Finite Group Databasebeta

← #34184541

Subgroups — #34184541

Order 20224.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773431C211yesmaximal, characteristic
2#23764228C211yesmaximal, characteristic
3#23761894C211yesmaximal, characteristic
4#23761829C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761233C211yesmaximal, characteristic, complemented
7C42×Q8×C79C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098881—21no—
10#19098848C412yescomplemented
11#19098763C412yescomplemented
12#19098702—21no—
13C4×#11929611—22no—
14#19098616C2211yescharacteristic
15#19098616C2211yescharacteristic
16#19098587C2211yescharacteristic
17C2×#14579089—21no—
18C2×#14579089—21no—
19#19098296—22no—
20Q8⋊(C79⋊C8)—22no—
21C2×C4×Q8×C79—21no—
22C2×C4×Q8×C79C2211yescharacteristic
23C4×C4⋊C4×C79—21no—
24C4×C4⋊C4×C79—21no—
25C4×C4⋊C4×C79—22no—
26C4×C4⋊C4×C79C2211yescharacteristic
27C43×C79—21no—
28C43×C79C2211yescharacteristic
29C2×#11929624—21no—
30C2×#11929611—22no—
31C2×#11929608—21no—
32#14579161C2×C411yescharacteristic
33#14579158C2×C411yescharacteristic
34#14579149—21no—
35#14579135—21no—
36#14579135—21no—
37C4×D316—21no—
38C4×D316—21no—
39#14579092—22no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)C2×C412yes—
44#14579089—42no—
45#14579089—42no—
46#14579085—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—28no—
54C4×Q8×C79—21no—
55C4×Q8×C79—21no—
56C4×Q8×C79C2×C412yes—
57C4×Q8×C79D412yes—
58C2×C4⋊C4×C79—22no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C42×C79—22no—
63C2×C42×C79—21no—
64C2×C42×C79—21no—
65C2×C42×C79—22no—
66C2×C42×C79—21no—
67C2×C42×C79—21no—
68C2×C42×C79D411yescharacteristic
69C2×C42×C79D411yescharacteristic
70C2×C42×C79C2311yescharacteristic
71#95×C79—22no—
72#95×C79D412yes—
73#11929624—41no—
74#11929624—41no—
75#11929624—41no—
76#11929624—41no—
77C2×Dic158—21no—
78C2×D316—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×C22⋊D79—41no—
82C2×C4×D79—41no—
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—24no—
94C2×Q8×C79—21no—
95C2×Q8×C79C22⋊C412yes—
96C22×C4×C79—21no—
97C22×C4×C79—21no—
98C22×C4×C79—21no—
99C22×C4×C79—21no—
100C22×C4×C79—22no—
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C4⋊C4×C79—28no—
105C4⋊C4×C79—24no—
106C4⋊C4×C79—22no—
107C4⋊C4×C79—22no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79C22⋊C412yes—
110C42×C79—28no—
111C42×C79—24no—
112C42×C79—22no—
113C42×C79—22no—
114C42×C79—21no—
115C42×C79—21no—
116C42×C79—21no—
117C42×C79—42no—
118C42×C79—44no—
119C42×C79—41no—
120C42×C79—41no—
121C42×C79SD1612yescomplemented
122C42×C79C22×C411yescharacteristic
123C42×C79C2×D411yescharacteristic
124C42×C79#4111yescharacteristic
125C42×C79#4111yescharacteristic
126C23×C79#12011yescharacteristic
127C22×D79—41no—
128Q8×C79—24no—
129Q8×C79—22no—
130Q8×C79#10414yescomplemented
131C2×C4×C79—22no—
132C2×C4×C79—24no—
133C2×C4×C79—22no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—24no—
139C2×C4×C79—22no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79—41no—
145C2×C4×C79—42no—
146C2×C4×C79C2×C22⋊C411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79C4×D411yescharacteristic
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12311yescharacteristic
152C2×C4×C79C2×SD1611yescharacteristic
153C2×C4×C79#13511yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C79#13711yescharacteristic
156C22⋊D79—81no—
157C22⋊D79—81no—
158C2×Dic79—41no—
159D316—42no—
160D316—41no—
161D316—41no—
162D316—81no—
163C4×D79—81no—
164C4×D79—81no—
165Dic158—41no—
166Dic158—41no—
167C79⋊C8—42no—
168C79⋊C8—81no—
169C79⋊C8—81no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79#38611yescharacteristic
173C22×C79#44811yescharacteristic
174C22×C79#44911yescharacteristic
175C2×D79—81no—
176C2×D79—81no—
177C2×D79—81no—
178C4×C79—24no—
179C4×C79—24no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—42no—
184C4×C79—44no—
185C4×C79—42no—
186C4×C79C2×#10412yes—
187C4×C79C4×SD1611yescharacteristic
188C4×C79#44011yescharacteristic
189C4×C79#46011yescharacteristic
190C4×C79#46411yescharacteristic
191Dic79—81no—
192#9234—791nomaximal, Sylow
193C2×C79—21no—
194C2×C79—21no—
195C2×C79#188011yescharacteristic
196C2×C79#189611yescharacteristic
197C2×C79#190611yescharacteristic
198D79—161no—
199#3103—791no—
