Finite Group Databasebeta

← #16923006

Subgroups — #16923006

Order 4352.

450 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 95 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#14528851C211yesmaximal, characteristic
3#14528763C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic, complemented
5#14526607C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1271C211yesmaximal, characteristic, complemented
8C2×#10587576C411yescharacteristic
9C2×C42×D17C2211yescharacteristic
10C2×#10587485—21no—
11C2×#10587491—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—
48#10587576—42no—
49C42×D17—42no—
50C17×#95—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587491—42no—
57#10587491—42no—
58#10587485—42no—
59C23×D17C22⋊C411yescharacteristic
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C411yescharacteristic
70C2×C4×D17—22no—
71C2×C4×D17—21no—
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C8×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17—41no—
85C2×C8×C17—41no—
86C2×C8×C17—41no—
87C2×C8×C17—41no—
88C4⋊C4×C17—21no—
89C4⋊C4×C17—21no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—41no—
93C42×C17—41no—
94#63227—42no—
95#63227—42no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17—42no—
99C4⋊Dic17—41no—
100#63225—42no—
101#63225—41no—
102#63225—41no—
103C4×Dic17—22no—
104C4×Dic17—22no—
105C4×Dic17—41no—
106C2×#3622—41no—
107C2×#3622—41no—
108C2×#3622—41no—
109C2×#3622—41no—
110C2×#3622—41no—
111C2×#3622—41no—
112#7731—171nomaximal, Sylow
113C23×C17C2×C22⋊C411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17—21no—
119C22×D17—21no—
120C22×D17#10112yes—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17#13111yescharacteristic
134C2×C4×C17#13111yescharacteristic
135C2×C4×C17#13511yescharacteristic
136C2×C4×C17#13511yescharacteristic
137C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C17#13711yescharacteristic
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17#9912yes—
154C4×D17—42no—
155C4×D17—42no—
156C4×D17—42no—
157C4×D17—42no—
158C4×D17—42no—
159C4×D17—42no—
160C8×C17—81no—
161C8×C17—81no—
162#3622—81no—
163#3622—81no—
164#2851—171no—
165#1817—171no—
166#1797—171no—
167#1271—171no—
168#1271—171no—
169#1271—171no—
170#1271D1711yescharacteristic, complemented
171C22×C17#40811yescharacteristic
172C22×C17C2×#9911yescharacteristic
173C22×C17C2×#10111yescharacteristic
174C22×C17#41611yescharacteristic
175C22×C17#41711yescharacteristic
176C22×C17#42111yescharacteristic
177C22×C17#42111yescharacteristic
178C2×D17—22no—
179C2×D17—22no—
180C2×D17—22no—
181C2×D17—41no—
182C2×D17—41no—
183C2×D17—41no—
184C2×D17—41no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—81no—
192C4×C17—81no—
193Dic17—22no—
194Dic17—41no—
195Dic17—41no—
196Dic17—81no—
197Dic17—81no—
198C2×#125—171no—
199C22×C42—171no—
200C2×C4⋊C8—341no—
201C2×C22⋊C8—341no—
202C2×C22⋊C8—341no—
203#389—171no—
204#389C2×D1711yescharacteristic
205#387—341no—
206#387—341no—
207#385—341no—
208#385—341no—
209#382—171no—
210#380—171no—
211#380—171no—
212#336—171no—
213#336—171no—
214#336—341no—
215#336C2×D1711yescharacteristic
216#336C2×D1711yescharacteristic
217C2×C17#147211yescharacteristic
218C2×C17#148211yescharacteristic
219C2×C17#149511yescharacteristic
220C2×C17#151311yescharacteristic
221C2×C17#151611yescharacteristic
222C2×C17#153411yescharacteristic
223C2×C17#155211yescharacteristic
224D17—42no—
225C23×C4—171no—
226C23×C4—171no—
227C23×C4—171no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—341no—
231C22×C8—341no—
232#125—682no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—171no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—341no—
239C2×C4⋊C4C4×D1711yescharacteristic
240C2×C4⋊C4C4×D1711yescharacteristic
241C2×C22⋊C4—341no—
242C2×C22⋊C4—341no—
243C2×C42—171no—
244C2×C42—171no—
245C2×C42—171no—
246C2×C42—682no—
247C2×C42C22×D1711yescharacteristic
248C4⋊C8—682no—
249C22⋊C8—682no—
250C22⋊C8—682no—
251#95—21no—
252#95—341no—
253#95—341no—
254#95—341no—
255#95—341no—
256#95—341no—
257C17#773111yescharacteristic, complemented, Sylow
258C24—171no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4—342no—
265C22×C4—342no—
266C22×C4—341no—
267C22×C4—341no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4—341no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4—681no—
275C22×C4—681no—
276C22×C4—681no—
277C22×C4—681no—
278C22×C4C2×C4×D1711yescharacteristic
279C22×C4D4×D1711yescharacteristic
280C22×C4D4×D1711yescharacteristic
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—681no—
288C2×C8—681no—
289C2×C8—681no—
290C2×C8—681no—
291C2×C8—681no—
292C2×C8—681no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—341no—
299C4⋊C4—341no—
300C4⋊C4—681no—
301C4⋊C4—682no—
302C4⋊C4—682no—
303C4⋊C4—681no—
304C4⋊C4—681no—
305C22⋊C4—682no—
306C22⋊C4—682no—
307C42—41no—
308C42—342no—
309C42—342no—
310C42—681no—
311C23—172no—
312C23—341no—
313C23—341no—
314C23—341no—
315C23—341no—
316C23—341no—
317C23—341no—
318C23C22⋊C4×D1711yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—172no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—682no—
341C2×C4—682no—
342C2×C4—682no—
343C2×C4—682no—
344C2×C4—682no—
345C2×C4—682no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4#1058758711yescharacteristic
353C2×C4#1058759311yescharacteristic
354C2×C4#1058759711yescharacteristic
355C2×C4#1058760111yescharacteristic
356C2×C4#1058760211yescharacteristic
357C2×C4#1058760511yescharacteristic
358C8—81no—
359C8—81no—
360C8—1361no—
361C8—1361no—
362C22—342no—
363C22—342no—
364C22—342no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22#1176277411yescharacteristic
370C22#99×D1711yescharacteristic
371C22#101×D1711yescharacteristic
372C22#1176280411yescharacteristic
373C22#1176280711yescharacteristic
374C22#1176282411yescharacteristic
375C22#1176282411yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—342no—
385C4—681no—
386C4—681no—
387C4—1361no—
388C4—1361no—
389C2—682no—
390C2#1452714611yescharacteristic
391C2#1452719111yescharacteristic
392C2#1452724611yescharacteristic
393C2#1452732411yescharacteristic
394C2#1452733911yescharacteristic
395C2#1452740911yescharacteristic
396C2#1452748311yescharacteristic
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#7731
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
170. #1271D17semidirect
257. C17#7731semidirect

Maximal subgroups

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

subgroupindex
#145370932
#145288512
#145287632
#145266082
#145266072
#145266062
C17×#12712
#773117