Finite Group Databasebeta

← #16931736

Subgroups — #16931736

Order 4352.

570 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 109 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#14537824C211yesmaximal, characteristic
2#14533738C211yesmaximal, characteristic
3#14528773C211yesmaximal, characteristic
4#14527469C211yesmaximal, characteristic, complemented
5#14526686C211yesmaximal, characteristic, complemented
6#14526685C211yesmaximal, characteristic, complemented
7C17×#1549C211yesmaximal, characteristic, complemented
8C2×#10587632—22no—
9C2×C4×D68C2211yescharacteristic
10#11762888—22no—
11#11762886—22no—
12#11762862C2211yescharacteristic
13#11762833C2211yescharacteristic
14C2×#10587494—21no—
15#11762771—22no—
16#11762744C414yes—
17#11762710—21no—
18#11762678—21no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762577—21no—
21#11762555—22no—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C17×#332—22no—
26C22×Q8×C17—21no—
27C22×D68D411yescharacteristic
28C22×C4×D17D412yes—
29C2×M16×C17—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17C2×C412yes—
36C2×C42×C17C2311yescharacteristic
37#10587632—48no—
38C2×#63227—24no—
39C2×#63227C2×C412yes—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17D411yescharacteristic
42C2×#63225—24no—
43C2×C4×Dic17—21no—
44C2×#63223—21no—
45C2×#63218—24no—
46C4×D68—21no—
47C4×D68—21no—
48C4×D68—21no—
49C4×D68—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8—21no—
52C17×C4⋊C8D412yes—
53C17×C22⋊C8—21no—
54C4×C8×C17—21no—
55#10587502C2×C412yes—
56C22⋊#3622—21no—
57#10587494—42no—
58#10587491—24no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622D412yes—
62C4×#3622—21no—
63C23×D17C22⋊C412yes—
64C2×Q8×C17—44no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C22×C4×C17—22no—
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—22no—
72C22×Dic17C22⋊C412yes—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68—21no—
79C2×C4×D17—22no—
80C2×C4×D17—22no—
81C2×C4×D17—44no—
82M16×C17—41no—
83M16×C17—41no—
84C2×C8×C17—22no—
85C2×C8×C17—41no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—22no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C42×C17C2×D411yescharacteristic
91C42×C17C2×D411yescharacteristic
92C42×C17#4111yescharacteristic
93C42×C17#4111yescharacteristic
94#63227—22no—
95#63227—22no—
96#63227—48no—
97C4⋊Dic17—24no—
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100#63225—48no—
101C4×Dic17—41no—
102C4×Dic17—41no—
103#63223—41no—
104#63223—41no—
105C2×#3622—22no—
106C2×#3622—41no—
107#63218—84no—
108#9403—171nomaximal, Sylow
109C23×C17C2×C22⋊C411yescharacteristic
110C22×D17—22no—
111C22×D17—22no—
112C22×D17—22no—
113C22×D17—42no—
114C22×D17—42no—
115Q8×C17—82no—
116Q8×C17—82no—
117C2×C4×C17—22no—
118C2×C4×C17—41no—
119C2×C4×C17—44no—
120C2×C4×C17—42no—
121C2×C4×C17C4×D411yescharacteristic
122C2×C4×C17C4×D411yescharacteristic
123C2×C4×C17#12011yescharacteristic
124C2×C4×C17#12111yescharacteristic
125C2×C4×C17#12111yescharacteristic
126C2×C4×C17#12311yescharacteristic
127C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C17#13711yescharacteristic
129C2×C4×C17#13711yescharacteristic
130C2×C4×C17#13711yescharacteristic
131C2×Dic17—24no—
132C2×Dic17—22no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136D68—41no—
137D68—41no—
138D68—41no—
139D68—41no—
140C4×D17—44no—
141C4×D17—82no—
142C8×C17—42no—
143C8×C17—81no—
144#3622—42no—
145#3622—81no—
146#2975—171no—
147#2542—171no—
148#1800—171no—
149#1549—171no—
150#1549D1711yescharacteristic, complemented
151#1303—171no—
152#1303—171no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17#38611yescharacteristic
156C22×C17#41711yescharacteristic
157C22×C17#41911yescharacteristic
158C2×D17—44no—
159C2×D17—42no—
160C2×D17—42no—
161C2×D17—42no—
162C2×D17—82no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—82no—
167C4×C17#45312yes—
168C4×C17#46811yescharacteristic
169C4×C17#46811yescharacteristic
170C4×C17#47011yescharacteristic
