Finite Group Databasebeta

← #34183104

Subgroups — #34183104

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098765C211yesmaximal, characteristic
2#23766351C211yesmaximal, characteristic
3#23764353C211yesmaximal, characteristic, complemented
4C2×#19098292C211yesmaximal, characteristic
5#23761283C211yesmaximal, characteristic, complemented
6#23761240C211yesmaximal, characteristic, complemented
7C79×#1990C211yesmaximal, characteristic, complemented
8C22×#11929611—21no—
9C2×C4×D316C2211yescharacteristic
10C2×#14579119—21no—
11#19098765—22no—
12#19098765C414yescomplemented
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C2×#14579085—21no—
16#19098464—21no—
17#19098453—22no—
18#19098420—21no—
19C4⋊C4×Dic79—21no—
20C4×C4⋊Dic79C2211yescharacteristic
21#19098325—21no—
22#19098325—21no—
23#19098292—22no—
24C2×C4⋊Q8×C79C2211yescharacteristic
25C2×C4×C8×C79C2211yescharacteristic
26#384×C79C2211yescharacteristic
27#340×C79—22no—
28C22×D316D411yescharacteristic
29C22×C4×D79—21no—
30C2×#11929611—24no—
31C2×#11929611—42no—
32C2×#11929611—42no—
33C22×C79⋊C8—21no—
34C22×C79⋊C8—21no—
35C4×D316—21no—
36C4×D316—21no—
37C4×D316C2×C412yes—
38C4×D316D412yes—
39#14579119—42no—
40#14579119—42no—
41C2×C4⋊Dic79D411yescharacteristic
42C2×C4⋊Dic79D411yescharacteristic
43C2×C4⋊Dic79D411yescharacteristic
44C2×(C2×D79)⋊C4—21no—
45C2×(C2×D79)⋊C4—22no—
46C2×(C2×D79)⋊C4—22no—
47C2×C4×Dic79—22no—
48C2×C4×Dic79—21no—
49C4⋊(C79⋊C8)—21no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)—21no—
52C4⋊(C79⋊C8)C2×C412yes—
53#14579085—24no—
54C79⋊#95—22no—
55C79⋊#95—21no—
56C22×Q8×C79—21no—
57C22×C8×C79—21no—
58C4⋊Q8×C79—21no—
59C4⋊Q8×C79—21no—
60C4⋊Q8×C79—21no—
61C4⋊Q8×C79C2×C412yes—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79—21no—
66C2×C4⋊C4×C79D412yes—
67C2×C42×C79C2311yescharacteristic
68C4×C8×C79—21no—
69C4×C8×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—21no—
79C22×Dic79—21no—
80C2×C4×D79—22no—
81C2×C4×D79—22no—
82C2×C4×D79—41no—
83C2×C4×D79—41no—
84C2×C4×D79—41no—
85C2×C4×D79—41no—
86#11929611—48no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96C4⋊Dic79—21no—
97C4⋊Dic79C22⋊C412yes—
98C4⋊Dic79SD1612yescomplemented
99(C2×D79)⋊C4—22no—
100(C2×D79)⋊C4—22no—
101(C2×D79)⋊C4—44no—
102(C2×D79)⋊C4—82no—
103C4×Dic79—44no—
104C4×Dic79—41no—
105C4×Dic79—41no—
106C2×Q8×C79—22no—
107C2×Q8×C79—41no—
108C2×Q8×C79—41no—
109C2×Q8×C79—41no—
110C2×Q8×C79—41no—
111C2×Q8×C79—41no—
112C22×C4×C79—22no—
113C22×C4×C79—41no—
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117C2×C8×C79—22no—
118C2×C8×C79—41no—
119C2×C8×C79—41no—
120C4⋊C4×C79—22no—
121C4⋊C4×C79—22no—
122C4⋊C4×C79—42no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79—41no—
127C4⋊C4×C79—41no—
128C4⋊C4×C79—41no—
129C42×C79C22×C411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79#4111yescharacteristic
132C42×C79#4111yescharacteristic
133C23×C79#12011yescharacteristic
134C22×D79—22no—
135C22×D79—41no—
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C22×D79—41no—
140C22×D79—41no—
141Q8×C79—42no—
142Q8×C79—42no—
143Q8×C79—81no—
144Q8×C79—81no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79—81no—
152C2×C4×C79—81no—
153C2×C4×C79C2×C22⋊C411yescharacteristic
154C2×C4×C79C4×D411yescharacteristic
155C2×C4×C79C4×D411yescharacteristic
156C2×C4×C79#12111yescharacteristic
