Finite Group Databasebeta

← #16936322

Subgroups — #16936322

Order 4352.

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

normal subgroups only

This group has 508 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#14539732C211yesmaximal, characteristic, complemented
2#14538581C211yesmaximal, characteristic
3#14527469C211yesmaximal, characteristic, complemented
4#14527186C211yesmaximal, characteristic, complemented
5#14527105C211yesmaximal, characteristic
6#14526681C211yesmaximal, characteristic, complemented
7C2×C17×#476C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763188—21no—
10#11763159—21no—
11#11763109—22no—
12#11763099—21no—
13#11763089—21no—
14#11763016—22no—
15#11762965—22no—
16#11762862C2211yescharacteristic
17#11762862C2211yescharacteristic
18#11762849—21no—
19#11762833C2211yescharacteristic
20#11762814—21no—
21#11762555—22no—
22#11762539—22no—
23#11762533—22no—
24C2×C17×C4⋊Q8C2211yescharacteristic
25C2×C4×D4×C17C2211yescharacteristic
26C17×#476—22no—
27C17×#476—22no—
28C2×C17×#106—21no—
29C2×C17×C4⋊C8C2211yescharacteristic
30C2×C17×#102—21no—
31C22×Q8×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C22×C8×C17—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×C4⋊Q8D412yes—
39C4×D4×C17—21no—
40C4×D4×C17—21no—
41C4×D4×C17—21no—
42C4×D4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17D411yescharacteristic
46C2×C22⋊C4×C17—21no—
47C2×C42×C17C2311yescharacteristic
48C2×#63231—24no—
49#10587612—21no—
50#10587610—21no—
51#10587610—21no—
52C2×#63223—21no—
53#10587556—21no—
54#10587555D411yescharacteristic
55#10587555D411yescharacteristic
56C4×Dic34—21no—
57C4×Dic34—21no—
58C17×#106—24no—
59C17×C4⋊C8—21no—
60C17×C4⋊C8D412yes—
61C17×#102—24no—
62#10587506—24no—
63#10587503—22no—
64#10587503—22no—
65C22⋊#3622—21no—
66C22⋊#3622D412yes—
67#10587479—22no—
68#10587478—22no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622—21no—
72C4⋊#3622D412yes—
73C4×#3622—21no—
74C4×#3622—21no—
75C24×C17—21no—
76C2×Q8×C17—22no—
77C2×Q8×C17—41no—
78C2×Q8×C17—82no—
79C2×D4×C17—21no—
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17Q1612yescomplemented
85C22×C4×C17—21no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17—41no—
89C22×C4×C17—82no—
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17#4111yescharacteristic
93C2×Dic34—21no—
94C2×Dic34—21no—
95C2×C8×C17—22no—
96C2×C8×C17—41no—
97C2×C8×C17—41no—
98C4⋊C4×C17—22no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C22⋊C4×C17—41no—
105C22⋊C4×C17—41no—
106C42×C17C2×D411yescharacteristic
107C42×C17C2×D411yescharacteristic
108C42×C17C2×D411yescharacteristic
109C42×C17C2×D411yescharacteristic
110C22⋊Dic17—41no—
111#63231—84no—
112C4⋊Dic17—21no—
113C4⋊Dic17—21no—
114C4⋊Dic17—21no—
115C4⋊Dic17—21no—
116#63225—41no—
117#63225—41no—
118C4×Dic17—41no—
119#63223—41no—
120#63223—41no—
121C2×#3622—22no—
122C2×#3622—22no—
123C2×#3622—41no—
124#10182—171nomaximal, Sylow
125C23×C17—22no—
126C23×C17—41no—
127C23×C17—41no—
128C23×C17—81no—
129C23×C17—81no—
130C23×C17#12111yescharacteristic
131Q8×C17—44no—
132Q8×C17—82no—
133D4×C17—22no—
134D4×C17—22no—
135D4×C17—41no—
136D4×C17—41no—
137C2×C4×C17—22no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—42no—
141C2×C4×C17—82no—
142C2×C4×C17—82no—
143C2×C4×C17—82no—
144C2×C4×C17—81no—
145C2×C4×C17#12011yescharacteristic
146C2×C4×C17#12011yescharacteristic
147C2×C4×C17#12011yescharacteristic
148C2×C4×C17#12111yescharacteristic
149C2×C4×C17#12111yescharacteristic
150C2×C4×C17#12711yescharacteristic
151C2×C4×C17C2×Q1611yescharacteristic
152C2×C4×C17#13511yescharacteristic
153C2×C4×C17#13711yescharacteristic
154C2×C4×C17#13711yescharacteristic
155C2×Dic17—41no—
156C2×Dic17—41no—
157Dic34—42no—
158Dic34—42no—
159C8×C17—42no—
160#3622—42no—
161#3622—42no—
162#3622—81no—
163#3209—171no—
164#3077—171no—
165C2×#476D1711yescharacteristic
166#1549—171no—
167#1481—171no—
168#1460—171no—
169#1301—171no—
170C22×C17—21no—
171C22×C17—21no—
172C22×C17—42no—
173C22×C17—82no—
174C22×C17—82no—
175C22×C17—81no—
176C22×C17—81no—
177C22×C17#39211yescharacteristic
178C22×C17#46711yescharacteristic
179C22×C17#47011yescharacteristic
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—82no—
