Finite Group Databasebeta

← #34184968

Subgroups — #34184968

Order 20224.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774341C211yesmaximal, characteristic
2#23764381C211yesmaximal, characteristic, complemented
3#23762068C211yesmaximal, characteristic
4#23761963C211yesmaximal, characteristic, complemented
5#23761258C211yesmaximal, characteristic, complemented
6#23761229C211yesmaximal, characteristic, complemented
7C79×#2328C211yesmaximal, characteristic
8#19099120C2211yescharacteristic
9#19098935C412yescomplemented
10#19098909C412yescomplemented
11#19098760—22no—
12#19098714—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16#19098323—21no—
17#19098323—21no—
18#19098307—22no—
19#19098299—22no—
20C2×#127×C79C2211yescharacteristic
21C2×C4×D4×C79—21no—
22#390×C79—21no—
23#386×C79—22no—
24#386×C79—22no—
25#382×C79C2211yescharacteristic
26#376×C79C2211yescharacteristic
27C2×#11929612—24no—
28C2×#11929608—21no—
29#14579161C2×C411yescharacteristic, complemented
30#14579161C2×C411yescharacteristic, complemented
31#14579153—21no—
32#14579140—21no—
33#14579140—21no—
34C4×Dic158—21no—
35C4×Dic158—21no—
36#14579120—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41#14579082—24no—
42C22⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)—21no—
44C4×C79⋊C8—21no—
45C4×C79⋊C8C2×C412yescomplemented
46C22×D4×C79—21no—
47C22×D4×C79—21no—
48C22×D4×C79D412yes—
49C23×C4×C79—22no—
50#127×C79—21no—
51#127×C79—21no—
52#127×C79C2×C412yes—
53#127×C79D412yes—
54C4×D4×C79—42no—
55C4×D4×C79—42no—
56C2×C4⋊C4×C79—21no—
57C2×C22⋊C4×C79—22no—
58C2×C22⋊C4×C79—22no—
59C2×C22⋊C4×C79—22no—
60C2×C42×C79D411yescharacteristic
61C2×C42×C79D411yescharacteristic
62C2×C42×C79C2311yescharacteristic
63#95×C79—21no—
64#95×C79—21no—
65#95×C79—21no—
66#95×C79D412yes—
67C2×Dic158—21no—
68C2×Dic158—21no—
69#11929612—84no—
70#11929608—41no—
71#11929608—41no—
72C2×C79⋊C8—22no—
73C2×C79⋊C8—41no—
74C2×C79⋊C8—41no—
75C4⋊Dic79—21no—
76C4⋊Dic79—21no—
77C4⋊Dic79—21no—
78C4⋊Dic79—21no—
79Dic79⋊C4—41no—
80Dic79⋊C4—41no—
81C22⋊Dic79—41no—
82C4×Dic79—41no—
83C24×C79—22no—
84C2×D4×C79—22no—
85C2×D4×C79—22no—
86C2×D4×C79—22no—
87C2×D4×C79—22no—
88C2×D4×C79—22no—
89C2×D4×C79—41no—
90C2×D4×C79—41no—
91C2×D4×C79—41no—
92C2×D4×C79—41no—
93C2×D4×C79—82no—
94C2×D4×C79—82no—
95C2×D4×C79C22⋊C414yes—
96C22×C4×C79—21no—
97C22×C4×C79—21no—
98C22×C4×C79—22no—
99C22×C4×C79—44no—
100C22×C4×C79—82no—
101C22×C4×C79—82no—
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C4⋊C4×C79—41no—
106C4⋊C4×C79—41no—
107C22⋊C4×C79—44no—
108C22⋊C4×C79—82no—
109C22⋊C4×C79—82no—
110C42×C79—21no—
111C42×C79—21no—
112C42×C79—21no—
113C42×C79SD1612yescomplemented
114C42×C79C22×C411yescharacteristic
115C42×C79C2×D411yescharacteristic
116C42×C79#4111yescharacteristic
117C42×C79#4111yescharacteristic
118C23×C79—24no—
119C23×C79—42no—
120C23×C79—42no—
121C23×C79—42no—
122C23×C79—42no—
123C23×C79—42no—
124C23×C79—42no—
125C23×C79#12011yescharacteristic
126D4×C79—44no—
127D4×C79—42no—
128D4×C79—42no—
129D4×C79—42no—
130D4×C79—42no—
131D4×C79—82no—
132D4×C79—82no—
133D4×C79—82no—
134C2×C4×C79—22no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—42no—
143C2×C4×C79—42no—
144C2×C4×C79—41no—
145C2×C4×C79—88no—
146C2×C4×C79C2×C22⋊C411yescharacteristic
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79C4×D411yescharacteristic
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79#12111yescharacteristic
151C2×C4×C79#12311yescharacteristic
152C2×C4×C79C2×SD1611yescharacteristic
153C2×C4×C79#13511yescharacteristic
154C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C79#13711yescharacteristic
156C2×Dic79—41no—
157C2×Dic79—41no—
158Dic158—42no—
159Dic158—81no—
160C79⋊C8—42no—
161C79⋊C8—81no—
162C79⋊C8—81no—
163C22×C79—21no—
164C22×C79—21no—
165C22×C79—44no—
166C22×C79—44no—
167C22×C79—44no—
168C22×C79—82no—
169C22×C79—82no—
170C22×C79—82no—
171C22×C79—82no—
172C22×C79#38611yescharacteristic
173C22×C79#45011yescharacteristic
174C22×C79#45211yescharacteristic
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—42no—
180C4×C79—81no—
181C4×C79—82no—
182C4×C79C4×SD1611yescharacteristic
183C4×C79#44011yescharacteristic
