Finite Group Databasebeta

← #34344283

Subgroups — #34344283

Order 20224.

1006 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×#19099291C211yesmaximal, characteristic
2C2×#19098966C211yesmaximal, characteristic
3C2×#19098845C211yesmaximal, characteristic
4#23764738C214yesmaximal
5#23764352C211yesmaximal, characteristic
6#23764337C211yesmaximal, characteristic
7#23763653C214yesmaximal
8C4×#14579084C211yesmaximal, characteristic
9C79×#2546C211yesmaximal, characteristic
10#19099291—24no—
11#19099291—24no—
12C2×#14579161C2211yescharacteristic
13C2×#14579159—21no—
14C2×#14579153C2212yes—
15C2×#14579151—22no—
16C2×C4×Dic158C2212yes—
17C2×C4×Dic158C2211yescharacteristic
18C2×#14579126—21no—
19#19098966—24no—
20#19098845—24no—
21C2×#14579094C2214yes—
22C2×#14579092C2212yes—
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×#14579084C2212yes—
25C2×#14579084C2212yes—
26C2×C4×C79⋊C8C2211yescharacteristic
27#19098325C2212yes—
28#19098325C2214yes—
29#19098318C2214yes—
30C2×C4⋊Q8×C79—21no—
31C2×C4⋊Q8×C79C2211yescharacteristic
32C2×C4×Q8×C79C2211yescharacteristic
33#398×C79—22no—
34#391×C79—21no—
35#389×C79C2214yes—
36C4×C4⋊C4×C79C2211yescharacteristic
37C22×Dic158—21no—
38C22×Dic158C2311yescharacteristic
39C22×C79⋊C8C2312yes—
40C22×C79⋊C8C2312yes—
41#14579161—22no—
42#14579161D414yes—
43#14579159—28no—
44#14579153—24no—
45#14579153—24no—
46#14579151—216no—
47C4×Dic158—22no—
48C4×Dic158—24no—
49C4×Dic158—24no—
50C4×Dic158D414yes—
51#14579126—28no—
52C2×C4⋊Dic79—21no—
53C2×C4⋊Dic79C2311yescharacteristic
54C2×C4⋊Dic79C2312yes—
55C2×Dic79⋊C4—22no—
56C2×Dic79⋊C4—21no—
57C2×Dic79⋊C4—22no—
58C2×Dic79⋊C4—22no—
59C2×C4×Dic79—22no—
60C2×C4×Dic79—21no—
61#14579094—28no—
62#14579092—24no—
63C4⋊(C79⋊C8)—22no—
64#14579084—24no—
65#14579084Q818yescomplemented
66C4×C79⋊C8—22no—
67C22×Q8×C79—21no—
68C22×Q8×C79D411yescharacteristic
69C4⋊Q8×C79—22no—
70C4⋊Q8×C79—22no—
71C4⋊Q8×C79—44no—
72C4×Q8×C79—22no—
73C4×Q8×C79—22no—
74C2×C4⋊C4×C79—24no—
75C2×C4⋊C4×C79—21no—
76C2×C4⋊C4×C79D411yescharacteristic
77C2×C4⋊C4×C79D412yes—
78C2×C4⋊C4×C79C2312yes—
79C2×C4⋊C4×C79C2312yes—
80C2×C42×C79—21no—
81C2×C42×C79C2312yes—
82C2×C42×C79C2311yescharacteristic
83#95×C79—21no—
84#95×C79—24no—
85C2×Dic158—24no—
86C2×Dic158—21no—
87C2×Dic158—22no—
88C2×Dic158—42no—
89C2×Dic158—42no—
90C2×Dic158C2×D412yes—
91C2×Dic158C2×Q814yes—
92C22×Dic79—21no—
93C22×Dic79—22no—
94C22×Dic79—21no—
95C2×C79⋊C8—22no—
96C2×C79⋊C8—22no—
97C2×C79⋊C8—22no—
98C2×C79⋊C8—24no—
99C2×C79⋊C8C2×Q814yes—
100C4⋊Dic79—24no—
101C4⋊Dic79—21no—
102C4⋊Dic79—22no—
103C4⋊Dic79C2×D412yes—
104C4⋊Dic79C2×D414yes—
105Dic79⋊C4—28no—
106Dic79⋊C4—28no—
107Dic79⋊C4—28no—
108Dic79⋊C4—24no—
109C4×Dic79—28no—
110C4×Dic79—24no—
111C2×Q8×C79—22no—
112C2×Q8×C79—22no—
113C2×Q8×C79—22no—
114C2×Q8×C79—42no—
115C2×Q8×C79—42no—
116C2×Q8×C79—42no—
117C22×C4×C79—22no—
118C22×C4×C79—22no—
