Finite Group Databasebeta

← #34183172

Subgroups — #34183172

Order 20224.

452 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772570C211yesmaximal, characteristic, complemented
2#23764233C211yesmaximal, characteristic, complemented
3#23761886C211yesmaximal, characteristic, complemented
4#23761869C211yesmaximal, characteristic, complemented
5#23761294C211yesmaximal, characteristic, complemented
6#23761290C211yesmaximal, characteristic, complemented
7C79×#3171C211yesmaximal, characteristic
8#19098694—21no—
9#19098641—21no—
10#19098635—21no—
11#19098635—21no—
12#19098624—21no—
13#19098620C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098590—21no—
16#19098587C2211yescharacteristic
17C2×#14579088—21no—
18C2×#14579088—21no—
19#19098409C2211yescharacteristic
20#19098300—21no—
21#19098297—21no—
22D4⋊(C79⋊C8)—21no—
23D4⋊(C79⋊C8)—21no—
24#532×C79C2211yescharacteristic
25#529×C79C2211yescharacteristic
26#501×C79C411yescharacteristic, complemented
27#496×C79C411yescharacteristic, complemented
28#490×C79C411yescharacteristic, complemented
29#489×C79C411yescharacteristic, complemented
30#469×C79—21no—
31#468×C79—21no—
32C4×SD16×C79—21no—
33C4×D8×C79—21no—
34C4×M16×C79C2211yescharacteristic
35C2×#11929608—21no—
36C2×#11929608D411yescharacteristic
37C2×#11929608D411yescharacteristic
38C22×C79⋊C8—21no—
39C2×C4×Dic79—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)D411yescharacteristic
46C4⋊(C79⋊C8)D411yescharacteristic
47#14579088—42no—
48#14579088—42no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)—21no—
53#14579072—21no—
54#14579072—21no—
55C4×C79⋊C8—21no—
56C4×C79⋊C8—21no—
57C79⋊#95—21no—
58C79⋊#95—21no—
59C2×#41×C79—21no—
60C2×SD16×C79—21no—
61C2×D8×C79—21no—
62C2×M16×C79—21no—
63C4⋊Q8×C79C2×C411yescharacteristic
64#127×C79C2×C411yescharacteristic
65#126×C79—21no—
66#125×C79C2×C411yescharacteristic
67#124×C79—21no—
68#124×C79C2×C411yescharacteristic
69#123×C79—21no—
70C22⋊Q8×C79D411yescharacteristic
71#121×C79—21no—
72#121×C79—21no—
73#121×C79D411yescharacteristic
74C4×Q8×C79D411yescharacteristic
75C4×D4×C79—21no—
76C4×D4×C79—21no—
77C4×D4×C79—21no—
78C4×D4×C79D411yescharacteristic
79#117×C79—21no—
80C2×C42×C79C2311yescharacteristic
81#107×C79—21no—
82#106×C79—21no—
83#103×C79—21no—
84#102×C79—21no—
85#102×C79—21no—
86#102×C79—21no—
87#97×C79C2×C411yescharacteristic
88#97×C79C2×C411yescharacteristic
89C4×C8×C79—21no—
90C22×Dic79—41no—
91#11929608—41no—
92#11929608—41no—
93#11929608—81no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—21no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—81no—
100C4×Dic79—22no—
101C4×Dic79—41no—
102#41×C79—41no—
103#41×C79—41no—
104#41×C79—41no—
105#41×C79—41no—
106C2×Q8×C79—21no—
107C2×Q8×C79C22⋊C411yescharacteristic
108C2×D4×C79—21no—
109C2×D4×C79—21no—
110C2×D4×C79—21no—
111C2×D4×C79—21no—
112C2×D4×C79—41no—
113C2×D4×C79C22⋊C411yescharacteristic
114C22×C4×C79—21no—
115C22×C4×C79—41no—
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79#4111yescharacteristic
119SD16×C79—42no—
120D8×C79—42no—
121M16×C79—42no—
122C2×C8×C79—21no—
123C2×C8×C79—21no—
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79C22⋊C411yescharacteristic
131C4⋊C4×C79C22⋊C411yescharacteristic
132C22⋊C4×C79—21no—
133C22⋊C4×C79—21no—
134C22⋊C4×C79—21no—
135C22⋊C4×C79—21no—
136C22⋊C4×C79—21no—
137C22⋊C4×C79—41no—
138C22⋊C4×C79—41no—
139C42×C79—21no—
140C42×C79C22×C411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C42×C79#4111yescharacteristic
144C23×C79—21no—
145C23×C79—41no—
146C23×C79#12111yescharacteristic
147Q8×C79—41no—
148Q8×C79—41no—
149Q8×C79—41no—
150D4×C79—41no—
151D4×C79—41no—
152D4×C79—41no—
153D4×C79—42no—
154D4×C79—41no—
155D4×C79—41no—
156D4×C79—81no—
157D4×C79—81no—
158D4×C79—81no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—81no—
170C2×C4×C79—81no—
171C2×C4×C79C2×C22⋊C411yescharacteristic
172C2×C4×C79C4×D411yescharacteristic
173C2×C4×C79C4×D411yescharacteristic
174C2×C4×C79#12011yescharacteristic
175C2×C4×C79#12111yescharacteristic
176C2×C4×C79#12311yescharacteristic
177C2×Dic79—42no—
178C2×Dic79—81no—
179C2×Dic79—81no—
180C8×C79—22no—
181C8×C79—41no—
182C79⋊C8—41no—
183C79⋊C8—41no—
184C79⋊C8—41no—
185C79⋊C8—41no—
186C79⋊C8—81no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79—41no—
193C22×C79—81no—
194C22×C79—81no—
195C22×C79—81no—
196C22×C79#38611yescharacteristic
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—21no—
200C4×C79—21no—
201C4×C79—21no—
202C4×C79—41no—
203C4×C79—41no—
204C4×C79—41no—
205C4×C79—81no—
206C4×C79#45311yescharacteristic
207C4×C79#45411yescharacteristic
208Dic79—82no—
209#8997—791nomaximal, Sylow
210C2×C79—41no—
211C2×C79—81no—
212C2×C79—161no—
