Finite Group Databasebeta

← #16936759

Subgroups — #16936759

Order 4352.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#14538577C211yesmaximal, characteristic
2#14533958C211yesmaximal, characteristic
3#14527469C211yesmaximal, characteristic, complemented
4#14527283C211yesmaximal, characteristic, complemented
5#14527174C211yesmaximal, characteristic
6#14526686C211yesmaximal, characteristic, complemented
7C17×#3230C211yesmaximal, characteristic, complemented
8C2×C4×D68C2211yescharacteristic
9#11763106—22no—
10#11763096—21no—
11#11763086—21no—
12#11762862C2211yescharacteristic
13#11762852—21no—
14#11762833C2211yescharacteristic
15#11762813—21no—
16#11762720—22no—
17#11762710—21no—
18#11762674—21no—
19#11762666—22no—
20C4×C4⋊Dic17C2211yescharacteristic
21#11762555—22no—
22#11762546—22no—
23#11762536—22no—
24C17×#533C2211yescharacteristic
25C2×C17×C4⋊Q8C2211yescharacteristic
26C17×#500—21no—
27C17×#487—21no—
28C17×#484—22no—
29C17×#477—22no—
30C17×#431C2211yescharacteristic
31C22×Q8×C17—21no—
32C22×D68D411yescharacteristic
33C22×C4×D17—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×C4⋊Q8D411yescharacteristic
39C17×C4⋊Q8D411yescharacteristic
40C17×#125—21no—
41C17×C22⋊Q8—21no—
42C17×C22⋊Q8—21no—
43C4×Q8×C17—21no—
44C4×Q8×C17—21no—
45C17×#117—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D412yes—
48C2×C42×C17C2311yescharacteristic
49C2×#63227—24no—
50C2×#63227—21no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17D411yescharacteristic
53C2×C4×Dic17—21no—
54C2×#63223—21no—
55C4×D68—21no—
56C4×D68—21no—
57C4×D68—21no—
58C4×D68—21no—
59C17×#107—24no—
60C17×C4⋊C8D412yes—
61C17×#103—24no—
62C17×C22⋊C8—21no—
63C4×C8×C17—21no—
64#10587502—24no—
65#10587502D412yes—
66C22⋊#3622—21no—
67#10587480—22no—
68#10587480—22no—
69#10587479—22no—
70#10587478—22no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4⋊#3622D412yes—
74C4×#3622—21no—
75C23×D17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—41no—
79C2×Q8×C17—41no—
80C2×Q8×C17—82no—
81C22×C4×C17—22no—
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×Dic17—21no—
86C22×Dic17—22no—
87C2×D68—21no—
88C2×D68—21no—
89C2×D68—21no—
90C2×D68—21no—
91C2×D68—21no—
92C2×D68Q1612yes—
93C2×C4×D17—44no—
94C2×C4×D17—41no—
95C2×C4×D17—41no—
96C2×C8×C17—22no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—22no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C42×C17—41no—
109C42×C17C2×D411yescharacteristic
110C42×C17C2×D411yescharacteristic
111C42×C17C2×D411yescharacteristic
112C42×C17#4111yescharacteristic
113#63227—41no—
114#63227—41no—
115#63227—84no—
116C4⋊Dic17—21no—
117C4⋊Dic17—21no—
118C4⋊Dic17—42no—
119C4×Dic17—41no—
120C4×Dic17—41no—
121#63223—41no—
122#63223—41no—
123C2×#3622—22no—
124C2×#3622—41no—
125#10249—171nomaximal, Sylow
126C23×C17#12011yescharacteristic
127C22×D17—22no—
128C22×D17—42no—
129C22×D17—42no—
130C22×D17—41no—
131C22×D17—41no—
132Q8×C17—42no—
133Q8×C17—81no—
134Q8×C17—82no—
135Q8×C17—82no—
136C2×C4×C17—22no—
137C2×C4×C17—41no—
138C2×C4×C17—44no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—42no—
142C2×C4×C17#12011yescharacteristic
143C2×C4×C17#12011yescharacteristic
144C2×C4×C17#12111yescharacteristic
145C2×C4×C17#12111yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12711yescharacteristic
148C2×C4×C17C2×Q1611yescharacteristic
149C2×C4×C17#13511yescharacteristic
150C2×C4×C17#13711yescharacteristic
151C2×C4×C17#13711yescharacteristic
152C2×Dic17—42no—
153C2×Dic17—42no—
154C2×Dic17—42no—
155C2×Dic17—41no—
156C2×Dic17—42no—
157D68—22no—
158D68—22no—
159D68—41no—
160D68—41no—
161C4×D17—82no—
162C4×D17—82no—
163C8×C17—42no—
164#3622—42no—
165#3622—81no—
166#3230D1711yescharacteristic, complemented
167#3077—171no—
168#2563—171no—
169#1549—171no—
170#1503—171no—
171#1478—171no—
172#1303—171no—
173C22×C17—21no—
174C22×C17—21no—
175C22×C17#39211yescharacteristic
176C22×C17#45111yescharacteristic
177C22×C17#45211yescharacteristic
178C2×D17—44no—
179C2×D17—42no—
180C2×D17—82no—
181C2×D17—81no—
182C2×D17—81no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—81no—
187C4×C17—82no—
188C4×C17—81no—
189C4×C17#45111yescharacteristic
190C4×C17#45211yescharacteristic
