Finite Group Databasebeta

← #34184101

Subgroups — #34184101

Order 20224.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772489C211yesmaximal, characteristic, complemented
2#23763230C211yesmaximal, characteristic, complemented
3#23762011C211yesmaximal, characteristic
4#23761895C211yesmaximal, characteristic
5#23761298C211yesmaximal, characteristic, complemented
6#23761257C211yesmaximal, characteristic, complemented
7C79×#3112C211yesmaximal, characteristic
8#19098694—22no—
9#19098694—21no—
10Q8×C79⋊C8—21no—
11#19098620—21no—
12#19098620C2211yescharacteristic
13#19098616—21no—
14#19098616C2211yescharacteristic
15C2×#14579085—21no—
16C2×#14579072C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098325—21no—
19#19098303—22no—
20#19098297—21no—
21#19098297—21no—
22#532×C79C2211yescharacteristic
23C2×C4×Q8×C79C2211yescharacteristic
24#480×C79—21no—
25#477×C79C411yescharacteristic, complemented
26#477×C79C411yescharacteristic, complemented
27#460×C79C411yescharacteristic, complemented
28#460×C79C411yescharacteristic, complemented
29#448×C79—21no—
30#440×C79—21no—
31#440×C79—21no—
32#432×C79C2211yescharacteristic
33C22×C79⋊C8—21no—
34C2×C4⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C4⋊(C79⋊C8)—22no—
37C4⋊(C79⋊C8)—22no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40#14579085—24no—
41C22⋊(C79⋊C8)—21no—
42C22⋊(C79⋊C8)D411yescharacteristic
43C22⋊(C79⋊C8)D411yescharacteristic
44#14579072—22no—
45#14579072—21no—
46#14579072—21no—
47#14579072D412yes—
48C4×C79⋊C8—22no—
49C4×C79⋊C8—21no—
50C79⋊#95—21no—
51C2×#41×C79—21no—
52C22×Q8×C79D411yescharacteristic
53C2×SD16×C79—21no—
54C2×SD16×C79—21no—
55C4⋊Q8×C79C2×C411yescharacteristic
56#127×C79C2×C411yescharacteristic
57#124×C79—21no—
58#121×C79—21no—
59#121×C79—21no—
60C4×Q8×C79—21no—
61C4×Q8×C79—21no—
62C4×Q8×C79C2×C411yescharacteristic
63C4×Q8×C79C2×C411yescharacteristic
64C4×Q8×C79D411yescharacteristic
65C4×Q8×C79D411yescharacteristic
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C2×C4⋊C4×C79—21no—
69C2×C4⋊C4×C79D411yescharacteristic
70C2×C42×C79—21no—
71C2×C42×C79C2311yescharacteristic
72#107×C79—21no—
73#107×C79—21no—
74C4⋊C8×C79C2×C411yescharacteristic
75C4⋊C8×C79C2×C411yescharacteristic
76#103×C79—21no—
77#103×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80C22⋊C8×C79—21no—
81#97×C79—21no—
82C22×Dic79—41no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—41no—
88C2×C79⋊C8—41no—
89C4⋊Dic79—22no—
90C4⋊Dic79—41no—
91C4×Dic79—81no—
92#41×C79—81no—
93#41×C79—81no—
94C2×Q8×C79—21no—
95C2×Q8×C79—22no—
96C2×Q8×C79—21no—
97C2×Q8×C79—41no—
98C2×Q8×C79C22⋊C411yescharacteristic
99C2×Q8×C79C22⋊C411yescharacteristic
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—41no—
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79—41no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79#4111yescharacteristic
110SD16×C79—42no—
111SD16×C79—42no—
112C2×C8×C79—21no—
113C2×C8×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—21no—
121C4⋊C4×C79C22⋊C411yescharacteristic
122C4⋊C4×C79C22⋊C411yescharacteristic
123C22⋊C4×C79—41no—
124C22⋊C4×C79—41no—
125C42×C79—21no—
126C42×C79—21no—
127C42×C79—21no—
128C42×C79—21no—
129C42×C79C22×C411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79C2×D411yescharacteristic
132C42×C79#4111yescharacteristic
133C23×C79—41no—
134C23×C79#12111yescharacteristic
135Q8×C79—21no—
136Q8×C79—21no—
137Q8×C79—21no—
138Q8×C79—42no—
139Q8×C79—81no—
140Q8×C79#10412yes—
141D4×C79—41no—
142D4×C79—41no—
143D4×C79—81no—
144D4×C79—81no—
145D4×C79—81no—
146D4×C79—81no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—41no—
156C2×C4×C79—42no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—82no—
161C2×C4×C79C2×C22⋊C411yescharacteristic
162C2×C4×C79C4×D411yescharacteristic
163C2×C4×C79C4×D411yescharacteristic
164C2×C4×C79#12011yescharacteristic
165C2×C4×C79#12111yescharacteristic
166C2×C4×C79#12311yescharacteristic
167C2×Dic79—42no—
168C2×Dic79—81no—
169C2×Dic79—81no—
170C8×C79—41no—
171C8×C79—41no—
172C79⋊C8—41no—
173C79⋊C8—41no—
174C79⋊C8—42no—
175C79⋊C8—42no—
176C22×C79—21no—
177C22×C79—21no—
178C22×C79—81no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79#38611yescharacteristic
182C22×C79#47111yescharacteristic
183C22×C79#47211yescharacteristic
184C4×C79—21no—
185C4×C79—22no—
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—41no—
191C4×C79—41no—
