Finite Group Databasebeta

← #34187612

Subgroups — #34187612

Order 20224.

514 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773826C211yesmaximal, characteristic
2C4⋊C4×C79⋊C8C211yesmaximal, characteristic, complemented
3#23762146C211yesmaximal, characteristic, complemented
4#23761894C211yesmaximal, characteristic, complemented
5#23761800C211yesmaximal, characteristic, complemented
6#23761799C211yesmaximal, characteristic, complemented
7C79×#1502C211yesmaximal, characteristic, complemented
8C2×#14579181—21no—
9C2×#14579181—21no—
10#19099113C2211yescharacteristic
11#19098945—21no—
12#19098945—21no—
13#19098942C411yescharacteristic, complemented
14#19098939C411yescharacteristic, complemented
15#19098937C411yescharacteristic, complemented
16#19098935C411yescharacteristic, complemented
17#19098616C2211yescharacteristic
18C2×C4⋊(C79⋊C8)C2211yescharacteristic
19C2×C4×C79⋊C8—21no—
20C2×C4×C79⋊C8C2211yescharacteristic
21#19098318—21no—
22#19098315—21no—
23#19098313—21no—
24#19098296—22no—
25#19098292—22no—
26D4⋊(C79⋊C8)—22no—
27#532×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C4×C4⋊C4×C79C2211yescharacteristic
30#327×C79—22no—
31C2×#11929624—21no—
32#14579181—42no—
33#14579181—42no—
34#14579181—42no—
35#14579181—42no—
36C2×#11929614—21no—
37C2×#11929614—21no—
38C2×#11929612—21no—
39C2×#11929612—21no—
40C2×#11929611—21no—
41C2×#11929611—21no—
42C2×#11929609—21no—
43C2×#11929609—21no—
44C22×C79⋊C8—21no—
45C22×C79⋊C8D411yescharacteristic
46C22×C79⋊C8D411yescharacteristic
47#14579161C2×C411yescharacteristic
48#14579158C2×C411yescharacteristic
49#14579149—21no—
50#14579135—21no—
51#14579135—21no—
52C4×D316—21no—
53C4×D316—21no—
54C4⋊(C79⋊C8)—21no—
55C4⋊(C79⋊C8)—21no—
56C4⋊(C79⋊C8)D412yes—
57C22⋊(C79⋊C8)—21no—
58C4×C79⋊C8—21no—
59C4×C79⋊C8—42no—
60C4×C79⋊C8C2×C411yescharacteristic
61C4×C79⋊C8C2×C411yescharacteristic
62C4×C79⋊C8D411yescharacteristic
63C4×C79⋊C8D411yescharacteristic
64C2×#41×C79—21no—
65C4⋊Q8×C79C2×C411yescharacteristic
66#127×C79C2×C411yescharacteristic
67#124×C79—21no—
68#121×C79—21no—
69#121×C79—21no—
70C4×D4×C79—21no—
71C4×D4×C79—21no—
72C2×C4⋊C4×C79—21no—
73C2×C42×C79—21no—
74C2×C42×C79C2311yescharacteristic
75C4⋊C8×C79—21no—
76C22⋊C8×C79—21no—
77C4×C8×C79—21no—
78#95×C79D412yes—
79#11929624—41no—
80#11929624—41no—
81#11929624—41no—
82#11929624—41no—
83C2×Dic158—21no—
84C2×D316—21no—
85C2×D316—21no—
86C2×D316—21no—
87C2×C22⋊D79—41no—
88C2×C4×D79—41no—
89#11929614—42no—
90#11929614—42no—
91#11929612—42no—
92#11929612—42no—
93#11929611—42no—
94#11929611—42no—
95#11929609—42no—
96#11929609—42no—
97C2×C79⋊C8—22no—
98C2×C79⋊C8—21no—
99C2×C79⋊C8—21no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—41no—
102C2×C79⋊C8—41no—
103C2×C79⋊C8C22⋊C411yescharacteristic
104C2×C79⋊C8C22⋊C411yescharacteristic
105C2×C79⋊C8C22⋊C411yescharacteristic
106C2×C79⋊C8C22⋊C411yescharacteristic
107C2×C79⋊C8D812yes—
108C2×C79⋊C8SD1612yes—
109C4⋊Dic79—21no—
110C4⋊Dic79—21no—
111(C2×D79)⋊C4—41no—
112(C2×D79)⋊C4—41no—
113#41×C79—41no—
114#41×C79—41no—
115#41×C79—41no—
116#41×C79—41no—
117C2×Q8×C79—21no—
118C2×D4×C79—21no—
119C2×D4×C79—21no—
120C2×D4×C79—21no—
121C2×D4×C79—41no—
122C22×C4×C79—22no—
123C22×C4×C79—41no—
124C22×C4×C79C2×D411yescharacteristic
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79#4111yescharacteristic
127C2×C8×C79—41no—
