Finite Group Databasebeta

← #34183859

Subgroups — #34183859

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771731C211yesmaximal, characteristic
2C2×#19098298C212yesmaximal
3#23761627C211yesmaximal, characteristic, complemented
4#23761280C212yesmaximal, complemented
5C2×#432×C79C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C2211yescharacteristic
8#19098465C411yescharacteristic, complemented
9#19098465C411yescharacteristic, complemented
10#19098461C411yescharacteristic, complemented
11#19098461C411yescharacteristic, complemented
12#19098454—21no—
13#19098454—21no—
14#19098442—21no—
15#19098423C2211yescharacteristic
16C2×C79⋊#95—21no—
17C2×C79⋊#95—21no—
18C2×C79⋊#95C2212yes—
19#19098298—24no—
20C22×C42×C79C2211yescharacteristic
21#432×C79—24no—
22C2×C4⋊C8×C79—21no—
23C2×C22⋊C8×C79C2212yes—
24C2×#97×C79—21no—
25C22×D316C2×C411yescharacteristic
26C22×C22⋊D79—21no—
27C23×Dic79—22no—
28#14579144—44no—
29#14579135—22no—
30#14579135—22no—
31C2×C4⋊Dic79C2×C411yescharacteristic
32C2×C4⋊Dic79C2×C411yescharacteristic
33C2×C4⋊Dic79C2×C411yescharacteristic
34C2×Dic79⋊C4—21no—
35C2×Dic79⋊C4—21no—
36C2×C22⋊Dic79—21no—
37C2×(C2×D79)⋊C4—21no—
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—21no—
41C2×(C2×D79)⋊C4—21no—
42C79⋊#95—24no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—22no—
47C79⋊#95—22no—
48C79⋊#95—42no—
49C23×C4×C79D411yescharacteristic
50C23×C4×C79D411yescharacteristic
51C23×C4×C79C2311yescharacteristic
52C22×C8×C79—22no—
53C2×C42×C79—22no—
54C2×C42×C79—22no—
55C2×C42×C79C2×C411yescharacteristic
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×C42×C79D411yescharacteristic
59C4⋊C8×C79—24no—
60C22⋊C8×C79—22no—
61C22⋊C8×C79D414yes—
62#97×C79—24no—
63C23×D79—21no—
64C2×D316—22no—
65C2×D316—41no—
66C2×D316—41no—
67C2×C22⋊D79—42no—
68C2×C22⋊D79—42no—
69C2×C22⋊D79—41no—
70C2×C22⋊D79—41no—
71C22×Dic79—24no—
72C22×Dic79—21no—
73C22×Dic79—21no—
74C22×Dic79—21no—
75C22×Dic79—42no—
76C22×Dic79—42no—
77C22×Dic79—42no—
78C22×Dic79—42no—
79C22×Dic79—42no—
80C22×Dic79—42no—
81C4⋊Dic79—22no—
82C4⋊Dic79—41no—
83C4⋊Dic79—41no—
84Dic79⋊C4—42no—
85Dic79⋊C4—42no—
86C22⋊Dic79—42no—
87(C2×D79)⋊C4—42no—
88(C2×D79)⋊C4—42no—
89(C2×D79)⋊C4—42no—
90(C2×D79)⋊C4—42no—
91(C2×D79)⋊C4—41no—
92(C2×D79)⋊C4—41no—
93C24×C79C2×D411yescharacteristic
94C22×C4×C79—22no—
95C22×C4×C79—22no—
96C22×C4×C79—21no—
97C22×C4×C79—22no—
98C22×C4×C79—21no—
99C22×C4×C79—22no—
100C22×C4×C79—41no—
101C22×C4×C79—41no—
102C22×C4×C79—41no—
103C22×C4×C79C22⋊C411yescharacteristic
104C22×C4×C79C22⋊C411yescharacteristic
105C22×C4×C79C22⋊C411yescharacteristic
106C22×C4×C79C22⋊C411yescharacteristic
107C22×C4×C79C22×C411yescharacteristic
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D412yes—
110C22×C4×C79#4112yes—
111C2×C8×C79—24no—
112C2×C8×C79—42no—
113C2×C8×C79—42no—
114C42×C79—22no—
115C42×C79—22no—
116C42×C79—21no—
117C42×C79—21no—
118C42×C79—21no—
119C42×C79—21no—
120C23×C79—21no—
121C23×C79—21no—
122C23×C79—21no—
123C23×C79—21no—
124C23×C79C2×C22⋊C411yescharacteristic
125C23×C79#12112yes—
126C23×C79AGL(1,ℤ/8)11yescharacteristic
127C23×C79AGL(1,ℤ/8)12yes—
128C23×C79#13711yescharacteristic
