Finite Group Databasebeta

← #34183135

Subgroups — #34183135

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773431C211yesmaximal, characteristic
2#23764319C211yesmaximal, characteristic
3C4×#14579085C211yesmaximal, characteristic, complemented
4#23761850C211yesmaximal, characteristic
5#23761301C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7Q8×M16×C79C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098881—21no—
10#19098848C411yescharacteristic, complemented
11#19098848C411yescharacteristic, complemented
12#19098763C411yescharacteristic, complemented
13#19098763C411yescharacteristic, complemented
14#19098702—21no—
15C4×#11929611—21no—
16C4×#11929611—21no—
17#19098616C2211yescharacteristic
18C2×#14579087—21no—
19C2×#14579087—21no—
20C2×#14579085—21no—
21C2×C4×C79⋊C8C2211yescharacteristic
22C4×C4⋊Dic79C2211yescharacteristic
23#19098325—21no—
24#19098290—22no—
25C2×C4×Q8×C79C2211yescharacteristic
26#446×C79—21no—
27#446×C79—22no—
28C8×Q8×C79—21no—
29#423×C79—21no—
30#423×C79C2211yescharacteristic
31C4×M16×C79—21no—
32C4×M16×C79C2211yescharacteristic
33C2×#11929624—21no—
34C2×#11929611—21no—
35C2×#11929611—21no—
36C22×C79⋊C8—21no—
37#14579161C2×C411yescharacteristic
38#14579158C2×C411yescharacteristic
39#14579149—21no—
40#14579135—21no—
41#14579135—21no—
42C4×D316—21no—
43C4×D316—21no—
44C2×C4⋊Dic79—21no—
45C2×C4×Dic79—21no—
46#14579092—21no—
47#14579092—21no—
48C4⋊(C79⋊C8)C2×C411yescharacteristic
49C4⋊(C79⋊C8)C2×C411yescharacteristic
50#14579087—42no—
51#14579087—42no—
52#14579085—24no—
53#14579085—21no—
54#14579085—21no—
55#14579081—21no—
56#14579081—21no—
57C22⋊(C79⋊C8)—21no—
58C4×C79⋊C8—21no—
59C4×C79⋊C8—21no—
60C4×C79⋊C8—21no—
61C79⋊#95—21no—
62C22×Q8×C79D411yescharacteristic
63C2×M16×C79—21no—
64C2×M16×C79D411yescharacteristic
65C2×M16×C79D411yescharacteristic
66C4×Q8×C79—21no—
67C4×Q8×C79—21no—
68C4×Q8×C79C2×C411yescharacteristic
69C4×Q8×C79C2×C411yescharacteristic
70C4×Q8×C79D411yescharacteristic
71C4×Q8×C79D411yescharacteristic
72C2×C4⋊C4×C79—21no—
73C2×C4⋊C4×C79D411yescharacteristic
74C2×C42×C79—21no—
75C2×C42×C79C2311yescharacteristic
76C4⋊C8×C79—22no—
77C4⋊C8×C79—22no—
78C4⋊C8×C79—21no—
79C4⋊C8×C79D412yes—
80#97×C79—22no—
81#97×C79—21no—
82C4×C8×C79—22no—
83C4×C8×C79—21no—
84#11929624—41no—
85#11929624—41no—
86#11929624—41no—
87#11929624—41no—
88C2×Dic158—21no—
89C2×D316—21no—
90C2×D316—21no—
91C2×D316—21no—
92C2×C22⋊D79—41no—
93C22×Dic79—41no—
94C2×C4×D79—41no—
95#11929611—42no—
96#11929611—42no—
97C2×C79⋊C8—22no—
98C2×C79⋊C8—21no—
99C2×C79⋊C8—21no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C4⋊Dic79—22no—
103C4⋊Dic79—21no—
104C4⋊Dic79—21no—
105C4⋊Dic79—41no—
106(C2×D79)⋊C4—41no—
107(C2×D79)⋊C4—41no—
108C4×Dic79—41no—
109C4×Dic79—41no—
110C2×Q8×C79—21no—
111C2×Q8×C79—22no—
112C2×Q8×C79—41no—
113C2×Q8×C79C22⋊C411yescharacteristic
114C2×Q8×C79C22⋊C411yescharacteristic
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120M16×C79—21no—
121M16×C79—21no—
122M16×C79—21no—
123M16×C79—41no—
124M16×C79—41no—
125M16×C79SD1612yescomplemented
126C2×C8×C79—21no—
127C2×C8×C79—21no—
128C2×C8×C79—22no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—21no—
133C4⋊C4×C79—21no—
134C4⋊C4×C79C22⋊C411yescharacteristic
135C4⋊C4×C79C22⋊C411yescharacteristic
136C42×C79—21no—
137C42×C79—21no—
138C42×C79—21no—
139C42×C79—21no—
140C42×C79C22×C411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C42×C79#4111yescharacteristic