157C2×C4×C79#12111yescharacteristic
158C2×C4×C79#12311yescharacteristic
159C2×C4×C79C2×SD1611yescharacteristic
160C2×C4×C79#13511yescharacteristic
161C2×C4×C79#13511yescharacteristic
162C2×C4×C79#13511yescharacteristic
163C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C79#13711yescharacteristic
167C2×Dic79—22no—
168C2×Dic79—22no—
169C2×Dic79—41no—
170C2×Dic79—41no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176D316—22no—
177D316—22no—
178D316—41no—
179D316—41no—
180C4×D79—42no—
181C4×D79—42no—
182C4×D79—42no—
183C4×D79—42no—
184C8×C79—42no—
185C79⋊C8—42no—
186C79⋊C8—81no—
187C22×C79#38611yescharacteristic
188C22×C79#44911yescharacteristic
189C22×C79#44911yescharacteristic
190C22×C79#45011yescharacteristic
191C22×C79#45211yescharacteristic
192C22×C79#45511yescharacteristic
193C22×C79#45611yescharacteristic
194C2×D79—42no—
195C2×D79—42no—
196C2×D79—42no—
197C2×D79—81no—
198C2×D79—81no—
199C2×D79—81no—
200C2×D79—81no—
201C4×C79—21no—
202C4×C79—21no—
203C4×C79—82no—
204C4×C79—82no—
205C4×C79#41611yescharacteristic
206C4×C79#41811yescharacteristic
207C4×C79C4×SD1611yescharacteristic
208C4×C79#44011yescharacteristic
209C4×C79#46111yescharacteristic
210C4×C79#46311yescharacteristic
211C4×C79#48211yescharacteristic
212C4×C79#48211yescharacteristic
213Dic79—42no—
214Dic79—42no—
215Dic79—81no—
216#8988—791nomaximal, Sylow
217C2×C79#164211yescharacteristic
218C2×C79#164611yescharacteristic
219C2×C79#165011yescharacteristic
220C2×C79#165411yescharacteristic
221C2×C79#187511yescharacteristic
222C2×C79#187711yescharacteristic
223C2×C79#188211yescharacteristic
224D79—82no—
225C2×#461—791no—
226#2354—791no—
227#1990D7911yescharacteristic, complemented
228#1988—791no—
229C2×D4⋊C8—791no—
230#1296—791no—
231#1279—791no—
232C79#898811yescharacteristic, complemented, Sylow
233C22×SD16—1581no—
234C2×C4⋊Q8C2×D7911yescharacteristic
235C2×C4×D4—791no—
236C2×#117—1581no—
237#461—794no—
238#461—1582no—
239C2×C4⋊C8—791no—
240C2×C4⋊C8—791no—
241C2×#103—1581no—
242C2×C4×C8C2×D7911yescharacteristic
243#390—1581no—
244#388—1582no—
245#384C2×D7911yescharacteristic
246#380—1581no—
247C4×C4⋊C4—791no—
248C4×C4⋊C4—1581no—
249#340—22no—
250#340—1581no—
251#340—1581no—
252D4⋊C8—1582no—
253C22×Q8—21no—
254C22×D4—791no—
255C23×C4—1581no—
256C2×SD16—3162no—
257C2×SD16—3162no—
258C2×SD16—1584no—
259C22×C8—21no—
260C22×C8—1581no—
261C22×C8—1581no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8C4×D7912yes—
266C4×D4—792no—
267C4×D4—792no—
268C4×D4—1581no—
269C4×D4—1581no—
270#117—3162no—
271#117—3162no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—791no—
277C2×C4⋊C4—791no—
278C2×C4⋊C4—791no—
279C2×C4⋊C4C22⋊D7912yes—
280C2×C22⋊C4—1582no—
281C2×C22⋊C4—1582no—
282C2×C22⋊C4—1581no—
283C2×C42—1582no—
284C2×C42—1581no—
285C2×C42C22×D7911yescharacteristic
286C4⋊C8—792no—
287C4⋊C8—1581no—
288C4⋊C8—1581no—
289C4⋊C8—1581no—
290#103—1584no—
291C4×C8—21no—
292C4×C8—21no—
293#95—1582no—
294#95—1581no—
295C24—1581no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×D4—792no—
303C2×D4—1581no—
304C2×D4—1581no—
305C2×D4—1581no—
306C2×D4—1581no—
307C2×D4—1581no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—3161no—
311C22×C4—3161no—
312C22×C4—3161no—
313C22×C4—3161no—
314C22×C4—1581no—
315C22×C4—1581no—
316C22×C4—1581no—
317C22×C4—1582no—
