Finite Group Databasebeta

← #34192085

Subgroups — #34192085

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×#19098765C211yesmaximal, characteristic
2#23764362C211yesmaximal, characteristic, complemented
3#23764193C211yesmaximal, characteristic
4#23763401C211yesmaximal, characteristic
5C2×#19098315C211yesmaximal, characteristic, complemented
6#23761284C211yesmaximal, characteristic, complemented
7C2×#332×C79C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C2×C4×D316C2211yescharacteristic
10#19098765—22no—
11#19098765—22no—
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579085—21no—
14C2×#14579083—21no—
15C2×(C2×D79)⋊C8—21no—
16C2×(C2×D79)⋊C8—21no—
17C2×C4×C79⋊C8C2211yescharacteristic
18#19098433C2211yescharacteristic
19#19098326—21no—
20#19098325—21no—
21#19098325—21no—
22#19098315—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×C4⋊C8×C79C2211yescharacteristic
26#332×C79—22no—
27C22×D316D411yescharacteristic
28C22×C4×D79—21no—
29C2×#11929611—24no—
30C2×#11929611—42no—
31C2×#11929611—42no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8D411yescharacteristic
34C22×C79⋊C8D411yescharacteristic
35C4×D316—21no—
36C4×D316—21no—
37C4×D316—21no—
38C4×D316—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79D411yescharacteristic
43C2×(C2×D79)⋊C4—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)D412yes—
46#14579085—24no—
47#14579083—42no—
48(C2×D79)⋊C8—42no—
49(C2×D79)⋊C8—42no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C22×Q8×C79—21no—
53C22×C8×C79—21no—
54C22×C8×C79—21no—
55C4⋊Q8×C79—21no—
56C4⋊Q8×C79—21no—
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79D412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C42×C79C2311yescharacteristic
62C4⋊C8×C79—21no—
63C4⋊C8×C79—21no—
64C4⋊C8×C79—21no—
65C4⋊C8×C79—21no—
66C23×D79—21no—
67C2×D316—21no—
68C2×D316—21no—
69C2×D316—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316D812yes—
73C22×Dic79—21no—
74C22×Dic79—41no—
75C2×C4×D79—41no—
76C2×C4×D79—41no—
77C2×C4×D79—41no—
78C2×C4×D79—41no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81#11929611—48no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—21no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8SD1612yescomplemented
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93C4⋊Dic79—41no—
94C4⋊Dic79—41no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—81no—
98(C2×D79)⋊C4—41no—
99(C2×D79)⋊C4—41no—
100C2×Q8×C79—22no—
101C2×Q8×C79—41no—
102C2×Q8×C79—41no—
103C2×Q8×C79—41no—
104C2×Q8×C79—41no—
105C2×Q8×C79—41no—
106C22×C4×C79—41no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79#4111yescharacteristic
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C2×C8×C79—41no—
113C2×C8×C79—41no—
114C2×C8×C79—41no—
115C2×C8×C79—41no—
116C4⋊C4×C79—22no—
117C4⋊C4×C79—22no—
118C4⋊C4×C79—41no—
119C4⋊C4×C79—41no—
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—
132Q8×C79—42no—
133Q8×C79—42no—
134Q8×C79—81no—
135Q8×C79—81no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—42no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79#12011yescharacteristic
142C2×C4×C79#12111yescharacteristic
143C2×C4×C79#12111yescharacteristic
144C2×C4×C79#12311yescharacteristic
145C2×C4×C79#12311yescharacteristic
146C2×C4×C79#12411yescharacteristic
147C2×C4×C79C2×D811yescharacteristic
148C2×C4×C79C2×SD1611yescharacteristic
149C2×C4×C79#13511yescharacteristic
150C2×C4×C79#13511yescharacteristic
151C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C79#13711yescharacteristic
154C2×C4×C79#13711yescharacteristic
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—81no—
159C2×Dic79—81no—
160C2×Dic79—81no—
161D316—22no—
162D316—22no—
163D316—41no—
164D316—41no—
165C4×D79—81no—
166C4×D79—81no—
167C4×D79—81no—
168C4×D79—81no—
169C8×C79—81no—
170C8×C79—81no—
171C79⋊C8—22no—
172C79⋊C8—42no—
173C79⋊C8—41no—
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—82no—
191C4×C79#44711yescharacteristic
192C4×C79#45211yescharacteristic
193C4×C79#46011yescharacteristic
194C4×C79#46411yescharacteristic
195C4×C79#46511yescharacteristic
196C4×C79#46811yescharacteristic
197C4×C79#48011yescharacteristic
198C4×C79#48211yescharacteristic
199Dic79—81no—
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×#335—791no—
