Finite Group Databasebeta

← #34183497

Subgroups — #34183497

Order 20224.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773960C211yesmaximal, characteristic
2#23764319C211yesmaximal, characteristic
3#23761899C211yesmaximal, characteristic, complemented
4#23761798C211yesmaximal, characteristic
5#23761239C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7C79×#2962C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098959C411yescharacteristic, complemented
10#19098937C411yescharacteristic, complemented
11#19098913C411yescharacteristic, complemented
12#19098911C411yescharacteristic, complemented
13#19098794—22no—
14C4×#11929611—22no—
15#19098616C2211yescharacteristic
16C2×#14579087—21no—
17C2×#14579087—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19C4×C4⋊Dic79C2211yescharacteristic
20#19098300—22no—
21#19098290—21no—
22#19098290—21no—
23#533×C79C2211yescharacteristic
24#446×C79—22no—
25#446×C79—21no—
26C8×Q8×C79—21no—
27#424×C79—21no—
28C2×C4⋊C8×C79C2211yescharacteristic
29#405×C79—21no—
30C4×M16×C79C2211yescharacteristic
31C2×#11929624—21no—
32C2×#11929611—22no—
33C22×C79⋊C8—21no—
34#14579161C2×C411yescharacteristic
35#14579158C2×C411yescharacteristic
36#14579149—21no—
37#14579135—21no—
38#14579135—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79—21no—
42C2×C4×Dic79—21no—
43#14579092—22no—
44C4⋊(C79⋊C8)—21no—
45#14579087—42no—
46#14579087—42no—
47#14579085—22no—
48#14579081—22no—
49C22⋊(C79⋊C8)—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C4×C79⋊C8C2×C411yescharacteristic
53C4×C79⋊C8C2×C411yescharacteristic
54C79⋊#95—21no—
55C2×M16×C79D411yescharacteristic
56C2×M16×C79D411yescharacteristic
57C22×C8×C79—21no—
58C4⋊Q8×C79C2×C411yescharacteristic
59C4⋊Q8×C79C2×C411yescharacteristic
60#125×C79—21no—
61C22⋊Q8×C79—21no—
62C22⋊Q8×C79D412yes—
63C4×Q8×C79—21no—
64C4×Q8×C79D412yes—
65#117×C79—21no—
66C2×C42×C79C2311yescharacteristic
67C4⋊C8×C79—22no—
68C4⋊C8×C79—22no—
69C4⋊C8×C79—21no—
70C4⋊C8×C79D411yescharacteristic
71C4⋊C8×C79D411yescharacteristic
72#97×C79—22no—
73#97×C79—21no—
74C4×C8×C79—22no—
75C4×C8×C79—21no—
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79#11929624—41no—
80C2×Dic158—21no—
81C2×D316—21no—
82C2×D316—21no—
83C2×D316—21no—
84C2×C22⋊D79—41no—
85C22×Dic79—41no—
86C2×C4×D79—41no—
87#11929611—44no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—42no—
91C2×C79⋊C8—41no—
92C4⋊Dic79—22no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79—41no—
96(C2×D79)⋊C4—41no—
97(C2×D79)⋊C4—41no—
98C4×Dic79—41no—
99C4×Dic79—41no—
100C2×Q8×C79—21no—
101C2×Q8×C79C22⋊C412yes—
102C22×C4×C79C2×D411yescharacteristic
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79#4111yescharacteristic
105M16×C79—21no—
106M16×C79—21no—
107M16×C79—21no—
108M16×C79SD1612yescomplemented
109C2×C8×C79—21no—
110C2×C8×C79—21no—
111C2×C8×C79—21no—
112C2×C8×C79—21no—
113C2×C8×C79—41no—
114C2×C8×C79—41no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—22no—
119C4⋊C4×C79—22no—
120C4⋊C4×C79C22⋊C412yes—
121C22⋊C4×C79—22no—
122C42×C79—22no—
123C42×C79C22×C411yescharacteristic
