Finite Group Databasebeta

← #34185842

Subgroups — #34185842

Order 20224.

414 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772396C211yesmaximal, characteristic
2#23762015C211yesmaximal, characteristic, complemented
3#23761964C211yesmaximal, characteristic, complemented
4#23761938C211yesmaximal, characteristic
5#23761299C211yesmaximal, characteristic, complemented
6#23761282C211yesmaximal, characteristic, complemented
7C79×#3181C211yesmaximal, 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#19098305—21no—
19#19098305—21no—
20#19098300—21no—
21#19098300—21no—
22#19098300—21no—
23#19098297—21no—
24#533×C79C2211yescharacteristic
25#529×C79C2211yescharacteristic
26#501×C79C411yescharacteristic, complemented
27#497×C79C411yescharacteristic, complemented
28#491×C79C411yescharacteristic, complemented
29#489×C79C411yescharacteristic, complemented
30#467×C79—21no—
31#465×C79—21no—
32#441×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×Q16×C79—21no—
58C2×SD16×C79—21no—
59C22×C8×C79—21no—
60C4⋊Q8×C79C2×C411yescharacteristic
61C4⋊Q8×C79C2×C411yescharacteristic
62#126×C79—21no—
63#125×C79—21no—
64#125×C79C2×C411yescharacteristic
65#124×C79C2×C411yescharacteristic
66#123×C79—21no—
67C22⋊Q8×C79—21no—
68C22⋊Q8×C79—21no—
69C22⋊Q8×C79D411yescharacteristic
70#121×C79D411yescharacteristic
71C4×Q8×C79—21no—
72C4×Q8×C79—21no—
73C4×Q8×C79D411yescharacteristic
74C4×D4×C79—21no—
75C4×D4×C79D411yescharacteristic
76#117×C79—21no—
77#117×C79—21no—
78C2×C42×C79C2311yescharacteristic
79#107×C79—21no—
80#106×C79—21no—
81#103×C79—21no—
82#103×C79—21no—
83#103×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—
98C2×Q8×C79—21no—
99C2×Q8×C79—21no—
100C2×Q8×C79C22⋊C411yescharacteristic
101C2×D4×C79—21no—
102C2×D4×C79C22⋊C411yescharacteristic
103C22×C4×C79—21no—
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C22×C4×C79#4111yescharacteristic
107Q16×C79—42no—
108SD16×C79—42no—
109C2×C8×C79—21no—
110C2×C8×C79—21no—
111C2×C8×C79—42no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—41no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79C22⋊C411yescharacteristic
123C4⋊C4×C79C22⋊C411yescharacteristic
124C22⋊C4×C79—21no—
125C22⋊C4×C79—21no—
126C22⋊C4×C79—21no—
127C22⋊C4×C79—21no—
128C22⋊C4×C79—21no—
129C22⋊C4×C79—41no—
130C42×C79—21no—
131C42×C79—41no—
132C42×C79C22×C411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79C2×D411yescharacteristic
135C42×C79#4111yescharacteristic
136C23×C79—21no—
137C23×C79#12111yescharacteristic
138Q8×C79—41no—
139Q8×C79—41no—
140Q8×C79—41no—
141Q8×C79—81no—
142D4×C79—41no—
143D4×C79—41no—
144D4×C79—41no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79C2×C22⋊C411yescharacteristic
157C2×C4×C79C4×D411yescharacteristic
158C2×C4×C79C4×D411yescharacteristic
159C2×C4×C79#12011yescharacteristic
160C2×C4×C79#12111yescharacteristic
161C2×C4×C79#12311yescharacteristic
162C2×Dic79—81no—
163C2×Dic79—81no—
164C2×Dic79—81no—
165C8×C79—22no—
166C8×C79—41no—
167C79⋊C8—22no—
168C79⋊C8—41no—
169C79⋊C8—41no—
170C79⋊C8—41no—
171C79⋊C8—81no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—41no—
175C22×C79—41no—
176C22×C79—41no—
177C22×C79#38611yescharacteristic
178C22×C79#47111yescharacteristic
179C22×C79#47311yescharacteristic
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—21no—
186C4×C79—41no—
187C4×C79—41no—
188C4×C79—41no—
189C4×C79—81no—
190C4×C79—81no—
191Dic79—161no—
192#9431—791nomaximal, Sylow
193C2×C79—21no—
194C2×C79—21no—
195C2×C79—81no—
196C2×C79#188411yescharacteristic
197C2×C79#190611yescharacteristic
198C2×C79#190811yescharacteristic
199#3181D7911yescharacteristic
200#2968—791no—
201#1531—791no—
202#1518—791no—
203#1512—791no—
204#1302—791no—
205#1295—791no—
206C79#943111yescharacteristic, complemented, Sylow
207#533C2×D7911yescharacteristic
208#529C2×D7911yescharacteristic
209#501Dic7911yescharacteristic, complemented
210#497Dic7911yescharacteristic, complemented
211#491Dic7911yescharacteristic, complemented
