Finite Group Databasebeta

← #34185849

Subgroups — #34185849

Order 20224.

436 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772396C211yesmaximal, characteristic, complemented
2#23762011C211yesmaximal, characteristic, complemented
3#23761960C211yesmaximal, characteristic, complemented
4#23761938C211yesmaximal, characteristic, complemented
5#23761299C211yesmaximal, characteristic, complemented
6#23761282C211yesmaximal, characteristic, complemented
7C79×#3180C211yesmaximal, characteristic
8#19098694—21no—
9#19098635—21no—
10#19098635—21no—
11D4×C79⋊C8—21no—
12#19098624—21no—
13#19098620C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098519—21no—
16C2×#14579072C2211yescharacteristic
17#19098429C2211yescharacteristic
18#19098303—21no—
19#19098303—21no—
20#19098300—21no—
21#19098297—21no—
22#19098297—21no—
23#19098297—21no—
24#532×C79C2211yescharacteristic
25#529×C79C2211yescharacteristic
26#501×C79C411yescharacteristic, complemented
27#496×C79C411yescharacteristic, complemented
28#490×C79C411yescharacteristic, complemented
29#489×C79C411yescharacteristic, complemented
30#466×C79—21no—
31#465×C79—21no—
32#442×C79—21no—
33#440×C79—21no—
34C2×#97×C79C2211yescharacteristic
35C2×#11929608—21no—
36C22×C79⋊C8—21no—
37C22×C79⋊C8—21no—
38C2×C4⋊Dic79—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C22⋊(C79⋊C8)D411yescharacteristic
47C22⋊(C79⋊C8)D411yescharacteristic
48#14579072—21no—
49#14579072—21no—
50#14579072—21no—
51#14579072D411yescharacteristic
52#14579072D411yescharacteristic
53C4×C79⋊C8—21no—
54C4×C79⋊C8—21no—
55C79⋊#95—21no—
56C79⋊#95—21no—
57C2×#41×C79—21no—
58C2×SD16×C79—21no—
59C2×D8×C79—21no—
60C22×C8×C79—21no—
61C4⋊Q8×C79C2×C411yescharacteristic
62#127×C79C2×C411yescharacteristic
63#126×C79—21no—
64#125×C79C2×C411yescharacteristic
65#124×C79—21no—
66#124×C79C2×C411yescharacteristic
67#123×C79—21no—
68C22⋊Q8×C79D411yescharacteristic
69#121×C79—21no—
70#121×C79—21no—
71#121×C79D411yescharacteristic
72C4×Q8×C79D411yescharacteristic
73C4×D4×C79—21no—
74C4×D4×C79—21no—
75C4×D4×C79—21no—
76C4×D4×C79D411yescharacteristic
77#117×C79—21no—
78C2×C42×C79C2311yescharacteristic
79#107×C79—21no—
80#106×C79—21no—
81#103×C79—21no—
82#102×C79—21no—
83#102×C79—21no—
84#102×C79—21no—
85#97×C79—21no—
86#97×C79C2×C411yescharacteristic
87#97×C79C2×C411yescharacteristic
88C22×Dic79—41no—
89#11929608—42no—
90C2×C79⋊C8—21no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—42no—
95C2×C79⋊C8—42no—
96C2×C79⋊C8—41no—
97C4⋊Dic79—81no—
98#41×C79—81no—
99#41×C79—81no—
100C2×Q8×C79—21no—
101C2×Q8×C79C22⋊C411yescharacteristic
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—41no—
107C2×D4×C79C22⋊C411yescharacteristic
108C22×C4×C79—21no—
109C22×C4×C79—41no—
110C22×C4×C79C2×D411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79#4111yescharacteristic
113SD16×C79—42no—
114D8×C79—42no—
115C2×C8×C79—21no—
116C2×C8×C79—21no—
117C2×C8×C79—42no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79C22⋊C411yescharacteristic
125C4⋊C4×C79C22⋊C411yescharacteristic
126C22⋊C4×C79—21no—
127C22⋊C4×C79—21no—
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—21no—
131C22⋊C4×C79—41no—
132C22⋊C4×C79—41no—
133C42×C79—21no—
134C42×C79C22×C411yescharacteristic
135C42×C79C2×D411yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79#4111yescharacteristic
138C23×C79—21no—
139C23×C79—41no—
140C23×C79#12111yescharacteristic
141Q8×C79—41no—
142Q8×C79—81no—
143D4×C79—41no—
144D4×C79—41no—
145D4×C79—41no—
146D4×C79—41no—
147D4×C79—41no—
148D4×C79—81no—
149D4×C79—81no—
150D4×C79—81no—
151D4×C79—81no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—82no—
163C2×C4×C79C2×C22⋊C411yescharacteristic
164C2×C4×C79C4×D411yescharacteristic
165C2×C4×C79C4×D411yescharacteristic
166C2×C4×C79#12011yescharacteristic
167C2×C4×C79#12111yescharacteristic
168C2×C4×C79#12311yescharacteristic
169C2×Dic79—81no—
170C2×Dic79—81no—
171C2×Dic79—81no—
172C8×C79—22no—
173C8×C79—41no—
174C79⋊C8—22no—
175C79⋊C8—41no—
176C79⋊C8—41no—
177C79⋊C8—41no—
178C79⋊C8—81no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—41no—
182C22×C79—41no—
183C22×C79—41no—
184C22×C79—81no—
185C22×C79—81no—
186C22×C79—81no—
187C22×C79#38611yescharacteristic
188C22×C79#47111yescharacteristic
189C22×C79#47211yescharacteristic
190C4×C79—21no—
191C4×C79—21no—
192C4×C79—21no—
193C4×C79—21no—
194C4×C79—21no—
195C4×C79—21no—
196C4×C79—41no—
197C4×C79—41no—
198C4×C79—41no—
199C4×C79—81no—
200Dic79—161no—
201#9432—791nomaximal, Sylow
202C2×C79—21no—
