Finite Group Databasebeta

← #34183179

Subgroups — #34183179

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#23772570C211yesmaximal, characteristic
2#23764382C211yesmaximal, characteristic, complemented
3#23761886C211yesmaximal, characteristic
4#23761873C211yesmaximal, characteristic, complemented
5#23761294C211yesmaximal, characteristic, complemented
6#23761290C211yesmaximal, characteristic, complemented
7C79×#3172C211yesmaximal, 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
17#19098409C2211yescharacteristic
18#19098322—21no—
19#19098322—21no—
20#19098300—21no—
21#19098297—21no—
22Q8⋊(C79⋊C8)—21no—
23Q8⋊(C79⋊C8)—21no—
24#533×C79C2211yescharacteristic
25#529×C79C2211yescharacteristic
26#501×C79C411yescharacteristic, complemented
27#497×C79C411yescharacteristic, complemented
28#491×C79C411yescharacteristic, complemented
29#489×C79C411yescharacteristic, complemented
30#470×C79—21no—
31#468×C79—21no—
32C4×Q16×C79—21no—
33C4×SD16×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
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51#14579072—21no—
52#14579072—21no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8—21no—
55C79⋊#95—21no—
56C79⋊#95—21no—
57C2×Q16×C79—21no—
58C2×SD16×C79—21no—
59C2×M16×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×C79C2×C411yescharacteristic
86#97×C79C2×C411yescharacteristic
87C4×C8×C79—21no—
88C22×Dic79—41no—
89#11929608—41no—
90#11929608—41no—
91#11929608—81no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—81no—
98C4×Dic79—22no—
99C4×Dic79—41no—
100C2×Q8×C79—21no—
101C2×Q8×C79—21no—
102C2×Q8×C79C22⋊C411yescharacteristic
103C2×D4×C79—21no—
104C2×D4×C79C22⋊C411yescharacteristic
105C22×C4×C79—21no—
106C22×C4×C79C2×D411yescharacteristic
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79#4111yescharacteristic
109Q16×C79—42no—
110SD16×C79—42no—
111M16×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×C79—21no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—41no—
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—
132C42×C79—21no—
133C42×C79—41no—
134C42×C79C22×C411yescharacteristic
135C42×C79C2×D411yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79#4111yescharacteristic
138C23×C79—21no—
139C23×C79#12111yescharacteristic
140Q8×C79—41no—
141Q8×C79—41no—
142Q8×C79—41no—
143Q8×C79—42no—
144D4×C79—41no—
145D4×C79—81no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79C2×C22⋊C411yescharacteristic
158C2×C4×C79C4×D411yescharacteristic
159C2×C4×C79C4×D411yescharacteristic
160C2×C4×C79#12011yescharacteristic
161C2×C4×C79#12111yescharacteristic
162C2×C4×C79#12311yescharacteristic
163C2×Dic79—42no—
164C2×Dic79—81no—
165C2×Dic79—81no—
166C8×C79—22no—
167C8×C79—41no—
168C79⋊C8—41no—
169C79⋊C8—41no—
170C79⋊C8—41no—
171C79⋊C8—41no—
172C79⋊C8—81no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—21no—
176C22×C79—41no—
177C22×C79—41no—
178C22×C79—41no—
179C22×C79#38611yescharacteristic
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—21no—
185C4×C79—41no—
186C4×C79—41no—
187C4×C79—41no—
188C4×C79—81no—
189C4×C79—81no—
190C4×C79#45511yescharacteristic
191C4×C79#45611yescharacteristic
192Dic79—82no—
193#8998—791nomaximal, Sylow
194C2×C79—41no—
195C2×C79—81no—
196C2×C79#188111yescharacteristic
197C2×C79#188211yescharacteristic
198C2×C79#188411yescharacteristic
199#3172D7911yescharacteristic
200#2995—791no—
201#1995—791no—
202#1500—791no—
203#1497—791no—
204#1300—791no—
205#1299—791no—
206C79#899811yescharacteristic, complemented, Sylow
207#533C2×D7911yescharacteristic
