Finite Group Databasebeta

← #34271026

Subgroups — #34271026

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773986C211yesmaximal, characteristic, complemented
2#23773471C212yesmaximal
3#23762954C212yesmaximal, complemented
4#23762947C212yesmaximal, complemented
5#23762918C211yesmaximal, characteristic, complemented
6#23762913C211yesmaximal, characteristic, complemented
7#23762894C211yesmaximal, characteristic, complemented
8#23762889C211yesmaximal, characteristic, complemented
9#23762148C211yesmaximal, characteristic, complemented
10#23762064C212yesmaximal, complemented
11C79×#2965C211yesmaximal, characteristic
12#19099120C2212yes—
13#19098968C2211yescharacteristic
14#19098943C2211yescharacteristic
15#19098914C2212yes—
16#19098904—22no—
17#19098848C2212yes—
18#19098843C2212yes—
19#19098804—21no—
20#19098804—21no—
21#19098773C2212yes—
22#19098768C2212yes—
23#19098725—22no—
24C4×Dic316—21no—
25C4×Dic316—21no—
26C4×#11929614—22no—
27C4×#11929612—22no—
28#19098616C2212yes—
29#19098316C2212yes—
30#19098314C2211yescharacteristic
31#19098314C2211yescharacteristic
32#19098310C2212yes—
33#19098310C2212yes—
34#19098308C2212yes—
35#19098308C2212yes—
36C4×#41×C79C2211yescharacteristic
37#446×C79C2211yescharacteristic
38#446×C79C2211yescharacteristic
39#436×C79C2211yescharacteristic
40#436×C79C2211yescharacteristic
41#435×C79—21no—
42#435×C79—21no—
43#431×C79—21no—
44#431×C79C2211yescharacteristic
45C4×M16×C79—21no—
46C2×C4×C8×C79C2211yescharacteristic
47C2×Dic316—21no—
48C2×Dic316—21no—
49C2×#11929614—22no—
50C2×#11929612—22no—
51#14579161C2312yes—
52#14579161C2312yes—
53#14579153—22no—
54#14579140—22no—
55#14579140—22no—
56C4×Dic158—22no—
57C4×Dic158—22no—
58#14579120—22no—
59#14579095—21no—
60#14579095—21no—
61#14579094—22no—
62#14579092—22no—
63C4⋊(C79⋊C8)C2312yes—
64C4⋊(C79⋊C8)C2312yes—
65#14579086—22no—
66#14579086—22no—
67#14579085—22no—
68#14579084—22no—
69#14579084—22no—
70#14579082—22no—
71C22⋊(C79⋊C8)—22no—
72C4×C79⋊C8—22no—
73C2×#41×C79D411yescharacteristic
74C2×M16×C79—21no—
75C2×M16×C79—21no—
76C22×C8×C79D411yescharacteristic
77C22×C8×C79D411yescharacteristic
78C4×Q8×C79D411yescharacteristic
79C4×Q8×C79C2311yescharacteristic
80C4×D4×C79—21no—
81C4×D4×C79—21no—
82C4×D4×C79D411yescharacteristic
83C4×D4×C79C2311yescharacteristic
84#117×C79—21no—
85#117×C79D411yescharacteristic
86C2×C42×C79—21no—
87C2×C42×C79C2311yescharacteristic
88C4⋊C8×C79—21no—
89C4⋊C8×C79—21no—
90C4⋊C8×C79C2311yescharacteristic
91C4⋊C8×C79C2311yescharacteristic
92C22⋊C8×C79—21no—
93C22⋊C8×C79—21no—
94C22⋊C8×C79—21no—
95#97×C79—21no—
96#97×C79—21no—
97C4×C8×C79—21no—
98C4×C8×C79D411yescharacteristic
99C4×C8×C79D411yescharacteristic
100C4×C8×C79C2311yescharacteristic
101C4×C8×C79C2311yescharacteristic
102C2×Dic158—22no—
103C2×Dic158—22no—
104Dic316—42no—
105Dic316—42no—
106#11929614—44no—
107#11929612—44no—
108C2×C79⋊C8—22no—
109C2×C79⋊C8—22no—
110C4⋊Dic79—22no—
111C4⋊Dic79—22no—
112C4⋊Dic79—22no—
113C4⋊Dic79—22no—
114Dic79⋊C4—42no—
115Dic79⋊C4—42no—
116C22⋊Dic79—42no—
117C4×Dic79—42no—
118#41×C79—42no—
119C2×Q8×C79C2×D411yescharacteristic
120C2×D4×C79—21no—
121C2×D4×C79C2×D411yescharacteristic
122C22×C4×C79—21no—
123C22×C4×C79—21no—
124C22×C4×C79—21no—
125C22×C4×C79C2×D411yescharacteristic
126C22×C4×C79C2×D411yescharacteristic
127C22×C4×C79#4111yescharacteristic
128M16×C79—42no—
129M16×C79—42no—
