Finite Group Databasebeta

← #34184513

Subgroups — #34184513

Order 20224.

570 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098763C211yesmaximal, characteristic
2#23764206C211yesmaximal, characteristic
3#23763405C211yesmaximal, characteristic, complemented
4C2×#19098306C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic, complemented
6#23761235C211yesmaximal, characteristic, complemented
7C2×#446×C79C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C2×#14579158C2211yescharacteristic
10#19098763—22no—
11#19098763C414yescomplemented
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579085—21no—
15C2×#14579083—21no—
16C2×#14579083—21no—
17C2×#14579081—21no—
18#19098433C2211yescharacteristic
19#19098325—21no—
20#19098306—22no—
21C2×C4×Q8×C79C2211yescharacteristic
22#446×C79—22no—
23#446×C79—22no—
24C2×C4⋊C8×C79—21no—
25C2×C4⋊C8×C79C2211yescharacteristic
26C2×#97×C79—21no—
27C2×C4×C8×C79C2211yescharacteristic
28C22×D316—21no—
29C22×D316—21no—
30C22×D316—21no—
31C2×#11929611—24no—
32C2×#11929611—42no—
33C2×#11929611—42no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36#14579158—21no—
37#14579158—21no—
38#14579158—21no—
39#14579158C2×C412yes—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)C2×C412yes—
47#14579085—42no—
48#14579083—42no—
49#14579083—42no—
50#14579081—24no—
51C22×Q8×C79D411yescharacteristic
52C22×C8×C79—21no—
53C22×C8×C79D411yescharacteristic
54C22×C8×C79D411yescharacteristic
55C4×Q8×C79—21no—
56C4×Q8×C79—21no—
57C4×Q8×C79C2×C412yes—
58C4×Q8×C79D412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79D411yescharacteristic
61C2×C42×C79—21no—
62C2×C42×C79C2311yescharacteristic
63C4⋊C8×C79—24no—
64C4⋊C8×C79—21no—
65C4⋊C8×C79D412yes—
66#97×C79—24no—
67C4×C8×C79—21no—
68C4×C8×C79—21no—
69C23×D79—41no—
70C2×D316—22no—
71C2×D316—22no—
72C2×D316—22no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—81no—
85C2×D316—81no—
86C22×Dic79—41no—
87#11929611—48no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C4⋊Dic79—41no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—81no—
99C2×Q8×C79—21no—
100C2×Q8×C79—21no—
101C2×Q8×C79—21no—
102C2×Q8×C79—21no—
103C2×Q8×C79—21no—
104C2×Q8×C79C22⋊C412yes—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79#4111yescharacteristic
110C2×C8×C79—21no—
111C2×C8×C79—21no—
112C2×C8×C79—22no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C2×C8×C79—21no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C2×C8×C79SD1612yescomplemented
119C4⋊C4×C79—22no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—22no—
122C4⋊C4×C79C22⋊C412yes—
123C42×C79—22no—
124C42×C79—22no—
125C42×C79C22×C411yescharacteristic
126C42×C79C2×D411yescharacteristic
127C42×C79C2×D411yescharacteristic
128C42×C79#4111yescharacteristic
129C23×C79#12111yescharacteristic
130C22×D79—42no—
131C22×D79—81no—
132C22×D79—81no—
133C22×D79—81no—
134C22×D79—81no—
135C22×D79—81no—
136C22×D79—81no—
137Q8×C79—22no—
138Q8×C79—22no—
139Q8×C79—41no—
140Q8×C79—41no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—22no—
144C2×C4×C79—22no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79C4×D411yescharacteristic
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79#12011yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79#12311yescharacteristic
