Finite Group Databasebeta

← #34182976

Subgroups — #34182976

Order 20224.

516 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772525C211yesmaximal, characteristic, complemented
2#23764229C211yesmaximal, characteristic, complemented
3C4×#14579088C211yesmaximal, characteristic, complemented
4#23761817C211yesmaximal, characteristic, complemented
5#23761291C211yesmaximal, characteristic, complemented
6#23761232C211yesmaximal, characteristic, complemented
7C79×#3106C211yesmaximal, characteristic
8#19098694—21no—
9#19098635—22no—
10D4×C79⋊C8—21no—
11#19098620—21no—
12#19098616C2211yescharacteristic
13#19098587C2211yescharacteristic
14C2×C4⋊(C79⋊C8)—21no—
15C2×#14579088—22no—
16C2×#14579088—21no—
17C4×#11929608C2211yescharacteristic
18C42×Dic79C2211yescharacteristic
19#19098322—21no—
20D4⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22#532×C79C2211yescharacteristic
23C4×#41×C79C2211yescharacteristic
24#482×C79—21no—
25#477×C79C411yescharacteristic, complemented
26#476×C79C411yescharacteristic, complemented
27#460×C79C411yescharacteristic, complemented
28#459×C79C411yescharacteristic, complemented
29#449×C79—21no—
30C4×SD16×C79—21no—
31C4×D8×C79—21no—
32#431×C79C2211yescharacteristic
33C2×#11929608—21no—
34C2×#11929608D412yes—
35C22×C79⋊C8—22no—
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—
45#14579088—44no—
46C22⋊(C79⋊C8)—22no—
47C22⋊(C79⋊C8)—21no—
48#14579072—22no—
49#14579072—21no—
50C4×C79⋊C8—21no—
51C4×C79⋊C8—21no—
52C2×#41×C79—21no—
53C2×#41×C79D411yescharacteristic
54C2×SD16×C79—21no—
55C2×D8×C79—21no—
56C4⋊Q8×C79C2×C411yescharacteristic
57#127×C79C2×C411yescharacteristic
58#124×C79—21no—
59#121×C79—21no—
60#121×C79—21no—
61C4×Q8×C79C2×C411yescharacteristic
62C4×Q8×C79D411yescharacteristic
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79C2×C411yescharacteristic
68C4×D4×C79D411yescharacteristic
69#117×C79—21no—
70#117×C79D411yescharacteristic
71C2×C42×C79—21no—
72C2×C42×C79C2311yescharacteristic
73#107×C79—21no—
74#106×C79—21no—
75C4⋊C8×C79C2×C411yescharacteristic
76C4⋊C8×C79C2×C411yescharacteristic
77#103×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80#102×C79—21no—
81C22⋊C8×C79—21no—
82C4×C8×C79—21no—
83C22×Dic79—41no—
84#11929608—22no—
85#11929608—42no—
86#11929608—41no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—22no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—82no—
92C4×Dic79—22no—
93C4×Dic79—22no—
94C4×Dic79—22no—
95C4×Dic79—41no—
96C4×Dic79—41no—
97C4×Dic79—41no—
98#41×C79—22no—
99#41×C79—41no—
100#41×C79—42no—
101#41×C79—42no—
102C2×Q8×C79—21no—
103C2×Q8×C79C22⋊C411yescharacteristic
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—21no—
107C2×D4×C79—21no—
108C2×D4×C79—41no—
109C2×D4×C79C22⋊C411yescharacteristic
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—21no—
113C22×C4×C79—41no—
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79#4111yescharacteristic
117SD16×C79—42no—
118D8×C79—42no—
119C2×C8×C79—21no—
120C2×C8×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—21no—
125C4⋊C4×C79C22⋊C411yescharacteristic
126C4⋊C4×C79C22⋊C411yescharacteristic
127C22⋊C4×C79—21no—
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—41no—
131C22⋊C4×C79—41no—
132C42×C79—21no—
133C42×C79—21no—
134C42×C79—21no—
135C42×C79C22×C411yescharacteristic
136C42×C79C2×D411yescharacteristic
137C42×C79C2×D411yescharacteristic
138C42×C79#4111yescharacteristic
139C23×C79—21no—
140C23×C79—41no—
141C23×C79#12111yescharacteristic
142Q8×C79—21no—
143Q8×C79—21no—
144Q8×C79—42no—
145D4×C79—21no—
146D4×C79—42no—
147D4×C79—42no—
148D4×C79—41no—
149D4×C79—41no—
150D4×C79—41no—
151D4×C79—41no—
152D4×C79—82no—
153D4×C79#10412yescomplemented
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—42no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—81no—
170C2×C4×C79—81no—
171C2×C4×C79C2×C22⋊C411yescharacteristic
172C2×C4×C79C4×D411yescharacteristic
173C2×C4×C79C4×D411yescharacteristic
174C2×C4×C79#12011yescharacteristic
175C2×C4×C79#12111yescharacteristic
176C2×C4×C79#12311yescharacteristic
177C2×Dic79—42no—
178C2×Dic79—42no—
179C2×Dic79—42no—
180C8×C79—41no—
181C8×C79—41no—
182C79⋊C8—42no—
183C79⋊C8—41no—
184C79⋊C8—42no—
185C79⋊C8—41no—
186C22×C79—21no—
187C22×C79—22no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79—81no—
193C22×C79—81no—
194C22×C79—81no—
195C22×C79#38611yescharacteristic
196C4×C79—21no—
197C4×C79—21no—
198C4×C79—21no—
199C4×C79—41no—
200C4×C79—41no—
201C4×C79—41no—
202C4×C79—81no—
203C4×C79C2×#10411yescharacteristic
204C4×C79#42111yescharacteristic
205C4×C79#44311yescharacteristic
206C4×C79#44411yescharacteristic
207C4×C79#45311yescharacteristic
208C4×C79#45411yescharacteristic
209Dic79—44no—
210#8969—791nomaximal, Sylow