208#529C2×D7911yescharacteristic
209#501Dic7911yescharacteristic, complemented
210#497Dic7911yescharacteristic, complemented
211#491Dic7911yescharacteristic, complemented
212#489Dic7911yescharacteristic, complemented
213#470—21no—
214#468—21no—
215#446—1581no—
216C4×Q16—21no—
217C4×SD16—21no—
218#436—1581no—
219#435—1581no—
220#435—1581no—
221#433—1581no—
222#432—791no—
223#431—791no—
224#424—1581no—
225#423—791no—
226C4×M16C2×D7911yescharacteristic
227#376—791no—
228#339—1581no—
229#339—1581no—
230#328—1581no—
231#327—1581no—
232Q8⋊C8—1581no—
233Q8⋊C8—1581no—
234C2×Q16—21no—
235C2×SD16—21no—
236C2×M16—21no—
237C2×M16—791no—
238C2×M16—791no—
239C2×M16—1581no—
240C22×C8—1581no—
241C4⋊Q8C2×Dic7911yescharacteristic
242C4⋊Q8C2×Dic7911yescharacteristic
243#126—21no—
244#125—21no—
245#125C2×Dic7911yescharacteristic
246#124C2×Dic7911yescharacteristic
247#123—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8C22⋊D7911yescharacteristic
251#121C22⋊D7911yescharacteristic
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C22⋊D7911yescharacteristic
255C4×D4—21no—
256C4×D4C22⋊D7911yescharacteristic
257#117—21no—
258#117—21no—
259C2×C42—1581no—
260C2×C42C22×D7911yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—791no—
264C4⋊C8—791no—
265C4⋊C8—1581no—
266C4⋊C8—1581no—
267C4⋊C8—1581no—
268C4⋊C8—1581no—
269C4⋊C8—1581no—
270#103—21no—
271#103—21no—
272#103—21no—
273#102—21no—
274C22⋊C8—1581no—
275C22⋊C8—1581no—
276C22⋊C8—1581no—
277C22⋊C8—1581no—
278#97—1581no—
279#97—1581no—
280#97C2×Dic7911yescharacteristic
281#97C2×Dic7911yescharacteristic
282C4×C8—21no—
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—42no—
300M16—3161no—
301M16—3161no—
302M16—6321no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—3161no—
306C2×C8—6321no—
307C2×C8—1581no—
308C2×C8—1581no—
309C2×C8—1581no—
310C2×C8—1581no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
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—
331C42—3161no—
332C42—1582no—
333C42C22×Dic7911yescharacteristic
334C42D4×D7911yescharacteristic
335C42C2×C22⋊D7911yescharacteristic
336C42#1192962311yescharacteristic
337C23—21no—
338C23#1457913311yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—42no—
343D4—41no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—3162no—
357C2×C4—6321no—
358C2×C4—6321no—
359C2×C4C2×C22⋊Dic7911yescharacteristic
360C2×C4D4×Dic7911yescharacteristic
361C2×C4D4×Dic7911yescharacteristic
362C2×C4#1457912911yescharacteristic
363C2×C4#1457913311yescharacteristic
364C2×C4#1457914211yescharacteristic
365C8—22no—
366C8—41no—
367C8—3161no—
368C8—3161no—
369C8—3161no—
370C8—3161no—
371C8—6321no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1909844311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—6322no—
390C4#1909873911yescharacteristic
391C4#1909874411yescharacteristic
392C2—41no—
393C2—81no—
394C2#2376376411yescharacteristic
395C2#2376377111yescharacteristic
396C2#2376378411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3172C2
socleC22×C79#386
prime radical#121×C79D4
cosocleC2×C42×C79C23
hypercenter#3172D79
hypocenterC79#8998
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #23764382C2semidirect
4. #23761873C2semidirect
5. #23761294C2semidirect
6. #23761290C2semidirect
26. #501×C79C4semidirect
27. #497×C79C4semidirect
28. #491×C79C4semidirect
29. #489×C79C4semidirect
206. C79#8998semidirect
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
#237725702
#237643822
#237618862
#237618732
#237612942
#237612902
C79×#31722
#899879