Finite Group Databasebeta

← #16938105

Subgroups — #16938105

Order 4352.

366 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 91 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#14538585C211yesmaximal, characteristic
2#14528800C211yesmaximal, characteristic
3#14527558C211yesmaximal, characteristic
4#14527348C211yesmaximal, characteristic
5#14527107C211yesmaximal, characteristic
6#14526643C211yesmaximal, characteristic
7C17×#1542C211yesmaximal, characteristic
8C2×C4×Dic34C2211yescharacteristic
9#11763113—22no—
10#11763101—21no—
11#11763091—21no—
12#11762862C2211yescharacteristic
13#11762852—21no—
14#11762833C2211yescharacteristic
15#11762817—21no—
16C2×#10587487—21no—
17#11762675C2211yescharacteristic
18#11762572—21no—
19#11762561—22no—
20#11762546—22no—
21#11762534—22no—
22C17×#533C2211yescharacteristic
23C17×#431C2211yescharacteristic
24C2×C17×C4⋊C8C2211yescharacteristic
25C17×#333—21no—
26C17×#332—21no—
27C22×Dic34D411yescharacteristic
28C22×C8×C17—21no—
29C17×C4⋊Q8D411yescharacteristic
30C17×C4⋊Q8D411yescharacteristic
31C17×#125—21no—
32C17×C22⋊Q8—21no—
33C17×C22⋊Q8—21no—
34C4×Q8×C17—21no—
35C4×Q8×C17—21no—
36C17×#117—21no—
37C2×C42×C17C2311yescharacteristic
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42C2×#63223—21no—
43C4×Dic34—21no—
44C4×Dic34—21no—
45C4×Dic34—21no—
46C4×Dic34—21no—
47C17×C4⋊C8—21no—
48C17×C4⋊C8—21no—
49C17×C4⋊C8—21no—
50C17×C22⋊C8—21no—
51C4×C8×C17—21no—
52#10587502—21no—
53#10587502—21no—
54C22⋊#3622D412yes—
55#10587487—42no—
56#10587481—22no—
57#10587481—22no—
58#10587479—22no—
59#10587478—22no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622D412yes—
63C4×#3622—21no—
64C2×Q8×C17—21no—
65C2×Q8×C17—21no—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17#4111yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—41no—
72C2×Dic34—21no—
73C2×Dic34—21no—
74C2×Dic34—21no—
75C2×Dic34—21no—
76C2×Dic34—21no—
77C2×Dic34Q1612yes—
78C2×C8×C17—41no—
79C2×C8×C17—41no—
80C2×C8×C17—41no—
81C2×C8×C17—41no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C42×C17—41no—
90C42×C17C2×D411yescharacteristic
91C42×C17C2×D411yescharacteristic
92C42×C17#4111yescharacteristic
93C42×C17#4111yescharacteristic
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98#63225—41no—
99#63225—41no—
100C4×Dic17—41no—
101C4×Dic17—41no—
102#63223—41no—
103#63223—41no—
104C2×#3622—22no—
105C2×#3622—22no—
106#10471—171nomaximal, Sylow
107C23×C17#12111yescharacteristic
108Q8×C17—81no—
109Q8×C17—81no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17#12011yescharacteristic
115C2×C4×C17#12111yescharacteristic
116C2×C4×C17#12111yescharacteristic
117C2×C4×C17#12311yescharacteristic
118C2×C4×C17#12311yescharacteristic
119C2×C4×C17#12411yescharacteristic
120C2×C4×C17C2×Q1611yescharacteristic
121C2×C4×C17#13511yescharacteristic
122C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C17#13711yescharacteristic
124C2×Dic17—22no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—81no—
129C2×Dic17—81no—
130C2×Dic17—81no—
131C2×Dic17—81no—
132Dic34—22no—
133Dic34—22no—
134Dic34—41no—
135Dic34—41no—
136C8×C17—81no—
137C8×C17—81no—
138#3622—42no—
139#3622—42no—
140#3078—171no—
141#1805—171no—
142#1571—171no—
143#1542D1711yescharacteristic
144#1518—171no—
145#1461—171no—
146#1286—171no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17#39411yescharacteristic
150C22×C17#46711yescharacteristic
151C22×C17#46811yescharacteristic
152C4×C17—21no—
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—81no—
157C4×C17—81no—
158C4×C17#44911yescharacteristic