129C22×D79—41no—
130C22×D79—41no—
131C22×D79—41no—
132C22×D79—41no—
133C22×D79—41no—
134C22×D79—41no—
135C22×D79—41no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79—44no—
154C2×C4×C79—42no—
155C2×C4×C79—42no—
156C2×C4×C79#10412yes—
157C2×C4×C79C4×D412yes—
158C2×C4×C79#12011yescharacteristic
159C2×C4×C79#12311yescharacteristic
160C22⋊D79—82no—
161C22⋊D79—82no—
162C2×Dic79—44no—
163C2×Dic79—44no—
164C2×Dic79—44no—
165C2×Dic79—41no—
166C2×Dic79—41no—
167C2×Dic79—41no—
168C2×Dic79—41no—
169C2×Dic79—41no—
170C2×Dic79—41no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—82no—
175C2×Dic79—82no—
176C2×Dic79—82no—
177C2×Dic79—82no—
178D316—82no—
179C8×C79—44no—
180C22×C79—22no—
181C22×C79—22no—
182C22×C79—22no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—22no—
188C22×C79#38611yescharacteristic
189C22×C79#41011yescharacteristic
190C22×C79#41011yescharacteristic
191C22×C79#41711yescharacteristic
192C22×C79#41811yescharacteristic
193C22×C79C2×#10411yescharacteristic
194C22×C79#42111yescharacteristic
195C22×C79#46812yes—
196C22×C79#46912yes—
197C2×D79—81no—
198C2×D79—81no—
199C2×D79—81no—
200C2×D79—81no—
201C2×D79—81no—
202C2×D79—81no—
203C2×D79—81no—
204C4×C79—22no—
205C4×C79—22no—
206C4×C79—21no—
207C4×C79—21no—
208C4×C79—41no—
209C4×C79—41no—
210C4×C79—41no—
211Dic79—84no—
212Dic79—81no—
213Dic79—81no—
214Dic79—81no—
215#9118—791nomaximal, Sylow
216C2×C79—22no—
217C2×C79—22no—
218C2×C79#150511yescharacteristic
219C2×C79#150911yescharacteristic
220C2×C79#151211yescharacteristic
221C2×C79#151411yescharacteristic
222C2×C79#187111yescharacteristic
223C2×C79#189511yescharacteristic
224C2×C79#189911yescharacteristic
225D79—161no—
226C2×#432D7911yescharacteristic, complemented
227#2854—791no—
228C2×#327—792no—
229#1434—791no—
230#1294—792no—
231C79#911811yescharacteristic, complemented, Sylow
232C2×#123—1581no—
233C2×#121—791no—
234C22×C42C2×D7911yescharacteristic
235#432—24no—
236C2×C4⋊C8—21no—
237C2×C22⋊C8C2×D7912yes—
238C2×#97—21no—
239#390—791no—
240#390—791no—
241#389—791no—
242#389—791no—
243#388—1581no—
244#388—1581no—
245#386—1581no—
246#380—791no—
247C2×#95—792no—
248C2×#95—1581no—
249C2×#95—1581no—
250#327—1584no—
251C22×D4—791no—
252C22×D4—1581no—
253C23×C4—1582no—
254C23×C4D31611yescharacteristic
255C23×C4D31611yescharacteristic
256C23×C4C22×D7911yescharacteristic
257C22×C8—22no—
258#123—3164no—
259#121—1582no—
260#121—1582no—
261C2×C4⋊C4—791no—
262C2×C4⋊C4—791no—
263C2×C4⋊C4—791no—
264C2×C4⋊C4—1581no—
265C2×C4⋊C4—1581no—
266C2×C22⋊C4—1581no—
267C2×C22⋊C4—1581no—
268C2×C22⋊C4—1581no—
269C2×C22⋊C4—1581no—
270C2×C22⋊C4—1581no—
271C2×C22⋊C4—1581no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C4×D7911yescharacteristic
275C2×C42C4×D7911yescharacteristic
276C2×C42C22⋊D7911yescharacteristic
277C2×C42C22⋊D7911yescharacteristic
278C4⋊C8—24no—
279C22⋊C8—22no—
280C22⋊C8C22⋊D7914yes—
281#97—24no—
282#95—3162no—
283#95—1584no—
284#95—1582no—
285#95—1582no—
286#95—1581no—
287#95—1581no—
288#95—1581no—
289C24—1581no—
290C24C2×D31611yescharacteristic
291C2×D4—3161no—
292C2×D4—3161no—
293C2×D4—3162no—
294C2×D4—3162no—
295C2×D4—3161no—
296C2×D4—3161no—
