Finite Group Databasebeta

← #34192089

Subgroups — #34192089

Order 20224.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098769C211yesmaximal, characteristic
2#23764361C211yesmaximal, characteristic, complemented
3#23764194C211yesmaximal, characteristic
4#23763401C211yesmaximal, characteristic
5C2×#19098310C211yesmaximal, characteristic, complemented
6#23761284C211yesmaximal, characteristic, complemented
7C2×#335×C79C211yesmaximal, characteristic, complemented
8C22×#11929613—21no—
9C2×#14579161C2211yescharacteristic
10C2×C4×D316C2211yescharacteristic
11#19098769—22no—
12#19098769—22no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579086—21no—
15C2×#14579083—21no—
16C2×(C2×D79)⋊C8—21no—
17C2×(C2×D79)⋊C8—21no—
18#19098433C2211yescharacteristic
19#19098326—21no—
20#19098326—21no—
21#19098326—21no—
22#19098310—22no—
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×C4×C8×C79C2211yescharacteristic
26#335×C79—22no—
27C22×Dic158—21no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C2×#11929613—24no—
31C2×#11929613—42no—
32C2×#11929613—42no—
33C22×C79⋊C8—21no—
34#14579161—21no—
35#14579161—21no—
36#14579161—21no—
37#14579161D412yes—
38C4×D316—21no—
39C4×D316—21no—
40C4×D316—21no—
41C4×D316—21no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79—21no—
44C2×C4⋊Dic79—21no—
45C2×C4⋊Dic79—21no—
46C2×C4⋊Dic79—21no—
47C2×C4⋊Dic79D411yescharacteristic
48C2×(C2×D79)⋊C4—21no—
49C4⋊(C79⋊C8)—21no—
50C4⋊(C79⋊C8)—21no—
51#14579086—24no—
52#14579083—42no—
53(C2×D79)⋊C8—42no—
54(C2×D79)⋊C8—42no—
55C22×C8×C79—21no—
56C22×C8×C79—21no—
57C22×C8×C79D411yescharacteristic
58C22×C8×C79D411yescharacteristic
59C2×C42×C79C2311yescharacteristic
60C4⋊C8×C79—21no—
61C4⋊C8×C79—21no—
62C4⋊C8×C79—21no—
63C4⋊C8×C79D412yes—
64C4×C8×C79—21no—
65C4×C8×C79—21no—
66C23×D79—21no—
67C2×Dic158—22no—
68C2×Dic158—41no—
69C2×Dic158—41no—
70C2×Dic158—41no—
71C2×Dic158—41no—
72C2×Dic158—41no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316—21no—
76C2×D316—21no—
77C2×D316—21no—
78C2×D316D812yes—
79C22×Dic79—21no—
80C22×Dic79—41no—
81C22×Dic79—41no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84C2×C4×D79—41no—
85C2×C4×D79—41no—
86C2×C4×D79—41no—
87C2×C4×D79—41no—
88#11929613—48no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C4⋊Dic79—21no—
93C4⋊Dic79—22no—
94C4⋊Dic79—22no—
95C4⋊Dic79—21no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—41no—
99C4⋊Dic79—41no—
100C4⋊Dic79—41no—
101C4⋊Dic79—41no—
102C4⋊Dic79—81no—
103(C2×D79)⋊C4—41no—
104(C2×D79)⋊C4—41no—
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79#4111yescharacteristic
108C2×C8×C79—22no—
109C2×C8×C79—21no—
110C2×C8×C79—21no—
111C2×C8×C79—21no—
112C2×C8×C79—21no—
113C2×C8×C79—21no—
114C2×C8×C79—41no—
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C2×C8×C79—41no—
119C2×C8×C79SD1612yescomplemented
120C42×C79C2×D411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C42×C79#4111yescharacteristic
124C23×C79#12111yescharacteristic
125C22×D79—22no—
126C22×D79—41no—
127C22×D79—41no—
128C22×D79—41no—
129C22×D79—41no—
130C22×D79—41no—
131C22×D79—41no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79#12011yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12111yescharacteristic
137C2×C4×C79#12311yescharacteristic
138C2×C4×C79#12311yescharacteristic
139C2×C4×C79#12411yescharacteristic