213C2×C79#188111yescharacteristic
214C2×C79#188311yescharacteristic
215C2×C79#188411yescharacteristic
216#3171D7911yescharacteristic
217#2995—791no—
218#1963—791no—
219#1500—791no—
220#1496—791no—
221#1300—791no—
222#1299—791no—
223C79#899711yescharacteristic, complemented, Sylow
224#532C2×D7911yescharacteristic
225#529C2×D7911yescharacteristic
226#501Dic7911yescharacteristic, complemented
227#496Dic7911yescharacteristic, complemented
228#490Dic7911yescharacteristic, complemented
229#489Dic7911yescharacteristic, complemented
230#469—21no—
231#468—21no—
232#446—1581no—
233C4×SD16—21no—
234C4×D8—21no—
235#436—1581no—
236#435—1581no—
237#435—1581no—
238#433—1581no—
239#432—791no—
240#431—791no—
241#424—1581no—
242#423—791no—
243C2×#104—1581no—
244C2×#104—1581no—
245C4×M16C2×D7911yescharacteristic
246#376—791no—
247#328—1581no—
248#327—1581no—
249D4⋊C8—1581no—
250D4⋊C8—1581no—
251C2×#41—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—21no—
255C2×M16—791no—
256C2×M16—791no—
257C2×M16—1581no—
258C22×C8—1581no—
259C4⋊Q8C2×Dic7911yescharacteristic
260#127C2×Dic7911yescharacteristic
261#126—21no—
262#125C2×Dic7911yescharacteristic
263#124—21no—
264#124C2×Dic7911yescharacteristic
265#123—21no—
266C22⋊Q8C22⋊D7911yescharacteristic
267#121—21no—
268#121—21no—
269#121C22⋊D7911yescharacteristic
270C4×Q8C22⋊D7911yescharacteristic
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4C22⋊D7911yescharacteristic
275#117—21no—
276C2×C42—1581no—
277C2×C42C22×D7911yescharacteristic
278#107—21no—
279#106—21no—
280C4⋊C8—791no—
281C4⋊C8—791no—
282C4⋊C8—1581no—
283C4⋊C8—1581no—
284C4⋊C8—1581no—
285C4⋊C8—1581no—
286C4⋊C8—1581no—
287#104—3162no—
288#104—3162no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293C22⋊C8—1581no—
294C22⋊C8—1581no—
295C22⋊C8—1581no—
296C22⋊C8—1581no—
297#97—1581no—
298#97—1581no—
299#97C2×Dic7911yescharacteristic
300#97C2×Dic7911yescharacteristic
301C4×C8—21no—
302C4×C8—1581no—
303C4×C8—1581no—
304#95—1581no—
305#95—1581no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310C2×Q8—21no—
311C2×Q8C22⋊Dic7911yescharacteristic
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4C22⋊Dic7911yescharacteristic
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—3161no—
321C22×C4D4×D7911yescharacteristic
322C22×C4C2×C22⋊D7911yescharacteristic
323C22×C4#1192962311yescharacteristic
324SD16—42no—
325D8—42no—
326M16—42no—
327M16—3161no—
328M16—3161no—
329M16—6321no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—3161no—
333C2×C8—6321no—
334C2×C8—1581no—
335C2×C8—1581no—
336C2×C8—1581no—
337C2×C8—1581no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4C22⋊Dic7911yescharacteristic
345C4⋊C4C22⋊Dic7911yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42—21no—
354C42—3161no—
355C42—1582no—
356C42C22×Dic7911yescharacteristic
357C42D4×D7911yescharacteristic
358C42C2×C22⋊D7911yescharacteristic
359C42#1192962311yescharacteristic
360C23—21no—
361C23—41no—
362C23#1457913311yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—42no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374D4—81no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—3162no—
388C2×C4—6321no—
389C2×C4—6321no—
390C2×C4C2×C22⋊Dic7911yescharacteristic
391C2×C4D4×Dic7911yescharacteristic
392C2×C4D4×Dic7911yescharacteristic
393C2×C4#1457912911yescharacteristic
394C2×C4#1457913311yescharacteristic
395C2×C4#1457914211yescharacteristic
396C8—22no—
397C8—41no—
398C8—3161no—
399C8—3161no—
400C8—3161no—
401C8—3161no—
402C8—6321no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#1909844311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—81no—
422C4—6322no—
423C4#1909872911yescharacteristic
424C4#1909873411yescharacteristic
425C2—41no—
426C2—81no—
427C2—161no—
428C2#2376376411yescharacteristic
429C2#2376377811yescharacteristic
430C2#2376378411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3171C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3171D79
hypocenterC79#8997
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3171

Complements

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

NH ≅ G/Ntype
1. #23772570C2semidirect
2. #23764233C2semidirect
3. #23761886C2semidirect
4. #23761869C2semidirect
5. #23761294C2semidirect
6. #23761290C2semidirect
26. #501×C79C4semidirect
27. #496×C79C4semidirect
28. #490×C79C4semidirect
29. #489×C79C4semidirect
223. C79#8997semidirect
226. #501Dic79semidirect
227. #496Dic79semidirect
228. #490Dic79semidirect
229. #489Dic79semidirect

Maximal subgroups

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

subgroupindex
#237725702
#237642332
#237618862
#237618692
#237612942
#237612902
C79×#31712
#899779