128C4⋊C4×C79—24no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C22⋊C4×C79—41no—
132C22⋊C4×C79—41no—
133C42×C79—41no—
134C42×C79—41no—
135C42×C79C22×C411yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79C2×D411yescharacteristic
138C42×C79#4111yescharacteristic
139C23×C79—41no—
140C23×C79#12111yescharacteristic
141C22×D79—41no—
142Q8×C79—41no—
143Q8×C79—41no—
144D4×C79—41no—
145D4×C79—41no—
146D4×C79—42no—
147D4×C79—81no—
148D4×C79—81no—
149D4×C79—81no—
150C2×C4×C79—22no—
151C2×C4×C79—24no—
152C2×C4×C79—41no—
153C2×C4×C79—42no—
154C2×C4×C79—42no—
155C2×C4×C79—41no—
156C2×C4×C79—81no—
157C2×C4×C79—81no—
158C2×C4×C79C2×C22⋊C411yescharacteristic
159C2×C4×C79C4×D411yescharacteristic
160C2×C4×C79C4×D411yescharacteristic
161C2×C4×C79#12111yescharacteristic
162C2×C4×C79#12411yescharacteristic
163C2×C4×C79#12711yescharacteristic
164C2×C4×C79C2×D811yescharacteristic
165C2×C4×C79C2×SD1611yescharacteristic
166C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
168C22⋊D79—81no—
169C22⋊D79—81no—
170C2×Dic79—41no—
171D316—41no—
172D316—41no—
173D316—42no—
174D316—81no—
175C4×D79—81no—
176C4×D79—81no—
177Dic158—41no—
178Dic158—41no—
179C8×C79—81no—
180C79⋊C8—21no—
181C79⋊C8—21no—
182C79⋊C8—42no—
183C79⋊C8—81no—
184C79⋊C8#10212yes—
185C79⋊C8#10212yes—
186C22×C79—21no—
187C22×C79—21no—
188C22×C79—81no—
189C22×C79—81no—
190C22×C79—81no—
191C22×C79#39011yescharacteristic
192C22×C79#45911yescharacteristic
193C22×C79#46011yescharacteristic
194C2×D79—81no—
195C2×D79—81no—
196C2×D79—81no—
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—21no—
200C4×C79—44no—
201C4×C79—81no—
202C4×C79C2×#10211yescharacteristic
203C4×C79#41811yescharacteristic
204C4×C79#44312yes—
205C4×C79#45911yescharacteristic
206C4×C79#46011yescharacteristic
207Dic79—81no—
208#9750—791nomaximal, Sylow
209C2×C79—21no—
210C2×C79—21no—
211C2×C79—161no—
212C2×C79#197411yescharacteristic
213C2×C79#197611yescharacteristic
214C2×C79#198211yescharacteristic
215D79—161no—
216#3147—791no—
217C8×C4⋊C4—791no—
218#1564—791no—
219#1502—791no—
220#1502D7911yescharacteristic, complemented
221#1480—791no—
222#1480—791no—
223C79#975011yescharacteristic, complemented, Sylow
224C2×#135—1581no—
225C2×#135—1581no—
226#532—791no—
227#532C2×D7911yescharacteristic
228#495—1581no—
229#495—1581no—
230#494—791no—
231#493—791no—
232#492—791no—
233#492—791no—
234#431—791no—
235#431C2×D7911yescharacteristic
236C2×C4⋊C8—791no—
237C2×C4×C8—791no—
238C2×C4×C8—1581no—
239C4×C4⋊C4C2×D7911yescharacteristic
240#336—1581no—
241#335—1581no—
242#334—1581no—
243#327—22no—
244#327—1582no—
245D4⋊C8—1582no—
246D4⋊C8—1582no—
247C2×#41—21no—
248C2×#41—1581no—
249#135—3162no—
250#135—3162no—
251#135—3162no—
252#135—3162no—
253C2×Q16—1581no—
254C2×Q16—1581no—
255C2×SD16—1581no—
256C2×SD16—1581no—
257C2×SD16—1581no—
258C2×SD16—1581no—
259C2×D8—1581no—
260C2×D8—1581no—
261C22×C8—791no—
262C22×C8—791no—
263C22×C8—1581no—
264C4⋊Q8—791no—
265C4⋊Q8C4×D7911yescharacteristic
266#127—791no—
267#127C4×D7911yescharacteristic
268#124—21no—
269#124—1581no—
270#121—21no—
271#121—21no—
272#121—1581no—
273#121—1581no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—1581no—
277C4×D4—1581no—
278C2×C4⋊C4—21no—
279C2×C42—21no—
280C2×C42C22×D7911yescharacteristic
281C4⋊C8—21no—
282C4⋊C8—792no—
283C4⋊C8—1581no—
284C4⋊C8—1581no—
285C22⋊C8—21no—
286C22⋊C8—1581no—
287C4×C8—21no—
