Finite Group Databasebeta

← #34183082

Subgroups — #34183082

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098840C211yesmaximal, characteristic
2#23766347C211yesmaximal, characteristic
3#23764362C211yesmaximal, characteristic, complemented
4C2×#19098290C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic, complemented
6#23761241C211yesmaximal, characteristic, complemented
7C79×#1981C211yesmaximal, characteristic, complemented
8C2×C4×D316C2211yescharacteristic
9C2×#14579119—21no—
10#19098840—22no—
11#19098840C414yescomplemented
12C2×#14579094—21no—
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579081—21no—
15C2×C4×C79⋊C8C2211yescharacteristic
16#19098465—21no—
17#19098455—22no—
18#19098433C2211yescharacteristic
19#19098420—21no—
20C4×C4⋊Dic79—21no—
21#19098325—22no—
22#19098290—22no—
23C2×C4⋊Q8×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×#103×C79—21no—
26C2×#103×C79—21no—
27C4×C4⋊C4×C79C2211yescharacteristic
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C22×C79⋊C8—21no—
31C22×C79⋊C8—21no—
32C4×D316—21no—
33C4×D316—21no—
34C4×D316C2×C412yes—
35C4×D316D412yes—
36#14579119—42no—
37#14579119—42no—
38C2×C4⋊Dic79—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79D411yescharacteristic
41C2×C4⋊Dic79D412yes—
42C2×(C2×D79)⋊C4—21no—
43C2×(C2×D79)⋊C4—22no—
44C2×(C2×D79)⋊C4—22no—
45C2×C4×Dic79—22no—
46#14579094—24no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)C2×C412yes—
49#14579081—24no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C79⋊#95—22no—
53C22×Q8×C79—21no—
54C22×C8×C79—21no—
55C4⋊Q8×C79—21no—
56C4⋊Q8×C79—21no—
57C4⋊Q8×C79—21no—
58C4⋊Q8×C79C2×C412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C42×C79—21no—
64C2×C42×C79C2311yescharacteristic
65C4⋊C8×C79—21no—
66C4⋊C8×C79—21no—
67#103×C79—42no—
68#103×C79—42no—
69#95×C79—21no—
70C23×D79—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×D316—21no—
76C2×D316C22⋊C412yes—
77C22×Dic79—21no—
78C22×Dic79—22no—
79C2×C4×D79—22no—
80C2×C4×D79—22no—
81C2×C4×D79—41no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84C2×C4×D79—41no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C4⋊Dic79—21no—
92C4⋊Dic79—42no—
93C4⋊Dic79—41no—
94C4⋊Dic79—41no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79C22⋊C412yes—
98(C2×D79)⋊C4—22no—
99(C2×D79)⋊C4—22no—
100(C2×D79)⋊C4—44no—
101(C2×D79)⋊C4—82no—
102C4×Dic79—44no—
103C2×Q8×C79—22no—
104C2×Q8×C79—41no—
105C2×Q8×C79—41no—
106C2×Q8×C79—41no—
107C2×Q8×C79—41no—
108C2×Q8×C79—41no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—41no—
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—22no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79D812yes—
126C42×C79—41no—
127C42×C79—41no—
128C42×C79C22×C411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79#4111yescharacteristic
132C23×C79#12011yescharacteristic
133C22×D79—22no—
134C22×D79—41no—
135C22×D79—41no—
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C22×D79—41no—
140Q8×C79—42no—
141Q8×C79—42no—
142Q8×C79—81no—
143Q8×C79—81no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—22no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—42no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—81no—
154C2×C4×C79—81no—
155C2×C4×C79C2×C22⋊C411yescharacteristic
156C2×C4×C79C4×D411yescharacteristic
157C2×C4×C79C4×D411yescharacteristic
