C17—21no—
60C2×D4×C17—21no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—42no—
68C22×C4×C17—81no—
69C22×C4×C17—81no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—82no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C42×C17—21no—
86C42×C17—21no—
87C22⋊Dic17—42no—
88C22⋊Dic17—41no—
89C22⋊Dic17—41no—
90C22⋊Dic17—81no—
91C22⋊Dic17—81no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94#63225—42no—
95#63225—41no—
96#63225—41no—
97#63225—81no—
98#63225—81no—
99#63223—82no—
100C2×#3622—42no—
101#11827—171nomaximal, Sylow
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17C2×C22⋊C411yescharacteristic
112C23×C17AGL(1,ℤ/8)11yescharacteristic
113C23×C17#13711yescharacteristic
114D4×C17—42no—
115D4×C17—42no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17—82no—
131C2×C4×C17—82no—
132C2×C4×C17—82no—
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17#9911yescharacteristic
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147#3622—82no—
148#2633—171no—
149#1799D1711yescharacteristic
150#1465—171no—
151#1352—172no—
152#1344—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—42no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17C2×#9911yescharacteristic
168C22×C17#41711yescharacteristic
169C22×C17#42111yescharacteristic
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—41no—
176C4×C17—162no—
177Dic17—81no—
178Dic17—81no—
179Dic17—81no—
180Dic17—81no—
181C2×#125—171no—
182C2×C4×D4C2×D1711yescharacteristic
183C2×#99—21no—
184#407—341no—
185#399—171no—
186#399—171no—
187#398—341no—
188#397—171no—
189#397—171no—
190#394—341no—
191#385—341no—
192#385—341no—
193#383—171no—
194#380—21no—
195#342C2×D1712yes—
196#324—341no—
197#323—172no—
198C2×C17—21no—
199C2×C17—21no—
200C2×C17—41no—
201C2×C17—41no—
202C2×C17—81no—
203C2×C17#151311yescharacteristic
204C2×C17#212311yescharacteristic
205C2×C17#212711yescharacteristic
206C22×D4C22⋊D1711yescharacteristic
207C23×C4D6811yescharacteristic
208C23×C4C22⋊D1711yescharacteristic
209C2×M16—681no—
210#125—681no—
211#125—681no—
212C4×D4—41no—
213C4×D4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4D6811yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4—341no—
226C2×C22⋊C4—341no—
227C2×C22⋊C4C4×D1711yescharacteristic
228C2×C22⋊C4C22×D1711yescharacteristic
229C2×C42C4×D1711yescharacteristic
230#99—42no—
231C22⋊C8—342no—
232C22⋊C8—681no—
233#95—41no—
234#95—171no—
235#95—171no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240C17#1182711yescharacteristic, complemented, Sylow
241C24#6322711yescharacteristic
242C24C2×C22⋊D1711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4—341no—
258C22×C4#6322711yescharacteristic
259C22×C4#6322711yescharacteristic
260C22×C4#6322711yescharacteristic
261C22×C4C2×C4×D1711yescharacteristic
262C22×C4C2×D6811yescharacteristic
263M16—1362no—
264C2×C8—682no—
265C4⋊C4—41no—
266C4⋊C4—82no—
267C4⋊C4—681no—
268C4⋊C4—681no—
269C4⋊C4—682no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—1361no—
273C4⋊C4—1361no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—682no—
279C22⋊C4—681no—
280C22⋊C4—681no—
281C22⋊C4—1361no—
282C22⋊C4—1361no—
283C42—21no—
284C42—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23C2×#6322711yescharacteristic
295C23#1058761811yescharacteristic
296C23#1058762811yescharacteristic
297D4—42no—
298D4—42no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4#1058747711yescharacteristic
329C2×C4#1058747711yescharacteristic
330C8—1362no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22C2×#1058747711yescharacteristic
346C22#1176280811yescharacteristic
347C22#1176282711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—162no—
355C4—1361no—
356C4—1361no—
357C4—1361no—
358C4—1361no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#1452732511yescharacteristic
365C2#1453043711yescharacteristic
366C2#1453045111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1799D17
hypocenterC17#11827
LCSG ▷ C22×C4×C17 ▷ C23×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
3. #14526807C2semidirect
4. #14526783C2semidirect
6. C2×#10587556C4semidirect
8. #11762745C4semidirect
11. #11762734C4semidirect
16. #11762677C4semidirect
240. C17#11827semidirect

Maximal subgroups

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

subgroupindex
#145346522
#145271222
#145268072
#145267832
C17×#17992
#1182717