155C2×C4×C79C2×SD1611yescharacteristic
156C2×C4×C79#13511yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C79#13711yescharacteristic
162C2×C4×C79#13711yescharacteristic
163C2×Dic79—81no—
164C2×Dic79—81no—
165C2×Dic79—81no—
166D316—42no—
167D316—42no—
168D316—42no—
169D316—42no—
170D316—82no—
171D316—82no—
172D316—81no—
173D316—81no—
174D316—81no—
175D316—81no—
176C8×C79—22no—
177C8×C79—41no—
178C8×C79—42no—
179C79⋊C8—42no—
180C79⋊C8—81no—
181C22×C79#38611yescharacteristic
182C22×C79#46011yescharacteristic
183C22×C79#46411yescharacteristic
184C22×C79#46811yescharacteristic
185C22×C79#46911yescharacteristic
186C22×C79#47111yescharacteristic
187C22×C79#47211yescharacteristic
188C2×D79—82no—
189C2×D79—82no—
190C2×D79—82no—
191C2×D79—161no—
192C2×D79—161no—
193C2×D79—161no—
194C2×D79—161no—
195C4×C79—21no—
196C4×C79—21no—
197C4×C79—42no—
198C4×C79—42no—
199C4×C79#41711yescharacteristic
200C4×C79#41811yescharacteristic
201C4×C79C4×SD1611yescharacteristic
202C4×C79#44011yescharacteristic
203C4×C79#44811yescharacteristic
204C4×C79#44911yescharacteristic
205C4×C79#46811yescharacteristic
206C4×C79#46911yescharacteristic
207Dic79—161no—
208#9230—791nomaximal, Sylow
209C2×C79#168611yescharacteristic
210C2×C79#168811yescharacteristic
211C2×C79#169811yescharacteristic
212C2×C79#170211yescharacteristic
213C2×C79#188011yescharacteristic
214C2×C79#189511yescharacteristic
215C2×C79#190711yescharacteristic
216D79—162no—
217C2×#460—791no—
218C2×#446D7911yescharacteristic, complemented
219#1957—791no—
220#1803—791no—
221C2×#331—791no—
222#1296—791no—
223#1277—791no—
224C79#923011yescharacteristic, complemented, Sylow
225C22×SD16—1581no—
226C2×#127—791no—
227C2×C4×Q8C2×D7911yescharacteristic
228#460—794no—
229#460—1582no—
230#446—22no—
231#446—22no—
232C2×C4⋊C8—21no—
233C2×C4⋊C8—791no—
234C2×C4⋊C8—791no—
235C2×C4⋊C8C2×D7911yescharacteristic
236C2×#103—1581no—
237C2×#102—1581no—
238C2×#102—1581no—
239C2×#102—1581no—
240C2×#97—21no—
241C2×C4×C8C2×D7911yescharacteristic
242#384—791no—
243#340—1581no—
244#331—1582no—
245C22×Q8D31611yescharacteristic
246C22×D4—1581no—
247C22×D4—1581no—
248C22×D4—1581no—
249C2×SD16—3162no—
250C2×SD16—3162no—
251C2×SD16—1584no—
252C22×C8—21no—
253C22×C8—1581no—
254C22×C8—1581no—
255C22×C8C22⋊D7911yescharacteristic
256C22×C8C22⋊D7911yescharacteristic
257#127—792no—
258#127—1581no—
259#127—1581no—
260#127—1581no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C4×D7912yes—
264C4×Q8C22⋊D7912yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—1581no—
267C2×C4⋊C4—1581no—
268C2×C4⋊C4—1581no—
269C2×C4⋊C4D31611yescharacteristic
270C2×C42—21no—
271C2×C42C22×D7911yescharacteristic
272C4⋊C8—24no—
273C4⋊C8—21no—
274C4⋊C8—792no—
275C4⋊C8—1581no—
276C4⋊C8—1581no—
277C4⋊C8—1581no—
278C4⋊C8C22⋊D7912yes—
279#103—3162no—
280#102—3162no—
281#102—3162no—
282#102—1584no—
283#97—24no—
284C4×C8—21no—
285C4×C8—21no—
286C24—3161no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8(C2×D79)⋊C412yes—
293C2×D4—3161no—
294C2×D4—3161no—
295C2×D4—3161no—
296C2×D4—3161no—
297C2×D4—3161no—
298C2×D4—3161no—
299C2×D4—3161no—
300C2×D4—3161no—
301C2×D4—3161no—
302C2×D4—3161no—
303C2×D4—3161no—
304C2×D4—6321no—
305C2×D4—6321no—
306C2×D4—1582no—
307C2×D4—1582no—
308C2×D4—1582no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—3161no—