318C22×C4—1582no—
319C22×C4D4×D7911yescharacteristic
320C22×C4D4×D7911yescharacteristic
321C22×C4C2×C22⋊D7911yescharacteristic
322SD16—3168no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—3161no—
327C2×C8—3161no—
328C2×C8—3161no—
329C2×C8—3161no—
330C2×C8—3161no—
331C2×C8—1582no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—792no—
342C4⋊C4—792no—
343C4⋊C4—1581no—
344C4⋊C4—1581no—
345C4⋊C4—1581no—
346C4⋊C4—1581no—
347C22⋊C4—3164no—
348C22⋊C4—6322no—
349C22⋊C4—1582no—
350C22⋊C4—1582no—
351C42—3164no—
352C42—3161no—
353C42—3161no—
354C42C2×C4×D7911yescharacteristic
355C42D4×D7911yescharacteristic
356C42#1192962311yescharacteristic
357C42#1192962411yescharacteristic
358C23—3161no—
359C23—3161no—
360C23—3161no—
361C23—3161no—
362C23—3161no—
363C23—3161no—
364C23—1582no—
365C23#1457912911yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370D4—3161no—
371D4—3161no—
372D4—1582no—
373D4—1582no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—3161no—
383C2×C4—3161no—
384C2×C4—3161no—
385C2×C4—3161no—
386C2×C4—3162no—
387C2×C4—3162no—
388C2×C4—3162no—
389C2×C4—3162no—
390C2×C4—3161no—
391C2×C4—3161no—
392C2×C4—3161no—
393C2×C4—1582no—
394C2×C4—1582no—
395C2×C4C22⋊C4×D7911yescharacteristic
396C2×C4(C22⋊D79)⋊C411yescharacteristic
397C2×C4C4×C22⋊D7911yescharacteristic
398C2×C4#1457913211yescharacteristic
399C2×C4#1457913411yescharacteristic
400C2×C4#1457914311yescharacteristic
401C2×C4SD16×D7911yescharacteristic
402C2×C4#1457918011yescharacteristic
403C2×C4#1457918211yescharacteristic
404C2×C4#1457918411yescharacteristic
405C2×C4#1457918511yescharacteristic
406C2×C4#1457918811yescharacteristic
407C2×C4#1457918811yescharacteristic
408C2×C4#1457919311yescharacteristic
409C8—42no—
410C8—3162no—
411C8—6321no—
412C22—3162no—
413C22—3162no—
414C22—3162no—
415C22—6321no—
416C22—6321no—
417C22—6321no—
418C22—6321no—
419C22#1909843911yescharacteristic
420C22#1909870511yescharacteristic
421C22#1909870711yescharacteristic
422C22#1909871211yescharacteristic
423C22#1909872411yescharacteristic
424C22#1909874011yescharacteristic
425C22#1909874511yescharacteristic
426C4—21no—
427C4—21no—
428C4—82no—
429C4—82no—
430C4—3162no—
431C4—3162no—
432C4—6321no—
433C4#1909855611yescharacteristic
434C4#1909856611yescharacteristic
435C4#1909864811yescharacteristic
436C4#1909866611yescharacteristic
437C4#1909876711yescharacteristic
438C4#1909878111yescharacteristic
439C4#1909888511yescharacteristic
440C4#1909888611yescharacteristic
441C2—6322no—
442C2#2376259711yescharacteristic
443C2#2376262511yescharacteristic
444C2#2376265311yescharacteristic
445C2#2376268111yescharacteristic
446C2#2376371911yescharacteristic
447C2#2376373111yescharacteristic
448C2#2376377211yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1990C2
socleC23×C79#120
prime radicalC22×D316D4
cosocleC2×C42×C79C23
hypercenter#1990D79
hypocenterC79#8988
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
3. #23764353C2semidirect
5. #23761283C2semidirect
6. #23761240C2semidirect
7. C79×#1990C2semidirect
12. #19098765C4semidirect
98. C4⋊Dic79SD16semidirect
227. #1990D79semidirect
232. C79#8988semidirect

Maximal subgroups

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

subgroupindex
C2×#190987652
#237663512
#237643532
C2×#190982922
#237612832
#237612402
C79×#19902
#898879