184C4×C79#46111yescharacteristic
185C4×C79#46311yescharacteristic
186Dic79—81no—
187Dic79—81no—
188#9295—791nomaximal, Sylow
189C2×C79—21no—
190C2×C79—21no—
191C2×C79—84no—
192C2×C79#187711yescharacteristic
193C2×C79#190411yescharacteristic
194C2×C79#191111yescharacteristic
195#3208—791no—
196#2328D7911yescharacteristic
197#1995—791no—
198#1545—791no—
199#1518—791no—
200#1286—791no—
201#1274—791no—
202C79#929511yescharacteristic, complemented, Sylow
203#533—791no—
204C2×#127C2×D7911yescharacteristic
205C2×C4×D4—21no—
206#492—792no—
207#486—792no—
208#460—1582no—
209#450—1582no—
210#431—791no—
211#431—791no—
212#423—791no—
213#390—21no—
214#386—22no—
215#386—22no—
216#382C2×D7911yescharacteristic
217#376C2×D7911yescharacteristic
218#339—1581no—
219#339—1581no—
220#331—1582no—
221#328—1582no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C22⋊D7912yes—
225C23×C4—22no—
226C2×SD16—1584no—
227C2×M16—1581no—
228C4⋊Q8—791no—
229C4⋊Q8—791no—
230#127—21no—
231#127—21no—
232#127C4×D7912yes—
233#127D31612yes—
234#125—1581no—
235C22⋊Q8—1581no—
236C22⋊Q8—1581no—
237C4×Q8—1581no—
238C4×Q8—1581no—
239C4×D4—42no—
240C4×D4—42no—
241#117—1581no—
242C2×C4⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42C22⋊D7911yescharacteristic
247C2×C42C22⋊D7911yescharacteristic
248C2×C42C22×D7911yescharacteristic
249C4⋊C8—1581no—
250C4⋊C8—1581no—
251C4⋊C8—1581no—
252C4⋊C8—1581no—
253#102—1584no—
254C22⋊C8—1581no—
255C22⋊C8—1581no—
256C4×C8—792no—
257C4×C8—1581no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95C22⋊D7912yes—
262C24—22no—
263C2×Q8—1581no—
264C2×Q8—1581no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4(C2×D79)⋊C414yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊D7911yescharacteristic
284C22×C4C2×C22⋊D7911yescharacteristic
285C22×C4C2×C22⋊D7911yescharacteristic
286SD16—6324no—
287M16—3161no—
288M16—3161no—
289C2×C8—3161no—
290C2×C8—3161no—
291C2×C8—1582no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—3161no—
295C4⋊C4—3161no—
296C4⋊C4—1581no—
297C4⋊C4—1581no—
298C4⋊C4—1581no—
299C4⋊C4—1581no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—3161no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—3161no—
308C42#1192961212yescomplemented
309C42C2×C4×D7911yescharacteristic
310C42C2×D31611yescharacteristic
311C42#1192962411yescharacteristic
312C42#1192962411yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1457913011yescharacteristic
321Q8—3162no—
322Q8—6321no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—88no—
343C2×C4—3161no—
344C2×C4—3161no—
345C2×C4C2×(C2×D79)⋊C411yescharacteristic
346C2×C4C4×C22⋊D7911yescharacteristic
347C2×C4C4×C22⋊D7911yescharacteristic
348C2×C4#1457913511yescharacteristic
349C2×C4#1457913511yescharacteristic
350C2×C4#1457914411yescharacteristic
351C2×C4C2×#1192961211yescharacteristic
352C2×C4#1457918111yescharacteristic
353C2×C4#1457918711yescharacteristic
354C2×C4#1457919111yescharacteristic
355C8—3162no—
356C8—6321no—
357C8—6321no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1909844211yescharacteristic
368C22#1909871411yescharacteristic
369C22#1909872511yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—82no—
376C4—81no—
377C4—6321no—
378C4—6321no—
379C4C4×#1192961211yescharacteristic
380C4#1909866311yescharacteristic
381C4#1909876811yescharacteristic
382C4#1909877811yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2376373411yescharacteristic
387C2#2376388711yescharacteristic
388C2#2376393211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2328C2
socleC22×C79#386
prime radicalC2×#127×C79C22
cosocleC2×C42×C79C23
hypercenter#2328D79
hypocenterC79#9295
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #23764381C2semidirect
4. #23761963C2semidirect
5. #23761258C2semidirect
6. #23761229C2semidirect
9. #19098935C4semidirect
10. #19098909C4semidirect
29. #14579161C2×C4semidirect
30. #14579161C2×C4semidirect
45. C4×C79⋊C8C2×C4semidirect
113. C42×C79SD16semidirect
202. C79#9295semidirect
308. C42#11929612semidirect

Maximal subgroups

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

subgroupindex
#237743412
#237643812
#237620682
#237619632
#237612582
#237612292
C79×#23282
#929579