Finite Group Databasebeta

← #34184073

Subgroups — #34184073

Order 20224.

450 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772420C211yesmaximal, characteristic
2#23763210C211yesmaximal, characteristic
3#23761995C211yesmaximal, characteristic, complemented
4#23761912C211yesmaximal, characteristic, complemented
5#23761292C211yesmaximal, characteristic, complemented
6#23761271C211yesmaximal, characteristic, complemented
7C79×#3116C211yesmaximal, characteristic
8Q8×C79⋊C8—21no—
9#19098641—22no—
10#19098641—21no—
11#19098620—21no—
12#19098620C2211yescharacteristic
13C2×#14579088—21no—
14C4×#11929608—21no—
15C4×#11929608C2211yescharacteristic
16C4×#11929608C2211yescharacteristic
17#19098409C2211yescharacteristic
18#19098322—21no—
19#19098305—21no—
20#19098305—21no—
21#19098300—22no—
22#533×C79C2211yescharacteristic
23C4×#41×C79C2211yescharacteristic
24#485×C79—21no—
25#477×C79C411yescharacteristic, complemented
26#476×C79C411yescharacteristic, complemented
27#462×C79C411yescharacteristic, complemented
28#461×C79C411yescharacteristic, complemented
29#452×C79—21no—
30#441×C79—21no—
31#440×C79—21no—
32#432×C79C2211yescharacteristic
33C2×#11929608—22no—
34C2×#11929608—21no—
35C2×#11929608—21no—
36C2×C4×Dic79—21no—
37C4⋊(C79⋊C8)—22no—
38C4⋊(C79⋊C8)—21no—
39#14579088—24no—
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)D412yes—
42#14579072—21no—
43#14579072—21no—
44#14579072D411yescharacteristic
45#14579072D411yescharacteristic
46C4×C79⋊C8—22no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C79⋊#95—21no—
50C79⋊#95—21no—
51C2×#41×C79D411yescharacteristic
52C2×Q16×C79—21no—
53C2×SD16×C79—21no—
54C4⋊Q8×C79C2×C411yescharacteristic
55C4⋊Q8×C79C2×C411yescharacteristic
56#125×C79—21no—
57C22⋊Q8×C79—21no—
58C22⋊Q8×C79—21no—
59C4×Q8×C79—21no—
60C4×Q8×C79—21no—
61C4×Q8×C79C2×C411yescharacteristic
62C4×Q8×C79D411yescharacteristic
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79C2×C411yescharacteristic
66C4×D4×C79D411yescharacteristic
67#117×C79—21no—
68#117×C79—21no—
69#117×C79D411yescharacteristic
70C2×C42×C79—21no—
71C2×C42×C79C2311yescharacteristic
72#107×C79—21no—
73#106×C79—21no—
74C4⋊C8×C79C2×C411yescharacteristic
75C4⋊C8×C79C2×C411yescharacteristic
76#103×C79—21no—
77#103×C79—21no—
78#103×C79—21no—
79#102×C79—21no—
80C22⋊C8×C79—21no—
81#97×C79—21no—
82C22×Dic79—41no—
83#11929608—22no—
84#11929608—41no—
85#11929608—42no—
86#11929608—44no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—41no—
91C4×Dic79—22no—
92C4×Dic79—41no—
93#41×C79—22no—
94#41×C79—41no—
95C2×Q8×C79—21no—
96C2×Q8×C79—21no—
97C2×Q8×C79C22⋊C411yescharacteristic
98C2×D4×C79—21no—
99C2×D4×C79C22⋊C411yescharacteristic
100C22×C4×C79—21no—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79#4111yescharacteristic
106Q16×C79—42no—
107SD16×C79—42no—
108C2×C8×C79—21no—
109C2×C8×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—41no—
117C4⋊C4×C79—41no—
118C4⋊C4×C79C22⋊C411yescharacteristic
119C4⋊C4×C79C22⋊C411yescharacteristic
120C22⋊C4×C79—21no—
121C22⋊C4×C79—21no—
122C22⋊C4×C79—21no—
123C22⋊C4×C79—41no—
124C42×C79—21no—
125C42×C79—21no—
126C42×C79—21no—
127C42×C79—41no—
128C42×C79C22×C411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79#4111yescharacteristic
132C23×C79—21no—
133C23×C79#12111yescharacteristic
134Q8×C79—21no—
135Q8×C79—41no—
136Q8×C79—41no—
137Q8×C79—81no—
138Q8×C79#10412yes—
139D4×C79—21no—
140D4×C79—21no—
141D4×C79—42no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—81no—
154C2×C4×C79C2×C22⋊C411yescharacteristic
155C2×C4×C79C4×D411yescharacteristic
156C2×C4×C79C4×D411yescharacteristic
157C2×C4×C79#12011yescharacteristic
158C2×C4×C79#12111yescharacteristic
159C2×C4×C79#12311yescharacteristic
160C2×Dic79—42no—
161C2×Dic79—81no—
162C2×Dic79—81no—
163C8×C79—41no—
164C8×C79—41no—
165C79⋊C8—22no—
166C79⋊C8—42no—
167C79⋊C8—41no—
168C79⋊C8—42no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—41no—
173C22×C79—41no—
174C22×C79—41no—
175C22×C79#38611yescharacteristic
176C4×C79—21no—
177C4×C79—22no—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—41no—
183C4×C79—41no—
184C4×C79—81no—
185C4×C79—81no—
186C4×C79C2×#10411yescharacteristic
187C4×C79#42111yescharacteristic
188Dic79—82no—
189#9160—791nomaximal, Sylow
190C2×C79—41no—