184C4×C17—81no—
185C4×C17#45111yescharacteristic
186C4×C17#45111yescharacteristic
187C4×C17#45211yescharacteristic
188C4×C17#45211yescharacteristic
189C4×C17#47112yes—
190Dic17—81no—
191Dic17—81no—
192#533—171no—
193C2×C4⋊Q8C2×D1711yescharacteristic
194C2×C4×D4C2×D1711yescharacteristic
195#494—341no—
196#487—341no—
197#478—342no—
198#476—22no—
199#476—22no—
200#476—341no—
201#474—341no—
202#462—342no—
203#451—342no—
204#431—171no—
205#431—171no—
206#428—341no—
207C2×#106—21no—
208#423—171no—
209C2×C4⋊C8C2×D1711yescharacteristic
210#418—341no—
211C2×#102—21no—
212#332—342no—
213Q8⋊C8—342no—
214#323—342no—
215C2×C17—21no—
216C2×C17—21no—
217C2×C17—82no—
218C2×C17#200111yescharacteristic
219C2×C17#200811yescharacteristic
220C2×C17#201911yescharacteristic
221C22×Q8—21no—
222C22×D4C22⋊D1711yescharacteristic
223C23×C4—21no—
224C2×Q16—344no—
225C2×M16—341no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—171no—
231C4⋊Q8—171no—
232C4⋊Q8C22⋊D1712yes—
233#125—341no—
234C22⋊Q8—341no—
235C22⋊Q8—341no—
236C4×Q8—341no—
237C4×Q8—341no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242#117—341no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C22⋊D1711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C42C22×D1711yescharacteristic
248#107—342no—
249#106—24no—
250#106—342no—
251C4⋊C8—21no—
252C4⋊C8—172no—
253C4⋊C8—341no—
254C4⋊C8—341no—
255C4⋊C8—341no—
256C4⋊C8C22⋊D1712yes—
257#103—344no—
258#102—24no—
259#102—342no—
260#102—342no—
261C22⋊C8—172no—
262C22⋊C8—341no—
263C4×C8—341no—
264C4×C8—341no—
265C17#1018211yescharacteristic, complemented, Sylow
266C24—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—82no—
270C2×Q8—341no—
271C2×Q8—341no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4#6323112yescomplemented
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4D4×D1711yescharacteristic
284C22×C4#6325411yescharacteristic
285C22×C4C2×C22⋊D1711yescharacteristic
286Q16—1364no—
287M16—681no—
288M16—681no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—342no—
293C2×C8—342no—
294C2×C8—681no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—341no—
302C4⋊C4—341no—
303C4⋊C4—341no—
304C4⋊C4—341no—
305C4⋊C4—681no—
306C4⋊C4—681no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—681no—
310C42—681no—
311C42D4×D1711yescharacteristic
312C42D4×D1711yescharacteristic
313C42C2×C22⋊D1711yescharacteristic
314C42C2×C22⋊D1711yescharacteristic
315C23—22no—
316C23—41no—
317C23—41no—
318C23—81no—
319C23—81no—
320C23#1058762511yescharacteristic
321Q8—44no—
322Q8—82no—
323Q8—682no—
324Q8—682no—
325D4—22no—
326D4—22no—
327D4—41no—
328D4—41no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4#1058762311yescharacteristic
340C2×C4#1058762311yescharacteristic
341C2×C4#1058762411yescharacteristic
342C2×C4#1058762511yescharacteristic
343C2×C4#1058762611yescharacteristic
344C2×C4#1058762811yescharacteristic
345C2×C4C2×#6323111yescharacteristic
346C2×C4#1058763611yescharacteristic
347C2×C4#1058763711yescharacteristic
348C2×C4#1058763911yescharacteristic
349C8—42no—
350C8—682no—
351C8—682no—
352C8—1361no—
353C22—21no—
354C22—21no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22#1176271711yescharacteristic
361C22#1176304811yescharacteristic
362C22#1176306511yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—81no—
368C4—1361no—
369C4—1361no—
370C4#1176296411yescharacteristic
371C4#1176296511yescharacteristic
372C4#1176296911yescharacteristic
373C4#1176297111yescharacteristic
374C4#1176306912yes—
375C2—21no—
376C2—21no—
377C2—82no—
378C2#1452979211yescharacteristic
379C2#1452983511yescharacteristic
380C2#1452989611yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#476C2
socleC22×C17#392
prime radicalC22×D4×C17D4
hypercenterC2×#476D17
hypocenterC17#10182
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #14539732C2semidirect
3. #14527469C2semidirect
4. #14527186C2semidirect
6. #14526681C2semidirect
84. C2×D4×C17Q16semidirect
265. C17#10182semidirect
277. C2×D4#63231semidirect

Maximal subgroups

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

subgroupindex
#145397322
#145385812
#145274692
#145271862
#145271052
#145266812
C2×C17×#4762
#1018217