159C4×C17#45111yescharacteristic
160C4×C17#48411yescharacteristic
161C4×C17#48511yescharacteristic
162Dic17—42no—
163Dic17—81no—
164Dic17—161no—
165#533C2×D1711yescharacteristic
166C2×C4×Q8—171no—
167#479—342no—
168#477—341no—
169#475—341no—
170#431—171no—
171#431C2×D1711yescharacteristic
172#428—341no—
173#423—171no—
174C2×C4⋊C8C2×D1711yescharacteristic
175#419—341no—
176C2×#103—341no—
177#382—171no—
178#340—341no—
179#335—342no—
180#333—21no—
181#332—21no—
182#328—342no—
183#324—342no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17#203611yescharacteristic
187C2×C17#204911yescharacteristic
188C2×C17#205311yescharacteristic
189C22×Q8—171no—
190C2×M16—341no—
191C22×C8—21no—
192C4⋊Q8C22⋊D1711yescharacteristic
193C4⋊Q8C22⋊D1711yescharacteristic
194#125—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—341no—
200C4×Q8—341no—
201C4×Q8—341no—
202C4×Q8—341no—
203#117—21no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C42—341no—
208C2×C42C22×D1711yescharacteristic
209#107—342no—
210#106—342no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—172no—
215C4⋊C8—341no—
216C4⋊C8—341no—
217#103—342no—
218#103—342no—
219#103—682no—
220C22⋊C8—21no—
221C22⋊C8—172no—
222C4×C8—21no—
223C4×C8—341no—
224#95—341no—
225#95—341no—
226C17#1047111yescharacteristic, complemented, Sylow
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—172no—
230C2×Q8—341no—
231C2×Q8—341no—
232C2×Q8—341no—
233C2×Q8—341no—
234C2×Q8—341no—
235C22×C4—341no—
236C22×C4—341no—
237C22×C4—681no—
238C22×C4#6325211yescharacteristic
239C22×C4D4×D1711yescharacteristic
240C22×C4D4×D1711yescharacteristic
241M16—681no—
242M16—681no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—342no—
248C2×C8—342no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C22⋊C4—41no—
262C42—41no—
263C42—681no—
264C42—681no—
265C42#6325211yescharacteristic
266C42D4×D1711yescharacteristic
267C42#6325411yescharacteristic
268C42C2×C22⋊D1711yescharacteristic
269C23#1058757011yescharacteristic
270Q8—81no—
271Q8—81no—
272Q8—342no—
273Q8—342no—
274Q8—681no—
275Q8—681no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—342no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—1361no—
285C2×C4—1361no—
286C2×C4—1361no—
287C2×C4—1361no—
288C2×C4#1058756911yescharacteristic
289C2×C4#1058757011yescharacteristic
290C2×C4#1058757111yescharacteristic
291C2×C4#1058760011yescharacteristic
292C2×C4#1058760211yescharacteristic
293C2×C4Q16×D1711yescharacteristic
294C2×C4#1058760411yescharacteristic
295C2×C4#1058761511yescharacteristic
296C2×C4#1058762311yescharacteristic
297C2×C4#1058762511yescharacteristic
298C8—81no—
299C8—81no—
300C8—682no—
301C8—682no—
302C22—21no—
303C22—21no—
304C22#1176272411yescharacteristic
305C22#1176304911yescharacteristic
306C22#1176305811yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—81no—
312C4—81no—
313C4—2721no—
314C4—682no—
315C4—1361no—
316C4#1176295311yescharacteristic
317C4#1176296311yescharacteristic
318C4#1176314311yescharacteristic
319C4#1176314811yescharacteristic
320C2—21no—
321C2—21no—
322C2#1452998511yescharacteristic
323C2#1453006411yescharacteristic
324C2#1453008811yescharacteristic
subgroupG/H
centerC22#11762724
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1542C2
socleC22×C17#394
prime radicalC23×C17#121
hypercenter#1542D17
hypocenterC17#10471
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
226. C17#10471semidirect

Maximal subgroups

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

subgroupindex
#145385852
#145288002
#145275582
#145273482
#145271072
#145266432
C17×#15422
#1047117