Finite Group Databasebeta

← #34184562

Subgroups — #34184562

Order 20224.

588 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
1#23773400C211yesmaximal, characteristic
2#23764228C211yesmaximal, characteristic, complemented
3#23761894C211yesmaximal, characteristic, complemented
4#23761825C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761233C211yesmaximal, characteristic, complemented
7C79×#2293C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098876—21no—
10#19098843C412yescomplemented
11#19098758C412yescomplemented
12#19098697—21no—
13C4×#11929609—22no—
14#19098616C2211yescharacteristic
15#19098616C2211yescharacteristic
16#19098587C2211yescharacteristic
17C2×#14579089—21no—
18C2×#14579089—21no—
19#19098296—22no—
20D4⋊(C79⋊C8)—22no—
21C2×C4×D4×C79C2211yescharacteristic
22C2×#117×C79—21no—
23#390×C79—21no—
24#389×C79—21no—
25#388×C79—22no—
26C4×C4⋊C4×C79C2211yescharacteristic
27C4×C22⋊C4×C79—21no—
28C43×C79C2211yescharacteristic
29C2×#11929624—21no—
30C2×#11929609—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#14579094—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#14579082—22no—
47#14579081—22no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C22×D4×C79D411yescharacteristic
53C23×C4×C79—21no—
54C4×D4×C79—21no—
55C4×D4×C79—21no—
56C4×D4×C79C2×C412yes—
57C4×D4×C79D412yes—
58#117×C79—42no—
59#117×C79—42no—
60C2×C4⋊C4×C79—22no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C22⋊C4×C79—21no—
64C2×C22⋊C4×C79—22no—
65C2×C22⋊C4×C79—22no—
66C2×C42×C79—21no—
67C2×C42×C79—21no—
68C2×C42×C79—21no—
69C2×C42×C79D411yescharacteristic
70C2×C42×C79D411yescharacteristic
71C2×C42×C79C2311yescharacteristic
72#95×C79—22no—
73#95×C79D412yes—
74#11929624—41no—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78C2×Dic158—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×C22⋊D79—41no—
83C2×C4×D79—41no—
84#11929609—44no—
85#11929608—41no—
86#11929608—41no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C4⋊Dic79—21no—
91C4⋊Dic79—21no—
92(C2×D79)⋊C4—41no—
93(C2×D79)⋊C4—41no—
94C24×C79—21no—
95C2×D4×C79—21no—
96C2×D4×C79—42no—
97C2×D4×C79C22⋊C412yes—
98C22×C4×C79—21no—
99C22×C4×C79—21no—
100C22×C4×C79—21no—
101C22×C4×C79—22no—
102C22×C4×C79—22no—
103C22×C4×C79—22no—
104C22×C4×C79—41no—
105C22×C4×C79—41no—
106C22×C4×C79—41no—
107C22×C4×C79—41no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C4⋊C4×C79—21no—
112C4⋊C4×C79—42no—
113C4⋊C4×C79—44no—
114C4⋊C4×C79C22⋊C412yes—
115C22⋊C4×C79—22no—
116C22⋊C4×C79—22no—
117C22⋊C4×C79—44no—
118C22⋊C4×C79—44no—
119C42×C79—21no—
120C42×C79—21no—
121C42×C79—21no—
122C42×C79—42no—
123C42×C79—42no—
124C42×C79—41no—
125C42×C79—41no—
126C42×C79D812yescomplemented
127C42×C79C22×C411yescharacteristic
128C42×C79C2×D411yescharacteristic
129C42×C79#4111yescharacteristic
130C42×C79#4111yescharacteristic
131C23×C79—22no—
132C23×C79—42no—
133C23×C79—42no—
134C23×C79—41no—
135C23×C79—41no—
136C23×C79#12011yescharacteristic
137C22×D79—41no—
138D4×C79—22no—
139D4×C79—22no—
140D4×C79—42no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
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×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—42no—
153C2×C4×C79—42no—
154C2×C4×C79—42no—
155C2×C4×C79—42no—
156C2×C4×C79—42no—
157C2×C4×C79—42no—
158C2×C4×C79—42no—
159C2×C4×C79C2×C22⋊C411yescharacteristic
160C2×C4×C79C4×D411yescharacteristic
161C2×C4×C79C4×D411yescharacteristic
162C2×C4×C79#12111yescharacteristic
163C2×C4×C79#12111yescharacteristic
164C2×C4×C79#12311yescharacteristic
165C2×C4×C79C2×D811yescharacteristic
166C2×C4×C79#13511yescharacteristic
167C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
169C22⋊D79—81no—
170C22⋊D79—81no—
171C2×Dic79—41no—
172D316—42no—
173D316—41no—
174D316—41no—
175D316—81no—
176C4×D79—81no—
177C4×D79—81no—
178Dic158—41no—
179Dic158—41no—
180C79⋊C8—42no—
181C79⋊C8—81no—
182C79⋊C8—81no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—42no—
186C22×C79—42no—
187C22×C79—42no—
188C22×C79—82no—
189C22×C79—82no—
190C22×C79#38611yescharacteristic
191C22×C79#44711yescharacteristic
192C22×C79#44911yescharacteristic
193C2×D79—81no—
194C2×D79—81no—
195C2×D79—81no—
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—42no—
200C4×C79—42no—
201C4×C79—42no—
202C4×C79—82no—
203C4×C79#42112yes—
204C4×C79C4×D811yescharacteristic
205C4×C79#44211yescharacteristic
206C4×C79#45911yescharacteristic
207C4×C79#46311yescharacteristic
208Dic79—81no—
209#9237—791nomaximal, Sylow
210C2×C79—21no—
211C2×C79—21no—