124C42×C79C2×D411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79#4111yescharacteristic
127C23×C79#12111yescharacteristic
128C22×D79—41no—
129Q8×C79—42no—
130Q8×C79—41no—
131Q8×C79—41no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—41no—
137C2×C4×C79C2×C22⋊C411yescharacteristic
138C2×C4×C79C4×D411yescharacteristic
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79#12011yescharacteristic
141C2×C4×C79#12111yescharacteristic
142C2×C4×C79#12311yescharacteristic
143C2×C4×C79C2×SD1611yescharacteristic
144C2×C4×C79#13511yescharacteristic
145C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C79#13711yescharacteristic
147C22⋊D79—81no—
148C22⋊D79—81no—
149C2×Dic79—42no—
150C2×Dic79—41no—
151C2×Dic79—81no—
152C2×Dic79—81no—
153D316—42no—
154D316—41no—
155D316—41no—
156D316—41no—
157D316—41no—
158C4×D79—81no—
159C4×D79—81no—
160Dic158—41no—
161Dic158—41no—
162C8×C79—22no—
163C8×C79—41no—
164C8×C79—41no—
165C8×C79—41no—
166C79⋊C8—22no—
167C79⋊C8—41no—
168C79⋊C8—81no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79#38611yescharacteristic
172C22×C79#46011yescharacteristic
173C22×C79#46411yescharacteristic
174C2×D79—81no—
175C2×D79—81no—
176C2×D79—81no—
177C4×C79—21no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—42no—
181C4×C79—42no—
182C4×C79C4×SD1611yescharacteristic
183C4×C79#44011yescharacteristic
184C4×C79#44311yescharacteristic
185C4×C79#44411yescharacteristic
186C4×C79#44811yescharacteristic
187C4×C79#44911yescharacteristic
188Dic79—82no—
189Dic79—81no—
190#9054—791nomaximal, Sylow
191C2×C79—21no—
192C2×C79—21no—
193C2×C79#187611yescharacteristic
194C2×C79#188011yescharacteristic
195C2×C79#188411yescharacteristic
196D79—161no—
197#3167—791no—
198#2962D7911yescharacteristic, complemented
199#1980—791no—
200#1503—791no—
201#1480—791no—
202#1279—791no—
203#1278—791no—
204C79#905411yescharacteristic, complemented, Sylow
205#533C2×D7911yescharacteristic
206#532—791no—
207#498—791no—
208#492—791no—
209#487—791no—
210#486—791no—
211#465—1582no—
212#446—22no—
213#446—21no—
214C8×Q8—21no—
215C4×SD16—1582no—
216#431—791no—
217#424—21no—
218C2×C4⋊C8C2×D7911yescharacteristic
219C2×#104—1581no—
220C2×#104—1581no—
221#405—21no—
222C4×M16C2×D7911yescharacteristic
223C2×C4×C8—791no—
224C4×C4⋊C4—791no—
225#328—1582no—
226D4⋊C8—1581no—
227D4⋊C8—1581no—
228C2×#41—1581no—
229C2×SD16—1582no—
230C2×M16C22⋊D7911yescharacteristic
231C2×M16C22⋊D7911yescharacteristic
232C22×C8—21no—
233C22×C8—1581no—
234C4⋊Q8—791no—
235C4⋊Q8C4×D7911yescharacteristic
236C4⋊Q8C4×D7911yescharacteristic
237#127—791no—
238#125—21no—
239#124—1581no—
240C22⋊Q8—21no—
241C22⋊Q8D31612yes—
242#121—1581no—
243#121—1581no—
244C4×Q8—21no—
245C4×Q8C22⋊D7912yes—
246C4×D4—1581no—
247C4×D4—1581no—
248#117—21no—
249C2×C4⋊C4—1581no—
250C2×C42—1581no—
251C2×C42C22×D7911yescharacteristic
252#106—1582no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—1581no—
257C4⋊C8C22⋊D7911yescharacteristic
258C4⋊C8C22⋊D7911yescharacteristic
259#104—3162no—
260#104—3162no—
261#103—1582no—
262#102—1582no—
263C22⋊C8—1581no—
264#97—22no—
265#97—21no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—791no—