297C2×D4—1582no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—3162no—
308C22×C4—3162no—
309C22×C4—3162no—
310C22×C4—3162no—
311C22×C4—3162no—
312C22×C4—3162no—
313C22×C4—1584no—
314C22×C4—1581no—
315C22×C4—1581no—
316C22×C4—1581no—
317C22×C4(C2×D79)⋊C411yescharacteristic
318C22×C4(C2×D79)⋊C411yescharacteristic
319C22×C4(C2×D79)⋊C411yescharacteristic
320C22×C4(C2×D79)⋊C411yescharacteristic
321C22×C4C2×C4×D7911yescharacteristic
322C22×C4C2×C22⋊D7911yescharacteristic
323C22×C4C2×C22⋊D7912yes—
324C22×C4#1192962412yes—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—3161no—
329C4⋊C4—3161no—
330C4⋊C4—3162no—
331C4⋊C4—3162no—
332C4⋊C4—1582no—
333C22⋊C4—3162no—
334C22⋊C4—3162no—
335C22⋊C4—3162no—
336C22⋊C4—3162no—
337C22⋊C4—3162no—
338C22⋊C4—3161no—
339C22⋊C4—3161no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—3161no—
351C23—3161no—
352C23—3161no—
353C23—3161no—
354C23—3161no—
355C23—3161no—
356C23—3161no—
357C23C2×(C2×D79)⋊C411yescharacteristic
358C23#1457913512yes—
359C23#1457918912yes—
360C23#1457918911yescharacteristic
361C23#1457919411yescharacteristic
362D4—6322no—
363D4—6322no—
364D4—6322no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
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—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—3164no—
386C2×C4—3164no—
387C2×C4—3164no—
388C2×C4—3161no—
389C2×C4—3161no—
390C2×C4—3161no—
391C2×C4—3161no—
392C2×C4—3161no—
393C2×C4—3161no—
394C2×C4—3161no—
395C2×C4—3161no—
396C2×C4—3161no—
397C2×C4—6322no—
398C2×C4—6322no—
399C2×C4—6322no—
400C2×C4—6322no—
401C2×C4#1457908712yes—
402C2×C4C4×C22⋊D7912yes—
403C2×C4#1457913011yescharacteristic
404C2×C4#1457914411yescharacteristic
405C8—44no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—6321no—
415C22—6321no—
416C22—6321no—
417C22—6321no—
418C22—6321no—
419C22—6321no—
420C22—6321no—
421C22#1909844211yescharacteristic
422C22#1909853611yescharacteristic
423C22#1909853611yescharacteristic
424C22#1909856511yescharacteristic
425C22#1909856911yescharacteristic
426C22C2×#1457908711yescharacteristic
427C22#1909857911yescharacteristic
428C22#1909880912yes—
429C22#1909881612yes—
430C4—21no—
431C4—22no—
432C4—21no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—6324no—
438C4—6321no—
439C4—6321no—
440C4—6321no—
441C2—22no—
442C2—22no—
443C2—12641no—
444C2#2376190911yescharacteristic
445C2#2376192611yescharacteristic
446C2#2376193911yescharacteristic
447C2#2376194811yescharacteristic
448C2#2376370211yescharacteristic
449C2#2376383811yescharacteristic
450C2#2376385811yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#432×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenterC2×#432D79
hypocenterC79#9118
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. #23761627C2semidirect
4. #23761280C2semidirect
5. C2×#432×C79C2semidirect
8. #19098465C4semidirect
9. #19098465C4semidirect
10. #19098461C4semidirect
11. #19098461C4semidirect
226. C2×#432D79semidirect
231. C79#9118semidirect

Maximal subgroups

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

subgroupindex
#237717312
C2×#190982982
#237616272
#237612802
C2×#432×C792
#911879