144C23×C79#12111yescharacteristic
145C22×D79—41no—
146Q8×C79—21no—
147Q8×C79—21no—
148Q8×C79—21no—
149Q8×C79—42no—
150Q8×C79#10412yescomplemented
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—41no—
158C2×C4×C79—42no—
159C2×C4×C79—81no—
160C2×C4×C79C2×C22⋊C411yescharacteristic
161C2×C4×C79C4×D411yescharacteristic
162C2×C4×C79C4×D411yescharacteristic
163C2×C4×C79#12011yescharacteristic
164C2×C4×C79#12111yescharacteristic
165C2×C4×C79#12311yescharacteristic
166C2×C4×C79C2×SD1611yescharacteristic
167C2×C4×C79#13511yescharacteristic
168C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C79#13711yescharacteristic
170C22⋊D79—81no—
171C22⋊D79—81no—
172C2×Dic79—42no—
173C2×Dic79—41no—
174C2×Dic79—81no—
175C2×Dic79—81no—
176D316—42no—
177D316—41no—
178D316—41no—
179D316—41no—
180D316—41no—
181C4×D79—81no—
182C4×D79—81no—
183Dic158—41no—
184Dic158—41no—
185C8×C79—22no—
186C8×C79—41no—
187C8×C79—42no—
188C79⋊C8—42no—
189C79⋊C8—41no—
190C79⋊C8—41no—
191C22×C79—21no—
192C22×C79—21no—
193C22×C79#38611yescharacteristic
194C22×C79#46011yescharacteristic
195C22×C79#46411yescharacteristic
196C2×D79—81no—
197C2×D79—81no—
198C2×D79—81no—
199C4×C79—21no—
200C4×C79—22no—
201C4×C79—21no—
202C4×C79—21no—
203C4×C79—41no—
204C4×C79—41no—
205C4×C79—41no—
206C4×C79C2×#10411yescharacteristic
207C4×C79#42111yescharacteristic
208C4×C79C4×SD1611yescharacteristic
209C4×C79#44011yescharacteristic
210C4×C79#44811yescharacteristic
211C4×C79#44911yescharacteristic
212Dic79—82no—
213Dic79—81no—
214#8992—791nomaximal, Sylow
215C2×C79—21no—
216C2×C79—21no—
217C2×C79#187111yescharacteristic
218C2×C79#188011yescharacteristic
219C2×C79#188111yescharacteristic
220D79—161no—
221#3103—791no—
222Q8×M16D7911yescharacteristic, complemented
223#1980—791no—
224C4×#103—791no—
225#1492—791no—
226#1303—791no—
227#1278—791no—
228C79#899211yescharacteristic, complemented, Sylow
229#532—791no—
230C2×C4×Q8C2×D7911yescharacteristic
231#481—1581no—
232#475—791no—
233#475—791no—
234#460—791no—
235#460—791no—
236#448—1581no—
237#446—21no—
238#446—22no—
239C8×Q8—21no—
240C4×SD16—1581no—
241C4×SD16—1581no—
242#431—791no—
243#423—21no—
244#423C2×D7911yescharacteristic
245C2×#104—1581no—
246C2×#104—1581no—
247C2×#103—1581no—
248C4×M16—21no—
249C4×M16C2×D7911yescharacteristic
250C2×C4×C8—791no—
251C4×C4⋊C4—791no—
252#340—1581no—
253D4⋊C8—1582no—
254C2×#41—1581no—
255C22×Q8D31611yescharacteristic
256C2×SD16—1581no—
257C2×SD16—1581no—
258C2×M16—21no—
259C2×M16C22⋊D7911yescharacteristic
260C2×M16C22⋊D7911yescharacteristic
261C22×C8—1581no—
262C4⋊Q8—791no—
263#127—791no—
264#124—1581no—
265#121—1581no—
266#121—1581no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4×D7911yescharacteristic
270C4×Q8C4×D7911yescharacteristic
271C4×Q8C22⋊D7911yescharacteristic
272C4×Q8C22⋊D7911yescharacteristic
273C4×D4—1581no—
274C4×D4—1581no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1581no—
277C2×C4⋊C4D31611yescharacteristic
278C2×C42—21no—
279C2×C42—1581no—
280C2×C42C22×D7911yescharacteristic
281#106—1581no—
282#106—1581no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—791no—
287C4⋊C8—791no—
288C4⋊C8C22⋊D7912yes—
289#104—3162no—
290#104—3162no—
291#103—1584no—
292#103—1581no—
293#103—1581no—
294#102—1581no—
295#102—1581no—
296C22⋊C8—1581no—
297#97—22no—
298#97—21no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—1581no—