269C4×C8—791no—
270C4×C8—1581no—
271C4×C8—1581no—
272#95—1581no—
273#41—3161no—
274#41—3161no—
275#41—3161no—
276#41—3161no—
277C2×Q8—21no—
278C2×Q8—1581no—
279C2×Q8(C2×D79)⋊C412yes—
280C2×D4—3161no—
281C2×D4—1581no—
282C2×D4—1581no—
283C2×D4—1581no—
284C22×C4—3161no—
285C22×C4—3161no—
286C22×C4C2×C22⋊D7911yescharacteristic
287C22×C4C2×D31611yescharacteristic
288C22×C4#1192962411yescharacteristic
289SD16—3164no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1192961112yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—3162no—
301C2×C8—3161no—
302C2×C8—1581no—
303C2×C8—1581no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—3161no—
310C4⋊C4—1582no—
311C4⋊C4—1581no—
312C4⋊C4—1581no—
313C4⋊C4(C2×D79)⋊C412yes—
314C22⋊C4—22no—
315C22⋊C4—3161no—
316C22⋊C4—3161no—
317C42—22no—
318C42—3161no—
319C42—3161no—
320C42C2×C4×D7911yescharacteristic
321C42C2×C22⋊D7911yescharacteristic
322C42C2×C22⋊D7911yescharacteristic
323C42#1192962411yescharacteristic
324C23—3161no—
325C23#1457913511yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—3161no—
330Q8—3161no—
331D4—3162no—
332D4—3161no—
333D4—3161no—
334D4—3161no—
335D4—3161no—
336D4—6321no—
337D4—6321no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—3162no—
344C2×C4—3161no—
345C2×C4—6321no—
346C2×C4—6321no—
347C2×C4—6321no—
348C2×C4—6321no—
349C2×C4C2×(C2×D79)⋊C411yescharacteristic
350C2×C4C4×C22⋊D7911yescharacteristic
351C2×C4C4×C22⋊D7911yescharacteristic
352C2×C4#1457913011yescharacteristic
353C2×C4#1457913511yescharacteristic
354C2×C4#1457914411yescharacteristic
355C2×C4C2×#1192961111yescharacteristic
356C2×C4#1457918111yescharacteristic
357C2×C4#1457918611yescharacteristic
358C2×C4#1457919211yescharacteristic
359C8—22no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—3161no—
364C8—6321no—
365C8—1582no—
366C22—21no—
367C22—21no—
368C22—6321no—
369C22—6321no—
370C22—6321no—
371C22#1909844211yescharacteristic
372C22#1909876311yescharacteristic
373C22#1909878511yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—6322no—
380C4—6321no—
381C4C4×#1192961111yescharacteristic
382C4#1909866111yescharacteristic
383C4#1909867811yescharacteristic
384C4#1909868311yescharacteristic
385C4#1909870211yescharacteristic
386C4#1909870811yescharacteristic
387C2—21no—
388C2—21no—
389C2—12641no—
390C2#2376372711yescharacteristic
391C2#2376375511yescharacteristic
392C2#2376378311yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2962C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2962D79
hypocenterC79#9054
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #23761899C2semidirect
5. #23761239C2semidirect
6. #23761238C2semidirect
7. C79×#2962C2semidirect
9. #19098959C4semidirect
10. #19098937C4semidirect
11. #19098913C4semidirect
12. #19098911C4semidirect
108. M16×C79SD16semidirect
198. #2962D79semidirect
204. C79#9054semidirect
293. M16#11929611semidirect

Maximal subgroups

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

subgroupindex
#237739602
#237643192
#237618992
#237617982
#237612392
#237612382
C79×#29622
#905479