200C42×Q8D7911yescharacteristic, complemented
201#1962—791no—
202#1502—791no—
203#1487—791no—
204#1303—791no—
205#1276—791no—
206C79#923411yescharacteristic, complemented, Sylow
207#532—791no—
208C2×C4×Q8—21no—
209C2×C4×Q8C2×D7911yescharacteristic
210#481—1581no—
211#475—792no—
212#460—792no—
213#448—1581no—
214C4×SD16—1582no—
215#431—791no—
216#431—791no—
217#423—791no—
218C2×#104—1581no—
219C2×#104—1581no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2×D7911yescharacteristic
224C43—21no—
225C43C2×D7911yescharacteristic
226#327—1582no—
227Q8⋊C8—1582no—
228C2×#41—1581no—
229C22×Q8C22⋊D7911yescharacteristic
230C2×SD16—1582no—
231C2×M16—1581no—
232C4⋊Q8—791no—
233#127—791no—
234#124—1581no—
235#121—1581no—
236#121—1581no—
237C4×Q8—28no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D7912yes—
241C4×Q8D31612yes—
242C4×D4—1581no—
243C4×D4—1581no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D7911yescharacteristic
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊D7911yescharacteristic
255C2×C42C22⋊D7911yescharacteristic
256C2×C42C22×D7911yescharacteristic
257#106—1582no—
258C4⋊C8—792no—
259C4⋊C8—1581no—
260C4⋊C8—1581no—
261C4⋊C8—1581no—
262#104—3162no—
263#104—3162no—
264#103—1582no—
265#102—1582no—
266C22⋊C8—1581no—
267C22⋊C8—1581no—
268C4×C8—1581no—
269C4×C8—1581no—
270#95—22no—
271#95C22⋊D7912yes—
272#41—3161no—
273#41—3161no—
274#41—3161no—
275#41—3161no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—1581no—
279C2×Q8(C2×D79)⋊C412yes—
280C2×D4—3161no—
281C2×D4—1581no—
282C2×D4—1581no—
283C2×D4—1581no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—3161no—
290C22×C4C2×C22⋊D7911yescharacteristic
291C22×C4C2×C22⋊D7911yescharacteristic
292C22×C4C2×C22⋊D7911yescharacteristic
293SD16—3164no—
294M16—3161no—
295M16—3161no—
296C2×C8—3161no—
297C2×C8—3161no—
298C2×C8—1582no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—1581no—
305C4⋊C4—1581no—
306C4⋊C4(C2×D79)⋊C412yes—
307C22⋊C4—3161no—
308C22⋊C4—3161no—
309C42—28no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#1192961112yescomplemented
321C42C2×C4×D7911yescharacteristic
322C42C2×D31611yescharacteristic
323C42#1192962411yescharacteristic
324C42#1192962411yescharacteristic
325C23—3161no—
326C23#1457913011yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—3161no—
330Q8—3161no—
331Q8#1457908914yescomplemented
332D4—3161no—
333D4—3161no—
334D4—3162no—
335D4—6321no—
336D4—6321no—
337D4—6321no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—3161no—
354C2×C4—6321no—
355C2×C4—6321no—
356C2×C4C2×(C2×D79)⋊C411yescharacteristic
357C2×C4C4×C22⋊D7911yescharacteristic
358C2×C4C4×C22⋊D7911yescharacteristic
359C2×C4#1457913511yescharacteristic
360C2×C4#1457913511yescharacteristic
361C2×C4#1457914411yescharacteristic
362C2×C4C2×#1192961111yescharacteristic
363C2×C4#1457918111yescharacteristic
364C2×C4#1457918611yescharacteristic
365C2×C4#1457919211yescharacteristic
366C8—3162no—
367C8—6321no—
368C8—6321no—
369C22—21no—
370C22—21no—
371C22—6321no—
372C22—6321no—
373C22—6321no—
374C22#1909844211yescharacteristic
375C22#1909870211yescharacteristic
376C22#1909870811yescharacteristic
377C4—24no—
378C4—24no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—6321no—
386C4C2×#1457908912yes—
387C4C4×#1192961111yescharacteristic
388C4#1909866111yescharacteristic
389C4#1909876311yescharacteristic
390C4#1909878511yescharacteristic
391C2—21no—
392C2—21no—
393C2—12641no—
394C2#2376375511yescharacteristic
395C2#2376384311yescharacteristic
396C2#2376390111yescharacteristic
subgroupG/H
centerC2×C4#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC42×Q8×C79C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenterC42×Q8D79
hypocenterC79#9234
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #23761829C2semidirect
5. #23761303C2semidirect
6. #23761233C2semidirect
7. C42×Q8×C79C2semidirect
10. #19098848C4semidirect
11. #19098763C4semidirect
121. C42×C79SD16semidirect
130. Q8×C79#104semidirect
200. C42×Q8D79semidirect
206. C79#9234semidirect
320. C42#11929611semidirect
331. Q8#14579089semidirect

Maximal subgroups

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

subgroupindex
#237734312
#237642282
#237618942
#237618292
#237613032
#237612332
C42×Q8×C792
#923479