Finite Group Databasebeta

← #16945018

Subgroups — #16945018

Order 4352.

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

normal subgroups only

This group has 448 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#14534526C211yesmaximal, characteristic
2#14527114C211yesmaximal, characteristic, complemented
3#14526789C212yesmaximal, complemented
4#14526786C212yesmaximal, complemented
5C17×#1821C211yesmaximal, characteristic, complemented
6C2×#10587615—21no—
7C2×#10587616C411yescharacteristic
8#11762773—21no—
9#11762739C411yescharacteristic
10#11762739C2211yescharacteristic
11#11762731—21no—
12#11762731—21no—
13#11762726C411yescharacteristic
14#11762726C2211yescharacteristic
15#11762688—21no—
16#11762669C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C22×C4⋊C4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×C22⋊D17—21no—
27C22×D68C2×C411yescharacteristic
28C2×M16×C17—22no—
29C2×C4⋊C4×C17—41no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C4⋊C4×C17C2311yescharacteristic
35C2×C22⋊Dic17—21no—
36#10587616—41no—
37#10587616—41no—
38#10587615—44no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17C2×C411yescharacteristic
46C2×#63225—21no—
47C2×#63225—21no—
48C2×#63223—41no—
49C17×C22⋊C8—41no—
50C17×C22⋊C8—41no—
51#10587502—21no—
52#10587502—21no—
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55C22⋊#3622—22no—
56C22⋊#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C23×D17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—22no—
66C22×C4×C17—41no—
67C22×C4×C17—81no—
68C22×C4×C17—81no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22×C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C2×C22⋊D17—42no—
76C2×C22⋊D17—42no—
77C2×C22⋊D17—41no—
78C2×C22⋊D17—41no—
79C22×Dic17—21no—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C2×D68—41no—
83C2×D68—81no—
84M16×C17—42no—
85M16×C17—42no—
86C2×C8×C17—42no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C22⋊Dic17—42no—
92#63227—42no—
93#63227—42no—
94#63227—42no—
95#63227—81no—
96#63227—81no—
97C4⋊Dic17—41no—
98C4⋊Dic17—81no—
99#63225—42no—
100#63225—42no—
101#63223—82no—
102C2×#3622—42no—
103#11805—171nomaximal, Sylow
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17#9911yescharacteristic
109C23×C17#9911yescharacteristic
110C23×C17C2×C22⋊C411yescharacteristic
111C22×D17—41no—
112C22×D17—41no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—41no—
116C22×D17—41no—
117C22×D17—41no—
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—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—81no—
135C2×C4×C17—81no—
136C2×C4×C17—81no—
137C2×C4×C17AGL(1,ℤ/8)12yes—
138C22⋊D17—82no—
139C22⋊D17—82no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149D68—82no—
150C8×C17—82no—
151#3622—82no—
152#2618—171no—
153#1821D1711yescharacteristic, complemented
154#1463—171no—
155#1346—172no—
156#1345—172no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17C2×#9911yescharacteristic
168C22×C17#41811yescharacteristic
169C22×C17#42111yescharacteristic
170C2×D17—81no—
171C2×D17—81no—
172C2×D17—81no—
173C2×D17—81no—
174C2×D17—81no—
175C2×D17—81no—
176C2×D17—81no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183Dic17—81no—
184Dic17—81no—
185Dic17—81no—
186C2×#123—341no—
187C2×#121—171no—
188C22×C4⋊C4C2×D1711yescharacteristic
189#407—21no—
190#407—341no—
191#398—171no—
192#398—171no—
193#396—341no—
194#396—341no—
195#394—171no—
196#394—171no—
197#386—341no—
198#380—171no—
199#343—21no—
200#343C2×D1712yes—
201#324—172no—
202#324—341no—
203C2×C17—21no—
204C2×C17—21no—
205C2×C17—41no—
206C2×C17—41no—
207C2×C17#151011yescharacteristic
208C2×C17#212111yescharacteristic
209C2×C17#213311yescharacteristic
210D17—161no—
211C22×D4—171no—
212C22×D4—341no—
213C23×C4C4×D1711yescharacteristic
214C23×C4D6811yescharacteristic
215C23×C4C22⋊D1711yescharacteristic
216C2×M16—22no—
217C2×M16—681no—
218#123—684no—
219#121—681no—
220#121—681no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4C4×D1711yescharacteristic
228C2×C4⋊C4D6811yescharacteristic
229C2×C4⋊C4C22⋊D1711yescharacteristic
230C2×C4⋊C4C22×D1711yescharacteristic
231C2×C22⋊C4—341no—
232C2×C22⋊C4—341no—
233C2×C22⋊C4—341no—
234C2×C22⋊C4—341no—
235C2×C22⋊C4—341no—
236C2×C22⋊C4—341no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—342no—
240C22⋊C8—681no—
241#95—171no—
242#95—171no—
243#95—341no—
244#95—341no—
245C17#1180511yescharacteristic, complemented, Sylow
246C24—341no—
247C24#6322711yescharacteristic
248C2×D4—681no—
249C2×D4—682no—
250C2×D4—682no—
251C2×D4—681no—
252C2×D4—681no—
253C2×D4—1361no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4#6322711yescharacteristic
268C22×C4#6322711yescharacteristic
269C22×C4#6322711yescharacteristic
270C22×C4C2×C4×D1711yescharacteristic
271C22×C4C2×D6811yescharacteristic
272C22×C4C2×C22⋊D1711yescharacteristic
273M16—42no—
274M16—42no—
275M16—1362no—
276C2×C8—42no—
277C2×C8—682no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—681no—
283C4⋊C4—682no—
284C4⋊C4—682no—
285C4⋊C4—1361no—
286C22⋊C4—682no—
287C22⋊C4—682no—
288C22⋊C4—682no—
289C22⋊C4—682no—
290C22⋊C4—1361no—
291C22⋊C4—1361no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—681no—
297C23—681no—
298C23—681no—
299C23—681no—
300C23—681no—
301C23—681no—
302C23—681no—
303C23#1058747711yescharacteristic
304C23#1058747711yescharacteristic
305C23C2×#6322711yescharacteristic
306D4—1362no—
307D4—1362no—
308D4—1362no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4—681no—
337C2×C4#1058763512yes—
338C8—82no—
339C8—1362no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—1361no—
351C22—1361no—
352C22—1361no—
353C22—1361no—
354C22—1361no—
355C22—1361no—
356C22—1361no—
357C22C2×#1058747711yescharacteristic
358C22#1176281311yescharacteristic
359C22#1176282711yescharacteristic
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—1361no—
367C4—1361no—
368C4—1361no—
369C2—21no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—2721no—
374C2#1452731211yescharacteristic
375C2#1453043111yescharacteristic
376C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1821C2
socleC22×C17C2×#99
prime radicalC2×#10587616C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1821D17
hypocenterC17#11805
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #14527114C2semidirect
3. #14526789C2semidirect
4. #14526786C2semidirect
5. C17×#1821C2semidirect
153. #1821D17semidirect
245. C17#11805semidirect

Maximal subgroups

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

subgroupindex
#145345262
#145271142
#145267892
#145267862
C17×#18212
#1180517