Finite Group Databasebeta

← #34184211

Subgroups — #34184211

Order 20224.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772872C211yesmaximal, characteristic
2#23761959C211yesmaximal, characteristic
3#23761894C211yesmaximal, characteristic
4#23761795C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761258C211yesmaximal, characteristic, complemented
7C79×#2351C211yesmaximal, characteristic, complemented
8C2×#14579192—21no—
9#19099113C2211yescharacteristic
10#19098785—22no—
11#19098773C412yescomplemented
12#19098763C412yescomplemented
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16#19098562—21no—
17#19098296—22no—
18#19098296—21no—
19#19098296—21no—
20Q8⋊(C79⋊C8)—22no—
21C2×C4×Q8×C79—21no—
22C2×C4×Q8×C79C2211yescharacteristic
23#389×C79—21no—
24#389×C79—22no—
25#382×C79—21no—
26#382×C79C2211yescharacteristic
27C4×C4⋊C4×C79—21no—
28C4×C4⋊C4×C79C2211yescharacteristic
29C2×#11929624—21no—
30#14579192—42no—
31#14579192—42no—
32C2×#11929614—22no—
33C2×#11929611—22no—
34C2×#11929608—21no—
35#14579161C2×C411yescharacteristic
36#14579158C2×C411yescharacteristic
37#14579149—21no—
38#14579135—21no—
39#14579135—21no—
40C4×D316—21no—
41C4×D316—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)C2×C412yes—
46#14579084—22no—
47#14579081—22no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C22×Q8×C79D411yescharacteristic
53C4×Q8×C79—21no—
54C4×Q8×C79—21no—
55C4×Q8×C79—42no—
56C4×Q8×C79—42no—
57C4×Q8×C79C2×C412yes—
58C4×Q8×C79D412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—22no—
61C2×C4⋊C4×C79—22no—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79D411yescharacteristic
65C2×C42×C79—21no—
66C2×C42×C79—22no—
67C2×C42×C79—21no—
68C2×C42×C79C2311yescharacteristic
69#95×C79—22no—
70#95×C79—21no—
71#95×C79D411yescharacteristic
72#95×C79D411yescharacteristic
73#95×C79D412yes—
74#11929624—41no—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78C2×Dic158—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×C22⋊D79—41no—
83C2×C4×D79—41no—
84#11929614—44no—
85#11929611—44no—
86#11929608—41no—
87#11929608—41no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—21no—
92C4⋊Dic79—21no—
93(C2×D79)⋊C4—41no—
94(C2×D79)⋊C4—41no—
95C2×Q8×C79—21no—
96C2×Q8×C79—21no—
97C2×Q8×C79—21no—
98C2×Q8×C79C22⋊C412yes—
99C2×Q8×C79D812yes—
100C2×Q8×C79SD1612yes—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—21no—
105C22×C4×C79—22no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C4⋊C4×C79—22no—
110C4⋊C4×C79—22no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—42no—
113C4⋊C4×C79—44no—
114C4⋊C4×C79—44no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79C22⋊C412yes—
118C42×C79—22no—
119C42×C79—22no—
120C42×C79—44no—
121C42×C79—41no—
122C42×C79—41no—
123C42×C79C22×C411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79#4111yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79#12011yescharacteristic
128C22×D79—41no—
129Q8×C79—22no—
130Q8×C79—22no—
131Q8×C79—22no—
132Q8×C79#10214yes—
133C2×C4×C79—22no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—42no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79—41no—
146C2×C4×C79—42no—
147C2×C4×C79—42no—
148C2×C4×C79—41no—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79C4×D411yescharacteristic
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79#12111yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12311yescharacteristic
155C2×C4×C79C2×D811yescharacteristic
156C2×C4×C79C2×SD1611yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79#13711yescharacteristic
159C22⋊D79—81no—
160C22⋊D79—81no—
161C2×Dic79—41no—
162D316—41no—
163D316—41no—
164D316—81no—
165D316—81no—
166C4×D79—81no—
167C4×D79—81no—
168Dic158—41no—
169Dic158—41no—
170C79⋊C8—42no—
171C79⋊C8—81no—
172C79⋊C8—81no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79#38611yescharacteristic
176C22×C79#44711yescharacteristic
177C22×C79#44811yescharacteristic
178C2×D79—81no—
179C2×D79—81no—
180C2×D79—81no—
181C4×C79—24no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—42no—
187C4×C79—42no—
188C4×C79—82no—
189C4×C79—81no—
190C4×C79C2×#10211yescharacteristic
191C4×C79#41711yescharacteristic
192C4×C79#48011yescharacteristic
193C4×C79#48111yescharacteristic
194Dic79—81no—
195#9180—791nomaximal, Sylow
196C2×C79—21no—
