Finite Group Databasebeta

← #34184225

Subgroups — #34184225

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 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×#2356C211yesmaximal, 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×C79C2211yescharacteristic
22#391×C79—21no—
23#387×C79—22no—
24#383×C79—21no—
25#382×C79C2211yescharacteristic
26C4×C4⋊C4×C79—21no—
27C4×C4⋊C4×C79C2211yescharacteristic
28#376×C79—21no—
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×C79C2×C412yes—
56C4×Q8×C79D412yes—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79D411yescharacteristic
61C2×C42×C79—21no—
62C2×C42×C79—22no—
63C2×C42×C79—21no—
64C2×C42×C79C2311yescharacteristic
65#95×C79—22no—
66#95×C79—22no—
67#95×C79—22no—
68#95×C79—21no—
69#95×C79D411yescharacteristic
70#95×C79D411yescharacteristic
71#95×C79D412yes—
72#11929624—41no—
73#11929624—41no—
74#11929624—41no—
75#11929624—41no—
76C2×Dic158—21no—
77C2×D316—21no—
78C2×D316—21no—
79C2×D316—21no—
80C2×C22⋊D79—41no—
81C2×C4×D79—41no—
82#11929614—44no—
83#11929611—44no—
84#11929608—41no—
85#11929608—41no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91(C2×D79)⋊C4—41no—
92(C2×D79)⋊C4—41no—
93C2×Q8×C79—21no—
94C2×Q8×C79—21no—
95C2×Q8×C79—21no—
96C2×Q8×C79—21no—
97C2×Q8×C79—21no—
98C2×Q8×C79C22⋊C412yes—
99C22×C4×C79—21no—
100C22×C4×C79—21no—
101C22×C4×C79—21no—
102C22×C4×C79—22no—
103C22×C4×C79—21no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C4⋊C4×C79—22no—
108C4⋊C4×C79—22no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79—42no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79C22⋊C412yes—
114C42×C79—22no—
115C42×C79—22no—
116C42×C79—44no—
117C42×C79—41no—
118C42×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79#4111yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79#12011yescharacteristic
124C22×D79—41no—
125Q8×C79—22no—
126Q8×C79—22no—
127Q8×C79—41no—
128Q8×C79—41no—
129C2×C4×C79—22no—
130C2×C4×C79—22no—
131C2×C4×C79—22no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—42no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—42no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—42no—
145C2×C4×C79C2×C22⋊C411yescharacteristic
146C2×C4×C79C4×D411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79#12311yescharacteristic
151C2×C4×C79#13511yescharacteristic
152C2×C4×C79#13511yescharacteristic
153C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C22⋊D79—81no—
156C22⋊D79—81no—
157C2×Dic79—41no—
158D316—41no—
159D316—41no—
160D316—81no—
161D316—81no—
162C4×D79—81no—
163C4×D79—81no—
164Dic158—41no—
165Dic158—41no—
166C79⋊C8—42no—
167C79⋊C8—81no—
168C79⋊C8—81no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79#38611yescharacteristic
172C22×C79#44911yescharacteristic
173C22×C79#44911yescharacteristic
174C2×D79—81no—
175C2×D79—81no—
176C2×D79—81no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—42no—
182C4×C79—42no—
183C4×C79—42no—
184C4×C79—81no—
185C4×C79—82no—
186C4×C79#41611yescharacteristic
187C4×C79#41811yescharacteristic
188C4×C79#48211yescharacteristic
189C4×C79#48211yescharacteristic
190Dic79—81no—
191#9182—791nomaximal, Sylow
192C2×C79—21no—
193C2×C79—21no—
194C2×C79#187511yescharacteristic
195C2×C79#190511yescharacteristic
196C2×C79#190611yescharacteristic
197D79—161no—
198#3037—791no—
199#2356D7911yescharacteristic, complemented
200#1517—791no—
201#1502—791no—
202#1479—791no—
203#1303—791no—
204#1286—791no—
205C79#918211yescharacteristic, complemented, Sylow
206C2×#137—1581no—
207#532—791no—
208C2×C4×Q8C2×D7911yescharacteristic
209#464—1582no—
210#462—792no—
211#460—792no—
212#431—791no—
213#431—791no—
