Finite Group Databasebeta

← #16936779

Subgroups — #16936779

Order 4352.

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

normal subgroups only

This group has 500 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#14539729C211yesmaximal, characteristic
2#14533996C211yesmaximal, characteristic
3#14527466C211yesmaximal, characteristic
4#14527283C211yesmaximal, characteristic
5#14527179C211yesmaximal, characteristic
6#14526685C211yesmaximal, characteristic
7C17×#3078C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9#11763188—21no—
10#11763161—21no—
11#11763019—22no—
12#11762964—22no—
13#11762862C2211yescharacteristic
14#11762862C2211yescharacteristic
15#11762748—22no—
16#11762716—21no—
17#11762715—22no—
18#11762674—21no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762557—22no—
21#11762546—22no—
22Q8⋊#3622—22no—
23C17×#533C2211yescharacteristic
24C2×C4×Q8×C17C2211yescharacteristic
25C17×#479—22no—
26C17×#477—21no—
27C17×#475—21no—
28C17×#428—21no—
29C17×#423C2211yescharacteristic
30C17×#419—21no—
31C22×Q8×C17D411yescharacteristic
32C22×Dic34D412yes—
33C2×M16×C17—21no—
34C17×C4⋊Q8D411yescharacteristic
35C17×C4⋊Q8D411yescharacteristic
36C17×#125—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C4×Q8×C17—21no—
43C4×Q8×C17—21no—
44C4×Q8×C17—21no—
45C17×#117—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C42×C17—21no—
49C2×C42×C17C2311yescharacteristic
50C2×#63231—24no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4×Dic17—21no—
55#10587555—21no—
56#10587555—21no—
57#10587555—21no—
58#10587555—21no—
59C17×#107—22no—
60C17×#106—22no—
61C17×C4⋊C8—21no—
62C17×C4⋊C8D412yes—
63C17×#103—22no—
64C17×#103—22no—
65#10587502—24no—
66#10587502—24no—
67#10587502D412yes—
68C22⋊#3622—21no—
69C22⋊#3622—21no—
70#10587481—24no—
71C4⋊#3622—21no—
72C4⋊#3622D412yes—
73C4×#3622—21no—
74C4×#3622—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—21no—
82C2×Q8×C17Q1612yescomplemented
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×Dic17—22no—
89C22×Dic17—22no—
90C2×Dic34—22no—
91C2×Dic34—22no—
92C2×Dic34—44no—
93M16×C17—41no—
94M16×C17—41no—
95C2×C8×C17—22no—
96C4⋊C4×C17—21no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C42×C17—41no—
108C42×C17—41no—
109C42×C17—41no—
110C42×C17C2×D411yescharacteristic
111C42×C17C2×D411yescharacteristic
112C42×C17C2×D411yescharacteristic
113C42×C17#4111yescharacteristic
114#63231—84no—
115C4⋊Dic17—42no—
116C4⋊Dic17—41no—
117C4⋊Dic17—41no—
118C4⋊Dic17—41no—
119C4⋊Dic17—41no—
120C4×Dic17—41no—
121C4×Dic17—41no—
122C2×#3622—22no—
123C2×#3622—41no—
124#10252—171nomaximal, Sylow
125C23×C17#12011yescharacteristic
126Q8×C17—22no—
127Q8×C17—22no—
128Q8×C17—41no—
129Q8×C17—41no—
130Q8×C17—42no—
131Q8×C17—81no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—42no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17#12011yescharacteristic
142C2×C4×C17#12011yescharacteristic
143C2×C4×C17#12111yescharacteristic
144C2×C4×C17#12111yescharacteristic
145C2×C4×C17#12111yescharacteristic
146C2×C4×C17#12711yescharacteristic
147C2×C4×C17C2×Q1611yescharacteristic
148C2×C4×C17#13511yescharacteristic
149C2×C4×C17#13711yescharacteristic
150C2×C4×C17#13711yescharacteristic
151C2×Dic17—44no—
152C2×Dic17—42no—
153C2×Dic17—42no—
154C2×Dic17—42no—
155C2×Dic17—42no—
156Dic34—44no—
157Dic34—82no—
158C8×C17—42no—
159#3622—42no—
160#3622—81no—
161#3209—171no—
162#3078D1711yescharacteristic
163#2568—171no—
164#1548—171no—
165#1503—171no—
166#1479—171no—
167#1303—171no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17#39211yescharacteristic
171C22×C17#45111yescharacteristic
172C22×C17#45211yescharacteristic
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—42no—
177C4×C17—81no—
178C4×C17—81no—
179C4×C17—81no—
180C4×C17#45111yescharacteristic
181C4×C17#45211yescharacteristic
