Finite Group Databasebeta

← #34181441

Subgroups — #34181441

Order 20224.

432 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772367C211yesmaximal, characteristic
2#23763420C211yesmaximal, characteristic
3C4×#14579094C211yesmaximal, characteristic, complemented
4#23762155C211yesmaximal, characteristic, complemented
5#23761303C211yesmaximal, characteristic, complemented
6#23761301C211yesmaximal, characteristic, complemented
7C79×#1806C211yesmaximal, characteristic
8#19098694C411yescharacteristic, complemented
9#19098694C411yescharacteristic, complemented
10#19098641C411yescharacteristic, complemented
11#19098641C411yescharacteristic, complemented
12#19098635—21no—
13#19098635—21no—
14#19098616—21no—
15#19098616C2211yescharacteristic
16C2×#14579094—21no—
17#19098587C2211yescharacteristic
18C2×#14579088—21no—
19C4×#11929608—21no—
20C2×C4×C79⋊C8C2211yescharacteristic
21C4×C4⋊Dic79C2211yescharacteristic
22#19098325—21no—
23#19098322—21no—
24#19098313—21no—
25#19098313—21no—
26C4×#41×C79C2211yescharacteristic
27#423×C79C2211yescharacteristic
28C2×#104×C79—21no—
29C4×C4⋊C4×C79C2211yescharacteristic
30#339×C79—21no—
31C2×#11929608—21no—
32C2×#11929608—21no—
33C2×#11929608—21no—
34C22×C79⋊C8D411yescharacteristic
35C22×C79⋊C8D411yescharacteristic
36C2×C4⋊Dic79—21no—
37C2×C4×Dic79—21no—
38#14579094—24no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)C2×C411yescharacteristic
44C4⋊(C79⋊C8)C2×C411yescharacteristic
45#14579088—42no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49#14579072—21no—
50#14579072—21no—
51C4×C79⋊C8—21no—
52C4×C79⋊C8C2×C411yescharacteristic
53C4×C79⋊C8C2×C411yescharacteristic
54C4×C79⋊C8D411yescharacteristic
55C4×C79⋊C8D411yescharacteristic
56C79⋊#95—21no—
57C2×#41×C79D411yescharacteristic
58C2×M16×C79—21no—
59C4×Q8×C79C2×C411yescharacteristic
60C4×Q8×C79Q811yescharacteristic
61C4×D4×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79C2×C411yescharacteristic
64C4×D4×C79Q811yescharacteristic
65#117×C79—21no—
66#117×C79D411yescharacteristic
67C2×C4⋊C4×C79—21no—
68C2×C42×C79—21no—
69C2×C42×C79—21no—
70C2×C42×C79C2311yescharacteristic
71C4⋊C8×C79—21no—
72C4⋊C8×C79—21no—
73#104×C79—42no—
74#95×C79—21no—
75C22×Dic79—41no—
76#11929608—42no—
77#11929608—42no—
78#11929608—41no—
79#11929608—41no—
80C2×C79⋊C8—22no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8C22⋊C411yescharacteristic
86C2×C79⋊C8C22⋊C411yescharacteristic
87C2×C79⋊C8C22⋊C411yescharacteristic
88C2×C79⋊C8C22⋊C411yescharacteristic
89C4⋊Dic79—22no—
90C4⋊Dic79—41no—
91C4×Dic79—41no—
92C4×Dic79—41no—
93#41×C79—21no—
94#41×C79—21no—
95#41×C79—21no—
96#41×C79—21no—
97C2×Q8×C79C4⋊C411yescharacteristic
98C2×D4×C79—21no—
99C2×D4×C79C4⋊C411yescharacteristic
100C22×C4×C79—21no—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79—41no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79#4111yescharacteristic
107M16×C79—41no—
108M16×C79—41no—
109C2×C8×C79—41no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—22no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79C4⋊C411yescharacteristic
115C4⋊C4×C79C4⋊C411yescharacteristic
