Finite Group Databasebeta

← #34272131

Subgroups — #34272131

Order 20224.

628 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#23773960C211yesmaximal, characteristic
2#23773522C211yesmaximal, characteristic
3C2×Q8×C79⋊C8C211yesmaximal, characteristic, complemented
4#23762856C212yesmaximal, complemented
5#23762846C212yesmaximal, complemented
6#23762780C212yesmaximal, complemented
7#23762750C212yesmaximal, complemented
8#23762161C211yesmaximal, characteristic, complemented
9#23762100C211yesmaximal, characteristic, complemented
10#23762096C211yesmaximal, characteristic
11C79×#3113C211yesmaximal, characteristic, complemented
12#19099113C2211yescharacteristic
13#19098959C2211yescharacteristic
14#19098937C2211yescharacteristic
15#19098913C2211yescharacteristic
16#19098911C2211yescharacteristic
17#19098876—21no—
18#19098858C2212yes—
19#19098794—21no—
20#19098794—21no—
21#19098763C2212yes—
22#19098702—21no—
23Q8×C79⋊C8—21no—
24Q8×C79⋊C8—21no—
25Q8×C79⋊C8C2212yes—
26Q8×C79⋊C8C2212yes—
27#19098661—22no—
28C4×#11929611—21no—
29C4×#11929611—21no—
30#19098620C2211yescharacteristic
31C2×C4⋊(C79⋊C8)—21no—
32C2×C4⋊(C79⋊C8)C2211yescharacteristic
33C2×C4×C79⋊C8—21no—
34C2×C4×C79⋊C8C2211yescharacteristic
35#19098315C2212yes—
36#19098313C2212yes—
37#19098311C2212yes—
38#19098309C2212yes—
39#19098309C2212yes—
40#19098306C2212yes—
41#533×C79C2211yescharacteristic
42C2×C4×Q8×C79C2211yescharacteristic
43#483×C79—21no—
44#475×C79C2212yes—
45#462×C79C2212yes—
46#451×C79—21no—
47#441×C79—22no—
48#432×C79C2211yescharacteristic
49C2×#11929624—21no—
50C2×#11929611—22no—
51C2×#11929611—21no—
52C2×#11929611—21no—
53C22×C79⋊C8—21no—
54C22×C79⋊C8D411yescharacteristic
55C22×C79⋊C8D411yescharacteristic
56#14579161C2311yescharacteristic
57#14579158C2311yescharacteristic
58#14579149—21no—
59#14579135—21no—
60#14579135—21no—
61C4×D316—21no—
62C4×D316—21no—
63#14579094—22no—
64#14579092—21no—
65#14579092—21no—
66C4⋊(C79⋊C8)—21no—
67C4⋊(C79⋊C8)—22no—
68C4⋊(C79⋊C8)—22no—
69C4⋊(C79⋊C8)C2312yes—
70C4⋊(C79⋊C8)C2312yes—
71#14579085—22no—
72#14579085—21no—
73#14579085—21no—
74#14579081—22no—
75#14579081—21no—
76#14579081—21no—
77C22⋊(C79⋊C8)—21no—
78#14579072—21no—
79C4×C79⋊C8—22no—
80C4×C79⋊C8—22no—
81C4×C79⋊C8D411yescharacteristic
82C4×C79⋊C8D411yescharacteristic
83C4×C79⋊C8C2311yescharacteristic
84C4×C79⋊C8C2311yescharacteristic
85C22×Q8×C79D411yescharacteristic
86C2×Q16×C79—22no—
87C4⋊Q8×C79C2311yescharacteristic
88C4⋊Q8×C79C2311yescharacteristic
89#125×C79—21no—
90C22⋊Q8×C79—21no—
91C22⋊Q8×C79—21no—
92C4×Q8×C79—21no—
93C4×Q8×C79—21no—
94C4×Q8×C79—21no—
95C4×Q8×C79—21no—
96C4×Q8×C79D412yes—
97C4×Q8×C79C2312yes—
98#117×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79D411yescharacteristic
101C2×C42×C79—21no—
102C2×C42×C79C2311yescharacteristic
103#106×C79—22no—
104C4⋊C8×C79C2312yes—
105#103×C79—22no—
106#103×C79—22no—
107C22⋊C8×C79—21no—
108#97×C79—21no—
109#11929624—82no—
110C2×Dic158—21no—
111C2×D316—21no—
112C2×D316—21no—
113C2×D316—21no—
114C2×C22⋊D79—41no—
115C2×C4×D79—41no—
116#11929611—42no—
117#11929611—42no—
118#11929611—44no—
119C2×C79⋊C8—22no—
120C2×C79⋊C8—22no—
121C2×C79⋊C8—22no—
122C2×C79⋊C8—41no—
123C2×C79⋊C8—42no—
124C2×C79⋊C8C2×D411yescharacteristic
125C2×C79⋊C8C2×D411yescharacteristic
126C2×C79⋊C8C2×D411yescharacteristic
127C2×C79⋊C8C2×D411yescharacteristic
128C4⋊Dic79—21no—
129C4⋊Dic79—21no—
