Finite Group Databasebeta

← #34183495

Subgroups — #34183495

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#23772349C211yesmaximal, characteristic, complemented
3#23764318C211yesmaximal, characteristic
4#23761800C211yesmaximal, characteristic
5#23761241C211yesmaximal, characteristic, complemented
6#23761236C211yesmaximal, characteristic, complemented
7C79×#1503C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098959C411yescharacteristic, complemented
10#19098937C411yescharacteristic, complemented
11#19098913C411yescharacteristic, complemented
12#19098911C411yescharacteristic, complemented
13#19098794—22no—
14#19098694—21no—
15#19098694—22no—
16Q8×C79⋊C8—21no—
17C4×#11929611—22no—
18#19098590—21no—
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×#14579089—21no—
21C2×#14579089—21no—
22#19098519—21no—
23C4×#11929608C2211yescharacteristic
24C2×C4×C79⋊C8C2211yescharacteristic
25#19098292—21no—
26#19098292—21no—
27#533×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29C4×C4⋊C4×C79C2211yescharacteristic
30#328×C79—22no—
31C2×#11929624—21no—
32C2×#11929611—22no—
33C2×#11929608D411yescharacteristic
34C2×#11929608D411yescharacteristic
35C22×C79⋊C8—21no—
36C22×C79⋊C8—21no—
37#14579161C2×C411yescharacteristic
38#14579158C2×C411yescharacteristic
39#14579149—21no—
40#14579135—21no—
41#14579135—21no—
42C4×D316—21no—
43C4×D316—21no—
44#14579092—22no—
45C4⋊(C79⋊C8)—22no—
46C4⋊(C79⋊C8)—22no—
47C4⋊(C79⋊C8)—21no—
48C4⋊(C79⋊C8)D411yescharacteristic
49C4⋊(C79⋊C8)D411yescharacteristic
50#14579089—42no—
51#14579089—42no—
52#14579085—22no—
53#14579081—22no—
54#14579072—22no—
55#14579072—21no—
56C4×C79⋊C8—22no—
57C4×C79⋊C8—21no—
58C4×C79⋊C8—21no—
59C4×C79⋊C8C2×C411yescharacteristic
60C4×C79⋊C8C2×C411yescharacteristic
61C4⋊Q8×C79C2×C411yescharacteristic
62C4⋊Q8×C79C2×C411yescharacteristic
63#125×C79—21no—
64C22⋊Q8×C79—21no—
65C22⋊Q8×C79D412yes—
66C4×Q8×C79—21no—
67C4×Q8×C79D412yes—
68#117×C79—21no—
69C2×C4⋊C4×C79—21no—
70C2×C42×C79—21no—
71C2×C42×C79C2311yescharacteristic
72C4⋊C8×C79—21no—
73C22⋊C8×C79—21no—
74C4×C8×C79—21no—
75#95×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—
85C2×C4×D79—41no—
86#11929611—44no—
87#11929608—21no—
88#11929608—21no—
89#11929608—21no—
90#11929608SD1612yescomplemented
91C2×C79⋊C8—21no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—42no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C4⋊Dic79—21no—
101C4⋊Dic79—21no—
102(C2×D79)⋊C4—41no—
103(C2×D79)⋊C4—41no—
104C2×Q8×C79—21no—
105C2×Q8×C79C22⋊C412yes—
106C22×C4×C79—41no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79#4111yescharacteristic
110C2×C8×C79—41no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—22no—
115C4⋊C4×C79—22no—
116C4⋊C4×C79—22no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79C22⋊C412yes—
119C22⋊C4×C79—22no—
120C42×C79—22no—
121C42×C79—41no—
122C42×C79—41no—
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×C79—42no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79C2×C22⋊C411yescharacteristic
141C2×C4×C79C4×D411yescharacteristic
142C2×C4×C79C4×D411yescharacteristic
143C2×C4×C79#12011yescharacteristic
144C2×C4×C79#12111yescharacteristic
145C2×C4×C79#12311yescharacteristic
146C2×C4×C79C2×SD1611yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C79#13711yescharacteristic
150C22⋊D79—81no—
151C22⋊D79—81no—
152C2×Dic79—41no—
153D316—42no—
154D316—41no—
155D316—41no—
156D316—41no—
157D316—41no—
158C4×D79—81no—
159C4×D79—81no—
160Dic158—41no—
161Dic158—41no—
162C8×C79—81no—
163C79⋊C8—22no—
164C79⋊C8—22no—
165C79⋊C8—41no—
166C79⋊C8—41no—
167C79⋊C8—41no—
168C79⋊C8—41no—
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×C79—82no—
183C4×C79C4×SD1611yescharacteristic
184C4×C79#44011yescharacteristic
185C4×C79#44311yescharacteristic
186C4×C79#44411yescharacteristic
187C4×C79#44811yescharacteristic
188C4×C79#44911yescharacteristic
