Finite Group Databasebeta

← #16922569

Subgroups — #16922569

Order 4352.

590 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 105 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#14531265C211yesmaximal, characteristic
2#14528476C211yesmaximal, characteristic
3#14528476C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic, complemented
5#14526607C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1271C211yesmaximal, characteristic, complemented
8C22×#63218—21no—
9C22×#63218—21no—
10C2×#10587578C411yescharacteristic
11C2×#10587578C411yescharacteristic
12C2×C42×D17C2211yescharacteristic
13#11762708C411yescharacteristic
14#11762707C2211yescharacteristic
15#11762705C411yescharacteristic
16#11762683—21no—
17#11762680—21no—
18C4⋊C4×Dic17—21no—
19#11762660—21no—
20#11762564C2211yescharacteristic
21#11762564C2211yescharacteristic
22#11762563—21no—
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×Dic17—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C22×#3622—21no—
49C22×#3622—21no—
50C2×#63218—24no—
51C2×#63218—24no—
52C2×#63218—42no—
53C2×#63218—42no—
54C2×#63218—42no—
55C2×#63218—42no—
56#10587578—22no—
57#10587578—22no—
58#10587578—42no—
59C42×D17—42no—
60C17×#95—21no—
61#10587502—21no—
62#10587502—21no—
63#10587502—21no—
64C23×D17C22⋊C411yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C22⋊C411yescharacteristic
74C22×Dic17C22⋊C411yescharacteristic
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—21no—
79C2×C4×D17—21no—
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82C2×C4×D17—21no—
83C2×C4×D17—21no—
84C2×C4×D17—22no—
85C2×C4×D17—22no—
86C2×C4×D17—22no—
87C2×C8×C17—22no—
88C2×C8×C17—22no—
89C2×C8×C17—41no—
90C2×C8×C17—41no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17M1612yescomplemented
98C42×C17—41no—
99#63227—24no—
100#63227—42no—
101C4⋊Dic17—21no—
102C4⋊Dic17—42no—
103C4⋊Dic17—41no—
104C4⋊Dic17M1612yescomplemented
105#63225—41no—
106#63225—41no—
107#63225—42no—
108C4×Dic17—24no—
109C4×Dic17—41no—
110C4×Dic17—41no—
111C4×Dic17—41no—
112C4×Dic17—42no—
113C2×#3622—22no—
114C2×#3622—22no—
115C2×#3622—41no—
116C2×#3622—41no—
117C2×#3622—41no—
118C2×#3622—41no—
119#63218—48no—
120#63218—48no—
121#7632—171nomaximal, Sylow
122C23×C17C2×C22⋊C411yescharacteristic
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17#9912yes—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—22no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17C2×M1611yescharacteristic
143C2×C4×C17C2×M1611yescharacteristic
144C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C17#13711yescharacteristic
147C2×C4×C17#13711yescharacteristic
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—22no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17#10112yes—
161C2×Dic17#10412yes—
162C2×Dic17#10412yes—
163C4×D17—24no—
164C4×D17—24no—
165C4×D17—24no—
166C4×D17—24no—
167C4×D17—24no—
168C4×D17—24no—
169C8×C17—42no—
170C8×C17—42no—
171#3622—42no—
172#3622—42no—
173#2296—171no—
174#1728—171no—
175#1728—171no—
176#1271—171no—
177#1271—171no—
178#1271—171no—
179#1271D1711yescharacteristic, complemented
180C22×C17#40711yescharacteristic
181C22×C17C2×#9911yescharacteristic
182C22×C17C2×#10111yescharacteristic
183C22×C17#41711yescharacteristic
184C22×C17#41711yescharacteristic
185C22×C17C2×#10411yescharacteristic
186C22×C17C2×#10411yescharacteristic
187C2×D17—22no—
188C2×D17—22no—
189C2×D17—22no—
190C2×D17—41no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—42no—
201C4×C17—81no—
202Dic17—22no—
203Dic17—22no—
204Dic17—22no—
205Dic17—42no—
206Dic17—81no—
207C22×M16—341no—
208C22×M16—341no—
209C2×#117—171no—
210C2×#117—171no—
211C22×C42—171no—
212#389—171no—
213#389—171no—
214#389—171no—
215#389C2×D1711yescharacteristic
216#385—341no—
217#385—341no—
218C4×C4⋊C4—341no—
219C4×C4⋊C4—341no—
220#336—171no—
221#336—171no—
222#336—341no—
223#336—341no—