130C2×C8×C79—22no—
131C2×C8×C79—21no—
132C2×C8×C79—21no—
133C2×C8×C79—41no—
134C2×C8×C79C2×D411yescharacteristic
135C2×C8×C79C2×D411yescharacteristic
136C2×C8×C79C2×D411yescharacteristic
137C2×C8×C79C2×D411yescharacteristic
138C4⋊C4×C79—21no—
139C4⋊C4×C79—21no—
140C4⋊C4×C79C2×D411yescharacteristic
141C4⋊C4×C79C2×D411yescharacteristic
142C22⋊C4×C79—21no—
143C22⋊C4×C79—21no—
144C22⋊C4×C79—21no—
145C42×C79—21no—
146C42×C79—21no—
147C42×C79—21no—
148C42×C79C2×D411yescharacteristic
149C42×C79C2×D411yescharacteristic
150C42×C79#4111yescharacteristic
151C42×C79C2411yescharacteristic
152C23×C79—21no—
153C23×C79#12111yescharacteristic
154Q8×C79—22no—
155D4×C79—22no—
156D4×C79—42no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—22no—
160C2×C4×C79—22no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—42no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—42no—
168C2×C4×C79#12111yescharacteristic
169C2×C4×C79#12111yescharacteristic
170C2×C4×C79#12111yescharacteristic
171C2×C4×C79C22×D411yescharacteristic
172C2×C4×C79C22×D411yescharacteristic
173C2×C4×C79C2×#4111yescharacteristic
174C2×Dic79—42no—
175C2×Dic79—42no—
176Dic158—42no—
177Dic158—42no—
178Dic158—42no—
179Dic158—42no—
180C8×C79—21no—
181C8×C79—21no—
182C8×C79—21no—
183C8×C79—41no—
184C8×C79—41no—
185C8×C79#13512yes—
186C79⋊C8—42no—
187C79⋊C8—42no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—41no—
192C22×C79—41no—
193C22×C79—41no—
194C22×C79C2×#12111yescharacteristic
195C4×C79—21no—
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—41no—
199C4×C79—41no—
200C4×C79—41no—
201C4×C79#47312yes—
202C4×C79C2×#13511yescharacteristic
203C4×C79#57511yescharacteristic
204C4×C79#57512yes—
205Dic79—82no—
206Dic79—82no—
207#19161—791nomaximal, Sylow
208C2×C79—41no—
209C2×C79—81no—
210C2×C79#303511yescharacteristic
211C2×C79#305411yescharacteristic
212C2×C79C2×#47311yescharacteristic
213#3170—791no—
214#3107—792no—
215#2965D7911yescharacteristic
216#1700—792no—
217#1699—792no—
218#1694—791no—
219#1693—791no—
220#1690—791no—
221#1689—791no—
222#1564—791no—
223#1544—792no—
224C79#1916111yescharacteristic, complemented, Sylow
225#533—792no—
226C4×#41C2×D7911yescharacteristic
227#500—791no—
228#494—791no—
229#487—792no—
230#485—1582no—
231#475—792no—
232#474—792no—
233#467—1581no—
234#467—1581no—
235#462—792no—
236#461—792no—
237#452—1582no—
238#446C2×D7911yescharacteristic
239#446C2×D7911yescharacteristic
240C4×Q16—1582no—
241C4×Q16—1581no—
242C4×Q16—1581no—
243C4×SD16—1582no—
244#436C2×D7911yescharacteristic
245#436C2×D7911yescharacteristic
246#435—21no—
247#435—21no—
248#431—21no—
249#431—792no—
250#431C2×D7911yescharacteristic
251C4×M16—21no—
252C2×C4×C8C2×D7911yescharacteristic
253#335—792no—
254#334—791no—
255#334—791no—
256#332—792no—
257#332—792no—
258#332—792no—
259#332—792no—
260C2×#41D31611yescharacteristic
261C2×Q16—1582no—
262C2×Q16—1581no—
263C2×Q16—1581no—
264C2×SD16—1582no—
265C2×M16—21no—
266C2×M16—21no—
267C22×C8C22⋊D7911yescharacteristic
268C22×C8C22⋊D7911yescharacteristic
269C4⋊Q8—792no—
270C4⋊Q8—792no—
271#125—1582no—
272C22⋊Q8—1582no—
273C22⋊Q8—1582no—
274C4×Q8—1582no—
275C4×Q8—1582no—
276C4×Q8D31611yescharacteristic
277C4×Q8C22×D7911yescharacteristic
278C4×D4—21no—
279C4×D4—21no—
280C4×D4D31611yescharacteristic
281C4×D4C22×D7911yescharacteristic
282#117—21no—
283#117—1582no—
284#117D31611yescharacteristic