212C2×C79—82no—
213C2×C79#187911yescharacteristic
214C2×C79#189711yescharacteristic
215C2×C79#190611yescharacteristic
216D79—161no—
217#3100—791no—
218#2293D7911yescharacteristic, complemented
219#1962—791no—
220#1502—791no—
221#1486—791no—
222#1303—791no—
223#1276—791no—
224C79#923711yescharacteristic, complemented, Sylow
225#532—791no—
226C2×C4×D4C2×D7911yescharacteristic
227C2×#117—21no—
228#480—1581no—
229#474—792no—
230#459—792no—
231#447—1581no—
232C4×D8—1582no—
233#431—791no—
234#431—791no—
235#423—791no—
236C2×#104—1581no—
237C2×#104—1581no—
238#390—21no—
239#389—21no—
240#388—22no—
241C4×C4⋊C4C2×D7911yescharacteristic
242C4×C22⋊C4—21no—
243C43C2×D7911yescharacteristic
244#327—1582no—
245D4⋊C8—1582no—
246C2×#41—1581no—
247C22×D4C22⋊D7911yescharacteristic
248C23×C4—21no—
249C2×D8—1582no—
250C2×M16—1581no—
251C4⋊Q8—791no—
252#127—791no—
253#124—1581no—
254#121—1581no—
255#121—1581no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—1581no—
259C4×D4—1581no—
260C4×D4C4×D7912yes—
261C4×D4D31612yes—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C22⋊D7911yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42C22⋊D7911yescharacteristic
274C2×C42C22⋊D7911yescharacteristic
275C2×C42C22×D7911yescharacteristic
276#107—1582no—
277C4⋊C8—792no—
278C4⋊C8—1581no—
279C4⋊C8—1581no—
280C4⋊C8—1581no—
281#104—3162no—
282#104—3162no—
283#102—1582no—
284#102—1582no—
285C22⋊C8—1581no—
286C22⋊C8—1581no—
287C4×C8—1581no—
288C4×C8—1581no—
289#95—22no—
290#95C22⋊D7912yes—
291C24—21no—
292#41—3161no—
293#41—3161no—
294#41—3161no—
295#41—3161no—
296C2×Q8—1581no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—3161no—
300C2×D4—1581no—
301C2×D4—1581no—
302C2×D4—1581no—
303C2×D4(C2×D79)⋊C412yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—3161no—
315C22×C4C2×C22⋊D7911yescharacteristic
316C22×C4C2×C22⋊D7911yescharacteristic
317C22×C4C2×C22⋊D7911yescharacteristic
318D8—3164no—
319M16—3161no—
320M16—3161no—
321C2×C8—3161no—
322C2×C8—3161no—
323C2×C8—1582no—
324C4⋊C4—21no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—1581no—
328C4⋊C4—1581no—
329C4⋊C4(C2×D79)⋊C412yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—3161no—
335C22⋊C4—3161no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—42no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42#1192960912yescomplemented
344C42C2×C4×D7911yescharacteristic
345C42C2×D31611yescharacteristic
346C42#1192962411yescharacteristic
347C42#1192962411yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—3161no—
354C23#1457913011yescharacteristic
355Q8—3161no—
356Q8—3161no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—3161no—
361D4—3161no—
362D4—3162no—
363D4—6321no—
364D4—6321no—
365D4—6321no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—3161no—
385C2×C4—6321no—
386C2×C4—6321no—
387C2×C4C2×(C2×D79)⋊C411yescharacteristic
388C2×C4C4×C22⋊D7911yescharacteristic
389C2×C4C4×C22⋊D7911yescharacteristic
390C2×C4#1457913511yescharacteristic
391C2×C4#1457913511yescharacteristic
392C2×C4#1457914411yescharacteristic
393C2×C4C2×#1192960911yescharacteristic
394C2×C4#1457918111yescharacteristic
395C2×C4#1457918611yescharacteristic
396C2×C4#1457918711yescharacteristic
397C8—3162no—
398C8—6321no—
399C8—6321no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—6321no—
408C22—6321no—
409C22—6321no—
410C22#1909844211yescharacteristic
411C22#1909869711yescharacteristic
412C22#1909870811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—6321no—
421C4#1909858112yes—
422C4C4×#1192960911yescharacteristic
423C4#1909867311yescharacteristic
424C4#1909875811yescharacteristic
425C4#1909877811yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—12641no—
430C2#2376374811yescharacteristic
431C2#2376384811yescharacteristic
432C2#2376390111yescharacteristic
subgroupG/H
centerC2×C4#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2293C2
socleC22×C79#386
prime radical#23773400C2
cosocleC2×C42×C79C23
hypercenter#2293D79
hypocenterC79#9237
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #23764228C2semidirect
3. #23761894C2semidirect
4. #23761825C2semidirect
5. #23761303C2semidirect
6. #23761233C2semidirect
7. C79×#2293C2semidirect
10. #19098843C4semidirect
11. #19098758C4semidirect
126. C42×C79D8semidirect
218. #2293D79semidirect
224. C79#9237semidirect
343. C42#11929609semidirect

Maximal subgroups

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

subgroupindex
#237734002
#237642282
#237618942
#237618252
#237613032
#237612332
C79×#22932
#923779