Finite Group Databasebeta

← #34182997

Subgroups — #34182997

Order 20224.

476 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772532C211yesmaximal, characteristic
2#23764378C211yesmaximal, characteristic, complemented
3C4×#14579088C211yesmaximal, characteristic
4#23761821C211yesmaximal, characteristic, complemented
5#23761291C211yesmaximal, characteristic, complemented
6#23761232C211yesmaximal, characteristic, complemented
7C79×#3107C211yesmaximal, characteristic
8Q8×C79⋊C8—21no—
9#19098641—22no—
10#19098641—21no—
11#19098616—21no—
12#19098616C2211yescharacteristic
13#19098587—21no—
14#19098587C2211yescharacteristic
15C2×#14579088—21no—
16C4×#11929608C2211yescharacteristic
17C42×Dic79C2211yescharacteristic
18#19098322—22no—
19#19098322—21no—
20Q8⋊(C79⋊C8)—21no—
21Q8⋊(C79⋊C8)—21no—
22#533×C79C2211yescharacteristic
23C4×#41×C79C2211yescharacteristic
24#485×C79—21no—
25#475×C79C411yescharacteristic, complemented
26#474×C79C411yescharacteristic, complemented
27#462×C79C411yescharacteristic, complemented
28#461×C79C411yescharacteristic, complemented
29#452×C79—21no—
30C4×Q16×C79—21no—
31C4×SD16×C79—21no—
32#431×C79C2211yescharacteristic
33C2×#11929608—22no—
34C2×#11929608—21no—
35C2×#11929608D412yes—
36C2×C4×Dic79—21no—
37C2×C4×Dic79—21no—
38C2×C4×Dic79—21no—
39C4⋊(C79⋊C8)—22no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42C4⋊(C79⋊C8)D411yescharacteristic
43C4⋊(C79⋊C8)D411yescharacteristic
44#14579088—24no—
45C22⋊(C79⋊C8)—22no—
46C22⋊(C79⋊C8)—21no—
47#14579072—21no—
48C4×C79⋊C8—22no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—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—
81C4×C8×C79—21no—
82C22×Dic79—41no—
83#11929608—22no—
84#11929608—42no—
85#11929608—41no—
86#11929608—82no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—41no—
91C4×Dic79—22no—
92C4×Dic79—22no—
93C4×Dic79—22no—
94C4×Dic79—41no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97#41×C79—22no—
98#41×C79—41no—
99C2×Q8×C79—21no—
100C2×Q8×C79—21no—
101C2×Q8×C79C22⋊C411yescharacteristic
102C2×D4×C79—21no—
103C2×D4×C79C22⋊C411yescharacteristic
104C22×C4×C79—21no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79C2×D411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79#4111yescharacteristic
110Q16×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—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—41no—
128C42×C79—21no—
129C42×C79—21no—
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—21no—
139Q8×C79—41no—
140Q8×C79—41no—
141Q8×C79—41no—
142Q8×C79—41no—
143Q8×C79#10412yes—
144D4×C79—21no—
145D4×C79—21no—
146D4×C79—42no—
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—21no—
156C2×C4×C79—42no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
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—42no—
170C2×Dic79—42no—
171C2×Dic79—42no—
172C8×C79—41no—
173C8×C79—41no—
174C79⋊C8—42no—
175C79⋊C8—42no—
176C79⋊C8—41no—
177C79⋊C8—41no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—41no—
182C22×C79—41no—
183C22×C79—41no—
184C22×C79#38611yescharacteristic
185C4×C79—21no—
186C4×C79—22no—
187C4×C79—21no—
188C4×C79—21no—
189C4×C79—41no—
190C4×C79—41no—
191C4×C79—81no—
192C4×C79—81no—
193C4×C79C2×#10411yescharacteristic
194C4×C79#42111yescharacteristic
195C4×C79#44311yescharacteristic
196C4×C79#44411yescharacteristic
197C4×C79#45511yescharacteristic
198C4×C79#45611yescharacteristic
199Dic79—44no—
200#8972—791nomaximal, Sylow
201C2×C79—41no—
202C2×C79—81no—
203C2×C79#187111yescharacteristic
204C2×C79#187611yescharacteristic
205C2×C79#188211yescharacteristic
206#3107D7911yescharacteristic
207#2991—791no—
208#1994—791no—
209C4×#104—791no—
210#1485—791no—
211#1299—791no—
212#1276—791no—
213C79#897211yescharacteristic, complemented, Sylow
214#533C2×D7911yescharacteristic
215C4×#41C2×D7911yescharacteristic
216#485—21no—
217#475Dic7911yescharacteristic, complemented
218#474Dic7911yescharacteristic, complemented
219#462Dic7911yescharacteristic, complemented