224#336C2×D1711yescharacteristic
225#336C2×D1711yescharacteristic
226C2×C17#146311yescharacteristic
227C2×C17#148911yescharacteristic
228C2×C17#148911yescharacteristic
229C2×C17#150411yescharacteristic
230C2×C17#150411yescharacteristic
231C2×C17#152611yescharacteristic
232C2×C17#156111yescharacteristic
233D17—42no—
234C23×C4—171no—
235C23×C4—171no—
236C23×C4—171no—
237C2×M16—344no—
238C2×M16—344no—
239C2×M16—682no—
240C2×M16—682no—
241C2×M16—682no—
242C2×M16—682no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—341no—
246C22×C8—341no—
247#117—342no—
248#117—342no—
249#117—682no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—171no—
252C2×C4⋊C4—171no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—341no—
255C2×C4⋊C4—341no—
256C2×C4⋊C4C4×D1711yescharacteristic
257C2×C4⋊C4C4×D1711yescharacteristic
258C2×C22⋊C4—341no—
259C2×C22⋊C4—341no—
260C2×C42—171no—
261C2×C42—171no—
262C2×C42—171no—
263C2×C42—341no—
264C2×C42—341no—
265C2×C42—682no—
266C2×C42C22×D1711yescharacteristic
267#95—21no—
268#95—341no—
269#95—341no—
270#95—341no—
271C17#763211yescharacteristic, complemented, Sylow
272C24—171no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—171no—
276C22×C4—171no—
277C22×C4—171no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—342no—
281C22×C4—342no—
282C22×C4—342no—
283C22×C4—342no—
284C22×C4—341no—
285C22×C4—341no—
286C22×C4—341no—
287C22×C4—341no—
288C22×C4—341no—
289C22×C4—341no—
290C22×C4—341no—
291C22×C4—341no—
292C22×C4C2×C4×D1711yescharacteristic
293C22×C4D4×D1711yescharacteristic
294C22×C4D4×D1711yescharacteristic
295M16—688no—
296M16—688no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—342no—
304C2×C8—342no—
305C2×C8—681no—
306C2×C8—681no—
307C2×C8—681no—
308C2×C8—681no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—172no—
314C4⋊C4—341no—
315C4⋊C4—681no—
316C4⋊C4—682no—
317C4⋊C4—682no—
318C4⋊C4—681no—
319C4⋊C4—681no—
320C4⋊C4#6321812yescomplemented
321C22⋊C4—344no—
322C22⋊C4—682no—
323C42—41no—
324C42—344no—
325C42—681no—
326C42—681no—
327C42—681no—
328C42—682no—
329C23—172no—
330C23—341no—
331C23—341no—
332C23—341no—
333C23—341no—
334C23—341no—
335C23—341no—
336C23C22⋊C4×D1711yescharacteristic
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—172no—
350C2×C4—172no—
351C2×C4—172no—
352C2×C4—344no—
353C2×C4—344no—
354C2×C4—344no—
355C2×C4—344no—
356C2×C4—344no—
357C2×C4—344no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—341no—
362C2×C4—341no—
363C2×C4—341no—
364C2×C4—342no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4C2×#6321811yescharacteristic
371C2×C4M16×D1711yescharacteristic
372C2×C4#1058759711yescharacteristic
373C2×C4#1058760011yescharacteristic
374C2×C4#1058760111yescharacteristic
375C2×C4#1058760411yescharacteristic
376C8—42no—
377C8—42no—
378C8—682no—
379C8—682no—
380C22—342no—
381C22—342no—
382C22—342no—
383C22—681no—
384C22—681no—
385C22—681no—
386C22—681no—
387C22#1176277111yescharacteristic
388C22#99×D1711yescharacteristic
389C22#101×D1711yescharacteristic
390C22#1176280711yescharacteristic
391C22#1176280911yescharacteristic
392C22#104×D1711yescharacteristic
393C22#104×D1711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—342no—
403C4—342no—
404C4—342no—
405C4—682no—
406C4—1361no—
407C2—682no—
408C2#1452711211yescharacteristic
409C2#1452722011yescharacteristic
410C2#1452722211yescharacteristic
411C2#1452728511yescharacteristic
412C2#1452728711yescharacteristic
413C2#1452737711yescharacteristic
414C2#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#7632
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
97. C4⋊C4×C17M16semidirect
104. C4⋊Dic17M16semidirect
179. #1271D17semidirect
271. C17#7632semidirect
320. C4⋊C4#63218semidirect

Maximal subgroups

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

subgroupindex
#145312652
#145284762
#145266082
#145266072
#145266062
C17×#12712
#763217