Finite Group Databasebeta

← #16922914

Subgroups — #16922914

Order 4352.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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#14537093C211yesmaximal, characteristic
2#14528763C211yesmaximal, characteristic
3#14528476C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic, complemented
5#14526607C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1271C211yesmaximal, characteristic, complemented
8C22×#63218—21no—
9C2×#10587576C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11C2×#10587485—21no—
12C2×#10587491—21no—
13#11762707C2211yescharacteristic
14#11762695—21no—
15#11762691—21no—
16#11762683—21no—
17#11762680—21no—
18#11762674C411yescharacteristic
19#11762666C411yescharacteristic
20#11762666C411yescharacteristic
21#11762564C2211yescharacteristic
22#11762564C2211yescharacteristic
23#11762563—21no—
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×C4×D17C2×C411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C42×C17C2311yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17C2×C411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C22×#3622—21no—
47C22×#3622—21no—
48C2×#63218—24no—
49C2×#63218—42no—
50C2×#63218—42no—
51#10587576—42no—
52C42×D17—42no—
53C17×#95—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59#10587491—42no—
60#10587485—42no—
61C23×D17C22⋊C411yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C2×C4×D17—22no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—22no—
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—21no—
80C2×C4×D17—22no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C8×C17—22no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C2×C8×C17—41no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C4⋊C4×C17—21no—
91C4⋊C4×C17—21no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—41no—
95C42×C17—41no—
96#63227—42no—
97#63227—42no—
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17—42no—
101C4⋊Dic17—41no—
102#63225—41no—
103#63225—41no—
104#63225—42no—
105C4×Dic17—22no—
106C4×Dic17—22no—
107C4×Dic17—41no—
108C2×#3622—22no—
109C2×#3622—41no—
110C2×#3622—41no—
111C2×#3622—41no—
112C2×#3622—41no—
113C2×#3622—41no—
114#63218—48no—
115#7711—171nomaximal, Sylow
116C23×C17C2×C22⋊C411yescharacteristic
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17#10412yes—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17#13111yescharacteristic
137C2×C4×C17#13111yescharacteristic
138C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C17#13711yescharacteristic
141C2×C4×C17#13711yescharacteristic
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17#9912yes—
156C2×Dic17#10112yes—
157C4×D17—24no—
158C4×D17—24no—
159C4×D17—42no—
160C4×D17—42no—
161C4×D17—42no—
162C4×D17—42no—
163C8×C17—42no—
164C8×C17—81no—
165#3622—42no—
166#3622—81no—
167#2851—171no—
168#1797—171no—
169#1728—171no—
170#1271—171no—
171#1271—171no—
172#1271—171no—
173#1271D1711yescharacteristic, complemented
174C22×C17#40811yescharacteristic
175C22×C17C2×#9911yescharacteristic
176C22×C17C2×#10111yescharacteristic
177C22×C17#41711yescharacteristic
178C22×C17#41711yescharacteristic
179C22×C17C2×#10411yescharacteristic
180C22×C17#42111yescharacteristic
181C2×D17—22no—
182C2×D17—22no—
183C2×D17—22no—
184C2×D17—41no—
185C2×D17—41no—
186C2×D17—41no—
187C2×D17—41no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—21no—
193C4×C17—21no—
194C4×C17—81no—
195C4×C17—81no—
196Dic17—22no—
197Dic17—22no—
198Dic17—41no—
199Dic17—81no—
200Dic17—81no—
201C22×M16—341no—
202C2×#125—171no—
203C22×C42—171no—
204C2×C4⋊C8—341no—
205C2×C22⋊C8—341no—
206#389—171no—
207#389C2×D1711yescharacteristic
208#387—341no—
209#387—341no—
210#385—341no—
211#385—341no—
212#382—171no—