140C2×C4×C79C2×D811yescharacteristic
141C2×C4×C79C2×SD1611yescharacteristic
142C2×C4×C79#13511yescharacteristic
143C2×C4×C79#13511yescharacteristic
144C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C79#13711yescharacteristic
147C2×C4×C79#13711yescharacteristic
148C2×Dic79—41no—
149C2×Dic79—42no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—81no—
153C2×Dic79—81no—
154C2×Dic79—81no—
155C2×Dic79—81no—
156C2×Dic79—81no—
157D316—22no—
158D316—22no—
159D316—41no—
160D316—41no—
161C4×D79—81no—
162C4×D79—81no—
163C4×D79—81no—
164C4×D79—81no—
165Dic158—42no—
166Dic158—42no—
167Dic158—81no—
168Dic158—81no—
169C8×C79—22no—
170C8×C79—42no—
171C8×C79—41no—
172C8×C79—81no—
173C79⋊C8—81no—
174C22×C79#39411yescharacteristic
175C22×C79#46111yescharacteristic
176C22×C79#46311yescharacteristic
177C22×C79#46611yescharacteristic
178C22×C79#46811yescharacteristic
179C22×C79#47111yescharacteristic
180C22×C79#47311yescharacteristic
181C2×D79—42no—
182C2×D79—42no—
183C2×D79—42no—
184C2×D79—81no—
185C2×D79—81no—
186C2×D79—81no—
187C2×D79—81no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79#44711yescharacteristic
191C4×C79#45211yescharacteristic
192C4×C79#46011yescharacteristic
193C4×C79#46411yescharacteristic
194C4×C79#46511yescharacteristic
195C4×C79#46811yescharacteristic
196C4×C79#48011yescharacteristic
197C4×C79#48211yescharacteristic
198Dic79—81no—
199Dic79—82no—
200Dic79—161no—
201#10476—791nomaximal, Sylow
202C2×C79#167311yescharacteristic
203C2×C79#168111yescharacteristic
204C2×C79#169111yescharacteristic
205C2×C79#169911yescharacteristic
206C2×C79#203311yescharacteristic
207C2×C79#205211yescharacteristic
208C2×C79#205511yescharacteristic
209D79—82no—
210C2×#461—791no—
211#1990—791no—
212#1954—791no—
213#1802—791no—
214C2×#335D7911yescharacteristic, complemented
215C2×#332—791no—
216#1296—791no—
217C79#1047611yescharacteristic, complemented, Sylow
218C22×SD16—1581no—
219C2×C4⋊Q8—791no—
220C2×C4×D4—791no—
221#461—1582no—
222#461—1582no—
223C2×C4⋊C8—791no—
224C2×C4⋊C8C2×D7911yescharacteristic
225C2×C4⋊C8C2×D7911yescharacteristic
226C2×#103—1581no—
227C2×#102—1581no—
228C2×C22⋊C8—1581no—
229C2×C22⋊C8—1581no—
230C2×C4×C8C2×D7911yescharacteristic
231#384—791no—
232#340—1581no—
233#340—1581no—
234#340—1581no—
235#335—22no—
236#332—1582no—
237C22×Q8—1581no—
238C22×D4—791no—
239C23×C4—1581no—
240C2×SD16—3162no—
241C2×SD16—3162no—
242C2×SD16—1584no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—1581no—
246C22×C8D31611yescharacteristic
247C22×C8D31611yescharacteristic
248C4⋊Q8—792no—
249C4⋊Q8—1581no—
250C4⋊Q8—1581no—
251C4⋊Q8—1581no—
252C4×D4—1581no—
253C4×D4—1581no—
254C4×D4—1581no—
255C4×D4—1581no—
256C2×C4⋊C4—791no—
257C2×C4⋊C4—1581no—
258C2×C4⋊C4—1581no—
259C2×C4⋊C4—1581no—
260C2×C4⋊C4—1581no—
261C2×C4⋊C4—1581no—
262C2×C22⋊C4—1581no—
263C2×C42C22×D7911yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269C4⋊C8C22⋊D7912yes—
270#103—1584no—
271#102—3162no—
272C22⋊C8—3162no—
273C22⋊C8—3162no—
274C4×C8—21no—
275C4×C8—21no—
276C24—1581no—
277C2×Q8—3161no—
278C2×Q8—3161no—
279C2×Q8—3161no—
280C2×Q8—3161no—
281C2×Q8—3161no—
282C2×Q8—1582no—
283C2×D4—792no—
284C2×D4—1581no—
285C2×D4—1581no—
286C2×D4—1581no—
287C2×D4—1581no—
288C2×D4—1581no—
289C22×C4—3161no—
290C22×C4—3161no—
291C22×C4—3161no—
292C22×C4—3161no—
293C22×C4—3161no—
294C22×C4—3161no—
295C22×C4—3161no—