116C22⋊C4×C79—21no—
117C22⋊C4×C79—21no—
118C22⋊C4×C79—21no—
119C42×C79—21no—
120C42×C79—21no—
121C42×C79—21no—
122C42×C79—41no—
123C42×C79—41no—
124C42×C79C22×C411yescharacteristic
125C42×C79C2×D411yescharacteristic
126C42×C79C2×Q811yescharacteristic
127C42×C79#4111yescharacteristic
128C23×C79—21no—
129C23×C79#12111yescharacteristic
130Q8×C79—21no—
131Q8×C79—21no—
132D4×C79—21no—
133D4×C79—21no—
134D4×C79—41no—
135D4×C79—41no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—41no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—42no—
148C2×C4×C79—81no—
149C2×C4×C79—81no—
150C2×C4×C79C2×C22⋊C411yescharacteristic
151C2×C4×C79C2×C4⋊C411yescharacteristic
152C2×C4×C79#11711yescharacteristic
153C2×C4×C79#12111yescharacteristic
154C2×C4×C79C22⋊Q811yescharacteristic
155C2×C4×C79C22⋊Q811yescharacteristic
156C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C79#13711yescharacteristic
159C2×C4×C79#13711yescharacteristic
160C2×Dic79—42no—
161C2×Dic79—81no—
162C2×Dic79—81no—
163C8×C79—81no—
164C79⋊C8—21no—
165C79⋊C8—21no—
166C79⋊C8—21no—
167C79⋊C8—41no—
168C79⋊C8—41no—
169C79⋊C8—81no—
170C79⋊C8#10412yescomplemented
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—41no—
174C22×C79—41no—
175C22×C79—41no—
176C22×C79#38011yescharacteristic
177C22×C79#46911yescharacteristic
178C22×C79#47011yescharacteristic
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—41no—
183C4×C79—41no—
184C4×C79—41no—
185C4×C79—82no—
186C4×C79C2×#10411yescharacteristic
187C4×C79#42111yescharacteristic
188C4×C79#42811yescharacteristic
189C4×C79#42811yescharacteristic
190C4×C79#46811yescharacteristic
191C4×C79#46811yescharacteristic
192Dic79—82no—
193#8668—791nomaximal, Sylow
194C2×C79—21no—
195C2×C79—21no—
196C2×C79—81no—
197C2×C79#180611yescharacteristic
198C2×C79#180911yescharacteristic
199C2×C79#181411yescharacteristic
200#2965—791no—
201#1806—791no—
202#1806D7911yescharacteristic
203C4×#107—791no—
204#1566—791no—
205#1303—791no—
206#1303—791no—
207C79#866811yescharacteristic, complemented, Sylow
208C4×#41C2×D7911yescharacteristic
209#446—791no—
210#446—791no—
211#436—791no—
212#436—791no—
213#435—1581no—
214#435—1581no—
215#431—791no—
216#431—1581no—
217C2×#107—1581no—
218#423—791no—
219#423C2×D7911yescharacteristic
220C2×#104—21no—
221C2×#104—1581no—
222C4×M16—1581no—
223C2×C4×C8—791no—
224C4×C4⋊C4—791no—
225C4×C4⋊C4C2×D7911yescharacteristic
226#340—1581no—
227#339—21no—
228#339—1581no—
229#334—1581no—
230#334—1581no—
231C2×#41C22⋊D7911yescharacteristic
232C2×M16—21no—
233C2×M16—1581no—
234C2×M16—1581no—
235C2×M16—1581no—
236C22×C8—791no—
237C22×C8—791no—
238C4×Q8Dic15811yescharacteristic
239C4×Q8C4×D7911yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4Dic15811yescharacteristic
243C4×D4C4×D7911yescharacteristic
244#117—21no—
245#117C22⋊D7911yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1581no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—1581no—
251C2×C42C22×D7911yescharacteristic
252#107—1584no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—791no—
256C4⋊C8—791no—