119C22×C4×C79C2×D412yes—
120C22×C4×C79C2×D412yes—
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79C2411yescharacteristic
124C4⋊C4×C79—22no—
125C4⋊C4×C79—24no—
126C4⋊C4×C79—22no—
127C4⋊C4×C79—22no—
128C4⋊C4×C79—22no—
129C4⋊C4×C79—48no—
130C4⋊C4×C79—42no—
131C4⋊C4×C79C2×Q814yes—
132C42×C79—22no—
133C42×C79—22no—
134C42×C79—42no—
135C42×C79C2×D412yes—
136C42×C79C2×D412yes—
137C23×C79C22×D411yescharacteristic
138Q8×C79—44no—
139Q8×C79—84no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—24no—
144C2×C4×C79—22no—
145C2×C4×C79—22no—
146C2×C4×C79—24no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79#12012yes—
153C2×C4×C79#12012yes—
154C2×C4×C79#13512yes—
155C2×C4×C79AGL(1,ℤ/8)12yes—
156C2×C4×C79C22×D411yescharacteristic
157C2×C4×C79C22×D411yescharacteristic
158C2×C4×C79C22×Q812yes—
159C2×C4×C792−1+412yes—
160C2×Dic79—22no—
161C2×Dic79—24no—
162C2×Dic79—22no—
163C2×Dic79—24no—
164C2×Dic79—42no—
165C2×Dic79—42no—
166C2×Dic79—42no—
167Dic158—22no—
168Dic158—24no—
169Dic158—48no—
170Dic158C4⋊Q814yescomplemented
171C79⋊C8—24no—
172C79⋊C8—42no—
173C22×C79C2×#12011yescharacteristic
174C22×C79D4×Q812yes—
175C22×C79C2×#13511yescharacteristic
176C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
177C22×C79#57812yes—
178C4×C79—22no—
179C4×C79—44no—
180C4×C79—42no—
181C4×C79—42no—
182C4×C79—82no—
183C4×C79#44914yes—
184C4×C79C2×C4⋊Q811yescharacteristic
185C4×C79#53611yescharacteristic
186C4×C79D4×Q812yes—
187Dic79—24no—
188Dic79—44no—
189Dic79—42no—
190#25840—791nomaximal, Sylow
191C2×C79#243411yescharacteristic
192C2×C79C2×#44911yescharacteristic
193C2×C79#301111yescharacteristic
194C2×C79#338312yes—
195C2×C79#338612yes—
196C2×#557—791no—
197C2×#500—791no—
198C2×#475—791no—
199#2546D7911yescharacteristic
200#2059—794no—
201#1988—791no—
202#1984—791no—
203#1863—794no—
204C4×#103—791no—
205C79#2584011yescharacteristic, complemented, Sylow
206#557—1584no—
207#557—1584no—
208C2×C4⋊Q8—21no—
209C2×C4⋊Q8—791no—
210C2×C4⋊Q8—1581no—
211C2×C4⋊Q8C2×D7911yescharacteristic
212C2×#125—792no—
213C2×#125—1582no—
214C2×C4×Q8—791no—
215C2×C4×Q8—792no—
216C2×C4×Q8—1581no—
217C2×C4×Q8C2×D7911yescharacteristic
218#500—1584no—
219#475—1584no—
220C2×#107—794no—
221C2×#106—792no—
222C2×C4⋊C8—791no—
223C2×#103—792no—
224C2×#103—792no—
225C2×C4×C8—791no—
226#398—22no—
227#391—21no—
228#389C2×D7914yes—
229C4×C4⋊C4C2×D7911yescharacteristic
230#340—792no—
231#340—794no—
232#336—794no—
233C22×Q8—21no—
234C22×Q8—791no—
235C22×Q8—1581no—
236C22×Q8C22⋊D7911yescharacteristic
237C22×C8—792no—
238C22×C8—792no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—44no—
242C4⋊Q8—794no—
243C4⋊Q8—1588no—
244C4⋊Q8—1582no—
245#125—1584no—
246#125—1584no—
247#125—15816no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—794no—
251C4×Q8—1584no—
252C4×Q8—1584no—
253C4×Q8—1588no—
254C4×Q8—1582no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—791no—