191C2×C79—81no—
192C2×C79#187111yescharacteristic
193C2×C79#190211yescharacteristic
194C2×C79#190611yescharacteristic
195#3116D7911yescharacteristic
196#2971—791no—
197#1759—791no—
198#1526—791no—
199#1506—791no—
200#1300—791no—
201#1291—791no—
202C79#916011yescharacteristic, complemented, Sylow
203#533C2×D7911yescharacteristic
204C4×#41C2×D7911yescharacteristic
205#485—21no—
206#477Dic7911yescharacteristic, complemented
207#476Dic7911yescharacteristic, complemented
208#462Dic7911yescharacteristic, complemented
209#461Dic7911yescharacteristic, complemented
210#452—21no—
211C8×Q8—1581no—
212#441—21no—
213#440—21no—
214#436—1582no—
215#436—1581no—
216#432—791no—
217#432—1581no—
218#432C2×D7911yescharacteristic
219C2×#104—1581no—
220C4×M16—791no—
221C4×M16—791no—
222C4×M16—1581no—
223#376—791no—
224#339—1581no—
225#330—1581no—
226#330—1581no—
227#328—1582no—
228C2×#41C22⋊D7911yescharacteristic
229C2×Q16—21no—
230C2×SD16—21no—
231C2×M16—1581no—
232C2×M16—1582no—
233C2×M16—1581no—
234C4⋊Q8C2×Dic7911yescharacteristic
235C4⋊Q8C2×Dic7911yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C2×Dic7911yescharacteristic
242C4×Q8C22⋊D7911yescharacteristic
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C2×Dic7911yescharacteristic
246C4×D4C22⋊D7911yescharacteristic
247#117—21no—
248#117—21no—
249#117C22⋊D7911yescharacteristic
250C2×C42—21no—
251C2×C42—1581no—
252C2×C42C22×D7911yescharacteristic
253#107—21no—
254#106—21no—
255C4⋊C8—1582no—
256C4⋊C8—1581no—
257C4⋊C8C2×Dic7911yescharacteristic
258C4⋊C8C2×Dic7911yescharacteristic
259#104—1584no—
260#103—21no—
261#103—21no—
262#103—21no—
263#102—21no—
264C22⋊C8—21no—
265C22⋊C8—792no—
266C22⋊C8—1582no—
267#97—21no—
268#97—791no—
269#97—791no—
270#97—1581no—
271#97—1581no—
272C4×C8—1582no—
273C4×C8—1581no—
274C4×C8—1581no—
275#95—1581no—
276#95—1581no—
277#41—22no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C22⋊Dic7911yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊Dic7911yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—3161no—
288C22×C4D4×D7911yescharacteristic
289C22×C4C2×C22⋊D7911yescharacteristic
290C22×C4#1192962311yescharacteristic
291Q16—42no—
292SD16—42no—
293M16—3161no—
294M16—3162no—
295M16—3164no—
296M16—1582no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—3161no—
300C2×C8—1581no—
301C2×C8—1582no—
302C2×C8—1581no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊Dic7911yescharacteristic
312C4⋊C4C22⋊Dic7911yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—41no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—3161no—
322C42—1582no—
323C42C22×Dic7911yescharacteristic
324C42D4×D7911yescharacteristic
325C42C2×C22⋊D7911yescharacteristic
326C42#1192962311yescharacteristic
327C23—21no—
328C23#1457913311yescharacteristic
329Q8—21no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8#1457908812yes—
334D4—21no—
335D4—21no—
336D4—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—3162no—
350C2×C4—6321no—
351C2×C4—6321no—
352C2×C4C2×C22⋊Dic7911yescharacteristic
353C2×C4D4×Dic7911yescharacteristic
354C2×C4D4×Dic7911yescharacteristic
355C2×C4#1457912911yescharacteristic
356C2×C4#1457913311yescharacteristic
357C2×C4#1457914211yescharacteristic
358C8—41no—
359C8—41no—
360C8—3162no—
361C8—3161no—
362C8—3162no—
363C8—1582no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22#1909844311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—6322no—
382C4C2×#1457908811yescharacteristic
383C4#1909858011yescharacteristic
384C2—41no—
385C2—81no—
386C2#2376370311yescharacteristic
387C2#2376387511yescharacteristic
388C2#2376390311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3116C2
socleC22×C79#386
prime radicalC2×#41×C79D4
cosocleC2×C42×C79C23
hypercenter#3116D79
hypocenterC79#9160
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
3. #23761995C2semidirect
4. #23761912C2semidirect
5. #23761292C2semidirect
6. #23761271C2semidirect
25. #477×C79C4semidirect
26. #476×C79C4semidirect
27. #462×C79C4semidirect
28. #461×C79C4semidirect
202. C79#9160semidirect
206. #477Dic79semidirect
207. #476Dic79semidirect
208. #462Dic79semidirect
209. #461Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724202
#237632102
#237619952
#237619122
#237612922
#237612712
C79×#31162
#916079