203C2×C79—21no—
204C2×C79—81no—
205C2×C79—161no—
206C2×C79#188411yescharacteristic
207C2×C79#190611yescharacteristic
208C2×C79#190711yescharacteristic
209#3180D7911yescharacteristic
210#2968—791no—
211#1530—791no—
212#1517—791no—
213#1512—791no—
214#1302—791no—
215#1295—791no—
216C79#943211yescharacteristic, complemented, Sylow
217#532C2×D7911yescharacteristic
218#529C2×D7911yescharacteristic
219#501Dic7911yescharacteristic, complemented
220#496Dic7911yescharacteristic, complemented
221#490Dic7911yescharacteristic, complemented
222#489Dic7911yescharacteristic, complemented
223#466—21no—
224#465—21no—
225#446—1581no—
226#442—21no—
227#440—21no—
228#435—1581no—
229#435—1581no—
230C8×D4—1581no—
231#433—1581no—
232#432—791no—
233#431—791no—
234#405—1581no—
235C2×#97—791no—
236C2×#97C2×D7911yescharacteristic
237#382—791no—
238#329—1581no—
239#329—1581no—
240#328—1581no—
241#327—1581no—
242#327—1581no—
243#327—1581no—
244C2×#41—21no—
245C2×SD16—21no—
246C2×D8—21no—
247C2×M16—1581no—
248C22×C8—21no—
249C22×C8—1581no—
250C22×C8—1581no—
251C4⋊Q8C2×Dic7911yescharacteristic
252#127C2×Dic7911yescharacteristic
253#126—21no—
254#125C2×Dic7911yescharacteristic
255#124—21no—
256#124C2×Dic7911yescharacteristic
257#123—21no—
258C22⋊Q8C22⋊D7911yescharacteristic
259#121—21no—
260#121—21no—
261#121C22⋊D7911yescharacteristic
262C4×Q8C22⋊D7911yescharacteristic
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4C22⋊D7911yescharacteristic
267#117—21no—
268C2×C4⋊C4—1581no—
269C2×C42C22×D7911yescharacteristic
270#107—21no—
271#106—21no—
272C4⋊C8—1581no—
273C4⋊C8—1581no—
274C4⋊C8—1581no—
275C4⋊C8—1581no—
276#103—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—791no—
281C22⋊C8—791no—
282C22⋊C8—1581no—
283C22⋊C8—1581no—
284C22⋊C8—1581no—
285#97—21no—
286#97—791no—
287#97—791no—
288#97—1581no—
289#97—1581no—
290#97—1581no—
291#97C2×Dic7911yescharacteristic
292#97C2×Dic7911yescharacteristic
293C4×C8—1581no—
294C4×C8—1581no—
295#95—1581no—
296#95—1581no—
297#41—81no—
298#41—81no—
299C2×Q8—21no—
300C2×Q8C22⋊Dic7911yescharacteristic
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4C22⋊Dic7911yescharacteristic
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—3161no—
310C22×C4D4×D7911yescharacteristic
311C22×C4C2×C22⋊D7911yescharacteristic
312C22×C4#1192962311yescharacteristic
313SD16—42no—
314D8—42no—
315M16—3162no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—42no—
319C2×C8—3162no—
320C2×C8—3162no—
321C2×C8—3161no—
322C2×C8—1581no—
323C2×C8—1581no—
324C2×C8—1581no—
325C2×C8—1581no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—6321no—
333C4⋊C4C22⋊Dic7911yescharacteristic
334C4⋊C4C22⋊Dic7911yescharacteristic
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—21no—
343C42C22×Dic7911yescharacteristic
344C42D4×D7911yescharacteristic
345C42C2×C22⋊D7911yescharacteristic
346C42#1192962311yescharacteristic
347C23—21no—
348C23—41no—
349C23#1457913311yescharacteristic
350Q8—41no—
351Q8—81no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—6321no—
373C2×C4—6321no—
374C2×C4—6321no—
375C2×C4C2×C22⋊Dic7911yescharacteristic
376C2×C4D4×Dic7911yescharacteristic
377C2×C4D4×Dic7911yescharacteristic
378C2×C4#1457912911yescharacteristic
379C2×C4#1457913311yescharacteristic
380C2×C4#1457914211yescharacteristic
381C8—22no—
382C8—41no—
383C8—3161no—
384C8—3161no—
385C8—3161no—
386C8—6321no—
387C8—1582no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1909844311yescharacteristic
397C22#1909882411yescharacteristic
398C22#1909883111yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4—12641no—
410C2—21no—
411C2—21no—
412C2—81no—
413C2—161no—
414C2#2376378411yescharacteristic
415C2#2376390311yescharacteristic
416C2#2376391011yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3180C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3180D79
hypocenterC79#9432
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3180

Complements

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

NH ≅ G/Ntype
1. #23772396C2semidirect
2. #23762011C2semidirect
3. #23761960C2semidirect
4. #23761938C2semidirect
5. #23761299C2semidirect
6. #23761282C2semidirect
26. #501×C79C4semidirect
27. #496×C79C4semidirect
28. #490×C79C4semidirect
29. #489×C79C4semidirect
216. C79#9432semidirect
219. #501Dic79semidirect
220. #496Dic79semidirect
221. #490Dic79semidirect
222. #489Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723962
#237620112
#237619602
#237619382
#237612992
#237612822
C79×#31802
#943279