215C2×#332D7911yescharacteristic, complemented
216#1296—791no—
217C79#1047611yescharacteristic, complemented, Sylow
218C22×SD16—1581no—
219C2×C4⋊Q8C2×D7911yescharacteristic
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×C8—791no—
231#384—791no—
232#340—1581no—
233#340—1581no—
234#340—1581no—
235#335—1582no—
236#332—22no—
237C22×Q8—21no—
238C22×D4—791no—
239C23×C4—1581no—
240C2×SD16—3162no—
241C2×SD16—3162no—
242C2×SD16—1584no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—791no—
246C22×C8—791no—
247C22×C8—1581no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22⋊D7912yes—
252C4×D4—1581no—
253C4×D4—1581no—
254C4×D4—1581no—
255C4×D4—1581no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—791no—
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—21no—
268C4⋊C8—792no—
269C4⋊C8—1581no—
270#103—1584no—
271#102—3162no—
272C22⋊C8—3162no—
273C22⋊C8—3162no—
274C4×C8—1581no—
275C4×C8—1581no—
276C24—1581no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—792no—
284C2×D4—1581no—
285C2×D4—1581no—
286C2×D4—1581no—
287C2×D4—1581no—
288C2×D4—1581no—
289C22×C4—41no—
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×C4D4×D7911yescharacteristic
300C22×C4#1192962411yescharacteristic
301SD16—3168no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—3161no—
309C2×C8—3161no—
310C2×C8—792no—
311C2×C8—1582no—
312C2×C8—1581no—
313C2×C8—1581no—
314C2×C8—1581no—
315C2×C8—1581no—
316C2×C8—1581no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—3161no—
322C4⋊C4—3161no—
323C4⋊C4—3161no—
324C4⋊C4—3161no—
325C4⋊C4—6321no—
326C4⋊C4—1581no—
327C4⋊C4—1581no—
328C22⋊C4—3161no—
329C22⋊C4—3161no—
330C42D4×D7911yescharacteristic
331C42C2×C22⋊D7911yescharacteristic
332C42#1192962311yescharacteristic
333C42#1192962411yescharacteristic
334C23—3161no—
335C23—3161no—
336C23—3161no—
337C23—3161no—
338C23—3161no—
339C23—3161no—
340C23—1582no—
341C23#1457913411yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346D4—3161no—
347D4—3161no—
348D4—1582no—
349D4—1582no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—3161no—
356C2×C4—3161no—
357C2×C4—3161no—
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#1457913311yescharacteristic
367C2×C4#1457913411yescharacteristic
368C2×C4#1457914311yescharacteristic
369C2×C4#1457914411yescharacteristic
370C2×C4#1457914711yescharacteristic
371C2×C4D8×D7911yescharacteristic
372C2×C4SD16×D7911yescharacteristic
373C2×C4#1457918211yescharacteristic
374C2×C4#1457918411yescharacteristic
375C2×C4#1457918511yescharacteristic
376C2×C4#1457918811yescharacteristic
377C2×C4#1457919011yescharacteristic
378C2×C4#1457919311yescharacteristic
379C8—81no—
380C8—81no—
381C8—3162no—
382C8—3161no—
383C8—1582no—
384C22—3162no—
385C22—3162no—
386C22—3162no—
387C22—6321no—
388C22—6321no—
389C22—6321no—
390C22—6321no—
391C22#1909847811yescharacteristic
392C22#1909876711yescharacteristic
393C22#1909878111yescharacteristic
394C22#1909879811yescharacteristic
395C22#1909880811yescharacteristic
396C22#1909882911yescharacteristic
397C22#1909883711yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—6321no—
402C4—12641no—
403C4#1909869511yescharacteristic
404C4#1909872211yescharacteristic
405C4#1909876111yescharacteristic
406C4#1909878711yescharacteristic
407C4#1909879511yescharacteristic
408C4#1909881211yescharacteristic
409C4#1909887511yescharacteristic
410C4#1909888611yescharacteristic
411C2—6322no—
412C2#2376278811yescharacteristic
413C2#2376284011yescharacteristic
414C2#2376289811yescharacteristic
415C2#2376295011yescharacteristic
416C2#2376458011yescharacteristic
417C2#2376469911yescharacteristic
418C2#2376471411yescharacteristic
subgroupG/H
centerC23#14579134
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#332×C79C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenterC2×#332D79
hypocenterC79#10476
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #23764362C2semidirect
5. C2×#19098315C2semidirect
6. #23761284C2semidirect
7. C2×#332×C79C2semidirect
90. C2×C79⋊C8SD16semidirect
215. C2×#332D79semidirect
217. C79#10476semidirect

Maximal subgroups

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

subgroupindex
C2×#190987652
#237643622
#237641932
#237634012
C2×#190983152
#237612842
C2×#332×C792
#1047679