Finite Group Databasebeta

← #34344250

Subgroups — #34344250

Order 20224.

1038 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099293C211yesmaximal, characteristic
2C2×#19098845C211yesmaximal, characteristic
3#23768389C211yesmaximal, characteristic
4#23764731C214yesmaximal
5#23764349C211yesmaximal, characteristic
6#23764336C211yesmaximal, characteristic
7#23763665C214yesmaximal
8C4×#14579086C211yesmaximal, characteristic
9C2×#498×C79C211yesmaximal, characteristic
10#19099293—24no—
11#19099293—24no—
12C2×#14579160—21no—
13C2×#14579160C2212yes—
14C2×#14579159—22no—
15C2×#14579159—21no—
16C2×C4×Dic158C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×#14579126—21no—
19C2×#14579126C2212yes—
20#19098845—24no—
21C2×#14579092C2212yes—
22C2×C4⋊(C79⋊C8)C2211yescharacteristic
23C2×#14579086C2212yes—
24C2×#14579084C2212yes—
25#19098494—22no—
26#19098470—21no—
27#19098459C2214yes—
28C4×C4⋊Dic79C2211yescharacteristic
29#19098326C2212yes—
30#19098325C2214yes—
31#19098317C2214yes—
32C2×C4⋊Q8×C79C2211yescharacteristic
33C2×C4⋊Q8×C79C2211yescharacteristic
34#498×C79—24no—
35C2×#106×C79C2214yes—
36C2×C4×C8×C79C2211yescharacteristic
37C22×Dic158—22no—
38C22×Dic158D411yescharacteristic
39C22×Dic158C2311yescharacteristic
40C22×C79⋊C8C2312yes—
41#14579160—28no—
42#14579160—24no—
43#14579160—24no—
44#14579159—216no—
45#14579159—44no—
46C4×Dic158—22no—
47C4×Dic158—22no—
48C4×Dic158—22no—
49C4×Dic158D414yes—
50#14579126—28no—
51#14579126—24no—
52#14579126—24no—
53C2×C4⋊Dic79D411yescharacteristic
54C2×C4⋊Dic79C2311yescharacteristic
55C2×C4⋊Dic79C2312yes—
56C2×Dic79⋊C4—21no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—22no—
59C2×Dic79⋊C4—24no—
60C2×Dic79⋊C4D412yes—
61C2×C4×Dic79—21no—
62C2×C4×Dic79—22no—
63C2×C4×Dic79—21no—
64C2×C4×Dic79C2312yes—
65#14579092—24no—
66C4⋊(C79⋊C8)—22no—
67#14579086Q818yescomplemented
68#14579084—24no—
69C79⋊#95—21no—
70C79⋊#95—24no—
71C22×Q8×C79—21no—
72C22×Q8×C79—21no—
73C22×C8×C79C2312yes—
74C4⋊Q8×C79—22no—
75C4⋊Q8×C79—22no—
76C4⋊Q8×C79—22no—
77C4⋊Q8×C79D414yes—
78C2×C4⋊C4×C79—21no—
79C2×C4⋊C4×C79—21no—
80C2×C4⋊C4×C79C2312yes—
81C2×C4⋊C4×C79C2312yes—
82C2×C42×C79C2311yescharacteristic
83#106×C79—28no—
84C4×C8×C79—22no—
85C2×Dic158—28no—
86C2×Dic158—22no—
87C2×Dic158—22no—
88C2×Dic158—21no—
89C2×Dic158—22no—
90C2×Dic158—84no—
91C2×Dic158SD1614yes—
92C2×Dic158C2×D412yes—
93C2×Dic158C2×Q814yes—
94C22×Dic79—21no—
95C22×Dic79—21no—
96C22×Dic79—22no—
97C22×Dic79C2×D412yes—
98C22×Dic79C2×D412yes—
99C2×C79⋊C8—22no—
100C2×C79⋊C8—24no—
101C4⋊Dic79—22no—
102C4⋊Dic79—21no—
103C4⋊Dic79—22no—
104C4⋊Dic79C2×D412yes—
105C4⋊Dic79C2×Q814yes—
106Dic79⋊C4—28no—
107Dic79⋊C4—28no—
108Dic79⋊C4—24no—
109Dic79⋊C4—24no—
110Dic79⋊C4—48no—
111C4×Dic79—28no—
112C4×Dic79—24no—
113C4×Dic79—22no—
114C4×Dic79—22no—
115C4×Dic79—42no—
116C2×Q8×C79—24no—
117C2×Q8×C79—42no—
118C2×Q8×C79—42no—
119C2×Q8×C79—42no—
120C2×Q8×C79—82no—
121C22×C4×C79—22no—
122C22×C4×C79—22no—