220#461Dic7911yescharacteristic, complemented
221#452—21no—
222C8×Q8—1581no—
223C4×Q16—21no—
224C4×SD16—21no—
225#436—1582no—
226#436—1581no—
227#431—791no—
228#431—1581no—
229#431C2×D7911yescharacteristic
230#423—791no—
231#423—1581no—
232C2×#104—1581no—
233C4×M16—791no—
234C43—791no—
235#339—1582no—
236#339—1581no—
237Q8⋊C8—1581no—
238Q8⋊C8—1581no—
239C2×#41C22⋊D7911yescharacteristic
240C2×Q16—21no—
241C2×SD16—21no—
242C2×M16—792no—
243C2×M16—1581no—
244C2×M16—1582no—
245C4⋊Q8C2×Dic7911yescharacteristic
246C4⋊Q8C2×Dic7911yescharacteristic
247#125—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C2×Dic7911yescharacteristic
253C4×Q8C22⋊D7911yescharacteristic
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C2×Dic7911yescharacteristic
257C4×D4C22⋊D7911yescharacteristic
258#117—21no—
259#117—21no—
260#117C22⋊D7911yescharacteristic
261C2×C42—21no—
262C2×C42—1581no—
263C2×C42—1581no—
264C2×C42—1581no—
265C2×C42C22×D7911yescharacteristic
266#107—21no—
267#106—21no—
268C4⋊C8—791no—
269C4⋊C8—791no—
270C4⋊C8—1582no—
271C4⋊C8—1581no—
272C4⋊C8—1581no—
273C4⋊C8C2×Dic7911yescharacteristic
274C4⋊C8C2×Dic7911yescharacteristic
275#104—1584no—
276#103—21no—
277#103—21no—
278#103—21no—
279#102—21no—
280C22⋊C8—21no—
281C22⋊C8—1582no—
282C22⋊C8—1581no—
283#97—1581no—
284C4×C8—21no—
285C4×C8—1582no—
286C4×C8—1581no—
287C4×C8—1581no—
288#41—22no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C22⋊Dic7911yescharacteristic
293C2×D4—21no—
294C2×D4C22⋊Dic7911yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—3161no—
299C22×C4D4×D7911yescharacteristic
300C22×C4C2×C22⋊D7911yescharacteristic
301C22×C4#1192962311yescharacteristic
302Q16—42no—
303SD16—42no—
304M16—3161no—
305M16—3162no—
306M16—6322no—
307M16—1582no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—3161no—
311C2×C8—1581no—
312C2×C8—1581no—
313C2×C8—1582no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4C22⋊Dic7911yescharacteristic
323C4⋊C4C22⋊Dic7911yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—3161no—
333C42—3161no—
334C42—3161no—
335C42—1582no—
336C42—1582no—
337C42—1582no—
338C42C22×Dic7911yescharacteristic
339C42D4×D7911yescharacteristic
340C42C2×C22⋊D7911yescharacteristic
341C42#1192962311yescharacteristic
342C23—21no—
343C23#1457913311yescharacteristic
344Q8—21no—
345Q8—41no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8#1457908812yes—
350D4—21no—
351D4—21no—
352D4—42no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—3162no—
370C2×C4—3162no—
371C2×C4—3162no—
372C2×C4C2×C22⋊Dic7911yescharacteristic
373C2×C4D4×Dic7911yescharacteristic
374C2×C4D4×Dic7911yescharacteristic
375C2×C4#1457912911yescharacteristic
376C2×C4#1457913311yescharacteristic
377C2×C4#1457914211yescharacteristic
378C8—41no—
379C8—41no—
380C8—3162no—
381C8—3162no—
382C8—3161no—
383C8—3161no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22#1909844311yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—3164no—
400C4C2×#1457908811yescharacteristic
401C4#1909858011yescharacteristic
402C4#1909868011yescharacteristic
403C4#1909868511yescharacteristic
404C4#1909873911yescharacteristic
405C4#1909874411yescharacteristic
406C2—41no—
407C2—81no—
408C2#2376370311yescharacteristic
409C2#2376372911yescharacteristic
410C2#2376377111yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3107C2
socleC22×C79#386
prime radicalC2×#41×C79D4
cosocleC2×C42×C79C23
hypercenter#3107D79
hypocenterC79#8972
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #23764378C2semidirect
4. #23761821C2semidirect
5. #23761291C2semidirect
6. #23761232C2semidirect
25. #475×C79C4semidirect
26. #474×C79C4semidirect
27. #462×C79C4semidirect
28. #461×C79C4semidirect
213. C79#8972semidirect
217. #475Dic79semidirect
218. #474Dic79semidirect
219. #462Dic79semidirect
220. #461Dic79semidirect

Maximal subgroups

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

subgroupindex
#237725322
#237643782
C4×#145790882
#237618212
#237612912
#237612322
C79×#31072
#897279