158C2×C4×C79#12111yescharacteristic
159C2×C4×C79#12111yescharacteristic
160C2×C4×C79#12311yescharacteristic
161C2×C4×C79C2×D811yescharacteristic
162C2×C4×C79#13511yescharacteristic
163C2×C4×C79#13511yescharacteristic
164C2×C4×C79#13511yescharacteristic
165C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C79#13711yescharacteristic
168C2×C4×C79#13711yescharacteristic
169C2×Dic79—22no—
170C2×Dic79—41no—
171C2×Dic79—41no—
172C2×Dic79—42no—
173C2×Dic79—42no—
174C2×Dic79—42no—
175D316—22no—
176D316—22no—
177D316—41no—
178D316—41no—
179C4×D79—42no—
180C4×D79—42no—
181C4×D79—42no—
182C4×D79—42no—
183C8×C79—81no—
184C79⋊C8—42no—
185C79⋊C8—42no—
186C22×C79#38611yescharacteristic
187C22×C79#44711yescharacteristic
188C22×C79#44911yescharacteristic
189C22×C79#45211yescharacteristic
190C22×C79#45211yescharacteristic
191C22×C79#45511yescharacteristic
192C22×C79#45611yescharacteristic
193C2×D79—42no—
194C2×D79—42no—
195C2×D79—42no—
196C2×D79—81no—
197C2×D79—81no—
198C2×D79—81no—
199C2×D79—81no—
200C4×C79—21no—
201C4×C79—21no—
202C4×C79—42no—
203C4×C79—82no—
204C4×C79—81no—
205C4×C79#41611yescharacteristic
206C4×C79#41911yescharacteristic
207C4×C79C4×D811yescharacteristic
208C4×C79#44211yescharacteristic
209C4×C79#45911yescharacteristic
210C4×C79#46311yescharacteristic
211C4×C79#48511yescharacteristic
212C4×C79#48511yescharacteristic
213Dic79—42no—
214Dic79—82no—
215#8984—791nomaximal, Sylow
216C2×C79#164111yescharacteristic
217C2×C79#164611yescharacteristic
218C2×C79#164711yescharacteristic
219C2×C79#165211yescharacteristic
220C2×C79#187311yescharacteristic
221C2×C79#187911yescharacteristic
222C2×C79#188211yescharacteristic
223D79—82no—
224C2×#474—791no—
225#2354—791no—
226#1990—791no—
227#1981D7911yescharacteristic, complemented
228C2×D4⋊C8—791no—
229#1296—791no—
230#1279—791no—
231C79#898411yescharacteristic, complemented, Sylow
232C2×C4⋊Q8C2×D7911yescharacteristic
233C2×C4×D4—791no—
234C2×#117—1581no—
235#474—794no—
236#474—1582no—
237C2×#107—1581no—
238C2×C4⋊C8—791no—
239C2×C4⋊C8C2×D7911yescharacteristic
240C2×#103—21no—
241C2×#103—21no—
242C2×#102—1581no—
243C2×C4×C8—791no—
244#390—1581no—
245#388—1582no—
246#384—791no—
247#380—1581no—
248C4×C4⋊C4—1581no—
249C4×C4⋊C4C2×D7911yescharacteristic
250#340—1582no—
251D4⋊C8—1582no—
252C22×Q8—21no—
253C22×D4—791no—
254C23×C4—1581no—
255C22×C8—21no—
256C22×C8—1581no—
257C22×C8—1581no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8C4×D7912yes—
262C4×D4—792no—
263C4×D4—792no—
264C4×D4—1581no—
265C4×D4—1581no—
266#117—3162no—
267#117—3162no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—792no—
271C2×C4⋊C4—791no—
272C2×C4⋊C4—1581no—
273C2×C4⋊C4—1581no—
274C2×C4⋊C4C22⋊D7911yescharacteristic
275C2×C4⋊C4C22⋊D7911yescharacteristic
276C2×C22⋊C4—1582no—
277C2×C22⋊C4—1582no—
278C2×C22⋊C4—1581no—
279C2×C42—21no—
280C2×C42—1582no—
281C2×C42C22×D7911yescharacteristic
282#107—1584no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—792no—
286C4⋊C8—1581no—
287#103—42no—
288#103—42no—
289#102—1584no—
290C4×C8—1581no—
291C4×C8—1581no—
292#95—21no—
293#95—1582no—
294C24—1581no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×D4—792no—
302C2×D4—1581no—
303C2×D4—1581no—
304C2×D4—1581no—
305C2×D4—1581no—
306C2×D4—1581no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—3161no—