212#489Dic7911yescharacteristic, complemented
213#467—21no—
214#465—21no—
215#446—1581no—
216#441—21no—
217#440—21no—
218#435—1581no—
219#435—1581no—
220C8×D4—1581no—
221#433—1581no—
222#432—791no—
223#431—791no—
224#405—1581no—
225C2×#97—791no—
226C2×#97C2×D7911yescharacteristic
227#382—791no—
228#330—1581no—
229#330—1581no—
230#328—1581no—
231#328—1581no—
232#328—1581no—
233#327—1581no—
234C2×Q16—21no—
235C2×SD16—21no—
236C2×M16—1581no—
237C22×C8—21no—
238C22×C8—1581no—
239C22×C8—1581no—
240C4⋊Q8C2×Dic7911yescharacteristic
241C4⋊Q8C2×Dic7911yescharacteristic
242#126—21no—
243#125—21no—
244#125C2×Dic7911yescharacteristic
245#124C2×Dic7911yescharacteristic
246#123—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8C22⋊D7911yescharacteristic
250#121C22⋊D7911yescharacteristic
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D7911yescharacteristic
254C4×D4—21no—
255C4×D4C22⋊D7911yescharacteristic
256#117—21no—
257#117—21no—
258C2×C4⋊C4—1581no—
259C2×C42C22×D7911yescharacteristic
260#107—21no—
261#106—21no—
262C4⋊C8—1581no—
263C4⋊C8—1581no—
264C4⋊C8—1581no—
265C4⋊C8—1581no—
266#103—21no—
267#103—21no—
268#103—21no—
269#102—21no—
270C22⋊C8—791no—
271C22⋊C8—791no—
272C22⋊C8—1581no—
273C22⋊C8—1581no—
274C22⋊C8—1581no—
275#97—21no—
276#97—791no—
277#97—791no—
278#97—1581no—
279#97—1581no—
280#97—1581no—
281#97C2×Dic7911yescharacteristic
282#97C2×Dic7911yescharacteristic
283C4×C8—1581no—
284C4×C8—1581no—
285#95—1581no—
286#95—1581no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic7911yescharacteristic
290C2×D4—21no—
291C2×D4C22⋊Dic7911yescharacteristic
292C22×C4—21no—
293C22×C4—3161no—
294C22×C4D4×D7911yescharacteristic
295C22×C4C2×C22⋊D7911yescharacteristic
296C22×C4#1192962311yescharacteristic
297Q16—42no—
298SD16—42no—
299M16—3162no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—42no—
303C2×C8—3162no—
304C2×C8—3162no—
305C2×C8—3161no—
306C2×C8—1581no—
307C2×C8—1581no—
308C2×C8—1581no—
309C2×C8—1581no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—6321no—
321C4⋊C4C22⋊Dic7911yescharacteristic
322C4⋊C4C22⋊Dic7911yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C42—21no—
330C42—41no—
331C42C22×Dic7911yescharacteristic
332C42D4×D7911yescharacteristic
333C42C2×C22⋊D7911yescharacteristic
334C42#1192962311yescharacteristic
335C23—21no—
336C23#1457913311yescharacteristic
337Q8—41no—
338Q8—41no—
339Q8—41no—
340Q8—81no—
341D4—41no—
342D4—41no—
343D4—41no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—6321no—
356C2×C4—6321no—
357C2×C4—6321no—
358C2×C4C2×C22⋊Dic7911yescharacteristic
359C2×C4D4×Dic7911yescharacteristic
360C2×C4D4×Dic7911yescharacteristic
361C2×C4#1457912911yescharacteristic
362C2×C4#1457913311yescharacteristic
363C2×C4#1457914211yescharacteristic
364C8—22no—
365C8—41no—
366C8—3161no—
367C8—3161no—
368C8—3161no—
369C8—6321no—
370C8—1582no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22#1909844311yescharacteristic
377C22#1909882811yescharacteristic
378C22#1909883611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—12641no—
391C2—21no—
392C2—21no—
393C2—81no—
394C2#2376378411yescharacteristic
395C2#2376390311yescharacteristic
396C2#2376391711yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3181C2
socleC22×C79#386
prime radical#121×C79D4
cosocleC2×C42×C79C23
hypercenter#3181D79
hypocenterC79#9431
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3181

Complements

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

NH ≅ G/Ntype
2. #23762015C2semidirect
3. #23761964C2semidirect
5. #23761299C2semidirect
6. #23761282C2semidirect
26. #501×C79C4semidirect
27. #497×C79C4semidirect
28. #491×C79C4semidirect
29. #489×C79C4semidirect
206. C79#9431semidirect
209. #501Dic79semidirect
210. #497Dic79semidirect
211. #491Dic79semidirect
212. #489Dic79semidirect

Maximal subgroups

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

subgroupindex
#237723962
#237620152
#237619642
#237619382
#237612992
#237612822
C79×#31812
#943179