312C22×C4C2×C22⋊D7911yescharacteristic
313C22×C4C2×D31611yescharacteristic
314C22×C4#1192962411yescharacteristic
315SD16—3168no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—3161no—
325C2×C8—3161no—
326C2×C8—3161no—
327C2×C8—3161no—
328C2×C8—3161no—
329C2×C8—1582no—
330C2×C8#1192961112yescomplemented
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—3161no—
335C4⋊C4—3161no—
336C4⋊C4—3161no—
337C4⋊C4—3161no—
338C4⋊C4—6321no—
339C4⋊C4(C2×D79)⋊C412yes—
340C42—22no—
341C42—22no—
342C42C2×C4×D7911yescharacteristic
343C42C2×C22⋊D7911yescharacteristic
344C42C2×C22⋊D7911yescharacteristic
345C42#1192962411yescharacteristic
346C23—3162no—
347C23—6321no—
348C23—6321no—
349C23—6321no—
350C23—6321no—
351C23—6321no—
352C23—6321no—
353C23#1457913511yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358D4—3162no—
359D4—3162no—
360D4—3162no—
361D4—3162no—
362D4—6322no—
363D4—6322no—
364D4—6321no—
365D4—6321no—
366D4—6321no—
367D4—6321no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—6321no—
377C2×C4—6321no—
378C2×C4—6321no—
379C2×C4C2×(C2×D79)⋊C411yescharacteristic
380C2×C4C4×C22⋊D7911yescharacteristic
381C2×C4C4×C22⋊D7911yescharacteristic
382C2×C4#1457913011yescharacteristic
383C2×C4#1457913511yescharacteristic
384C2×C4#1457914411yescharacteristic
385C2×C4C2×#1192961111yescharacteristic
386C2×C4#1457918111yescharacteristic
387C2×C4#1457918611yescharacteristic
388C2×C4#1457918911yescharacteristic
389C2×C4#1457918911yescharacteristic
390C2×C4#1457918911yescharacteristic
391C2×C4#1457919211yescharacteristic
392C2×C4#1457919411yescharacteristic
393C8—22no—
394C8—41no—
395C8—42no—
396C8—3162no—
397C8—6321no—
398C22—6322no—
399C22—6322no—
400C22—6322no—
401C22—12641no—
402C22—12641no—
403C22—12641no—
404C22—12641no—
405C22#1909844211yescharacteristic
406C22#1909876311yescharacteristic
407C22#1909878511yescharacteristic
408C22#1909880911yescharacteristic
409C22#1909881611yescharacteristic
410C22#1909882611yescharacteristic
411C22#1909883311yescharacteristic
412C4—21no—
413C4—21no—
414C4—42no—
415C4—42no—
416C4—12641no—
417C4#1909856511yescharacteristic
418C4#1909856911yescharacteristic
419C4C4×#1192961111yescharacteristic
420C4#1909866111yescharacteristic
421C4#1909870211yescharacteristic
422C4#1909870811yescharacteristic
423C4#1909880911yescharacteristic
424C4#1909881611yescharacteristic
425C2—12642no—
426C2#2376287211yescharacteristic
427C2#2376288411yescharacteristic
428C2#2376294011yescharacteristic
429C2#2376296811yescharacteristic
430C2#2376375511yescharacteristic
431C2#2376383811yescharacteristic
432C2#2376390811yescharacteristic
subgroupG/H
centerC23#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#446×C79C2
socleC23×C79#121
prime radicalC2×#14579158C22
cosocleC2×C42×C79C23
hypercenterC2×#446D79
hypocenterC79#9230
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
3. #23763405C2semidirect
5. #23761284C2semidirect
6. #23761235C2semidirect
7. C2×#446×C79C2semidirect
11. #19098763C4semidirect
118. C2×C8×C79SD16semidirect
218. C2×#446D79semidirect
224. C79#9230semidirect
330. C2×C8#11929611semidirect

Maximal subgroups

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

subgroupindex
C2×#190987632
#237642062
#237634052
C2×#190983062
#237612842
#237612352
C2×#446×C792
#923079