192C4×C79—41no—
193C4×C79—81no—
194C4×C79C2×#10411yescharacteristic
195C4×C79#42111yescharacteristic
196Dic79—82no—
197#9164—791nomaximal, Sylow
198C2×C79—21no—
199C2×C79—21no—
200C2×C79—161no—
201C2×C79#187111yescharacteristic
202C2×C79#190511yescharacteristic
203C2×C79#190711yescharacteristic
204#3112D7911yescharacteristic
205#2984—791no—
206#1763—791no—
207#1530—791no—
208#1502—791no—
209#1302—791no—
210#1285—791no—
211C79#916411yescharacteristic, complemented, Sylow
212#532C2×D7911yescharacteristic
213C2×C4×Q8C2×D7911yescharacteristic
214#480—21no—
215#477Dic7911yescharacteristic, complemented
216#477Dic7911yescharacteristic, complemented
217#460Dic7911yescharacteristic, complemented
218#460Dic7911yescharacteristic, complemented
219#448—21no—
220#446—1582no—
221#446—1581no—
222C8×Q8—1581no—
223#440—21no—
224#440—21no—
225#432—791no—
226#432—1581no—
227#432C2×D7911yescharacteristic
228#431—791no—
229#431—1581no—
230C2×#103—1581no—
231C2×#97—791no—
232C4×C4⋊C4—791no—
233#340—1581no—
234#329—1582no—
235#327—1581no—
236#327—1581no—
237C2×#41—21no—
238C22×Q8C22⋊D7911yescharacteristic
239C2×SD16—21no—
240C2×SD16—21no—
241C22×C8—1581no—
242C4⋊Q8C2×Dic7911yescharacteristic
243#127C2×Dic7911yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C2×Dic7911yescharacteristic
250C4×Q8C2×Dic7911yescharacteristic
251C4×Q8C22⋊D7911yescharacteristic
252C4×Q8C22⋊D7911yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—1581no—
257C2×C4⋊C4C22⋊D7911yescharacteristic
258C2×C42—21no—
259C2×C42—1581no—
260C2×C42C22×D7911yescharacteristic
261#107—21no—
262#107—21no—
263C4⋊C8—1582no—
264C4⋊C8—1582no—
265C4⋊C8—1581no—
266C4⋊C8—1581no—
267C4⋊C8C2×Dic7911yescharacteristic
268C4⋊C8C2×Dic7911yescharacteristic
269#103—21no—
270#103—21no—
271#103—1584no—
272#102—21no—
273#102—21no—
274C22⋊C8—21no—
275C22⋊C8—791no—
276C22⋊C8—791no—
277C22⋊C8—1581no—
278#97—21no—
279#97—792no—
280#97—1582no—
281#97—1581no—
282#97—1581no—
283C4×C8—1582no—
284C4×C8—1581no—
285#95—1581no—
286#41—81no—
287#41—81no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8C22⋊Dic7911yescharacteristic
293C2×Q8C22⋊Dic7911yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—3161no—
302C22×C4D4×D7911yescharacteristic
303C22×C4C2×C22⋊D7911yescharacteristic
304C22×C4#1192962311yescharacteristic
305SD16—42no—
306SD16—42no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—3161no—
310C2×C8—3161no—
311C2×C8—1582no—
312C2×C8—1581no—
313C2×C8—1582no—
314C2×C8—1581no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—3161no—
323C4⋊C4—1582no—
324C4⋊C4C22⋊Dic7911yescharacteristic
325C4⋊C4C22⋊Dic7911yescharacteristic
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—6321no—
333C42C22×Dic7911yescharacteristic
334C42D4×D7911yescharacteristic
335C42C2×C22⋊D7911yescharacteristic
336C42#1192962311yescharacteristic
337C23—41no—
338C23#1457913311yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—81no—
344Q8#1457908812yes—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—3162no—
366C2×C4—6321no—
367C2×C4—6321no—
368C2×C4C2×C22⋊Dic7911yescharacteristic
369C2×C4D4×Dic7911yescharacteristic
370C2×C4D4×Dic7911yescharacteristic
371C2×C4#1457912911yescharacteristic
372C2×C4#1457913311yescharacteristic
373C2×C4#1457914211yescharacteristic
374C8—41no—
375C8—41no—
376C8—3162no—
377C8—3161no—
378C8—3162no—
379C8—3161no—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#1909844311yescharacteristic
386C22#1909882411yescharacteristic
387C22#1909883111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—6322no—
399C4C2×#1457908811yescharacteristic
400C4#1909858011yescharacteristic
401C2—21no—
402C2—21no—
403C2—161no—
404C2#2376370311yescharacteristic
405C2#2376389611yescharacteristic
406C2#2376391011yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3112C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3112D79
hypocenterC79#9164
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. #23772489C2semidirect
2. #23763230C2semidirect
5. #23761298C2semidirect
6. #23761257C2semidirect
25. #477×C79C4semidirect
26. #477×C79C4semidirect
27. #460×C79C4semidirect
28. #460×C79C4semidirect
211. C79#9164semidirect
215. #477Dic79semidirect
216. #477Dic79semidirect
217. #460Dic79semidirect
218. #460Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724892
#237632302
#237620112
#237618952
#237612982
#237612572
C79×#31122
#916479