Finite Group Databasebeta

← #16943778

Subgroups — #16943778

Order 4352.

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

normal subgroups only

This group has 470 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
1C4×#10587621C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526731C212yesmaximal, complemented
4#14526713C212yesmaximal, complemented
5C17×#1466C211yesmaximal, characteristic
6C2×#10587621C411yescharacteristic
7C22×C4×Dic17C411yescharacteristic
8#11762773—21no—
9#11762697—21no—
10#11762690—21no—
11#11762689C411yescharacteristic
12#11762681C2211yescharacteristic
13#11762670C2211yescharacteristic
14C4×#63225—21no—
15C4×C22⋊Dic17—21no—
16C4×C22⋊Dic17C411yescharacteristic
17#11762535—21no—
18#11762533C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#432—21no—
21C17×#323—21no—
22C17×#323C2212yes—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×Dic17C2×C411yescharacteristic
27C23×Dic17C2×C411yescharacteristic
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17C2311yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17C2×C411yescharacteristic
37C2×C22⋊Dic17C2×C411yescharacteristic
38#10587621—22no—
39#10587621—22no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17—41no—
46C2×C4×Dic17—41no—
47C2×#63223—41no—
48C17×C4⋊C8—41no—
49C17×C22⋊C8—22no—
50C17×#97—41no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C22⋊#3622—22no—
57C22⋊#3622—41no—
58C24×C17C22⋊C411yescharacteristic
59C24×C17C22×C411yescharacteristic
60C2×D4×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—21no—
80C22×Dic17—42no—
81C22×Dic17—41no—
82C22×Dic17—41no—
83C22×Dic17—41no—
84C22×Dic17M1612yescomplemented
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—21no—
93C22⋊Dic17—42no—
94C22⋊Dic17—41no—
95C22⋊Dic17M1612yescomplemented
96#63225—24no—
97#63225—42no—
98C4×Dic17—42no—
99C4×Dic17—42no—
100C4×Dic17—41no—
101C4×Dic17—41no—
102#63223—82no—
103C2×#3622—42no—
104#11526—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17C2×C22⋊C411yescharacteristic
115C23×C17C2×M1611yescharacteristic
116C23×C17C2×M1611yescharacteristic
117D4×C17—42no—
118D4×C17—42no—
119C2×C4×C17—22no—
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—42no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—81no—
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17#9911yescharacteristic
135C2×Dic17—22no—
136C2×Dic17—22no—
137C2×Dic17—22no—
138C2×Dic17—41no—
139C2×Dic17—42no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—42no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—42no—
147C2×Dic17—42no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C8×C17—82no—
152#3622—82no—
153C4×#123—171no—
154#1466D1711yescharacteristic
155#1465—171no—
156#1326—172no—
157#1317—172no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—42no—
165C22×C17—42no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—81no—
172C22×C17#40711yescharacteristic
173C22×C17C2×#9911yescharacteristic
174C22×C17#41311yescharacteristic
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17—41no—
180C4×C17#35212yes—
181Dic17—42no—
182Dic17—42no—
183Dic17—42no—
184Dic17—81no—
185C2×#123—171no—
186C2×C4×D4C2×D1711yescharacteristic
187C22×C42—171no—
188#432—21no—
189#407—341no—
190#388—341no—
191#387—341no—
192#386—171no—
193#385—171no—
194#381—171no—
195C4×C4⋊C4—341no—
196C4×C22⋊C4—171no—
197C4×C22⋊C4—341no—
198#324—341no—
199#323—21no—
200#323—172no—
201#323C2×D1712yes—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17—41no—
205C2×C17—41no—
206C2×C17—81no—
207C2×C17#146511yescharacteristic
208C2×C17C2×#35211yescharacteristic
209C2×C17#212011yescharacteristic
210C22×D4C4×D1711yescharacteristic
211C23×C4—171no—
212C23×C4—171no—
213C23×C4C4×D1711yescharacteristic
214C23×C4D6811yescharacteristic
215C2×M16—681no—
216#123—342no—
217#123—342no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4D6811yescharacteristic
223C2×C22⋊C4—171no—
224C2×C22⋊C4—171no—
225C2×C22⋊C4—341no—
226C2×C22⋊C4—341no—
227C2×C22⋊C4C22⋊D1711yescharacteristic
228C2×C22⋊C4C22×D1711yescharacteristic
229C2×C42—341no—
230C2×C42—341no—
231C2×C42—341no—
232C2×C42—681no—
233C2×C42—681no—
234C2×C42C22⋊D1711yescharacteristic
235C4⋊C8—41no—
236C22⋊C8—22no—
237C22⋊C8—342no—
238C22⋊C8—681no—
239#97—41no—
240#95—341no—
241#95—341no—
242#95—341no—
243#95—341no—
244#95—341no—
245C17#1152611yescharacteristic, complemented, Sylow
246C24#6322711yescharacteristic
247C24C2×C4×D1711yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—172no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—682no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4#6322711yescharacteristic
269C22×C4#6322711yescharacteristic
270C22×C4#6322711yescharacteristic
271C22×C4C2×D6811yescharacteristic
272C22×C4C2×C22⋊D1711yescharacteristic
273M16—1362no—
274C2×C8—42no—
275C2×C8—682no—
276C4⋊C4—41no—
277C4⋊C4—344no—
278C4⋊C4—682no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—172no—
282C22⋊C4—344no—
283C22⋊C4—341no—
284C22⋊C4—682no—
285C22⋊C4—681no—
286C42—21no—
287C42—21no—
288C42—682no—
289C42—682no—
290C42—681no—
291C42—681no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23M16×D1711yescharacteristic
302C23M16×D1711yescharacteristic
303C23C2×#6322711yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—342no—
321C2×C4—342no—
322C2×C4—342no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—682no—
332C2×C4—682no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4#1058747711yescharacteristic
337C2×C4#1058747711yescharacteristic
338C8—82no—
339C8—1362no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22#1176277211yescharacteristic
355C22C2×#1058747711yescharacteristic
356C22#1176279111yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—682no—
363C4—682no—
364C4—682no—
365C4—1361no—
366C4#1176260012yes—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2#1452712011yescharacteristic
373C2C2×#1176260011yescharacteristic
374C2#1453042311yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC24×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1466C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊C4×C17C23
hypercenter#1466D17
hypocenterC17#11526
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #14526731C2semidirect
4. #14526713C2semidirect
84. C22×Dic17M16semidirect
95. C22⋊Dic17M16semidirect
245. C17#11526semidirect

Maximal subgroups

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

subgroupindex
C4×#105876212
#145271222
#145267312
#145267132
C17×#14662
#1152617