191C4×C17#45412yes—
192C4×C17#46711yescharacteristic
193C4×C17#47011yescharacteristic
194Dic17—82no—
195Dic17—81no—
196#533C2×D1711yescharacteristic
197C2×C4⋊Q8C2×D1711yescharacteristic
198C2×C4×D4—171no—
199#500—21no—
200#487—21no—
201#484—22no—
202#478—342no—
203#477—22no—
204#476—341no—
205#474—341no—
206#431—171no—
207#431C2×D1711yescharacteristic
208#428—341no—
209#423—171no—
210#418—341no—
211#393—342no—
212#390—341no—
213#382—341no—
214#380—342no—
215C4×C4⋊C4—171no—
216#332—342no—
217#328—342no—
218D4⋊C8—342no—
219C2×C17—21no—
220C2×C17—21no—
221C2×C17#200111yescharacteristic
222C2×C17#200311yescharacteristic
223C2×C17#200811yescharacteristic
224D17—82no—
225C22×Q8—21no—
226C22×D4—171no—
227C23×C4—341no—
228C2×M16—341no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22⋊D1711yescharacteristic
234C4⋊Q8C22⋊D1711yescharacteristic
235#125—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—341no—
241C4×D4—341no—
242C4×D4—341no—
243C4×D4—341no—
244#117—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—171no—
247C2×C4⋊C4—341no—
248C2×C4⋊C4C22⋊D1712yes—
249C2×C22⋊C4—341no—
250C2×C22⋊C4—344no—
251C2×C42—341no—
252C2×C42C22×D1711yescharacteristic
253#107—24no—
254#107—342no—
255#106—342no—
256C4⋊C8—172no—
257C4⋊C8—341no—
258C4⋊C8—341no—
259C4⋊C8C22⋊D1712yes—
260#103—24no—
261#102—342no—
262#102—342no—
263C22⋊C8—21no—
264C22⋊C8—341no—
265C4×C8—21no—
266C4×C8—341no—
267#95—172no—
268#95—344no—
269C17#1024911yescharacteristic, complemented, Sylow
270C24—341no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×D4—172no—
277C2×D4—341no—
278C2×D4—341no—
279C2×D4—341no—
280C2×D4—341no—
281C2×D4—341no—
282C22×C4—22no—
283C22×C4—342no—
284C22×C4—341no—
285C22×C4—684no—
286C22×C4—681no—
287C22×C4—681no—
288C22×C4D4×D1711yescharacteristic
289C22×C4D4×D1711yescharacteristic
290C22×C4C2×C22⋊D1711yescharacteristic
291M16—681no—
292M16—681no—
293C2×C8—22no—
294C2×C8—342no—
295C2×C8—681no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—341no—
307C4⋊C4—341no—
308C4⋊C4—682no—
309C22⋊C4—41no—
310C22⋊C4—681no—
311C22⋊C4—681no—
312C22⋊C4—1364no—
313C42—41no—
314C42—681no—
315C42—681no—
316C42D4×D1711yescharacteristic
317C42#6325411yescharacteristic
318C42C2×C22⋊D1711yescharacteristic
319C42C2×C22⋊D1711yescharacteristic
320C23—342no—
321C23—682no—
322C23—682no—
323C23—681no—
324C23—681no—
325C23#1058762311yescharacteristic
326Q8—42no—
327Q8—81no—
328Q8—82no—
329Q8—82no—
330D4—342no—
331D4—342no—
332D4—681no—
333D4—681no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—682no—
341C2×C4—682no—
342C2×C4—682no—
343C2×C4—682no—
344C2×C4—681no—
345C2×C4—1362no—
346C2×C4—1362no—
347C2×C4Q16×D1711yescharacteristic
348C2×C4#1058760411yescharacteristic
349C2×C4#1058760411yescharacteristic
350C2×C4#1058760511yescharacteristic
351C2×C4#1058762311yescharacteristic
352C2×C4#1058762411yescharacteristic
353C2×C4#1058762511yescharacteristic
354C2×C4#1058762511yescharacteristic
355C2×C4#1058762611yescharacteristic
356C2×C4#1058763911yescharacteristic
357C8—42no—
358C8—682no—
359C8—1361no—
360C22—21no—
361C22—21no—
362C22—684no—
363C22—682no—
364C22—1362no—
365C22—1361no—
366C22—1361no—
367C22#1176271711yescharacteristic
368C22#1176296311yescharacteristic
369C22#1176297011yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—81no—
374C4—82no—
375C4—81no—
376C4—1362no—
377C4—1361no—
378C4#1176296311yescharacteristic
379C4#1176297011yescharacteristic
380C4#1176298212yes—
381C4#1176304711yescharacteristic
382C4#1176306411yescharacteristic
383C2—21no—
384C2—21no—
385C2—1362no—
386C2#1452979111yescharacteristic
387C2#1452980411yescharacteristic
388C2#1452983511yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#3230C2
socleC22×C17#392
prime radicalC22×D68D4
hypercenter#3230D17
hypocenterC17#10249
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
3. #14527469C2semidirect
4. #14527283C2semidirect
6. #14526686C2semidirect
7. C17×#3230C2semidirect
166. #3230D17semidirect
269. C17#10249semidirect

Maximal subgroups

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

subgroupindex
#145385772
#145339582
#145274692
#145272832
#145271742
#145266862
C17×#32302
#1024917