288C4×C8—3162no—
289C4×C8—791no—
290C4×C8—791no—
291C4×C8—791no—
292C4×C8—791no—
293C4×C8—1581no—
294#95C22⋊D7912yes—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—3161no—
300#41—3161no—
301#41—3161no—
302#41—3161no—
303C2×Q8—21no—
304C2×Q8—1581no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—3161no—
310C2×D4—1581no—
311C2×D4—1581no—
312C2×D4—1581no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—3161no—
316C22×C4D4×D7911yescharacteristic
317C22×C4C2×C22⋊D7911yescharacteristic
318C22×C4#1192962311yescharacteristic
319Q16—3162no—
320Q16—3162no—
321SD16—3162no—
322SD16—3162no—
323SD16—3162no—
324SD16—3162no—
325D8—3162no—
326D8—3162no—
327C2×C8—41no—
328C2×C8—3161no—
329C2×C8—3161no—
330C2×C8—3161no—
331C2×C8—791no—
332C2×C8—791no—
333C2×C8—792no—
334C2×C8—792no—
335C2×C8—791no—
336C2×C8—791no—
337C2×C8—1582no—
338C2×C8—1581no—
339C2×C8—1581no—
340C4⋊C4—24no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—1581no—
344C4⋊C4—1581no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—3161no—
348C22⋊C4—3161no—
349C42—41no—
350C42—41no—
351C42C2×C4×D7911yescharacteristic
352C42D4×D7911yescharacteristic
353C42D4×D7911yescharacteristic
354C42#1192962411yescharacteristic
355C23—41no—
356C23—3161no—
357C23#1457913311yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—3161no—
361Q8—3161no—
362D4—42no—
363D4—41no—
364D4—41no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—3162no—
369D4—3161no—
370D4—3161no—
371D4—6321no—
372D4—6321no—
373D4—6321no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—3161no—
383C2×C4—6321no—
384C2×C4—6321no—
385C2×C4C22⋊C4×D7911yescharacteristic
386C2×C4(C22⋊D79)⋊C411yescharacteristic
387C2×C4C4×C22⋊D7911yescharacteristic
388C2×C4#1457913411yescharacteristic
389C2×C4#1457914711yescharacteristic
390C2×C4#1457915711yescharacteristic
391C2×C4D8×D7911yescharacteristic
392C2×C4SD16×D7911yescharacteristic
393C2×C4#1457918511yescharacteristic
394C2×C4#1457918811yescharacteristic
395C8—81no—
396C8—3162no—
397C8—792no—
398C8—792no—
399C8—6321no—
400C8—1581no—
401C8—1581no—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—6321no—
408C22—6321no—
409C22—6321no—
410C22#1909846311yescharacteristic
411C22#1909875611yescharacteristic
412C22#1909876111yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—81no—
418C4—6321no—
419C4#102×D7911yescharacteristic
420C4#1909856611yescharacteristic
421C4#1909867712yes—
422C4#1909875711yescharacteristic
423C4#1909876211yescharacteristic
424C2—21no—
425C2—21no—
426C2—161no—
427C2—12641no—
428C2#2376428511yescharacteristic
429C2#2376429511yescharacteristic
430C2#2376432511yescharacteristic
subgroupG/H
centerC2×C4#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1502C2
socleC22×C79#390
prime radical#23773826C2
cosocleC2×C42×C79C23
hypercenter#1502D79
hypocenterC79#9750
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C79⋊C8C2semidirect
3. #23762146C2semidirect
4. #23761894C2semidirect
5. #23761800C2semidirect
6. #23761799C2semidirect
7. C79×#1502C2semidirect
13. #19098942C4semidirect
14. #19098939C4semidirect
15. #19098937C4semidirect
16. #19098935C4semidirect
220. #1502D79semidirect
223. C79#9750semidirect

Maximal subgroups

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

subgroupindex
#237738262
C4⋊C4×C79⋊C82
#237621462
#237618942
#237618002
#237617992
C79×#15022
#975079