130(C2×D79)⋊C4—41no—
131(C2×D79)⋊C4—41no—
132C2×Q8×C79—24no—
133C2×Q8×C79—21no—
134C2×Q8×C79—21no—
135C2×Q8×C79—21no—
136C2×Q8×C79C2×D412yes—
137C22×C4×C79—21no—
138C22×C4×C79—21no—
139C22×C4×C79C2×D411yescharacteristic
140C22×C4×C79C2×D411yescharacteristic
141C22×C4×C79#4111yescharacteristic
142Q16×C79—44no—
143C2×C8×C79—22no—
144C4⋊C4×C79—22no—
145C4⋊C4×C79—21no—
146C4⋊C4×C79—22no—
147C4⋊C4×C79—21no—
148C4⋊C4×C79—21no—
149C4⋊C4×C79—21no—
150C4⋊C4×C79—21no—
151C4⋊C4×C79—41no—
152C4⋊C4×C79—41no—
153C4⋊C4×C79C2×D412yes—
154C22⋊C4×C79—41no—
155C42×C79—22no—
156C42×C79—22no—
157C42×C79—41no—
158C42×C79C2×D411yescharacteristic
159C42×C79C2×D411yescharacteristic
160C42×C79#4111yescharacteristic
161C42×C79C2411yescharacteristic
162C23×C79#12111yescharacteristic
163C22×D79—41no—
164Q8×C79—24no—
165Q8×C79—22no—
166Q8×C79—42no—
167Q8×C79—41no—
168Q8×C79—41no—
169Q8×C79#13514yes—
170C2×C4×C79—24no—
171C2×C4×C79—22no—
172C2×C4×C79—22no—
173C2×C4×C79—22no—
174C2×C4×C79—21no—
175C2×C4×C79—21no—
176C2×C4×C79—22no—
177C2×C4×C79—42no—
178C2×C4×C79—41no—
179C2×C4×C79—41no—
180C2×C4×C79#12111yescharacteristic
181C2×C4×C79#12111yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×C4×C79C22×D411yescharacteristic
184C2×C4×C79C22×D411yescharacteristic
185C2×C4×C79C2×#4111yescharacteristic
186C22⋊D79—82no—
187C2×Dic79—41no—
188D316—42no—
189D316—41no—
190D316—41no—
191D316—81no—
192C4×D79—82no—
193Dic158—81no—
194C8×C79—42no—
195C79⋊C8—21no—
196C79⋊C8—21no—
197C79⋊C8—21no—
198C79⋊C8—42no—
199C79⋊C8—42no—
200C79⋊C8#13712yes—
201C22×C79—21no—
202C22×C79—21no—
203C22×C79#47112yes—
204C22×C79C2×#12111yescharacteristic
205C2×D79—81no—
206C2×D79—81no—
207C2×D79—81no—
208C4×C79—21no—
209C4×C79—24no—
210C4×C79—21no—
211C4×C79—21no—
212C4×C79—21no—
213C4×C79—42no—
214C4×C79—81no—
215C4×C79—81no—
216C4×C79C2×#13512yes—
217C4×C79C2×#13711yescharacteristic
218C4×C79#57511yescharacteristic
219Dic79—81no—
220#19263—791nomaximal, Sylow
221C2×C79—22no—
222C2×C79#303711yescharacteristic
223C2×C79#305011yescharacteristic
224C2×C79C2×#47111yescharacteristic
225D79—161no—
226#3167—791no—
227#3113D7911yescharacteristic, complemented
228#3112—791no—
229C2×C8×Q8—791no—
230#1684—792no—
231#1682—792no—
232#1672—792no—
233#1666—792no—
234#1568—791no—
235#1552—791no—
236#1551—791no—
237C79#1926311yescharacteristic, complemented, Sylow
238#533C2×D7911yescharacteristic
239#532—791no—
240C2×C4×Q8C2×D7911yescharacteristic
241#498—791no—
242#492—791no—
243#487—791no—
244#486—791no—
245#483—21no—
246#480—1581no—
247#477—792no—
248#475C2×D7912yes—
249#465—1581no—
250#465—1581no—
251#462C2×D7912yes—
252#460—792no—
253#451—21no—
254#448—1581no—
255C8×Q8—792no—
256C8×Q8—792no—
257C8×Q8—1581no—
258C8×Q8—1581no—
259#441—22no—
260#440—1582no—
261C4×SD16—1581no—
262C4×SD16—1581no—
263#432—791no—
264#432C2×D7911yescharacteristic
265C2×C4⋊C8—791no—
266C2×C4⋊C8—1581no—
267C2×C4×C8—791no—
268C2×C4×C8—1581no—
269#335—792no—
270#334—792no—
271#333—792no—
272#332—792no—
273#332—792no—
274#331—792no—
275C2×#41—1581no—
276C22×Q8C22⋊D7911yescharacteristic
277C2×Q16—22no—
278C2×SD16—1581no—
279C2×SD16—1581no—
280C2×SD16—1582no—
281C22×C8—791no—
282C22×C8—791no—
283C22×C8—1581no—
284C4⋊Q8—791no—
285C4⋊Q8C22×D7911yescharacteristic
286C4⋊Q8C22×D7911yescharacteristic
287#127—791no—
288#125—21no—
289#124—1581no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292#121—1581no—
