Finite Group Databasebeta

← #16922285

Subgroups — #16922285

Order 4352.

596 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531314C211yesmaximal, characteristic
2#14528541C211yesmaximal, characteristic
3D17×#336C211yesmaximal, characteristic
4#14526609C211yesmaximal, characteristic, complemented
5#14526609C211yesmaximal, characteristic, complemented
6#14526607C211yesmaximal, characteristic, complemented
7C17×#1271C211yesmaximal, characteristic, complemented
8C22×C8×D17—21no—
9C2×#10587583C411yescharacteristic
10C2×#10587573C411yescharacteristic
11C2×C42×D17C2211yescharacteristic
12C2×#10587491—21no—
13C2×C8×Dic17—21no—
14#11762715C2211yescharacteristic
15#11762692—21no—
16#11762680—21no—
17#11762666C411yescharacteristic
18C4⋊C4×Dic17—21no—
19C4⋊C4×Dic17C411yescharacteristic
20#11762656—21no—
21#11762564C2211yescharacteristic
22#11762564C2211yescharacteristic
23#11762563—21no—
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×C4×D17C2×C411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×Dic34C2×C411yescharacteristic
31C22×C8×C17—21no—
32C22×C8×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C42×C17C2311yescharacteristic
37C2×#63227—21no—
38C2×#63227—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C22×#3622—21no—
48C22×#3622—21no—
49C2×C8×D17—24no—
50C2×C8×D17—42no—
51C2×C8×D17—42no—
52#10587583—22no—
53#10587583C814yescomplemented
54#10587573—42no—
55C42×D17—42no—
56C17×#95—21no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61#10587491—24no—
62C8×Dic17—42no—
63C23×D17C22⋊C411yescharacteristic
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C22⋊C411yescharacteristic
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17C2×C812yes—
83C2×C4×D17M1612yes—
84C2×C4×D17SD1612yes—
85C2×C4×D17Q1612yes—
86C2×Dic34—21no—
87C2×Dic34—41no—
88C2×Dic34—41no—
89C2×Dic34C2×C812yescomplemented
90C2×C8×C17—22no—
91C2×C8×C17—22no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—41no—
96C4⋊C4×C17—21no—
97C4⋊C4×C17—41no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17C2×C812yescomplemented
101C42×C17—41no—
102#63227—24no—
103#63227—42no—
104C4⋊Dic17—24no—
105C4⋊Dic17—41no—
106#63225—42no—
107C4×Dic17—22no—
108C4×Dic17—22no—
109C4×Dic17—41no—
110C4×Dic17—42no—
111C4×Dic17—42no—
112C2×#3622—22no—
113C2×#3622—22no—
114C2×#3622—41no—
115C2×#3622—41no—
116C2×#3622—41no—
117C2×#3622—41no—
118C8×D17—48no—
119#7566—171nomaximal, Sylow
120C23×C17C2×C22⋊C411yescharacteristic
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17C22⋊C812yes—
126C22×D17#10312yes—
127C22×D17#10412yes—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—22no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17C22×C811yescharacteristic
141C2×C4×C17C2×M1611yescharacteristic
142C2×C4×C17C2×SD1611yescharacteristic
143C2×C4×C17C2×Q1611yescharacteristic
144C2×C4×C17#13511yescharacteristic
145C2×C4×C17#13511yescharacteristic
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—22no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17C22⋊C812yes—
158C2×Dic17#9912yes—
159C2×Dic17#10312yes—
160C2×Dic17#10412yes—
161C4×D17—24no—
162C4×D17—24no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—42no—
166C4×D17—42no—
167Dic34—42no—
168Dic34—42no—
169C8×C17—42no—
170C8×C17—42no—
171#3622—42no—
172#3622—42no—
173C4×C22⋊Q8—171no—
174C4×C22⋊C8—171no—
175C2×#336—171no—
176#1272—171no—
177#1272—171no—
178#1271—171no—
179#1271D1711yescharacteristic, complemented
180C22×C17C2×C22⋊C811yescharacteristic
181C22×C17C2×#9911yescharacteristic
182C22×C17#41311yescharacteristic
183C22×C17C2×#10311yescharacteristic
184C22×C17#41611yescharacteristic
185C22×C17C2×#10411yescharacteristic
186C22×C17C2×#10411yescharacteristic
187C2×D17—22no—
188C2×D17—22no—
189C2×D17—22no—
190C2×D17—22no—
191C2×D17—22no—
192C2×D17—22no—
193C2×D17Q8⋊C814yes—
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—42no—
204Dic17—41no—
205Dic17—81no—
206Dic17#32814yes—
207C23×C8—341no—
208C2×C22⋊Q8—171no—
209C2×C4×Q8—171no—
210C22×C42—171no—
211C2×C22⋊C8—341no—
212C2×C4×C8—341no—
213#391—171no—
214#389C2×D1711yescharacteristic
215#387—341no—
216#385—341no—