211C2×C79—41no—
212C2×C79—42no—
213C2×C79—161no—
214C2×C79#187111yescharacteristic
215C2×C79#187611yescharacteristic
216C2×C79#188311yescharacteristic
217#3106D7911yescharacteristic
218#2990—791no—
219#1962—791no—
220C4×#104—791no—
221#1484—791no—
222#1299—791no—
223#1276—791no—
224C79#896911yescharacteristic, complemented, Sylow
225#532C2×D7911yescharacteristic
226C4×#41C2×D7911yescharacteristic
227#482—21no—
228#477Dic7911yescharacteristic, complemented
229#476Dic7911yescharacteristic, complemented
230#460Dic7911yescharacteristic, complemented
231#459Dic7911yescharacteristic, complemented
232#449—21no—
233#446—1581no—
234C4×SD16—21no—
235C4×D8—21no—
236#435—1582no—
237C8×D4—1581no—
238#432—1581no—
239#431—791no—
240#431C2×D7911yescharacteristic
241#423—791no—
242C2×C4⋊C8—1581no—
243C2×#104—1582no—
244C2×#104—1581no—
245C4×M16—791no—
246C43—791no—
247#339—1581no—
248D4⋊C8—1581no—
249D4⋊C8—1581no—
250C2×#41—21no—
251C2×#41C22⋊D7911yescharacteristic
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—792no—
255C2×M16—1581no—
256C22×C8—1582no—
257C4⋊Q8C2×Dic7911yescharacteristic
258#127C2×Dic7911yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×Q8C2×Dic7911yescharacteristic
263C4×Q8C22⋊D7911yescharacteristic
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4C2×Dic7911yescharacteristic
269C4×D4C22⋊D7911yescharacteristic
270#117—21no—
271#117C22⋊D7911yescharacteristic
272C2×C42—21no—
273C2×C42—1581no—
274C2×C42—1581no—
275C2×C42—1581no—
276C2×C42C22×D7911yescharacteristic
277#107—21no—
278#106—21no—
279C4⋊C8—791no—
280C4⋊C8—791no—
281C4⋊C8—1582no—
282C4⋊C8—1581no—
283C4⋊C8—1581no—
284C4⋊C8C2×Dic7911yescharacteristic
285C4⋊C8C2×Dic7911yescharacteristic
286#104—3164no—
287#104—1584no—
288#103—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292C22⋊C8—21no—
293C22⋊C8—1582no—
294C22⋊C8—1581no—
295#97—1582no—
296#97—1581no—
297C4×C8—21no—
298C4×C8—1581no—
299C4×C8—1581no—
300#41—22no—
301#41—41no—
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×Q8C22⋊Dic7911yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4C22⋊Dic7911yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—3161no—
317C22×C4D4×D7911yescharacteristic
318C22×C4C2×C22⋊D7911yescharacteristic
319C22×C4#1192962311yescharacteristic
320SD16—42no—
321D8—42no—
322M16—3161no—
323M16—3162no—
324M16—1582no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—3161no—
328C2×C8—6322no—
329C2×C8—1581no—
330C2×C8—1581no—
331C2×C8—1582no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4C22⋊Dic7911yescharacteristic
337C4⋊C4C22⋊Dic7911yescharacteristic
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—3161no—
347C42—3161no—
348C42—3161no—
349C42—1582no—
350C42—1582no—
351C42—1582no—
352C42C22×Dic7911yescharacteristic
353C42D4×D7911yescharacteristic
354C42C2×C22⋊D7911yescharacteristic
355C42#1192962311yescharacteristic
356C23—21no—
357C23—41no—
358C23#1457913311yescharacteristic
359Q8—21no—
360Q8—21no—
361Q8—42no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—82no—
370D4#1457908812yescomplemented
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—3162no—
389C2×C4—3162no—
390C2×C4—3162no—
391C2×C4C2×C22⋊Dic7911yescharacteristic
392C2×C4D4×Dic7911yescharacteristic
393C2×C4D4×Dic7911yescharacteristic
394C2×C4#1457912911yescharacteristic
395C2×C4#1457913311yescharacteristic
396C2×C4#1457914211yescharacteristic
397C8—41no—
398C8—41no—
399C8—3162no—
400C8—3161no—
401C8—3162no—
402C8—3161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#1909844311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—3164no—
421C4C2×#1457908811yescharacteristic
422C4#1909858011yescharacteristic
423C4#1909868011yescharacteristic
424C4#1909868511yescharacteristic
425C4#1909872911yescharacteristic
426C4#1909873411yescharacteristic
427C2—41no—
428C2—42no—
429C2—161no—
430C2#2376370311yescharacteristic
431C2#2376372911yescharacteristic
432C2#2376377811yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3106C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3106D79
hypocenterC79#8969
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #23772525C2semidirect
2. #23764229C2semidirect
3. C4×#14579088C2semidirect
4. #23761817C2semidirect
5. #23761291C2semidirect
6. #23761232C2semidirect
25. #477×C79C4semidirect
26. #476×C79C4semidirect
27. #460×C79C4semidirect
28. #459×C79C4semidirect
153. D4×C79#104semidirect
224. C79#8969semidirect
228. #477Dic79semidirect
229. #476Dic79semidirect
230. #460Dic79semidirect
231. #459Dic79semidirect
370. D4#14579088semidirect

Maximal subgroups

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

subgroupindex
#237725252
#237642292
C4×#145790882
#237618172
#237612912
#237612322
C79×#31062
#896979