171C4×C17#47011yescharacteristic
172Dic17—44no—
173Dic17—42no—
174C2×C4⋊Q8C2×D1711yescharacteristic
175C2×C22⋊Q8—342no—
176C2×C4×D4—171no—
177#436—342no—
178#436—342no—
179#431—171no—
180#431C2×D1711yescharacteristic
181#423—171no—
182#423C2×D1711yescharacteristic
183C2×#100—341no—
184#407—342no—
185#399—174no—
186#390—341no—
187#384—341no—
188C4×C4⋊C4—171no—
189#343—341no—
190#332—22no—
191#332—342no—
192C2×C17—21no—
193C2×C17—21no—
194C2×C17#188711yescharacteristic
195C2×C17#189411yescharacteristic
196C2×C17#191511yescharacteristic
197D17—82no—
198C22×Q8—21no—
199C22×D4—171no—
200C23×C4—172no—
201C2×M16—21no—
202C2×M16—344no—
203C2×M16—341no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C22⋊Q8—688no—
209C4×D4—341no—
210C4×D4—341no—
211C4×D4—341no—
212C4×D4—341no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—344no—
217C2×C4⋊C4C4×D1712yes—
218C2×C22⋊C4—172no—
219C2×C22⋊C4—344no—
220C2×C42—341no—
221C2×C42C22×D1711yescharacteristic
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—172no—
225C4⋊C8—341no—
226C4⋊C8—341no—
227C4⋊C8C22⋊D1712yes—
228#100—682no—
229C22⋊C8—21no—
230C22⋊C8—344no—
231C22⋊C8—341no—
232C4×C8—21no—
233C4×C8—341no—
234#95—172no—
235C17#940311yescharacteristic, complemented, Sylow
236C24—172no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×D4—341no—
241C2×D4—341no—
242C2×D4—341no—
243C2×D4—341no—
244C2×D4—341no—
245C2×D4—341no—
246C22×C4—22no—
247C22×C4—172no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—684no—
252C22×C4C2×C4×D1711yescharacteristic
253C22×C4D4×D1711yescharacteristic
254C22×C4D4×D1711yescharacteristic
255M16—41no—
256M16—41no—
257M16—681no—
258M16—681no—
259M16—1364no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—342no—
263C2×C8—681no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—344no—
269C4⋊C4—341no—
270C4⋊C4—341no—
271C4⋊C4—688no—
272C22⋊C4—342no—
273C22⋊C4—342no—
274C22⋊C4—688no—
275C42—681no—
276C42—681no—
277C42#6325211yescharacteristic
278C42D4×D1711yescharacteristic
279C42#6325411yescharacteristic
280C42C2×C22⋊D1711yescharacteristic
281C23—342no—
282C23—342no—
283C23—342no—
284C23—682no—
285C23—682no—
286C23C22⋊C4×D1711yescharacteristic
287Q8—82no—
288Q8—82no—
289D4—681no—
290D4—681no—
291D4—681no—
292D4—681no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—344no—
298C2×C4—342no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—684no—
302C2×C4—682no—
303C2×C4—1362no—
304C2×C4#1058756711yescharacteristic
305C2×C4#1058756911yescharacteristic
306C2×C4#1058757011yescharacteristic
307C2×C4#1058760011yescharacteristic
308C2×C4#1058760111yescharacteristic
309C2×C4#1058760411yescharacteristic
310C2×C4#1058760411yescharacteristic
311C2×C4C4×C22⋊D1711yescharacteristic
312C2×C4#1058762311yescharacteristic
313C2×C4#1058762411yescharacteristic
314C8—42no—
315C8—81no—
316C8—682no—
317C8—1361no—
318C22—21no—
319C22—21no—
320C22—684no—
321C22—682no—
322C22—682no—
323C22—682no—
324C22—1362no—
325C22#1176268511yescharacteristic
326C22#1176280911yescharacteristic
327C22#1176281611yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—82no—
332C4—684no—
333C4—682no—
334C4#1176297512yes—
335C4#1176305211yescharacteristic
336C4#1176305811yescharacteristic
337C4#1176306411yescharacteristic
338C4#1176306611yescharacteristic
339C2—21no—
340C2—21no—
341C2—1362no—
342C2#1452917911yescharacteristic
343C2#1452921411yescharacteristic
344C2#1452933311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1549C2
socleC22×C17#386
prime radicalC22×D68D4
cosocleC2×C42×C17C23
hypercenter#1549D17
hypocenterC17#9403
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. #14527469C2semidirect
5. #14526686C2semidirect
6. #14526685C2semidirect
7. C17×#1549C2semidirect
150. #1549D17semidirect
235. C17#9403semidirect

Maximal subgroups

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

subgroupindex
#145378242
#145337382
#145287732
#145274692
#145266862
#145266852
C17×#15492
#940317