217#380—171no—
218C4×C4⋊C4—171no—
219C4×C4⋊C4—341no—
220C4×C22⋊C4—341no—
221#336—171no—
222#336—171no—
223#336—341no—
224#336C2×D1711yescharacteristic
225#336C2×D1711yescharacteristic
226C2×C17#145911yescharacteristic
227C2×C17C2×Q8⋊C811yescharacteristic
228C2×C17#147611yescharacteristic
229C2×C17C2×#32811yescharacteristic
230C2×C17#150111yescharacteristic
231C2×C17#152411yescharacteristic
232C2×C17#154211yescharacteristic
233D17—24no—
234C22×Q8—171no—
235C23×C4—171no—
236C23×C4—171no—
237C23×C4—171no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—344no—
241C22×C8—341no—
242C22×C8—341no—
243C22×C8—682no—
244C22×C8—682no—
245C22⋊Q8—174no—
246C22⋊Q8—342no—
247C4×Q8—682no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—171no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4C4×D1711yescharacteristic
253C2×C4⋊C4C4×D1711yescharacteristic
254C2×C22⋊C4—341no—
255C2×C22⋊C4—341no—
256C2×C42—171no—
257C2×C42—171no—
258C2×C42—171no—
259C2×C42—341no—
260C2×C42—341no—
261C2×C42—682no—
262C2×C42C22×D1711yescharacteristic
263C22⋊C8—344no—
264C4×C8—682no—
265#95—21no—
266#95—341no—
267#95—341no—
268#95—341no—
269#95—341no—
270C17#756611yescharacteristic, complemented, Sylow
271C24—171no—
272C2×Q8—172no—
273C2×Q8—341no—
274C2×Q8—681no—
275C2×Q8—681no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—171no—
279C22×C4—171no—
280C22×C4—172no—
281C22×C4—172no—
282C22×C4—172no—
283C22×C4—172no—
284C22×C4—171no—
285C22×C4—342no—
286C22×C4—342no—
287C22×C4—342no—
288C22×C4—341no—
289C22×C4—341no—
290C22×C4—341no—
291C22×C4—341no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4C2×C4×D1711yescharacteristic
296C22×C4D4×D1711yescharacteristic
297C22×C4D4×D1711yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—342no—
305C2×C8—342no—
306C2×C8—688no—
307C2×C8—681no—
308C2×C8—681no—
309C2×C8—681no—
310C2×C8—681no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—344no—
316C4⋊C4—681no—
317C4⋊C4—682no—
318C4⋊C4C8×D1712yescomplemented
319C22⋊C4—344no—
320C22⋊C4—682no—
321C42—41no—
322C42—342no—
323C42—342no—
324C42—681no—
325C42—682no—
326C42—682no—
327C23—172no—
328C23—172no—
329C23—172no—
330C23—341no—
331C23—341no—
332C23—341no—
333C23—341no—
334C23C22⋊C4×D1711yescharacteristic
335Q8—682no—
336Q8—682no—
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—172no—
353C2×C4—344no—
354C2×C4—344no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—342no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—341no—
361C2×C4—682no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4C2×C8×D1711yescharacteristic
371C2×C4M16×D1711yescharacteristic
372C2×C4#1058759811yescharacteristic
373C2×C4SD16×D1711yescharacteristic
374C2×C4#1058760211yescharacteristic
375C2×C4Q16×D1711yescharacteristic
376C8—42no—
377C8—42no—
378C8—682no—
379C8—682no—
380C22—174no—
381C22—342no—
382C22—342no—
383C22—342no—
384C22—342no—
385C22—342no—
386C22—342no—
387C22C22⋊C8×D1711yescharacteristic
388C22#99×D1711yescharacteristic
389C22#1176279211yescharacteristic
390C22#103×D1711yescharacteristic
391C22#1176280211yescharacteristic
392C22#104×D1711yescharacteristic
393C22#104×D1711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—174no—
403C4—342no—
404C4—681no—
405C4—682no—
406C4—1361no—
407C2—344no—
408C2#1452710111yescharacteristic
409C2D17×Q8⋊C811yescharacteristic
410C2#1452716311yescharacteristic
411C2D17×#32811yescharacteristic
412C2#1452727211yescharacteristic
413C2#1452737111yescharacteristic
414C2#1452744011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC2×C4×C17C22×C8
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1271D17
hypocenterC17#7566
LCSG ▷ C2×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. #14526609C2semidirect
5. #14526609C2semidirect
6. #14526607C2semidirect
7. C17×#1271C2semidirect
53. #10587583C8semidirect
89. C2×Dic34C2×C8semidirect
100. C4⋊C4×C17C2×C8semidirect
179. #1271D17semidirect
270. C17#7566semidirect
318. C4⋊C4C8×D17semidirect

Maximal subgroups

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

subgroupindex
#145313142
#145285412
D17×#3362
#145266092
#145266072
C17×#12712
#756617