182C4×C17#45512yes—
183C4×C17#46711yescharacteristic
184C4×C17#47011yescharacteristic
185Dic17—82no—
186Dic17—82no—
187#533C2×D1711yescharacteristic
188C2×C4⋊Q8—171no—
189C2×C4×Q8C2×D1711yescharacteristic
190#494—341no—
191#487—341no—
192#479—22no—
193#477—21no—
194#475—21no—
195#462—342no—
196#451—342no—
197#431—171no—
198#431—171no—
199#428—21no—
200#423C2×D1711yescharacteristic
201#419—21no—
202#400—342no—
203#391—342no—
204#391—341no—
205#382—341no—
206C4×C4⋊C4—171no—
207#333—342no—
208#328—342no—
209Q8⋊C8—342no—
210C2×C17—21no—
211C2×C17—21no—
212C2×C17#200111yescharacteristic
213C2×C17#200411yescharacteristic
214C2×C17#200911yescharacteristic
215C22×Q8—172no—
216C22×Q8C22⋊D1711yescharacteristic
217C2×Q16—344no—
218C2×M16—21no—
219C4⋊Q8—341no—
220C4⋊Q8—341no—
221C4⋊Q8—341no—
222C4⋊Q8—341no—
223C4⋊Q8C22⋊D1711yescharacteristic
224C4⋊Q8C22⋊D1711yescharacteristic
225#125—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234#117—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—341no—
239C2×C4⋊C4C22⋊D1711yescharacteristic
240C2×C42—21no—
241C2×C42—341no—
242C2×C42C22×D1711yescharacteristic
243#107—22no—
244#106—22no—
245C4⋊C8—21no—
246C4⋊C8—172no—
247C4⋊C8—341no—
248C4⋊C8C22⋊D1712yes—
249#103—22no—
250#103—22no—
251#103—344no—
252C22⋊C8—341no—
253C22⋊C8—341no—
254C4×C8—341no—
255C4×C8—341no—
256#95—172no—
257#95—344no—
258#95—344no—
259C17#1025211yescharacteristic, complemented, Sylow
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—342no—
268C2×Q8—342no—
269C2×Q8—684no—
270C2×Q8#6323112yescomplemented
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—342no—
274C22×C4—342no—
275C22×C4D4×D1711yescharacteristic
276C22×C4D4×D1711yescharacteristic
277C22×C4C2×C22⋊D1711yescharacteristic
278Q16—1364no—
279M16—41no—
280M16—41no—
281C2×C8—22no—
282C2×C8—342no—
283C2×C8—681no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—682no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—681no—
298C4⋊C4—681no—
299C22⋊C4—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—681no—
304C42—681no—
305C42D4×D1711yescharacteristic
306C42#6325411yescharacteristic
307C42C2×C22⋊D1711yescharacteristic
308C42C2×C22⋊D1711yescharacteristic
309C23#1058762311yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—41no—
313Q8—41no—
314Q8—42no—
315Q8—81no—
316Q8—684no—
317Q8—1362no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—682no—
328C2×C4—684no—
329C2×C4—682no—
330C2×C4—682no—
331C2×C4—682no—
332C2×C4#1058762311yescharacteristic
333C2×C4#1058762411yescharacteristic
334C2×C4#1058762511yescharacteristic
335C2×C4#1058762511yescharacteristic
336C2×C4#1058762611yescharacteristic
337C2×C4#1058762811yescharacteristic
338C2×C4C2×#6323111yescharacteristic
339C2×C4#1058763611yescharacteristic
340C2×C4#1058763711yescharacteristic
341C2×C4#1058763911yescharacteristic
342C8—42no—
343C8—682no—
344C8—1361no—
345C22—21no—
346C22—21no—
347C22#1176271711yescharacteristic
348C22#1176296411yescharacteristic
349C22#1176296911yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—1362no—
358C4—1362no—
359C4#1176296511yescharacteristic
360C4#1176297111yescharacteristic
361C4#1176298512yes—
362C4#1176304811yescharacteristic
363C4#1176306511yescharacteristic
364C2—21no—
365C2—21no—
366C2#1452979211yescharacteristic
367C2#1452981011yescharacteristic
368C2#1452984111yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#3078C2
socleC22×C17#392
prime radicalC23×C17#120
hypercenter#3078D17
hypocenterC17#10252
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
82. C2×Q8×C17Q16semidirect
259. C17#10252semidirect
270. C2×Q8#63231semidirect

Maximal subgroups

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

subgroupindex
#145397292
#145339962
#145274662
#145272832
#145271792
#145266852
C17×#30782
#1025217