293#121—1581no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8C22⋊D7912yes—
299C4×Q8C22×D7912yes—
300C4×D4—1581no—
301C4×D4—1581no—
302#117—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C22⋊D7911yescharacteristic
305C2×C42—21no—
306C2×C42C22×D7911yescharacteristic
307#107—1582no—
308#106—22no—
309#106—1581no—
310#106—1581no—
311C4⋊C8—792no—
312C4⋊C8—792no—
313C4⋊C8—1582no—
314C4⋊C8—1582no—
315C4⋊C8—1581no—
316C4⋊C8C22×D7912yes—
317#103—22no—
318#103—22no—
319#103—1581no—
320#103—1581no—
321#103—1582no—
322#102—1581no—
323#102—1581no—
324#102—1582no—
325C22⋊C8—21no—
326C22⋊C8—1581no—
327#97—21no—
328#97—1581no—
329C4×C8—791no—
330C4×C8—791no—
331C4×C8—791no—
332C4×C8—791no—
333C4×C8—1582no—
334C4×C8—1582no—
335#41—6322no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—1581no—
341C2×Q8C2×C22⋊D7912yes—
342C2×D4—3161no—
343C2×D4—1581no—
344C2×D4—1581no—
345C2×D4—1581no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—3161no—
349C22×C4D4×D7911yescharacteristic
350C22×C4C2×C22⋊D7911yescharacteristic
351C22×C4#1192962311yescharacteristic
352Q16—44no—
353SD16—3164no—
354SD16—3162no—
355SD16—3162no—
356C2×C8—22no—
357C2×C8—3162no—
358C2×C8—3161no—
359C2×C8—791no—
360C2×C8—791no—
361C2×C8—791no—
362C2×C8—791no—
363C2×C8—1582no—
364C2×C8—1582no—
365C2×C8—1582no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1581no—
376C4⋊C4—1581no—
377C4⋊C4C2×C22⋊D7912yes—
378C22⋊C4—41no—
379C22⋊C4—3161no—
380C22⋊C4—3161no—
381C42—22no—
382C42—22no—
383C42—41no—
384C42D4×D7911yescharacteristic
385C42C2×C22⋊D7911yescharacteristic
386C42#1192962311yescharacteristic
387C42C23×D7911yescharacteristic
388C23—3161no—
389C23#1457913311yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—42no—
393Q8—41no—
394Q8—41no—
395Q8—6321no—
396Q8#1457918114yes—
397D4—3162no—
398D4—3161no—
399D4—3161no—
400D4—6321no—
401D4—6322no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—3161no—
413C2×C4—6322no—
414C2×C4#1457913311yescharacteristic
415C2×C4#1457913311yescharacteristic
416C2×C4#1457913311yescharacteristic
417C2×C4C2×D4×D7911yescharacteristic
418C2×C4C22×C22⋊D7911yescharacteristic
419C2×C4C2×#1192962311yescharacteristic
420C8—42no—
421C8—3162no—
422C8—3162no—
423C8—792no—
424C8—1581no—
425C8—1581no—
426C8—1581no—
427C22—21no—
428C22—21no—
429C22—6321no—
430C22—6321no—
431C22—6321no—
432C22#1909882812yes—
433C22C2×#1457913311yescharacteristic
434C4—24no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—6321no—
443C4C2×#1457918112yes—
444C4#137×D7911yescharacteristic
445C4#1909936211yescharacteristic
446C2—22no—
447C2—12641no—
448C2#2377286911yescharacteristic
449C2#2377297811yescharacteristic
450C2C2×#1909882811yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3113C2
socleC22×C79C2×#121
prime radical#19099113C22
hypercenter#3113D79
hypocenterC79#19263
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
3. C2×Q8×C79⋊C8C2semidirect
4. #23762856C2semidirect
5. #23762846C2semidirect
6. #23762780C2semidirect
7. #23762750C2semidirect
8. #23762161C2semidirect
9. #23762100C2semidirect
11. C79×#3113C2semidirect
227. #3113D79semidirect
237. C79#19263semidirect

Maximal subgroups

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

subgroupindex
#237739602
#237735222
C2×Q8×C79⋊C82
#237628562
#237628462
#237627802
#237627502
#237621612
#237621002
#237620962
C79×#31132
#1926379