Finite Group Databasebeta

← #16944853

Subgroups — #16944853

Order 4352.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#14534164C211yesmaximal, characteristic
2#14527114C211yesmaximal, characteristic
3#14526813C212yesmaximal, complemented
4#14526789C212yesmaximal, complemented
5C2×C17×#343C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic
7#11762773—21no—
8#11762739C411yescharacteristic
9#11762739C2211yescharacteristic
10#11762729C411yescharacteristic
11#11762729C2211yescharacteristic
12#11762704—21no—
13#11762704—21no—
14#11762682—21no—
15#11762682—21no—
16C4×C22⋊Dic17C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C22×M16×C17—21no—
20C22×C4⋊C4×C17C2211yescharacteristic
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—24no—
28C2×M16×C17—22no—
29C2×M16×C17—41no—
30C2×M16×C17—42no—
31C22×C8×C17—41no—
32C2×C4⋊C4×C17—41no—
33C2×C4⋊C4×C17—41no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C4⋊C4×C17C2311yescharacteristic
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41#10587610—41no—
42#10587610—41no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×#63225—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×C4×Dic17—21no—
49C2×#63223—41no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56C22⋊#3622—22no—
57C22⋊#3622—41no—
58C24×C17C22⋊C411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—22no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
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×C4×C17C2×D411yescharacteristic
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—21no—
80C2×Dic34—41no—
81C2×Dic34—81no—
82M16×C17—24no—
83M16×C17—44no—
84M16×C17—42no—
85C2×C8×C17—44no—
86C2×C8×C17—42no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C22⋊Dic17—24no—
92C22⋊Dic17—41no—
93C22⋊Dic17—41no—
94C4⋊Dic17—41no—
95#63225—42no—
96#63225—42no—
97#63225—41no—
98#63225—41no—
99#63225—81no—
100#63225—81no—
101C4×Dic17—81no—
102#63223—82no—
103C2×#3622—42no—
104#11768—171nomaximal, Sylow
105C23×C17—22no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—41no—
109C23×C17#9911yescharacteristic
110C23×C17#9911yescharacteristic
111C23×C17C2×C22⋊C411yescharacteristic
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
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—21no—
123C2×C4×C17—21no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—81no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17#10412yes—
133C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C17#13711yescharacteristic
135C2×Dic17—22no—
136C2×Dic17—22no—
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—
147Dic34—82no—
148C8×C17—44no—
149#3622—82no—
150#2583—171no—
151C2×#343D1711yescharacteristic
152#1463—171no—
153#1354—172no—
154#1346—172no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—42no—
163C22×C17—42no—
164C22×C17#35612yes—
165C22×C17C2×#9911yescharacteristic
166C22×C17#41711yescharacteristic
167C22×C17C2×#10411yescharacteristic
168C4×C17—21no—
169C4×C17—22no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175Dic17—42no—
176Dic17—42no—
177Dic17—81no—
178Dic17—81no—
179C22×M16—21no—
180C2×C22⋊Q8—171no—
181C22×C4⋊C4C2×D1711yescharacteristic
182#407—341no—
183#398—171no—
184#398—171no—
185#395—171no—
186#395—171no—
187#389—341no—
188#389—341no—
189#385—341no—
190#385—341no—
191C4×C22⋊C4—171no—
192#343—21no—
193#343C2×D1712yes—
194#324—172no—
195#324—341no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17—22no—
199C2×C17—41no—
200C2×C17#150411yescharacteristic
201C2×C17#212111yescharacteristic
202C2×C17C2×#35611yescharacteristic
203C22×Q8—171no—
204C23×C4C4×D1711yescharacteristic
205C23×C4D6811yescharacteristic
206C23×C4C22⋊D1711yescharacteristic
207C2×M16—24no—
208C2×M16—22no—
209C2×M16—41no—
210C2×M16—42no—
211C2×M16—681no—
212C22×C8—41no—
213C22⋊Q8—681no—
214C22⋊Q8—681no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4C4×D1711yescharacteristic
224C2×C4⋊C4D6811yescharacteristic
225C2×C4⋊C4C22⋊D1711yescharacteristic
226C2×C4⋊C4C22×D1711yescharacteristic
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C42—341no—
230C22⋊C8—342no—
231C22⋊C8—681no—
232#95—171no—
233#95—171no—
234#95—341no—
235#95—341no—
236#95—341no—
237#95—341no—
238C17#1176811yescharacteristic, complemented, Sylow
239C24#6322711yescharacteristic
240C2×Q8—681no—
241C2×Q8—1361no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4#6322711yescharacteristic
258C22×C4#6322711yescharacteristic
259C22×C4#6322711yescharacteristic
260C22×C4C2×C4×D1711yescharacteristic
261C22×C4C2×D6811yescharacteristic
262C22×C4C2×C22⋊D1711yescharacteristic
263M16—24no—
264M16—44no—
265M16—42no—
266M16—1362no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—682no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—681no—
275C4⋊C4—682no—
276C4⋊C4—682no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—1361no—
280C4⋊C4—1361no—
281C22⋊C4—344no—
282C22⋊C4—681no—
283C22⋊C4—681no—
284C42—1361no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1058747711yescharacteristic
290C23#1058747711yescharacteristic
291C23C2×#6322711yescharacteristic
292Q8—1362no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—342no—
314C2×C4—342no—
315C2×C4—681no—
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#1058747612yes—
326C2×C4#1058761811yescharacteristic
327C2×C4#1058762811yescharacteristic
328C8—44no—
329C8—1362no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1176260712yes—
340C22C2×#1058747711yescharacteristic
341C22#1176280811yescharacteristic
342C22C2×#1058747611yescharacteristic
343C4—21no—
344C4—22no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—682no—
351C4—682no—
352C4—1361no—
353C4—1361no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#1452728611yescharacteristic
359C2#1453043111yescharacteristic
360C2C2×#1176260711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC2×C17×#343C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenterC2×#343D17
hypocenterC17#11768
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. #14526813C2semidirect
4. #14526789C2semidirect
238. C17#11768semidirect

Maximal subgroups

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

subgroupindex
#145341642
#145271142
#145268132
#145267892
C2×C17×#3432
#1176817