Finite Group Databasebeta

← #16936781

Subgroups — #16936781

Order 4352.

500 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 101 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#14538585C211yesmaximal, characteristic
3#14533999C211yesmaximal, characteristic
4#14527464C211yesmaximal, characteristic
5#14527180C211yesmaximal, characteristic
6#14526687C211yesmaximal, characteristic
7C17×#1503C211yesmaximal, characteristic
8C2×#10587555C2211yescharacteristic
9C2×C4×Dic34C2211yescharacteristic
10#11763188—21no—
11#11763161—21no—
12#11763113—22no—
13#11763101—21no—
14#11763091—21no—
15#11763019—22no—
16#11762964—22no—
17#11762862C2211yescharacteristic
18#11762852—21no—
19#11762833C2211yescharacteristic
20#11762817—21no—
21#11762749—22no—
22#11762715—21no—
23#11762714—22no—
24#11762675—21no—
25#11762557—22no—
26#11762541—22no—
27C17×#533C2211yescharacteristic
28C17×#431C2211yescharacteristic
29C4×C4⋊C4×C17C2211yescharacteristic
30C17×#328—22no—
31C22×Dic34D411yescharacteristic
32C22×Dic34D412yes—
33C17×C4⋊Q8D411yescharacteristic
34C17×C4⋊Q8D411yescharacteristic
35C17×#125—21no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C17×#117—21no—
41C2×C4⋊C4×C17—21no—
42C2×C42×C17—21no—
43C2×C42×C17C2311yescharacteristic
44C2×#63231—24no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47C2×C4⋊Dic17D411yescharacteristic
48C2×#63225—21no—
49C2×C4×Dic17—21no—
50C2×#63223—21no—
51#10587555—21no—
52#10587555—21no—
53#10587555—21no—
54#10587555—21no—
55C4×Dic34—21no—
56C4×Dic34—21no—
57C4×Dic34—21no—
58C4×Dic34—21no—
59C17×C4⋊C8—21no—
60C17×C22⋊C8—21no—
61C4×C8×C17—21no—
62C17×#95D412yes—
63#10587502—24no—
64#10587502—24no—
65C22⋊#3622—21no—
66#10587481—24no—
67#10587481—22no—
68#10587481—22no—
69#10587479—22no—
70#10587478—22no—
71C4⋊#3622—21no—
72C4⋊#3622D412yes—
73C4⋊#3622D412yes—
74C4×#3622—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C22×C4×C17—22no—
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×Dic17—21no—
82C22×Dic17—22no—
83C22×Dic17—21no—
84C2×Dic34—21no—
85C2×Dic34—21no—
86C2×Dic34—21no—
87C2×Dic34—21no—
88C2×Dic34—21no—
89C2×Dic34—22no—
90C2×Dic34—22no—
91C2×Dic34—44no—
92C2×Dic34Q1612yescomplemented
93C2×C8×C17—41no—
94C4⋊C4×C17—21no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17—21no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—42no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C22⋊C4×C17—41no—
102C42×C17—41no—
103C42×C17—41no—
104C42×C17—41no—
105C42×C17C2×D411yescharacteristic
106C42×C17C2×D411yescharacteristic
107C42×C17C2×D411yescharacteristic
108C42×C17#4111yescharacteristic
109#63231—84no—
110C4⋊Dic17—21no—
111C4⋊Dic17—21no—
112C4⋊Dic17—41no—
113C4⋊Dic17—41no—
114C4⋊Dic17—41no—
115C4⋊Dic17—41no—
116#63225—41no—
117#63225—41no—
118C4×Dic17—41no—
119C4×Dic17—41no—
120#63223—41no—
121#63223—41no—
122C2×#3622—22no—
123C2×#3622—22no—
124#10252—171nomaximal, Sylow
125C23×C17#12011yescharacteristic
126Q8×C17—42no—
127Q8×C17—81no—
128C2×C4×C17—22no—
129C2×C4×C17—41no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—42no—
135C2×C4×C17#12011yescharacteristic
136C2×C4×C17#12011yescharacteristic
137C2×C4×C17#12111yescharacteristic
138C2×C4×C17#12111yescharacteristic
139C2×C4×C17#12111yescharacteristic
140C2×C4×C17#12711yescharacteristic
141C2×C4×C17C2×Q1611yescharacteristic
142C2×C4×C17#13511yescharacteristic
143C2×C4×C17#13711yescharacteristic
144C2×C4×C17#13711yescharacteristic
145C2×Dic17—22no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—44no—
149C2×Dic17—42no—
150C2×Dic17—42no—
151C2×Dic17—41no—
152Dic34—22no—
153Dic34—22no—
154Dic34—41no—
155Dic34—41no—
156Dic34—44no—
157Dic34—82no—
158C8×C17—81no—
159#3622—42no—
160#3622—42no—
161#3209—171no—
162#3078—171no—
163#2568—171no—
164#1548—171no—
165#1503D1711yescharacteristic
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—82no—
177C4×C17—81no—
178C4×C17—81no—
179C4×C17#45111yescharacteristic
180C4×C17#45211yescharacteristic
181C4×C17#45512yes—