285C2×C42—21no—
286C2×C42C22×D7911yescharacteristic
287#107—1581no—
288#107—1581no—
289#107—1582no—
290#106—1582no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—792no—
294C4⋊C8—792no—
295C4⋊C8C22×D7911yescharacteristic
296C4⋊C8C22×D7911yescharacteristic
297#103—1582no—
298#103—1582no—
299#103—1582no—
300#103—1582no—
301#103—1582no—
302#102—1582no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—1582no—
307#97—21no—
308#97—21no—
309C4×C8—21no—
310C4×C8—1582no—
311C4×C8C22⋊D7911yescharacteristic
312C4×C8C22⋊D7911yescharacteristic
313C4×C8C22×D7911yescharacteristic
314C4×C8C22×D7911yescharacteristic
315#41—42no—
316C2×Q8—1582no—
317C2×Q8—1582no—
318C2×Q8C2×D31611yescharacteristic
319C2×D4—21no—
320C2×D4C2×D31611yescharacteristic
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4C2×C22⋊D7911yescharacteristic
325C22×C4C2×D31611yescharacteristic
326C22×C4#1192962411yescharacteristic
327Q16—3162no—
328Q16—3164no—
329Q16—3162no—
330SD16—3164no—
331M16—42no—
332M16—42no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—41no—
337C2×C8—1582no—
338C2×C8—1582no—
339C2×C8C2×C22⋊D7911yescharacteristic
340C2×C8C2×C22⋊D7911yescharacteristic
341C2×C8C2×C22⋊D7911yescharacteristic
342C2×C8C2×C22⋊D7911yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—3162no—
346C4⋊C4—3162no—
347C4⋊C4—1582no—
348C4⋊C4—1582no—
349C4⋊C4—1582no—
350C4⋊C4—1582no—
351C4⋊C4C2×D31611yescharacteristic
352C4⋊C4C2×D31611yescharacteristic
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—3162no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—3162no—
361C42C2×C22⋊D7911yescharacteristic
362C42C2×D31611yescharacteristic
363C42#1192962411yescharacteristic
364C42C23×D7911yescharacteristic
365C23—21no—
366C23#1457913511yescharacteristic
367Q8—22no—
368Q8—3162no—
369Q8—3162no—
370Q8—3162no—
371Q8—3162no—
372D4—22no—
373D4—42no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—3162no—
386C2×C4—3162no—
387C2×C4#1457913511yescharacteristic
388C2×C4#1457913511yescharacteristic
389C2×C4#1457913511yescharacteristic
390C2×C4C22×C22⋊D7911yescharacteristic
391C2×C4C22×D31611yescharacteristic
392C2×C4C2×#1192962411yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—41no—
397C8—41no—
398C8—3162no—
399C8—3162no—
400C8#1457918112yes—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22C2×#1457913511yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—6322no—
415C4—6322no—
416C4#1909883812yes—
417C4C2×#1457918111yescharacteristic
418C4#1909936111yescharacteristic
419C4#1909936312yes—
420C2—41no—
421C2—81no—
422C2#2377285511yescharacteristic
423C2#2377300911yescharacteristic
424C2C2×#1909883811yescharacteristic
subgroupG/H
centerC2×C4#14579135
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#2965C2
socleC22×C79C2×#121
prime radicalC2×#41×C79D4
hypercenter#2965D79
hypocenterC79#19161
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #23773986C2semidirect
3. #23762954C2semidirect
4. #23762947C2semidirect
5. #23762918C2semidirect
6. #23762913C2semidirect
7. #23762894C2semidirect
8. #23762889C2semidirect
9. #23762148C2semidirect
10. #23762064C2semidirect
224. C79#19161semidirect

Maximal subgroups

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

subgroupindex
#237739862
#237734712
#237629542
#237629472
#237629182
#237629132
#237628942
#237628892
#237621482
#237620642
C79×#29652
#1916179