123C22×C4×C79C2×D411yescharacteristic
124C22×C4×C79C2×D411yescharacteristic
125C22×C4×C79C2411yescharacteristic
126C2×C8×C79—22no—
127C2×C8×C79—22no—
128C2×C8×C79C2×Q814yes—
129C4⋊C4×C79—24no—
130C4⋊C4×C79—22no—
131C4⋊C4×C79—22no—
132C4⋊C4×C79—22no—
133C4⋊C4×C79—42no—
134C4⋊C4×C79C2×D414yes—
135C42×C79C2×D412yes—
136C42×C79C2×D412yes—
137C23×C79C22×D411yescharacteristic
138Q8×C79—84no—
139Q8×C79—84no—
140C2×C4×C79—22no—
141C2×C4×C79—24no—
142C2×C4×C79—44no—
143C2×C4×C79—42no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79#12012yes—
147C2×C4×C79#12012yes—
148C2×C4×C79C2×SD1612yes—
149C2×C4×C79#13712yes—
150C2×C4×C79C22×D411yescharacteristic
151C2×C4×C79C22×D411yescharacteristic
152C2×C4×C79C22×Q812yes—
153C2×C4×C792−1+412yes—
154C2×Dic79—22no—
155C2×Dic79—22no—
156C2×Dic79—24no—
157C2×Dic79—22no—
158C2×Dic79—22no—
159C2×Dic79—22no—
160C2×Dic79—24no—
161C2×Dic79—24no—
162C2×Dic79—42no—
163C2×Dic79—42no—
164C2×Dic79—42no—
165C2×Dic79C2×SD1614yes—
166Dic158—22no—
167Dic158—24no—
168Dic158—28no—
169Dic158—88no—
170Dic158C4⋊Q814yescomplemented
171C8×C79—24no—
172C79⋊C8—42no—
173C22×C79C2×#12011yescharacteristic
174C22×C79D4×Q812yes—
175C22×C79C22×SD1611yescharacteristic
176C22×C79C2×#13711yescharacteristic
177C22×C79#57712yes—
178C4×C79—22no—
179C4×C79—44no—
180C4×C79—82no—
181C4×C79#44814yes—
182C4×C79C2×C4⋊Q811yescharacteristic
183C4×C79#53611yescharacteristic
184C4×C79D4×Q812yes—
185Dic79—24no—
186Dic79—24no—
187Dic79—44no—
188Dic79—42no—
189Dic79—42no—
190#25837—791nomaximal, Sylow
191C2×C79#243411yescharacteristic
192C2×C79C2×#44811yescharacteristic
193C2×C79#300911yescharacteristic
194C2×C79Q8×SD1612yes—
195C2×C79#338712yes—
196C2×Q82—791no—
197C2×#498D7911yescharacteristic
198C2×#475—791no—
199#2546—791no—
200#2058—794no—
201#1987—791no—
202#1984—791no—
203#1865—794no—
204C4×#103—791no—
205C79#2583711yescharacteristic, complemented, Sylow
206Q82—1584no—
207Q82—1584no—
208C2×C4⋊Q8—792no—
209C2×C4⋊Q8—1581no—
210C2×C4⋊Q8—1582no—
211C2×C4⋊Q8—1581no—
212C2×C4⋊Q8C2×D7911yescharacteristic
213C2×C4⋊Q8C2×D7911yescharacteristic
214C2×C4×Q8—791no—
215C2×C4×Q8—792no—
216C2×C4×Q8—791no—
217C2×C4×Q8—1581no—
218#498—24no—
219#475—1584no—
220C2×#106—792no—
221C2×#106C2×D7914yes—
222C2×C4⋊C8—791no—
223C2×#103—792no—
224C2×#103—792no—
225C2×C4×C8C2×D7911yescharacteristic
226#398—1582no—
227#391—1581no—
228#389—794no—
229C4×C4⋊C4—791no—
230#340—792no—
231#340—794no—
232#336—794no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—791no—
236C22×Q8—791no—
237C22×Q8—1582no—
238C22×C8—792no—
239C22×C8C22×D7912yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—3164no—
244C4⋊Q8—1584no—
245C4⋊Q8—1588no—
246C4⋊Q8—15816no—
247C4⋊Q8—1584no—
248C4⋊Q8C22⋊D7914yes—
249C4×Q8—794no—
250C4×Q8—1584no—
251C4×Q8—1588no—
252C4×Q8—1584no—
253C4×Q8—1582no—
254C4×Q8—1582no—
255C4×Q8—1582no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—791no—
259C2×C4⋊C4—792no—