302C4×C8—1581no—
303C4×C8—1581no—
304#95—1581no—
305#41—3161no—
306#41—3161no—
307#41—3161no—
308#41—3161no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—1581no—
313C2×Q8(C2×D79)⋊C411yescharacteristic
314C2×Q8(C2×D79)⋊C411yescharacteristic
315C2×D4—3161no—
316C2×D4—1581no—
317C2×D4—1581no—
318C2×D4—1581no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—3161no—
322C22×C4—3161no—
323C22×C4C2×C22⋊D7911yescharacteristic
324C22×C4C2×D31611yescharacteristic
325C22×C4#1192962411yescharacteristic
326SD16—3162no—
327SD16—3162no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16#1192961112yescomplemented
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—3161no—
338C2×C8—3161no—
339C2×C8—1582no—
340C2×C8—1581no—
341C2×C8—1581no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—3161no—
348C4⋊C4—1582no—
349C4⋊C4—1581no—
350C4⋊C4—1581no—
351C4⋊C4(C2×D79)⋊C411yescharacteristic
352C4⋊C4(C2×D79)⋊C411yescharacteristic
353C22⋊C4—3161no—
354C22⋊C4—3161no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—3161no—
360C42—3161no—
361C42C2×C4×D7911yescharacteristic
362C42C2×C22⋊D7911yescharacteristic
363C42C2×C22⋊D7911yescharacteristic
364C42#1192962411yescharacteristic
365C23—3161no—
366C23#1457913511yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—3161no—
372Q8—3161no—
373Q8#1457908712yescomplemented
374D4—3162no—
375D4—3161no—
376D4—3161no—
377D4—3161no—
378D4—3161no—
379D4—6321no—
380D4—6321no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—3162no—
391C2×C4—3161no—
392C2×C4—6321no—
393C2×C4—6321no—
394C2×C4—6321no—
395C2×C4—6321no—
396C2×C4C2×(C2×D79)⋊C411yescharacteristic
397C2×C4C4×C22⋊D7911yescharacteristic
398C2×C4C4×C22⋊D7911yescharacteristic
399C2×C4#1457913011yescharacteristic
400C2×C4#1457913511yescharacteristic
401C2×C4#1457914411yescharacteristic
402C2×C4C2×#1192961111yescharacteristic
403C2×C4#1457918111yescharacteristic
404C2×C4#1457918611yescharacteristic
405C2×C4#1457919211yescharacteristic
406C8—22no—
407C8—41no—
408C8—42no—
409C8—3162no—
410C8—3161no—
411C8—3161no—
412C22—21no—
413C22—21no—
414C22—6321no—
415C22—6321no—
416C22—6321no—
417C22#1909844211yescharacteristic
418C22#1909876311yescharacteristic
419C22#1909878511yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—6322no—
428C4—6321no—
429C4C2×#1457908711yescharacteristic
430C4#1909857911yescharacteristic
431C4C4×#1192961111yescharacteristic
432C4#1909866111yescharacteristic
433C4#1909870211yescharacteristic
434C4#1909870811yescharacteristic
435C2—21no—
436C2—21no—
437C2—12641no—
438C2#2376370211yescharacteristic
439C2#2376375511yescharacteristic
440C2#2376376211yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingQ8×M16×C79C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenterQ8×M16D79
hypocenterC79#8992
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C4×#14579085C2semidirect
5. #23761301C2semidirect
6. #23761238C2semidirect
7. Q8×M16×C79C2semidirect
10. #19098848C4semidirect
11. #19098848C4semidirect
12. #19098763C4semidirect
13. #19098763C4semidirect
125. M16×C79SD16semidirect
150. Q8×C79#104semidirect
222. Q8×M16D79semidirect
228. C79#8992semidirect
333. M16#11929611semidirect
373. Q8#14579087semidirect

Maximal subgroups

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

subgroupindex
#237734312
#237643192
C4×#145790852
#237618502
#237613012
#237612382
Q8×M16×C792
#899279