214#423—791no—
215#417—1581no—
216#391—21no—
217#387—22no—
218#383—21no—
219#382C2×D7911yescharacteristic
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2×D7911yescharacteristic
222#376—21no—
223#327—1581no—
224#327—1581no—
225#327—1582no—
226Q8⋊C8—1582no—
227C2×#41—1581no—
228C22×Q8C22⋊D7911yescharacteristic
229#137—3162no—
230#137—3162no—
231C2×Q16—1582no—
232C2×SD16—1582no—
233C2×M16—1581no—
234C4⋊Q8—791no—
235#127—791no—
236#124—1581no—
237#121—1581no—
238#121—1581no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C4×D7912yes—
242C4×Q8D31612yes—
243C4×D4—1581no—
244C4×D4—1581no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22⋊D7911yescharacteristic
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42C22×D7911yescharacteristic
253C4⋊C8—792no—
254C4⋊C8—1581no—
255C4⋊C8—1581no—
256C4⋊C8—1581no—
257#103—1582no—
258#102—1582no—
259C22⋊C8—1581no—
260C22⋊C8—1581no—
261C4×C8—1581no—
262C4×C8—1581no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95C22⋊D7911yescharacteristic
268#95C22⋊D7911yescharacteristic
269#95C22⋊D7912yes—
270#41—3161no—
271#41—3161no—
272#41—3161no—
273#41—3161no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—1581no—
280C2×Q8(C2×D79)⋊C412yes—
281C2×D4—3161no—
282C2×D4—1581no—
283C2×D4—1581no—
284C2×D4—1581no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—3161no—
291C22×C4C2×C22⋊D7911yescharacteristic
292C22×C4C2×C22⋊D7911yescharacteristic
293C22×C4C2×C22⋊D7911yescharacteristic
294Q16—3164no—
295SD16—3164no—
296M16—3161no—
297M16—3161no—
298C2×C8—3161no—
299C2×C8—3161no—
300C2×C8—1582no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1581no—
308C4⋊C4—1581no—
309C4⋊C4(C2×D79)⋊C412yes—
310C22⋊C4—3161no—
311C22⋊C4—3161no—
312C42—22no—
313C42—22no—
314C42—44no—
315C42—41no—
316C42—41no—
317C42C2×C4×D7911yescharacteristic
318C42C2×D31611yescharacteristic
319C42#1192962411yescharacteristic
320C42#1192962411yescharacteristic
321C23—3161no—
322C23#1457913011yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—41no—
326Q8—41no—
327Q8—3161no—
328Q8—3161no—
329D4—3161no—
330D4—3161no—
331D4—6321no—
332D4—6321no—
333D4—6321no—
334D4—6321no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—3161no—
352C2×C4—6321no—
353C2×C4—6321no—
354C2×C4C2×(C2×D79)⋊C411yescharacteristic
355C2×C4C4×C22⋊D7911yescharacteristic
356C2×C4C4×C22⋊D7911yescharacteristic
357C2×C4#1457913511yescharacteristic
358C2×C4#1457913511yescharacteristic
359C2×C4#1457914411yescharacteristic
360C2×C4#1457918111yescharacteristic
361C2×C4#1457918111yescharacteristic
362C2×C4#1457918611yescharacteristic
363C2×C4#1457918611yescharacteristic
364C8—3162no—
365C8—6321no—
366C8—6321no—
367C22—21no—
368C22—21no—
369C22—6321no—
370C22—6321no—
371C22—6321no—
372C22#1909844211yescharacteristic
373C22#1909870811yescharacteristic
374C22#1909870811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—6321no—
385C4#1909855711yescharacteristic
386C4#1909856711yescharacteristic
387C4#1909888711yescharacteristic
388C4#1909888711yescharacteristic
389C2—21no—
390C2—21no—
391C2—12641no—
392C2#2376372211yescharacteristic
393C2#2376389411yescharacteristic
394C2#2376390111yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2356C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2356D79
hypocenterC79#9182
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
4. #23761795C2semidirect
5. #23761303C2semidirect
6. #23761258C2semidirect
7. C79×#2356C2semidirect
11. #19098773C4semidirect
12. #19098763C4semidirect
199. #2356D79semidirect
205. C79#9182semidirect

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×#23562
#918279