311C22×C4—3161no—
312C22×C4—3161no—
313C22×C4—3161no—
314C22×C4—1582no—
315C22×C4—1581no—
316C22×C4—1582no—
317C22×C4—1582no—
318C22×C4D4×D7911yescharacteristic
319C22×C4D4×D7911yescharacteristic
320C22×C4C2×C22⋊D7911yescharacteristic
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—3161no—
325C2×C8—3161no—
326C2×C8—3161no—
327C2×C8—3161no—
328C2×C8—1582no—
329C2×C8—1582no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—3162no—
338C4⋊C4—3161no—
339C4⋊C4—3161no—
340C4⋊C4—3161no—
341C4⋊C4—3161no—
342C4⋊C4—792no—
343C4⋊C4—1581no—
344C4⋊C4#1192960912yes—
345C22⋊C4—3164no—
346C22⋊C4—6322no—
347C22⋊C4—1582no—
348C22⋊C4—1582no—
349C42—41no—
350C42—41no—
351C42—3164no—
352C42C2×C4×D7911yescharacteristic
353C42D4×D7911yescharacteristic
354C42#1192962311yescharacteristic
355C42#1192962411yescharacteristic
356C23—3161no—
357C23—3161no—
358C23—3161no—
359C23—3161no—
360C23—3161no—
361C23—3161no—
362C23—1582no—
363C23#1457912911yescharacteristic
364Q8—42no—
365Q8—42no—
366Q8—81no—
367Q8—81no—
368D4—3161no—
369D4—3161no—
370D4—1582no—
371D4—1582no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—3161no—
384C2×C4—3161no—
385C2×C4—3162no—
386C2×C4—3162no—
387C2×C4—3162no—
388C2×C4—3162no—
389C2×C4—3162no—
390C2×C4—3162no—
391C2×C4—3162no—
392C2×C4—1582no—
393C2×C4C22⋊C4×D7911yescharacteristic
394C2×C4(C22⋊D79)⋊C411yescharacteristic
395C2×C4C4×C22⋊D7911yescharacteristic
396C2×C4#1457913211yescharacteristic
397C2×C4#1457913411yescharacteristic
398C2×C4#1457914311yescharacteristic
399C2×C4C2×#1192960911yescharacteristic
400C2×C4#1457918111yescharacteristic
401C2×C4#1457918211yescharacteristic
402C2×C4#1457918411yescharacteristic
403C2×C4#1457918611yescharacteristic
404C2×C4#1457918711yescharacteristic
405C2×C4#1457919011yescharacteristic
406C2×C4#1457919311yescharacteristic
407C8—81no—
408C8—3162no—
409C8—3162no—
410C22—3162no—
411C22—3162no—
412C22—3162no—
413C22—6321no—
414C22—6321no—
415C22—6321no—
416C22—6321no—
417C22#1909843911yescharacteristic
418C22#1909869611yescharacteristic
419C22#1909870611yescharacteristic
420C22#1909872211yescharacteristic
421C22#1909872411yescharacteristic
422C22#1909873911yescharacteristic
423C22#1909874411yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—82no—
428C4—81no—
429C4—3162no—
430C4—6322no—
431C4#1909855811yescharacteristic
432C4#1909857011yescharacteristic
433C4C4×#1192960911yescharacteristic
434C4#1909867311yescharacteristic
435C4#1909875511yescharacteristic
436C4#1909877511yescharacteristic
437C4#1909890211yescharacteristic
438C4#1909890311yescharacteristic
439C2—6322no—
440C2#2376259111yescharacteristic
441C2#2376262611yescharacteristic
442C2#2376263311yescharacteristic
443C2#2376266811yescharacteristic
444C2#2376370911yescharacteristic
445C2#2376375111yescharacteristic
446C2#2376376611yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1981C2
socleC23×C79#120
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1981D79
hypocenterC79#8984
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
3. #23764362C2semidirect
5. #23761284C2semidirect
6. #23761241C2semidirect
7. C79×#1981C2semidirect
11. #19098840C4semidirect
227. #1981D79semidirect
231. C79#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#190988402
#237663472
#237643622
C2×#190982902
#237612842
#237612412
C79×#19812
#898479