213#380—171no—
214#380—171no—
215#336—171no—
216#336—171no—
217#336—341no—
218#336C2×D1711yescharacteristic
219#336C2×D1711yescharacteristic
220C2×C17#147211yescharacteristic
221C2×C17#148311yescharacteristic
222C2×C17#149511yescharacteristic
223C2×C17#150411yescharacteristic
224C2×C17#151311yescharacteristic
225C2×C17#152811yescharacteristic
226C2×C17#156111yescharacteristic
227D17—42no—
228C23×C4—171no—
229C23×C4—171no—
230C23×C4—171no—
231C2×M16—344no—
232C2×M16—682no—
233C2×M16—682no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—341no—
237C22×C8—341no—
238#125—682no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—171no—
241C2×C4⋊C4—171no—
242C2×C4⋊C4—341no—
243C2×C4⋊C4—341no—
244C2×C4⋊C4—341no—
245C2×C4⋊C4C4×D1711yescharacteristic
246C2×C4⋊C4C4×D1711yescharacteristic
247C2×C22⋊C4—341no—
248C2×C22⋊C4—341no—
249C2×C42—171no—
250C2×C42—171no—
251C2×C42—171no—
252C2×C42—682no—
253C2×C42C22×D1711yescharacteristic
254C4⋊C8—682no—
255C22⋊C8—682no—
256#95—21no—
257#95—341no—
258#95—341no—
259#95—341no—
260#95—341no—
261#95—341no—
262C17#771111yescharacteristic, complemented, Sylow
263C24—171no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—342no—
270C22×C4—342no—
271C22×C4—342no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4—341no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4—681no—
281C22×C4—681no—
282C22×C4—681no—
283C22×C4C2×C4×D1711yescharacteristic
284C22×C4D4×D1711yescharacteristic
285C22×C4D4×D1711yescharacteristic
286M16—688no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—342no—
294C2×C8—681no—
295C2×C8—681no—
296C2×C8—681no—
297C2×C8—681no—
298C2×C8—681no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—341no—
305C4⋊C4—341no—
306C4⋊C4—681no—
307C4⋊C4—682no—
308C4⋊C4—682no—
309C4⋊C4—681no—
310C4⋊C4—681no—
311C22⋊C4—682no—
312C22⋊C4—682no—
313C42—41no—
314C42—342no—
315C42—342no—
316C42—681no—
317C23—172no—
318C23—341no—
319C23—341no—
320C23—341no—
321C23—341no—
322C23—341no—
323C23—341no—
324C23C22⋊C4×D1711yescharacteristic
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—172no—
338C2×C4—172no—
339C2×C4—344no—
340C2×C4—344no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—682no—
349C2×C4—682no—
350C2×C4—682no—
351C2×C4—682no—
352C2×C4—681no—
353C2×C4—681no—
354C2×C4—681no—
355C2×C4—681no—
356C2×C4—681no—
357C2×C4—681no—
358C2×C4#1058758711yescharacteristic
359C2×C4#1058759311yescharacteristic
360C2×C4#1058759711yescharacteristic
361C2×C4#1058760011yescharacteristic
362C2×C4#1058760111yescharacteristic
363C2×C4#1058760411yescharacteristic
364C8—42no—
365C8—81no—
366C8—682no—
367C8—1361no—
368C22—342no—
369C22—342no—
370C22—342no—
371C22—681no—
372C22—681no—
373C22—681no—
374C22—681no—
375C22#1176277411yescharacteristic
376C22#99×D1711yescharacteristic
377C22#101×D1711yescharacteristic
378C22#1176280711yescharacteristic
379C22#1176280911yescharacteristic
380C22#104×D1711yescharacteristic
381C22#1176282411yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—342no—
391C4—342no—
392C4—681no—
393C4—1361no—
394C4—1361no—
395C2—682no—
396C2#1452714611yescharacteristic
397C2#1452719611yescharacteristic
398C2#1452724611yescharacteristic
399C2#1452728511yescharacteristic
400C2#1452732611yescharacteristic
401C2#1452738411yescharacteristic
402C2#1452752011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1271D17
hypocenterC17#7711
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #14526608C2semidirect
5. #14526607C2semidirect
6. #14526606C2semidirect
7. C17×#1271C2semidirect
173. #1271D17semidirect
262. C17#7711semidirect

Maximal subgroups

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

subgroupindex
#145370932
#145287632
#145284762
#145266082
#145266072
#145266062
C17×#12712
#771117