260C2×C4⋊C4—791no—
261C2×C4⋊C4—792no—
262C2×C4⋊C4—1584no—
263C2×C4⋊C4—1582no—
264C2×C4⋊C4—1582no—
265C2×C4⋊C4—1581no—
266C2×C4⋊C4C22×D7912yes—
267C2×C4⋊C4C22×D7912yes—
268C2×C42—792no—
269C2×C42—1581no—
270C2×C42—1582no—
271C2×C42—1581no—
272C2×C42C22×D7911yescharacteristic
273#106—28no—
274#106—1584no—
275C4⋊C8—1582no—
276#103—798no—
277#103—1584no—
278C4×C8—22no—
279#95—1584no—
280#95—1581no—
281C2×Q8—24no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—82no—
286C2×Q8—794no—
287C2×Q8—794no—
288C2×Q8—792no—
289C2×Q8—6324no—
290C2×Q8—1588no—
291C2×Q8—1582no—
292C2×Q8—1582no—
293C2×Q8—1582no—
294C2×Q8—1581no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—792no—
298C22×C4—792no—
299C22×C4—1582no—
300C22×C4—1581no—
301C22×C4—1581no—
302C22×C4D4×D7911yescharacteristic
303C22×C4D4×D7911yescharacteristic
304C22×C4C23×D7911yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—1584no—
308C2×C8—1582no—
309C2×C8Q8×D7914yes—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—3168no—
316C4⋊C4—794no—
317C4⋊C4—792no—
318C4⋊C4—1588no—
319C4⋊C4—1588no—
320C4⋊C4—1582no—
321C4⋊C4—1584no—
322C4⋊C4—1584no—
323C4⋊C4—1582no—
324C4⋊C4—1581no—
325C4⋊C4C2×C22⋊D7914yes—
326C42—3162no—
327C42—1588no—
328C42—1584no—
329C42—1582no—
330C42—1582no—
331C42D4×D7912yes—
332C42C2×C22⋊D7912yes—
333C23C2×D4×D7911yescharacteristic
334Q8—84no—
335Q8—84no—
336Q8—794no—
337Q8—6328no—
338Q8—1582no—
339Q8—1588no—
340Q8—1584no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—3162no—
348C2×C4—3162no—
349C2×C4—3162no—
350C2×C4—794no—
351C2×C4—1584no—
352C2×C4—1582no—
353C2×C4—1582no—
354C2×C4—1584no—
355C2×C4—1584no—
356C2×C4—1582no—
357C2×C4—1582no—
358C2×C4—1582no—
359C2×C4#1457912912yes—
360C2×C4#1457912912yes—
361C2×C4SD16×D7912yes—
362C2×C4#1457919012yes—
363C2×C4C2×D4×D7911yescharacteristic
364C2×C4C22×C22⋊D7911yescharacteristic
365C2×C4C2×Q8×D7912yes—
366C2×C4#1457920912yes—
367C8—24no—
368C8—3162no—
369C22C2×#1457912911yescharacteristic
370C22#1909922812yes—
371C22C2×SD16×D7911yescharacteristic
372C22C2×#1457919011yescharacteristic
373C22#1909936912yes—
374C4—22no—
375C4—44no—
376C4—82no—
377C4—3162no—
378C4—3162no—
379C4—3164no—
380C4—1584no—
381C4—1584no—
382C4#1909870014yes—
383C4C4⋊Q8×D7911yescharacteristic
384C4#1909913811yescharacteristic
385C4Q8×C22⋊D7912yes—
386C2#2376704111yescharacteristic
387C2C2×#1909870011yescharacteristic
388C2#2377267311yescharacteristic
389C2#2377625712yes—
390C2#2377633712yes—
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#498×C79C2
socle, prime radicalC23×C79C22×D4
hypercenterC2×#498D79
hypocenterC79#25837
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
67. #14579086Q8semidirect
170. Dic158C4⋊Q8semidirect
205. C79#25837semidirect

Maximal subgroups

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

subgroupindex
C2×#190992932
C2×#190988452
#237683892
#237647312
#237643492
#237643362
#237636652
C4×#145790862
C2×#498×C792
#2583779