258C2×C4⋊C4—792no—
259C2×C4⋊C4—1581no—
260C2×C4⋊C4—1582no—
261C2×C4⋊C4—1582no—
262C2×C4⋊C4—1582no—
263C2×C4⋊C4—1581no—
264C2×C4⋊C4C22⋊D7911yescharacteristic
265C2×C4⋊C4C22⋊D7912yes—
266C2×C4⋊C4C22×D7912yes—
267C2×C4⋊C4C22×D7912yes—
268C2×C42—21no—
269C2×C42—1582no—
270C2×C42—1581no—
271C2×C42C22×D7912yes—
272C2×C42C22×D7911yescharacteristic
273#107—1588no—
274#106—1584no—
275C4⋊C8—1582no—
276#103—798no—
277#103—1584no—
278C4×C8—1582no—
279#95—21no—
280#95—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—3162no—
288C2×Q8—3162no—
289C2×Q8—794no—
290C2×Q8—792no—
291C2×Q8—1582no—
292C2×Q8—1584no—
293C2×Q8—1581no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—1581no—
297C22×C4—1582no—
298C22×C4—1581no—
299C22×C4C2×C22⋊D7912yes—
300C22×C4C2×C22⋊D7912yes—
301C22×C4C2×C22⋊D7911yescharacteristic
302C22×C4C2×C22⋊D7911yescharacteristic
303C22×C4C23×D7911yescharacteristic
304C2×C8—794no—
305C2×C8—1584no—
306C2×C8—1582no—
307C2×C8—1582no—
308C2×C8—1582no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—48no—
315C4⋊C4—42no—
316C4⋊C4—792no—
317C4⋊C4—794no—
318C4⋊C4—1588no—
319C4⋊C4—1588no—
320C4⋊C4—1588no—
321C4⋊C4—1584no—
322C4⋊C4—1584no—
323C4⋊C4—1581no—
324C4⋊C4—1582no—
325C4⋊C4Q8×D7914yes—
326C42—22no—
327C42—22no—
328C42—42no—
329C42—1588no—
330C42—1584no—
331C42D4×D7912yes—
332C42D4×D7912yes—
333C23C22×C22⋊D7911yescharacteristic
334Q8—44no—
335Q8—84no—
336Q8—3168no—
337Q8—794no—
338Q8—1582no—
339Q8—1584no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3162no—
353C2×C4—3162no—
354C2×C4—3162no—
355C2×C4—1582no—
356C2×C4—1584no—
357C2×C4—1584no—
358C2×C4—1582no—
359C2×C4#1457912912yes—
360C2×C4#1457912912yes—
361C2×C4#1457918112yes—
362C2×C4#1457918712yes—
363C2×C4C2×D4×D7911yescharacteristic
364C2×C4C2×D4×D7911yescharacteristic
365C2×C4C2×Q8×D7912yes—
366C2×C4#1457920912yes—
367C8—3162no—
368C8—1584no—
369C22C2×#1457912911yescharacteristic
370C22Q8×C22⋊D7912yes—
371C22C2×#1457918111yescharacteristic
372C22C2×#1457918711yescharacteristic
373C22#1909937812yes—
374C4—22no—
375C4—42no—
376C4—44no—
377C4—42no—
378C4—82no—
379C4—3162no—
380C4—3164no—
381C4—1584no—
382C4#1909870614yes—
383C4C4⋊Q8×D7911yescharacteristic
384C4#1909913811yescharacteristic
385C4#1909922812yes—
386C2#2376704111yescharacteristic
387C2C2×#1909870611yescharacteristic
388C2#2377269311yescharacteristic
389C2#2377628712yes—
390C2#2377632112yes—
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2546C2
socle, prime radicalC23×C79C22×D4
hypercenter#2546D79
hypocenterC79#25840
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
65. #14579084Q8semidirect
170. Dic158C4⋊Q8semidirect
205. C79#25840semidirect

Maximal subgroups

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

subgroupindex
C2×#190992912
C2×#190989662
C2×#190988452
#237647382
#237643522
#237643372
#237636532
C4×#145790842
C79×#25462
#2584079