257C4⋊C8—1581no—
258C4⋊C8—1581no—
259C4⋊C8—1581no—
260C4⋊C8—1581no—
261#104—42no—
262#104—3162no—
263C22⋊C8—1581no—
264C22⋊C8—1581no—
265C22⋊C8—1581no—
266#97—1581no—
267#97—1581no—
268C4×C8—791no—
269C4×C8—791no—
270C4×C8—791no—
271C4×C8—791no—
272C4×C8—1581no—
273#95—21no—
274#95—1581no—
275#41—21no—
276#41—21no—
277#41—21no—
278#41—21no—
279C2×Q8Dic79⋊C411yescharacteristic
280C2×D4—21no—
281C2×D4Dic79⋊C411yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—3161no—
287C22×C4D4×D7911yescharacteristic
288C22×C4C2×C22⋊D7911yescharacteristic
289C22×C4#1192962311yescharacteristic
290M16—41no—
291M16—41no—
292M16—3162no—
293M16—3162no—
294M16—3161no—
295M16—3161no—
296C2×C8—41no—
297C2×C8—3161no—
298C2×C8—3161no—
299C2×C8—791no—
300C2×C8—791no—
301C2×C8—791no—
302C2×C8—791no—
303C2×C8—1581no—
304C2×C8—1581no—
305C2×C8—1582no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—3161no—
311C4⋊C4—1582no—
312C4⋊C4Dic79⋊C411yescharacteristic
313C4⋊C4Dic79⋊C411yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—41no—
322C42—3161no—
323C42—3161no—
324C42C2×C4×D7911yescharacteristic
325C42D4×D7911yescharacteristic
326C42C2×Dic15811yescharacteristic
327C42#1192962311yescharacteristic
328C23—21no—
329C23#1457913311yescharacteristic
330Q8—21no—
331Q8—21no—
332D4—21no—
333D4—21no—
334D4—41no—
335D4—41no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—3162no—
351C2×C4—6321no—
352C2×C4—6321no—
353C2×C4C22⋊C4×D7911yescharacteristic
354C2×C4C2×Dic79⋊C411yescharacteristic
355C2×C4#1457911711yescharacteristic
356C2×C4#1457913311yescharacteristic
357C2×C4#1457913811yescharacteristic
358C2×C4#1457913811yescharacteristic
359C2×C4#1457918611yescharacteristic
360C2×C4#1457918711yescharacteristic
361C2×C4#1457919111yescharacteristic
362C2×C4#1457919211yescharacteristic
363C8—81no—
364C8—3161no—
365C8—3161no—
366C8—792no—
367C8—6321no—
368C8—1581no—
369C8—1581no—
370C8—1581no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22#1909842111yescharacteristic
377C22#1909881411yescharacteristic
378C22#1909881911yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4—6322no—
387C4#104×D7911yescharacteristic
388C4#1909857811yescharacteristic
389C4#1909860311yescharacteristic
390C4#1909860611yescharacteristic
391C4#1909880711yescharacteristic
392C4#1909881111yescharacteristic
393C2—21no—
394C2—21no—
395C2—81no—
396C2#2376341811yescharacteristic
397C2#2376343111yescharacteristic
398C2#2376345211yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1806C2
socleC22×C79#380
prime radicalC2×#41×C79D4
cosocleC2×C42×C79C23
hypercenter#1806D79
hypocenterC79#8668
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
3. C4×#14579094C2semidirect
4. #23762155C2semidirect
5. #23761303C2semidirect
6. #23761301C2semidirect
8. #19098694C4semidirect
9. #19098694C4semidirect
10. #19098641C4semidirect
11. #19098641C4semidirect
170. C79⋊C8#104semidirect
207. C79#8668semidirect

Maximal subgroups

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

subgroupindex
#237723672
#237634202
C4×#145790942
#237621552
#237613032
#237613012
C79×#18062
#866879