197C2×C79—21no—
198C2×C79#187411yescharacteristic
199C2×C79#190511yescharacteristic
200C2×C79#190611yescharacteristic
201D79—161no—
202#3037—791no—
203#2351D7911yescharacteristic, complemented
204#1517—791no—
205#1502—791no—
206#1479—791no—
207#1303—791no—
208#1286—791no—
209C79#918011yescharacteristic, complemented, Sylow
210C2×#137—1581no—
211#532—791no—
212C2×C4×Q8—21no—
213C2×C4×Q8C2×D7911yescharacteristic
214#464—1582no—
215#462—792no—
216#460—792no—
217#431—791no—
218#431—791no—
219#423—791no—
220#417—1581no—
221#389—21no—
222#389—22no—
223#382—21no—
224#382C2×D7911yescharacteristic
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2×D7911yescharacteristic
227#327—1581no—
228#327—1581no—
229#327—1582no—
230Q8⋊C8—1582no—
231C2×#41—1581no—
232C22×Q8C22⋊D7911yescharacteristic
233#137—3162no—
234#137—3162no—
235C2×Q16—1582no—
236C2×SD16—1582no—
237C2×M16—1581no—
238C4⋊Q8—791no—
239#127—791no—
240#124—1581no—
241#121—1581no—
242#121—1581no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8C4×D7912yes—
248C4×Q8D31612yes—
249C4×D4—1581no—
250C4×D4—1581no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D7911yescharacteristic
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42C22×D7911yescharacteristic
261C4⋊C8—792no—
262C4⋊C8—1581no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265#103—1582no—
266#102—1582no—
267C22⋊C8—1581no—
268C22⋊C8—1581no—
269C4×C8—1581no—
270C4×C8—1581no—
271#95—22no—
272#95—21no—
273#95C22⋊D7911yescharacteristic
274#95C22⋊D7911yescharacteristic
275#95C22⋊D7912yes—
276#41—3161no—
277#41—3161no—
278#41—3161no—
279#41—3161no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—1581no—
284C2×Q8(C2×D79)⋊C412yes—
285C2×Q8#1192960912yes—
286C2×Q8#1192961112yes—
287C2×D4—3161no—
288C2×D4—1581no—
289C2×D4—1581no—
290C2×D4—1581no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—3161no—
297C22×C4C2×C22⋊D7911yescharacteristic
298C22×C4C2×C22⋊D7911yescharacteristic
299C22×C4C2×C22⋊D7911yescharacteristic
300Q16—3164no—
301SD16—3164no—
302M16—3161no—
303M16—3161no—
304C2×C8—3161no—
305C2×C8—3161no—
306C2×C8—1582no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1581no—
316C4⋊C4—1581no—
317C4⋊C4(C2×D79)⋊C412yes—
318C22⋊C4—3161no—
319C22⋊C4—3161no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—41no—
324C42—41no—
325C42C2×C4×D7911yescharacteristic
326C42C2×D31611yescharacteristic
327C42#1192962411yescharacteristic
328C42#1192962411yescharacteristic
329C23—3161no—
330C23#1457913011yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—3161no—
335Q8—3161no—
336Q8#1457908114yes—
337D4—3161no—
338D4—3161no—
339D4—6321no—
340D4—6321no—
341D4—6321no—
342D4—6321no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—3161no—
360C2×C4—6321no—
361C2×C4—6321no—
362C2×C4C2×(C2×D79)⋊C411yescharacteristic
363C2×C4C4×C22⋊D7911yescharacteristic
364C2×C4C4×C22⋊D7911yescharacteristic
365C2×C4#1457913511yescharacteristic
366C2×C4#1457913511yescharacteristic
367C2×C4#1457914411yescharacteristic
368C2×C4C2×#1192960911yescharacteristic
369C2×C4C2×#1192961111yescharacteristic
370C2×C4#1457918711yescharacteristic
371C2×C4#1457919211yescharacteristic
372C8—3162no—
373C8—6321no—
374C8—6321no—
375C22—21no—
376C22—21no—
377C22—6321no—
378C22—6321no—
379C22—6321no—
380C22#1909844211yescharacteristic
381C22#1909869711yescharacteristic
382C22#1909870211yescharacteristic
383C4—24no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—81no—
392C4—6321no—
393C4C2×#1457908111yescharacteristic
394C4#1909856211yescharacteristic
395C4#1909887611yescharacteristic
396C4#1909888111yescharacteristic
397C2—21no—
398C2—21no—
399C2—12641no—
400C2#2376371711yescharacteristic
401C2#2376389411yescharacteristic
402C2#2376390111yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2351C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2351D79
hypocenterC79#9180
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #23761795C2semidirect
5. #23761303C2semidirect
6. #23761258C2semidirect
7. C79×#2351C2semidirect
11. #19098773C4semidirect
12. #19098763C4semidirect
203. #2351D79semidirect
209. C79#9180semidirect

Maximal subgroups

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

subgroupindex
#237728722
#237619592
#237618942
#237617952
#237613032
#237612582
C79×#23512
#918079