296C22×C4—3161no—
297C22×C4—1581no—
298C22×C4D4×D7911yescharacteristic
299C22×C4C2×D31611yescharacteristic
300C22×C4#1192962511yescharacteristic
301SD16—3168no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3161no—
314C2×C8—3161no—
315C2×C8—3161no—
316C2×C8#1192961312yescomplemented
317C4⋊C4—3161no—
318C4⋊C4—3161no—
319C4⋊C4—3161no—
320C4⋊C4—3161no—
321C4⋊C4—3161no—
322C4⋊C4—3161no—
323C4⋊C4—6321no—
324C4⋊C4—1581no—
325C4⋊C4—1582no—
326C4⋊C4—1581no—
327C4⋊C4—1582no—
328C22⋊C4—3161no—
329C22⋊C4—3161no—
330C42D4×D7911yescharacteristic
331C42C2×C22⋊D7911yescharacteristic
332C42#1192962411yescharacteristic
333C42#1192962511yescharacteristic
334C23—3161no—
335C23—3161no—
336C23—3161no—
337C23—3161no—
338C23—3161no—
339C23—3161no—
340C23—1582no—
341C23#1457913611yescharacteristic
342Q8—3162no—
343Q8—3162no—
344Q8—6321no—
345Q8—6321no—
346D4—3161no—
347D4—3161no—
348D4—1582no—
349D4—1582no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—3161no—
353C2×C4—3162no—
354C2×C4—3161no—
355C2×C4—3161no—
356C2×C4—6321no—
357C2×C4—6321no—
358C2×C4—6321no—
359C2×C4—6321no—
360C2×C4—6321no—
361C2×C4—6321no—
362C2×C4—6321no—
363C2×C4—6321no—
364C2×C4—6321no—
365C2×C4#1457912911yescharacteristic
366C2×C4#1457913511yescharacteristic
367C2×C4#1457913611yescharacteristic
368C2×C4#1457914611yescharacteristic
369C2×C4#1457914611yescharacteristic
370C2×C4#1457915011yescharacteristic
371C2×C4D8×D7911yescharacteristic
372C2×C4C2×#1192961311yescharacteristic
373C2×C4#1457918311yescharacteristic
374C2×C4#1457918411yescharacteristic
375C2×C4#1457918511yescharacteristic
376C2×C4#1457918911yescharacteristic
377C2×C4#1457919311yescharacteristic
378C2×C4#1457919411yescharacteristic
379C8—22no—
380C8—42no—
381C8—41no—
382C8—81no—
383C8—6321no—
384C22—3162no—
385C22—3162no—
386C22—3162no—
387C22—6321no—
388C22—6321no—
389C22—6321no—
390C22—6321no—
391C22#1909848011yescharacteristic
392C22#1909876911yescharacteristic
393C22#1909877911yescharacteristic
394C22#1909880011yescharacteristic
395C22#1909881011yescharacteristic
396C22#1909882711yescharacteristic
397C22#1909883911yescharacteristic
398C4—21no—
399C4—21no—
400C4—6322no—
401C4—6321no—
402C4—12641no—
403C4#1909869511yescharacteristic
404C4#1909872411yescharacteristic
405C4#1909876411yescharacteristic
406C4#1909878611yescharacteristic
407C4#1909879211yescharacteristic
408C4#1909880911yescharacteristic
409C4#1909887811yescharacteristic
410C4#1909889011yescharacteristic
411C2—6322no—
412C2#2376279011yescharacteristic
413C2#2376283811yescharacteristic
414C2#2376290011yescharacteristic
415C2#2376294811yescharacteristic
416C2#2376458411yescharacteristic
417C2#2376470111yescharacteristic
418C2#2376471811yescharacteristic
subgroupG/H
centerC23#14579136
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#335×C79C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenterC2×#335D79
hypocenterC79#10476
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #23764361C2semidirect
5. C2×#19098310C2semidirect
6. #23761284C2semidirect
7. C2×#335×C79C2semidirect
119. C2×C8×C79SD16semidirect
214. C2×#335D79semidirect
217. C79#10476semidirect
316. C2×C8#11929613semidirect

Maximal subgroups

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

subgroupindex
C2×#190987692
#237643612
#237641942
#237634012
C2×#190983102
#237612842
C2×#335×C792
#1047679