182C4×C17#46711yescharacteristic
183C4×C17#47011yescharacteristic
184Dic17—42no—
185Dic17—82no—
186Dic17—81no—
187#533C2×D1711yescharacteristic
188C2×C4⋊Q8—171no—
189C2×C4×Q8—171no—
190#494—341no—
191#487—341no—
192#479—342no—
193#477—341no—
194#475—341no—
195#462—342no—
196#451—342no—
197#431—171no—
198#431C2×D1711yescharacteristic
199#428—341no—
200#423—171no—
201#419—341no—
202#400—342no—
203#391—341no—
204#391—342no—
205#382—341no—
206C4×C4⋊C4C2×D1711yescharacteristic
207#333—342no—
208#328—22no—
209Q8⋊C8—342no—
210C2×C17—21no—
211C2×C17—21no—
212C2×C17#200111yescharacteristic
213C2×C17#200411yescharacteristic
214C2×C17#200911yescharacteristic
215C22×Q8—171no—
216C22×Q8—172no—
217C2×Q16—344no—
218C2×M16—341no—
219C4⋊Q8—341no—
220C4⋊Q8—341no—
221C4⋊Q8—341no—
222C4⋊Q8—341no—
223C4⋊Q8D6811yescharacteristic
224C4⋊Q8D6811yescharacteristic
225#125—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—341no—
231C4×Q8—341no—
232C4×Q8—341no—
233C4×Q8—341no—
234#117—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—171no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—341no—
239C2×C4⋊C4—341no—
240C2×C42—21no—
241C2×C42—341no—
242C2×C42C22×D1711yescharacteristic
243#107—342no—
244#106—342no—
245C4⋊C8—21no—
246C4⋊C8—172no—
247C4⋊C8—172no—
248C4⋊C8—341no—
249#103—342no—
250#103—342no—
251#103—344no—
252C22⋊C8—21no—
253C22⋊C8—341no—
254C4×C8—21no—
255C4×C8—341no—
256#95—344no—
257#95—344no—
258#95C22⋊D1712yes—
259C17#1025211yescharacteristic, complemented, Sylow
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—172no—
263C2×Q8—341no—
264C2×Q8—341no—
265C2×Q8—342no—
266C2×Q8—341no—
267C2×Q8—342no—
268C2×Q8—341no—
269C2×Q8—341no—
270C2×Q8—684no—
271C22×C4—22no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4—342no—
275C22×C4D4×D1711yescharacteristic
276C22×C4D4×D1711yescharacteristic
277C22×C4C2×C22⋊D1711yescharacteristic
278Q16—1364no—
279M16—681no—
280M16—681no—
281C2×C8—41no—
282C2×C8—342no—
283C2×C8—342no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—341no—
292C4⋊C4—341no—
293C4⋊C4—681no—
294C4⋊C4—681no—
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—
305C42C2×D6811yescharacteristic
306C42#6325211yescharacteristic
307C42D4×D1711yescharacteristic
308C42D4×D1711yescharacteristic
309C23#1058762311yescharacteristic
310Q8—42no—
311Q8—81no—
312Q8—342no—
313Q8—342no—
314Q8—681no—
315Q8—681no—
316Q8—684no—
317Q8—1362no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—342no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—684no—
329C2×C4—682no—
330C2×C4—681no—
331C2×C4—682no—
332C2×C4#1058756811yescharacteristic
333C2×C4#1058756811yescharacteristic
334C2×C4#1058757011yescharacteristic
335C2×C4#1058757011yescharacteristic
336C2×C4#1058760111yescharacteristic
337C2×C4#1058760211yescharacteristic
338C2×C4Q16×D1711yescharacteristic
339C2×C4#1058760411yescharacteristic
340C2×C4#1058761611yescharacteristic
341C2×C4#1058762611yescharacteristic
342C8—81no—
343C8—682no—
344C8—682no—
345C22—21no—
346C22—21no—
347C22#1176271811yescharacteristic
348C22#1176296311yescharacteristic
349C22#1176296811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—81no—
355C4—81no—
356C4—682no—
357C4—1362no—
358C4—1361no—
359C4#1176296611yescharacteristic
360C4#1176297411yescharacteristic
361C4#1176298812yes—
362C4#1176304911yescharacteristic
363C4#1176306611yescharacteristic
364C2—21no—
365C2—21no—
366C2#1452979311yescharacteristic
367C2#1452981211yescharacteristic
368C2#1452984311yescharacteristic
subgroupG/H
centerC22#11762718
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1503C2
socleC22×C17#392
prime radicalC23×C17#120
hypercenter#1503D17
hypocenterC17#10252
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
92. C2×Dic34Q16semidirect
259. C17#10252semidirect

Maximal subgroups

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

subgroupindex
#145397292
#145385852
#145339992
#145274642
#145271802
#145266872
C17×#15032
#1025217