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#2962—791no—
199#1980—791no—
200#1503D7911yescharacteristic, complemented
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—1581no—
213#446—1582no—
214C8×Q8—1581no—
215C4×SD16—1582no—
216#431C2×D7911yescharacteristic
217#424—1581no—
218C2×C4⋊C8—791no—
219C2×#104—1581no—
220C2×#104—1581no—
221#405—1581no—
222C4×M16—791no—
223C2×C4×C8—791no—
224C4×C4⋊C4C2×D7911yescharacteristic
225#328—22no—
226D4⋊C8—1581no—
227D4⋊C8—1581no—
228C2×#41—1581no—
229C2×SD16—1582no—
230C2×M16—791no—
231C2×M16—791no—
232C22×C8—1581no—
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⋊Q8C22⋊D7912yes—
242#121—1581no—
243#121—1581no—
244C4×Q8—21no—
245C4×Q8D31612yes—
246C4×D4—1581no—
247C4×D4—1581no—
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42C22×D7911yescharacteristic
252#106—1582no—
253C4⋊C8—21no—
254C4⋊C8—791no—
255C4⋊C8—791no—
256C4⋊C8—1582no—
257C4⋊C8—1582no—
258C4⋊C8—1581no—
259#104—3162no—
260#104—3162no—
261#103—1582no—
262#102—1582no—
263C22⋊C8—21no—
264#97—1582no—
265#97—1581no—
266C4×C8—21no—
267C4×C8—791no—
268C4×C8—791no—
269C4×C8—1582no—
270C4×C8—1581no—
271C4×C8—1581no—
272#95—21no—
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—41no—
285C22×C4—3161no—
286C22×C4D4×D7911yescharacteristic
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4#1192962311yescharacteristic
289SD16—3164no—
290M16—792no—
291M16—1581no—
292M16—1581no—
293M16—1581no—
294C2×C8—41no—
295C2×C8—3161no—
296C2×C8—3161no—
297C2×C8—3162no—
298C2×C8—1581no—
299C2×C8—1581no—
300C2×C8—1581no—
301C2×C8—1581no—
302C2×C8—1581no—
303C2×C8—1581no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1581no—
312C4⋊C4—1581no—
313C4⋊C4(C2×D79)⋊C412yes—
314C22⋊C4—22no—
315C22⋊C4—3161no—
316C22⋊C4—3161no—
317C42—22no—
318C42—41no—
319C42—41no—
320C42C2×C4×D7911yescharacteristic
321C42D4×D7911yescharacteristic
322C42C2×D31611yescharacteristic
323C42#1192962311yescharacteristic
324C23—3161no—
325C23#1457913311yescharacteristic
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—42no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—3161no—
347C2×C4—6321no—
348C2×C4—6321no—
349C2×C4C2×(C2×D79)⋊C411yescharacteristic
350C2×C4(C22⋊D79)⋊C411yescharacteristic
351C2×C4(C22⋊D79)⋊C411yescharacteristic
352C2×C4#1457913111yescharacteristic
353C2×C4#1457913311yescharacteristic
354C2×C4#1457914511yescharacteristic
355C2×C4SD16×D7911yescharacteristic
356C2×C4#1457918411yescharacteristic
357C2×C4#1457918811yescharacteristic
358C2×C4#1457919311yescharacteristic
359C8—81no—
360C8—3161no—
361C8—3161no—
362C8—3161no—
363C8—3161no—
364C8—1582no—
365C8—1582no—
366C22—21no—
367C22—21no—
368C22—6321no—
369C22—6321no—
370C22—6321no—
371C22#1909844011yescharacteristic
372C22#1909876111yescharacteristic
373C22#1909878711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—6321no—
381C4#1909864811yescharacteristic
382C4#1909866611yescharacteristic
383C4#1909867711yescharacteristic
384C4#1909868211yescharacteristic
385C4#1909870311yescharacteristic
386C4#1909871111yescharacteristic
387C2—21no—
388C2—21no—
389C2—12641no—
390C2#2376372511yescharacteristic
391C2#2376375311yescharacteristic
392C2#2376378111yescharacteristic
subgroupG/H
centerC2×C4#14579131
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1503C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1503D79
hypocenterC79#9054
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #23772349C2semidirect
5. #23761241C2semidirect
6. #23761236C2semidirect
7. C79×#1503C2semidirect
9. #19098959C4semidirect
10. #19098937C4semidirect
11. #19098913C4semidirect
12. #19098911C4semidirect
90. #11929608SD16semidirect
200. #1503D79semidirect
204. C79#9054semidirect

Maximal subgroups

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

subgroupindex
#237739602
#237723492
#237643182
#237618002
#237612412
#237612362
C79×#15032
#905479